Termination proof of /tmp/tmpd9i9Q4/log.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_log(TRUE, x, y) → +@z(1@z, log(/@z(-@z(x, y), y), y))
Cond_log1(TRUE, y) → 0@z
log(x, y) → Cond_log(&&(>=@z(x, 2@z), >=@z(y, 2@z)), x, y)
log(1@z, y) → Cond_log1(>=@z(y, 2@z), y)

The set Q consists of the following terms:

Cond_log(TRUE, x0, x1)
Cond_log1(TRUE, x0)
log(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_log(TRUE, x, y) → +@z(1@z, log(/@z(-@z(x, y), y), y))
Cond_log1(TRUE, y) → 0@z
log(x, y) → Cond_log(&&(>=@z(x, 2@z), >=@z(y, 2@z)), x, y)
log(1@z, y) → Cond_log1(>=@z(y, 2@z), y)

The integer pair graph contains the following rules and edges:

(0): LOG(1@z, y[0]) → COND_LOG1(>=@z(y[0], 2@z), y[0])
(1): COND_LOG(TRUE, x[1], y[1]) → LOG(/@z(-@z(x[1], y[1]), y[1]), y[1])
(2): LOG(x[2], y[2]) → COND_LOG(&&(>=@z(x[2], 2@z), >=@z(y[2], 2@z)), x[2], y[2])

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

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

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

The set Q consists of the following terms:

Cond_log(TRUE, x0, x1)
Cond_log1(TRUE, x0)
log(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

          ↳ IDependencyGraphProof

I DP problem:
The following domains are used:

z

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

(0): LOG(1@z, y[0]) → COND_LOG1(>=@z(y[0], 2@z), y[0])
(1): COND_LOG(TRUE, x[1], y[1]) → LOG(/@z(-@z(x[1], y[1]), y[1]), y[1])
(2): LOG(x[2], y[2]) → COND_LOG(&&(>=@z(x[2], 2@z), >=@z(y[2], 2@z)), x[2], y[2])

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

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

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

The set Q consists of the following terms:

Cond_log(TRUE, x0, x1)
Cond_log1(TRUE, x0)
log(x0, x1)

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

↳ ITRS

  ↳ ITRStoIDPProof

    ↳ IDP

      ↳ UsableRulesProof

        ↳ 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): COND_LOG(TRUE, x[1], y[1]) → LOG(/@z(-@z(x[1], y[1]), y[1]), y[1])
(2): LOG(x[2], y[2]) → COND_LOG(&&(>=@z(x[2], 2@z), >=@z(y[2], 2@z)), x[2], y[2])

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

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

The set Q consists of the following terms:

Cond_log(TRUE, x0, x1)
Cond_log1(TRUE, x0)
log(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_LOG(TRUE, x[1], y[1]) → LOG(/@z(-@z(x[1], y[1]), y[1]), y[1]) the following chains were created:

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

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

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

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

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

(6)    (-1 + (2)y[2] + (-1)x[2] ≥ 0∧(2)y[2] ≥ 0∧(2)x[2] + (-2)y[2] ≥ 0∧-2 + x[2] ≥ 0∧y[2] + -2 ≥ 0 ⇒ (U^Increasing(LOG(/@z(-@z(x[1], y[1]), y[1]), y[1])), ≥)∧-2 + (-1)Bound + x[2] ≥ 0∧-1 + x[2] ≥ 0)

(7)    ((2)y[2] ≥ 0∧-2 + x[2] ≥ 0∧-1 + x[2] ≥ 0∧y[2] + -2 ≥ 0∧-1 + (-2)x[2] + (2)y[2] ≥ 0 ⇒ (U^Increasing(LOG(/@z(-@z(x[1], y[1]), y[1]), y[1])), ≥)∧-2 + (-1)Bound + x[2] ≥ 0∧-1 + x[2] ≥ 0)

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

We simplified constraint (6) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(9)    (-1 + y[2] + (-1)x[2] ≥ 0∧(2)y[2] ≥ 0∧(2)x[2] ≥ 0∧-2 + y[2] + x[2] ≥ 0∧y[2] + -2 ≥ 0 ⇒ (U^Increasing(LOG(/@z(-@z(x[1], y[1]), y[1]), y[1])), ≥)∧-2 + (-1)Bound + y[2] + x[2] ≥ 0∧-1 + y[2] + x[2] ≥ 0)

