Termination proof of /tmp/tmpJPtuTT/terminate.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, i, j, k) → eval(j, +@z(i, 1@z), -@z(k, 1@z))
eval(i, j, k) → Cond_eval(&&(<=@z(i, 100@z), <=@z(j, k)), i, j, k)

The set Q consists of the following terms:

Cond_eval(TRUE, x0, x1, x2)
eval(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_eval(TRUE, i, j, k) → eval(j, +@z(i, 1@z), -@z(k, 1@z))
eval(i, j, k) → Cond_eval(&&(<=@z(i, 100@z), <=@z(j, k)), i, j, k)

The integer pair graph contains the following rules and edges:

(0): EVAL(i[0], j[0], k[0]) → COND_EVAL(&&(<=@z(i[0], 100@z), <=@z(j[0], k[0])), i[0], j[0], k[0])
(1): COND_EVAL(TRUE, i[1], j[1], k[1]) → EVAL(j[1], +@z(i[1], 1@z), -@z(k[1], 1@z))

(0) -> (1), if ((k[0] →^* k[1])∧(i[0] →^* i[1])∧(j[0] →^* j[1])∧(&&(<=@z(i[0], 100@z), <=@z(j[0], k[0])) →^* TRUE))

(1) -> (0), if ((+@z(i[1], 1@z) →^* j[0])∧(-@z(k[1], 1@z) →^* k[0])∧(j[1] →^* i[0]))

The set Q consists of the following terms:

Cond_eval(TRUE, x0, x1, x2)
eval(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): EVAL(i[0], j[0], k[0]) → COND_EVAL(&&(<=@z(i[0], 100@z), <=@z(j[0], k[0])), i[0], j[0], k[0])
(1): COND_EVAL(TRUE, i[1], j[1], k[1]) → EVAL(j[1], +@z(i[1], 1@z), -@z(k[1], 1@z))

(0) -> (1), if ((k[0] →^* k[1])∧(i[0] →^* i[1])∧(j[0] →^* j[1])∧(&&(<=@z(i[0], 100@z), <=@z(j[0], k[0])) →^* TRUE))

(1) -> (0), if ((+@z(i[1], 1@z) →^* j[0])∧(-@z(k[1], 1@z) →^* k[0])∧(j[1] →^* i[0]))

The set Q consists of the following terms:

Cond_eval(TRUE, x0, x1, x2)
eval(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 EVAL(i, j, k) → COND_EVAL(&&(<=@z(i, 100@z), <=@z(j, k)), i, j, k) the following chains were created:

We consider the chain EVAL(i[0], j[0], k[0]) → COND_EVAL(&&(<=@z(i[0], 100@z), <=@z(j[0], k[0])), i[0], j[0], k[0]) which results in the following constraint:
(1)    (EVAL(i[0], j[0], k[0])≥NonInfC∧EVAL(i[0], j[0], k[0])≥COND_EVAL(&&(<=@z(i[0], 100@z), <=@z(j[0], k[0])), i[0], j[0], k[0])∧(U^Increasing(COND_EVAL(&&(<=@z(i[0], 100@z), <=@z(j[0], k[0])), i[0], j[0], k[0])), ≥))

We simplified constraint (1) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(2)    ((U^Increasing(COND_EVAL(&&(<=@z(i[0], 100@z), <=@z(j[0], k[0])), i[0], j[0], k[0])), ≥)∧0 ≥ 0∧0 ≥ 0)

We simplified constraint (2) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(3)    ((U^Increasing(COND_EVAL(&&(<=@z(i[0], 100@z), <=@z(j[0], k[0])), i[0], j[0], k[0])), ≥)∧0 ≥ 0∧0 ≥ 0)

We simplified constraint (3) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(4)    (0 ≥ 0∧(U^Increasing(COND_EVAL(&&(<=@z(i[0], 100@z), <=@z(j[0], k[0])), i[0], j[0], k[0])), ≥)∧0 ≥ 0)

We simplified constraint (4) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint:
(5)    (0 ≥ 0∧0 = 0∧(U^Increasing(COND_EVAL(&&(<=@z(i[0], 100@z), <=@z(j[0], k[0])), i[0], j[0], k[0])), ≥)∧0 ≥ 0∧0 = 0∧0 = 0∧0 = 0∧0 = 0∧0 = 0)

For Pair COND_EVAL(TRUE, i, j, k) → EVAL(j, +@z(i, 1@z), -@z(k, 1@z)) the following chains were created:

