Termination proof of /tmp/tmpq8T-xf/04.itrs

Termination of the given ITRSProblem could successfully be proven:

↳ ITRS

  ↳ ITRStoIDPProof

ITRS problem:
The following domains are used:

z

The TRS R consists of the following rules:

Cond_eval(TRUE, x, y) → eval(y, x)
eval(x, y) → Cond_eval(>@z(x, y), x, y)

The set Q consists of the following terms:

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

Added dependency pairs

↳ ITRS

  ↳ ITRStoIDPProof

    ↳ IDP

      ↳ UsableRulesProof

I DP problem:
The following domains are used:

z

The ITRS R consists of the following rules:

Cond_eval(TRUE, x, y) → eval(y, x)
eval(x, y) → Cond_eval(>@z(x, y), x, y)

The integer pair graph contains the following rules and edges:

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

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

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

The set Q consists of the following terms:

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

As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [LPAR04] we can delete all non-usable rules [FROCOS05] from R.

↳ ITRS

  ↳ ITRStoIDPProof

    ↳ IDP

      ↳ UsableRulesProof

        ↳ IDP

          ↳ IDPNonInfProof

I DP problem:
The following domains are used:

z

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

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

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

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

The set Q consists of the following terms:

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

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

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

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

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

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

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

(8)    (x[1] ≥ 0∧y[1] ≥ 0 ⇒ (2)x[1] ≥ 0∧(U^Increasing(EVAL(y[0], x[0])), ≥)∧1 + (-1)Bound + x[1] ≥ 0)

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

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

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

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

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

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

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

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

The following pairs are in P_>:

COND_EVAL(TRUE, x[0], y[0]) → EVAL(y[0], x[0])

The following pairs are in P_bound:

COND_EVAL(TRUE, x[0], y[0]) → EVAL(y[0], x[0])

The following pairs are in P_≥:

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

There are no usable rules.

↳ ITRS

  ↳ ITRStoIDPProof

    ↳ IDP

      ↳ UsableRulesProof

        ↳ 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): EVAL(x[1], y[1]) → COND_EVAL(>@z(x[1], y[1]), x[1], y[1])

The set Q consists of the following terms:

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

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