Termination proof of /tmp/tmpCuPTd1/A17.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(TRUE, x, y) → 1@z
cond(FALSE, x, y) → *@z(2@z, log(x, *@z(y, y)))
logNat(TRUE, x, y) → cond(<=@z(x, y), x, y)
log(x, y) → logNat(&&(>=@z(x, 0@z), >=@z(y, 2@z)), x, y)

The set Q consists of the following terms:

cond(TRUE, x0, x1)
cond(FALSE, x0, x1)
logNat(TRUE, x0, x1)
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(TRUE, x, y) → 1@z
cond(FALSE, x, y) → *@z(2@z, log(x, *@z(y, y)))
logNat(TRUE, x, y) → cond(<=@z(x, y), x, y)
log(x, y) → logNat(&&(>=@z(x, 0@z), >=@z(y, 2@z)), x, y)

The integer pair graph contains the following rules and edges:

(0): COND(FALSE, x[0], y[0]) → LOG(x[0], *@z(y[0], y[0]))
(1): LOGNAT(TRUE, x[1], y[1]) → COND(<=@z(x[1], y[1]), x[1], y[1])
(2): LOG(x[2], y[2]) → LOGNAT(&&(>=@z(x[2], 0@z), >=@z(y[2], 2@z)), x[2], y[2])

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

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

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

The set Q consists of the following terms:

cond(TRUE, x0, x1)
cond(FALSE, x0, x1)
logNat(TRUE, x0, x1)
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

          ↳ 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(FALSE, x[0], y[0]) → LOG(x[0], *@z(y[0], y[0]))
(1): LOGNAT(TRUE, x[1], y[1]) → COND(<=@z(x[1], y[1]), x[1], y[1])
(2): LOG(x[2], y[2]) → LOGNAT(&&(>=@z(x[2], 0@z), >=@z(y[2], 2@z)), x[2], y[2])

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

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

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

The set Q consists of the following terms:

cond(TRUE, x0, x1)
cond(FALSE, x0, x1)
logNat(TRUE, x0, x1)
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(FALSE, x, y) → LOG(x, *@z(y, y)) the following chains were created:

We consider the chain LOG(x[2], y[2]) → LOGNAT(&&(>=@z(x[2], 0@z), >=@z(y[2], 2@z)), x[2], y[2]), LOGNAT(TRUE, x[1], y[1]) → COND(<=@z(x[1], y[1]), x[1], y[1]), COND(FALSE, x[0], y[0]) → LOG(x[0], *@z(y[0], y[0])) which results in the following constraint:
(1)    (y[2]=y[1]∧x[2]=x[1]∧<=@z(x[1], y[1])=FALSE∧y[1]=y[0]∧x[1]=x[0]∧&&(>=@z(x[2], 0@z), >=@z(y[2], 2@z))=TRUE ⇒ COND(FALSE, x[0], y[0])≥NonInfC∧COND(FALSE, x[0], y[0])≥LOG(x[0], *@z(y[0], y[0]))∧(U^Increasing(LOG(x[0], *@z(y[0], y[0]))), ≥))

We simplified constraint (1) using rules (III), (IDP_BOOLEAN) which results in the following new constraint:
(2)    (<=@z(x[2], y[2])=FALSE∧>=@z(x[2], 0@z)=TRUE∧>=@z(y[2], 2@z)=TRUE ⇒ COND(FALSE, x[2], y[2])≥NonInfC∧COND(FALSE, x[2], y[2])≥LOG(x[2], *@z(y[2], y[2]))∧(U^Increasing(LOG(x[0], *@z(y[0], y[0]))), ≥))

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

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

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

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

We simplified constraint (6) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(7)    (y[2] ≥ 0∧x[2] ≥ 0∧3 + y[2] + x[2] ≥ 0 ⇒ (U^Increasing(LOG(x[0], *@z(y[0], y[0]))), ≥)∧(-1)Bound + x[2] ≥ 0∧1 + (3)y[2] + y[2]² ≥ 0)

For Pair LOGNAT(TRUE, x, y) → COND(<=@z(x, y), x, y) the following chains were created:

We consider the chain LOGNAT(TRUE, x[1], y[1]) → COND(<=@z(x[1], y[1]), x[1], y[1]) which results in the following constraint:
(8)    (LOGNAT(TRUE, x[1], y[1])≥NonInfC∧LOGNAT(TRUE, x[1], y[1])≥COND(<=@z(x[1], y[1]), x[1], y[1])∧(U^Increasing(COND(<=@z(x[1], y[1]), x[1], y[1])), ≥))