We consider the chain EVAL(i[0], j[0], k[0]) → COND_EVAL(&&(<=@z(i[0], 100@z), <=@z(j[0], k[0])), i[0], j[0], k[0]), COND_EVAL(TRUE, i[1], j[1], k[1]) → EVAL(j[1], +@z(i[1], 1@z), -@z(k[1], 1@z)), EVAL(i[0], j[0], k[0]) → COND_EVAL(&&(<=@z(i[0], 100@z), <=@z(j[0], k[0])), i[0], j[0], k[0]) which results in the following constraint:
(6)    (j[1]=i[0]1∧-@z(k[1], 1@z)=k[0]1∧i[0]=i[1]∧&&(<=@z(i[0], 100@z), <=@z(j[0], k[0]))=TRUE∧k[0]=k[1]∧j[0]=j[1]∧+@z(i[1], 1@z)=j[0]1 ⇒ COND_EVAL(TRUE, i[1], j[1], k[1])≥NonInfC∧COND_EVAL(TRUE, i[1], j[1], k[1])≥EVAL(j[1], +@z(i[1], 1@z), -@z(k[1], 1@z))∧(U^Increasing(EVAL(j[1], +@z(i[1], 1@z), -@z(k[1], 1@z))), ≥))

We simplified constraint (6) using rules (III), (IV), (IDP_BOOLEAN) which results in the following new constraint:
(7)    (<=@z(i[0], 100@z)=TRUE∧<=@z(j[0], k[0])=TRUE ⇒ COND_EVAL(TRUE, i[0], j[0], k[0])≥NonInfC∧COND_EVAL(TRUE, i[0], j[0], k[0])≥EVAL(j[0], +@z(i[0], 1@z), -@z(k[0], 1@z))∧(U^Increasing(EVAL(j[1], +@z(i[1], 1@z), -@z(k[1], 1@z))), ≥))

We simplified constraint (7) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(8)    (100 + (-1)i[0] ≥ 0∧k[0] + (-1)j[0] ≥ 0 ⇒ (U^Increasing(EVAL(j[1], +@z(i[1], 1@z), -@z(k[1], 1@z))), ≥)∧-1 + (-1)Bound + k[0] + (-1)j[0] + (-1)i[0] ≥ 0∧1 ≥ 0)

We simplified constraint (8) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(9)    (100 + (-1)i[0] ≥ 0∧k[0] + (-1)j[0] ≥ 0 ⇒ (U^Increasing(EVAL(j[1], +@z(i[1], 1@z), -@z(k[1], 1@z))), ≥)∧-1 + (-1)Bound + k[0] + (-1)j[0] + (-1)i[0] ≥ 0∧1 ≥ 0)

We simplified constraint (9) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(10)    (100 + (-1)i[0] ≥ 0∧k[0] + (-1)j[0] ≥ 0 ⇒ -1 + (-1)Bound + k[0] + (-1)j[0] + (-1)i[0] ≥ 0∧(U^Increasing(EVAL(j[1], +@z(i[1], 1@z), -@z(k[1], 1@z))), ≥)∧1 ≥ 0)

We simplified constraint (10) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(11)    (100 + (-1)i[0] ≥ 0∧j[0] ≥ 0 ⇒ -1 + (-1)Bound + j[0] + (-1)i[0] ≥ 0∧(U^Increasing(EVAL(j[1], +@z(i[1], 1@z), -@z(k[1], 1@z))), ≥)∧1 ≥ 0)

We simplified constraint (11) using rule (IDP_SMT_SPLIT) which results in the following new constraints:
(12)    (100 + (-1)i[0] ≥ 0∧j[0] ≥ 0∧k[0] ≥ 0 ⇒ -1 + (-1)Bound + j[0] + (-1)i[0] ≥ 0∧(U^Increasing(EVAL(j[1], +@z(i[1], 1@z), -@z(k[1], 1@z))), ≥)∧1 ≥ 0)

(13)    (100 + (-1)i[0] ≥ 0∧j[0] ≥ 0∧k[0] ≥ 0 ⇒ -1 + (-1)Bound + j[0] + (-1)i[0] ≥ 0∧(U^Increasing(EVAL(j[1], +@z(i[1], 1@z), -@z(k[1], 1@z))), ≥)∧1 ≥ 0)

We simplified constraint (12) using rule (IDP_SMT_SPLIT) which results in the following new constraints:
(14)    (100 + (-1)i[0] ≥ 0∧j[0] ≥ 0∧k[0] ≥ 0∧i[0] ≥ 0 ⇒ -1 + (-1)Bound + j[0] + (-1)i[0] ≥ 0∧(U^Increasing(EVAL(j[1], +@z(i[1], 1@z), -@z(k[1], 1@z))), ≥)∧1 ≥ 0)

(15)    (100 + i[0] ≥ 0∧j[0] ≥ 0∧k[0] ≥ 0∧i[0] ≥ 0 ⇒ -1 + (-1)Bound + j[0] + i[0] ≥ 0∧(U^Increasing(EVAL(j[1], +@z(i[1], 1@z), -@z(k[1], 1@z))), ≥)∧1 ≥ 0)

