Termination proof of /tmp/tmp9EkYUl/A07.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:

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

The set Q consists of the following terms:

g(TRUE, x0, x1, x2)
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:

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

The integer pair graph contains the following rules and edges:

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

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

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

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

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

The set Q consists of the following terms:

g(TRUE, x0, x1, x2)
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): G(TRUE, x[0], y[0], z[0]) → F(>@z(x[0], z[0]), x[0], +@z(y[0], 1@z), z[0])
(1): F(TRUE, x[1], y[1], z[1]) → G(>@z(x[1], y[1]), x[1], y[1], z[1])
(2): G(TRUE, x[2], y[2], z[2]) → F(>@z(x[2], z[2]), x[2], y[2], +@z(z[2], 1@z))

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

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

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

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

The set Q consists of the following terms:

g(TRUE, x0, x1, x2)
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 G(TRUE, x, y, z) → F(>@z(x, z), x, +@z(y, 1@z), z) the following chains were created:

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

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

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

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

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

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

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

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

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

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

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

We simplified constraint (12) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(13)    (0 ≥ 0∧0 ≥ 0∧(U^Increasing(G(>@z(x[1], y[1]), x[1], y[1], z[1])), ≥))

We simplified constraint (13) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint:
(14)    (0 = 0∧0 = 0∧0 ≥ 0∧0 = 0∧(U^Increasing(G(>@z(x[1], y[1]), x[1], y[1], z[1])), ≥)∧0 = 0∧0 = 0∧0 ≥ 0∧0 = 0)

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

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

We simplified constraint (15) using rules (III), (IV) which results in the following new constraint:
(16)    (>@z(x[1], y[1])=TRUE∧>@z(x[1], z[2])=TRUE ⇒ G(TRUE, x[1], y[1], z[2])≥NonInfC∧G(TRUE, x[1], y[1], z[2])≥F(>@z(x[1], z[2]), x[1], y[1], +@z(z[2], 1@z))∧(U^Increasing(F(>@z(x[2], z[2]), x[2], y[2], +@z(z[2], 1@z))), ≥))

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

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

We simplified constraint (18) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(19)    (x[1] + -1 + (-1)y[1] ≥ 0∧x[1] + -1 + (-1)z[2] ≥ 0 ⇒ (U^Increasing(F(>@z(x[2], z[2]), x[2], y[2], +@z(z[2], 1@z))), ≥)∧1 + (-1)Bound + (-1)z[2] + x[1] ≥ 0∧0 ≥ 0)

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

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

We simplified constraint (21) using rule (IDP_SMT_SPLIT) which results in the following new constraints:
(22)    (x[1] ≥ 0∧z[2] ≥ 0∧y[1] ≥ 0 ⇒ (U^Increasing(F(>@z(x[2], z[2]), x[2], y[2], +@z(z[2], 1@z))), ≥)∧2 + (-1)Bound + z[2] ≥ 0∧0 ≥ 0)

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

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

G(TRUE, x, y, z) → F(>@z(x, z), x, +@z(y, 1@z), z)
- (z[0] ≥ 0∧x[1] ≥ 0∧y[1] ≥ 0 ⇒ 0 ≥ 0∧0 ≥ 0∧(U^Increasing(F(>@z(x[0], z[0]), x[0], +@z(y[0], 1@z), z[0])), ≥))
- (z[0] ≥ 0∧x[1] ≥ 0∧y[1] ≥ 0 ⇒ 0 ≥ 0∧0 ≥ 0∧(U^Increasing(F(>@z(x[0], z[0]), x[0], +@z(y[0], 1@z), z[0])), ≥))
F(TRUE, x, y, z) → G(>@z(x, y), x, y, z)
- (0 = 0∧0 = 0∧0 ≥ 0∧0 = 0∧(U^Increasing(G(>@z(x[1], y[1]), x[1], y[1], z[1])), ≥)∧0 = 0∧0 = 0∧0 ≥ 0∧0 = 0)
G(TRUE, x, y, z) → F(>@z(x, z), x, y, +@z(z, 1@z))
- (x[1] ≥ 0∧z[2] ≥ 0∧y[1] ≥ 0 ⇒ (U^Increasing(F(>@z(x[2], z[2]), x[2], y[2], +@z(z[2], 1@z))), ≥)∧2 + (-1)Bound + z[2] ≥ 0∧0 ≥ 0)
- (x[1] ≥ 0∧z[2] ≥ 0∧y[1] ≥ 0 ⇒ (U^Increasing(F(>@z(x[2], z[2]), x[2], y[2], +@z(z[2], 1@z))), ≥)∧2 + (-1)Bound + z[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(F(x₁, x₂, x₃, x₄)) = 1 + (-1)x₄ + x₂
POL(TRUE) = -1
POL(G(x₁, x₂, x₃, x₄)) = 1 + (-1)x₄ + x₂
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_>:

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

The following pairs are in P_bound:

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

The following pairs are in P_≥:

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

            ↳ IDP

              ↳ IDPNonInfProof

I DP problem:
The following domains are used:

z

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

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

The set Q consists of the following terms:

g(TRUE, x0, x1, x2)
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 G(TRUE, x[0], y[0], z[0]) → F(>@z(x[0], z[0]), x[0], +@z(y[0], 1@z), z[0]) the following chains were created:

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

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

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

We simplified constraint (5) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(6)    (x[1] ≥ 0∧y[1] + x[1] + (-1)z[0] ≥ 0 ⇒ 0 ≥ 0∧(U^Increasing(F(>@z(x[0], z[0]), x[0], +@z(y[0], 1@z), z[0])), ≥)∧3 + (-1)Bound + x[1] ≥ 0)

We simplified constraint (6) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(7)    (x[1] ≥ 0∧z[0] ≥ 0 ⇒ 0 ≥ 0∧(U^Increasing(F(>@z(x[0], z[0]), x[0], +@z(y[0], 1@z), z[0])), ≥)∧3 + (-1)Bound + x[1] ≥ 0)

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

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

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

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

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

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

We simplified constraint (12) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(13)    (0 ≥ 0∧0 ≥ 0∧(U^Increasing(G(>@z(x[1], y[1]), x[1], y[1], z[1])), ≥))

We simplified constraint (13) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint:
(14)    (0 = 0∧(U^Increasing(G(>@z(x[1], y[1]), x[1], y[1], z[1])), ≥)∧0 = 0∧0 = 0∧0 ≥ 0∧0 = 0∧0 ≥ 0∧0 = 0∧0 = 0)

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

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

POL(F(x₁, x₂, x₃, x₄)) = 2 + (-1)x₃ + x₂
POL(TRUE) = -1
POL(G(x₁, x₂, x₃, x₄)) = 2 + (-1)x₃ + x₂
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_>:

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

The following pairs are in P_bound:

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

The following pairs are in P_≥:

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

            ↳ 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(TRUE, x[1], y[1], z[1]) → G(>@z(x[1], y[1]), x[1], y[1], z[1])

The set Q consists of the following terms:

g(TRUE, x0, x1, x2)
f(TRUE, x0, x1, x2)

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