We simplified constraint (7) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(10)    ((2)y[2] ≥ 0∧-1 + x[2] ≥ 0∧x[2] ≥ 0∧y[2] + -2 ≥ 0∧-3 + (-2)x[2] + (2)y[2] ≥ 0 ⇒ (U^Increasing(LOG(/@z(-@z(x[1], y[1]), y[1]), y[1])), ≥)∧-1 + (-1)Bound + x[2] ≥ 0∧x[2] ≥ 0)

We simplified constraint (8) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(11)    (4 + (2)y[2] ≥ 0∧4 + (2)y[2] + (2)x[2] ≥ 0∧2 + (2)y[2] + x[2] ≥ 0∧y[2] ≥ 0∧x[2] ≥ 0 ⇒ (U^Increasing(LOG(/@z(-@z(x[1], y[1]), y[1]), y[1])), ≥)∧2 + (-1)Bound + (2)y[2] + x[2] ≥ 0∧2 + (2)y[2] ≥ 0)

We simplified constraint (9) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(12)    (1 + y[2] + (-1)x[2] ≥ 0∧4 + (2)y[2] ≥ 0∧(2)x[2] ≥ 0∧y[2] + x[2] ≥ 0∧y[2] ≥ 0 ⇒ (U^Increasing(LOG(/@z(-@z(x[1], y[1]), y[1]), y[1])), ≥)∧(-1)Bound + y[2] + x[2] ≥ 0∧1 + y[2] + x[2] ≥ 0)

We simplified constraint (10) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(13)    ((2)y[2] ≥ 0∧x[2] ≥ 0∧1 + x[2] ≥ 0∧y[2] + -2 ≥ 0∧-5 + (-2)x[2] + (2)y[2] ≥ 0 ⇒ (U^Increasing(LOG(/@z(-@z(x[1], y[1]), y[1]), y[1])), ≥)∧(-1)Bound + x[2] ≥ 0∧1 + x[2] ≥ 0)

We simplified constraint (13) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(14)    (4 + (2)y[2] ≥ 0∧x[2] ≥ 0∧1 + x[2] ≥ 0∧y[2] ≥ 0∧-1 + (-2)x[2] + (2)y[2] ≥ 0 ⇒ (U^Increasing(LOG(/@z(-@z(x[1], y[1]), y[1]), y[1])), ≥)∧(-1)Bound + x[2] ≥ 0∧1 + x[2] ≥ 0)

For Pair LOG(x[2], y[2]) → COND_LOG(&&(>=@z(x[2], 2@z), >=@z(y[2], 2@z)), x[2], y[2]) the following chains were created:

We consider the chain LOG(x[2], y[2]) → COND_LOG(&&(>=@z(x[2], 2@z), >=@z(y[2], 2@z)), x[2], y[2]) which results in the following constraint:
(15)    (LOG(x[2], y[2])≥NonInfC∧LOG(x[2], y[2])≥COND_LOG(&&(>=@z(x[2], 2@z), >=@z(y[2], 2@z)), x[2], y[2])∧(U^Increasing(COND_LOG(&&(>=@z(x[2], 2@z), >=@z(y[2], 2@z)), x[2], y[2])), ≥))

We simplified constraint (15) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(16)    ((U^Increasing(COND_LOG(&&(>=@z(x[2], 2@z), >=@z(y[2], 2@z)), x[2], y[2])), ≥)∧0 ≥ 0∧0 ≥ 0)

We simplified constraint (16) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(17)    ((U^Increasing(COND_LOG(&&(>=@z(x[2], 2@z), >=@z(y[2], 2@z)), x[2], y[2])), ≥)∧0 ≥ 0∧0 ≥ 0)

We simplified constraint (17) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(18)    (0 ≥ 0∧0 ≥ 0∧(U^Increasing(COND_LOG(&&(>=@z(x[2], 2@z), >=@z(y[2], 2@z)), x[2], y[2])), ≥))

We simplified constraint (18) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint:
(19)    (0 = 0∧0 ≥ 0∧0 = 0∧0 = 0∧0 = 0∧(U^Increasing(COND_LOG(&&(>=@z(x[2], 2@z), >=@z(y[2], 2@z)), x[2], y[2])), ≥)∧0 ≥ 0)