We simplified constraint (8) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(9)    ((U^Increasing(COND(<=@z(x[1], y[1]), x[1], y[1])), ≥)∧0 ≥ 0∧0 ≥ 0)

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

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

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

For Pair LOG(x, y) → LOGNAT(&&(>=@z(x, 0@z), >=@z(y, 2@z)), x, y) the following chains were created:

We consider the chain LOG(x[2], y[2]) → LOGNAT(&&(>=@z(x[2], 0@z), >=@z(y[2], 2@z)), x[2], y[2]) which results in the following constraint:
(13)    (LOG(x[2], y[2])≥NonInfC∧LOG(x[2], y[2])≥LOGNAT(&&(>=@z(x[2], 0@z), >=@z(y[2], 2@z)), x[2], y[2])∧(U^Increasing(LOGNAT(&&(>=@z(x[2], 0@z), >=@z(y[2], 2@z)), x[2], y[2])), ≥))

We simplified constraint (13) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(14)    ((U^Increasing(LOGNAT(&&(>=@z(x[2], 0@z), >=@z(y[2], 2@z)), x[2], y[2])), ≥)∧0 ≥ 0∧0 ≥ 0)

We simplified constraint (14) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(15)    ((U^Increasing(LOGNAT(&&(>=@z(x[2], 0@z), >=@z(y[2], 2@z)), x[2], y[2])), ≥)∧0 ≥ 0∧0 ≥ 0)

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

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

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

COND(FALSE, x, y) → LOG(x, *@z(y, y))
- (y[2] ≥ 0∧x[2] ≥ 0∧3 + y[2] + x[2] ≥ 0 ⇒ (U^Increasing(LOG(x[0], *@z(y[0], y[0]))), ≥)∧(-1)Bound + x[2] ≥ 0∧1 + (3)y[2] + y[2]² ≥ 0)
LOGNAT(TRUE, x, y) → COND(<=@z(x, y), x, y)
- (0 = 0∧0 ≥ 0∧0 = 0∧0 = 0∧(U^Increasing(COND(<=@z(x[1], y[1]), x[1], y[1])), ≥)∧0 ≥ 0∧0 = 0)
LOG(x, y) → LOGNAT(&&(>=@z(x, 0@z), >=@z(y, 2@z)), x, y)
- (0 = 0∧0 = 0∧0 = 0∧(U^Increasing(LOGNAT(&&(>=@z(x[2], 0@z), >=@z(y[2], 2@z)), x[2], y[2])), ≥)∧0 ≥ 0∧0 = 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(<=@z(x₁, x₂)) = -1
POL(LOGNAT(x₁, x₂, x₃)) = -1 + x₂ + (-1)x₃
POL(>=@z(x₁, x₂)) = -1
POL(0@z) = 0
POL(*@z(x₁, x₂)) = x₁·x₂
POL(TRUE) = -1
POL(&&(x₁, x₂)) = 1
POL(2@z) = 2
POL(LOG(x₁, x₂)) = -1 + x₁ + (-1)x₂
POL(FALSE) = -1
POL(COND(x₁, x₂, x₃)) = -1 + x₂ + (-1)x₃
POL(undefined) = -1

The following pairs are in P_>:

COND(FALSE, x[0], y[0]) → LOG(x[0], *@z(y[0], y[0]))

The following pairs are in P_bound:

COND(FALSE, x[0], y[0]) → LOG(x[0], *@z(y[0], y[0]))

The following pairs are in P_≥:

LOGNAT(TRUE, x[1], y[1]) → COND(<=@z(x[1], y[1]), x[1], y[1])
LOG(x[2], y[2]) → LOGNAT(&&(>=@z(x[2], 0@z), >=@z(y[2], 2@z)), x[2], y[2])

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

*@z¹ ↔
&&(FALSE, FALSE)¹ → FALSE¹
&&(TRUE, TRUE)¹ → TRUE¹
&&(TRUE, FALSE)¹ → FALSE¹
&&(FALSE, TRUE)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:

(1): LOGNAT(TRUE, x[1], y[1]) → COND(<=@z(x[1], y[1]), x[1], y[1])
(2): LOG(x[2], y[2]) → LOGNAT(&&(>=@z(x[2], 0@z), >=@z(y[2], 2@z)), x[2], y[2])

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

The set Q consists of the following terms:

cond(TRUE, x0, x1)
cond(FALSE, x0, x1)
logNat(TRUE, x0, x1)
log(x0, x1)

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