We simplified constraint (13) using rule (IDP_SMT_SPLIT) which results in the following new constraints:
(16)    (100 + (-1)i[0] ≥ 0∧j[0] ≥ 0∧k[0] ≥ 0∧i[0] ≥ 0 ⇒ -1 + (-1)Bound + j[0] + (-1)i[0] ≥ 0∧(U^Increasing(EVAL(j[1], +@z(i[1], 1@z), -@z(k[1], 1@z))), ≥)∧1 ≥ 0)

(17)    (100 + i[0] ≥ 0∧j[0] ≥ 0∧k[0] ≥ 0∧i[0] ≥ 0 ⇒ -1 + (-1)Bound + j[0] + i[0] ≥ 0∧(U^Increasing(EVAL(j[1], +@z(i[1], 1@z), -@z(k[1], 1@z))), ≥)∧1 ≥ 0)

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

EVAL(i, j, k) → COND_EVAL(&&(<=@z(i, 100@z), <=@z(j, k)), i, j, k)
- (0 ≥ 0∧0 = 0∧(U^Increasing(COND_EVAL(&&(<=@z(i[0], 100@z), <=@z(j[0], k[0])), i[0], j[0], k[0])), ≥)∧0 ≥ 0∧0 = 0∧0 = 0∧0 = 0∧0 = 0∧0 = 0)
COND_EVAL(TRUE, i, j, k) → EVAL(j, +@z(i, 1@z), -@z(k, 1@z))
- (100 + (-1)i[0] ≥ 0∧j[0] ≥ 0∧k[0] ≥ 0∧i[0] ≥ 0 ⇒ -1 + (-1)Bound + j[0] + (-1)i[0] ≥ 0∧(U^Increasing(EVAL(j[1], +@z(i[1], 1@z), -@z(k[1], 1@z))), ≥)∧1 ≥ 0)
- (100 + i[0] ≥ 0∧j[0] ≥ 0∧k[0] ≥ 0∧i[0] ≥ 0 ⇒ -1 + (-1)Bound + j[0] + i[0] ≥ 0∧(U^Increasing(EVAL(j[1], +@z(i[1], 1@z), -@z(k[1], 1@z))), ≥)∧1 ≥ 0)
- (100 + (-1)i[0] ≥ 0∧j[0] ≥ 0∧k[0] ≥ 0∧i[0] ≥ 0 ⇒ -1 + (-1)Bound + j[0] + (-1)i[0] ≥ 0∧(U^Increasing(EVAL(j[1], +@z(i[1], 1@z), -@z(k[1], 1@z))), ≥)∧1 ≥ 0)
- (100 + i[0] ≥ 0∧j[0] ≥ 0∧k[0] ≥ 0∧i[0] ≥ 0 ⇒ -1 + (-1)Bound + j[0] + i[0] ≥ 0∧(U^Increasing(EVAL(j[1], +@z(i[1], 1@z), -@z(k[1], 1@z))), ≥)∧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(-@z(x₁, x₂)) = x₁ + (-1)x₂
POL(<=@z(x₁, x₂)) = -1
POL(100@z) = 100
POL(TRUE) = 0
POL(&&(x₁, x₂)) = 0
POL(+@z(x₁, x₂)) = x₁ + x₂
POL(COND_EVAL(x₁, x₂, x₃, x₄)) = -1 + x₄ + (-1)x₃ + (-1)x₂ + (-1)x₁
POL(EVAL(x₁, x₂, x₃)) = -1 + x₃ + (-1)x₂ + (-1)x₁
POL(FALSE) = 0
POL(1@z) = 1
POL(undefined) = -1

The following pairs are in P_>:

COND_EVAL(TRUE, i[1], j[1], k[1]) → EVAL(j[1], +@z(i[1], 1@z), -@z(k[1], 1@z))

The following pairs are in P_bound:

COND_EVAL(TRUE, i[1], j[1], k[1]) → EVAL(j[1], +@z(i[1], 1@z), -@z(k[1], 1@z))

The following pairs are in P_≥:

EVAL(i[0], j[0], k[0]) → COND_EVAL(&&(<=@z(i[0], 100@z), <=@z(j[0], k[0])), i[0], j[0], k[0])

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

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

↳ 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:

(0): EVAL(i[0], j[0], k[0]) → COND_EVAL(&&(<=@z(i[0], 100@z), <=@z(j[0], k[0])), i[0], j[0], k[0])

The set Q consists of the following terms:

Cond_eval(TRUE, x0, x1, x2)
eval(x0, x1, x2)

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