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

COND_LOG(TRUE, x[1], y[1]) → LOG(/@z(-@z(x[1], y[1]), y[1]), y[1])
- (4 + (2)y[2] ≥ 0∧4 + (2)y[2] + (2)x[2] ≥ 0∧2 + (2)y[2] + x[2] ≥ 0∧y[2] ≥ 0∧x[2] ≥ 0 ⇒ (U^Increasing(LOG(/@z(-@z(x[1], y[1]), y[1]), y[1])), ≥)∧2 + (-1)Bound + (2)y[2] + x[2] ≥ 0∧2 + (2)y[2] ≥ 0)
- (1 + y[2] + (-1)x[2] ≥ 0∧4 + (2)y[2] ≥ 0∧(2)x[2] ≥ 0∧y[2] + x[2] ≥ 0∧y[2] ≥ 0 ⇒ (U^Increasing(LOG(/@z(-@z(x[1], y[1]), y[1]), y[1])), ≥)∧(-1)Bound + y[2] + x[2] ≥ 0∧1 + y[2] + x[2] ≥ 0)
- (4 + (2)y[2] ≥ 0∧x[2] ≥ 0∧1 + x[2] ≥ 0∧y[2] ≥ 0∧-1 + (-2)x[2] + (2)y[2] ≥ 0 ⇒ (U^Increasing(LOG(/@z(-@z(x[1], y[1]), y[1]), y[1])), ≥)∧(-1)Bound + x[2] ≥ 0∧1 + x[2] ≥ 0)
LOG(x[2], y[2]) → COND_LOG(&&(>=@z(x[2], 2@z), >=@z(y[2], 2@z)), x[2], y[2])
- (0 = 0∧0 ≥ 0∧0 = 0∧0 = 0∧0 = 0∧(U^Increasing(COND_LOG(&&(>=@z(x[2], 2@z), >=@z(y[2], 2@z)), x[2], y[2])), ≥)∧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(-@z(x₁, x₂)) = x₁ + (-1)x₂
POL(>=@z(x₁, x₂)) = -1
POL(TRUE) = 1
POL(&&(x₁, x₂)) = 1
POL(2@z) = 2
POL(LOG(x₁, x₂)) = -1 + x₁
POL(FALSE) = 1
POL(COND_LOG(x₁, x₂, x₃)) = -1 + x₂ + (-1)x₁
POL(undefined) = -1

Polynomial Interpretations with Context Sensitive Arithemetic Replacement
POL(Term^CSAR-Mode @ Context)

POL(/@z(x₁, y[2])¹ @ {LOG_2/0}) = max{x₁, (-1)x₁} + (-1)min{max{x₂, (-1)x₂} + -1, max{x₁, (-1)x₁}}

The following pairs are in P_>:

LOG(x[2], y[2]) → COND_LOG(&&(>=@z(x[2], 2@z), >=@z(y[2], 2@z)), x[2], y[2])

The following pairs are in P_bound:

COND_LOG(TRUE, x[1], y[1]) → LOG(/@z(-@z(x[1], y[1]), y[1]), y[1])

The following pairs are in P_≥:

COND_LOG(TRUE, x[1], y[1]) → LOG(/@z(-@z(x[1], y[1]), y[1]), y[1])

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

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

↳ ITRS

  ↳ ITRStoIDPProof

    ↳ IDP

      ↳ UsableRulesProof

        ↳ IDP

          ↳ IDependencyGraphProof

            ↳ 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:

(2): LOG(x[2], y[2]) → COND_LOG(&&(>=@z(x[2], 2@z), >=@z(y[2], 2@z)), x[2], y[2])

The set Q consists of the following terms:

Cond_log(TRUE, x0, x1)
Cond_log1(TRUE, x0)
log(x0, x1)

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

↳ ITRS

  ↳ ITRStoIDPProof

    ↳ IDP

      ↳ UsableRulesProof

        ↳ IDP

          ↳ IDependencyGraphProof

            ↳ IDP

              ↳ IDPNonInfProof

                ↳ AND

                  ↳ IDP

                  ↳ 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): COND_LOG(TRUE, x[1], y[1]) → LOG(/@z(-@z(x[1], y[1]), y[1]), y[1])

The set Q consists of the following terms:

Cond_log(TRUE, x0, x1)
Cond_log1(TRUE, x0)
log(x0, x1)

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