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(-@z(x, 1@z), y)
eval(x, y) → Cond_eval(&&(&&(>@z(x, 0@z), !(=@z(x, 0@z))), <=@z(x, y)), x, y)
Cond_eval1(TRUE, x, y) → eval(y, y)
eval(x, y) → Cond_eval1(&&(&&(>@z(x, 0@z), !(=@z(x, 0@z))), >@z(x, y)), x, y)

The set Q consists of the following terms:

Cond_eval(TRUE, x0, x1)
eval(x0, x1)
Cond_eval1(TRUE, 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(-@z(x, 1@z), y)
eval(x, y) → Cond_eval(&&(&&(>@z(x, 0@z), !(=@z(x, 0@z))), <=@z(x, y)), x, y)
Cond_eval1(TRUE, x, y) → eval(y, y)
eval(x, y) → Cond_eval1(&&(&&(>@z(x, 0@z), !(=@z(x, 0@z))), >@z(x, y)), x, y)

The integer pair graph contains the following rules and edges:

(0): EVAL(x[0], y[0]) → COND_EVAL1(&&(&&(>@z(x[0], 0@z), !(=@z(x[0], 0@z))), >@z(x[0], y[0])), x[0], y[0])
(1): EVAL(x[1], y[1]) → COND_EVAL(&&(&&(>@z(x[1], 0@z), !(=@z(x[1], 0@z))), <=@z(x[1], y[1])), x[1], y[1])
(2): COND_EVAL(TRUE, x[2], y[2]) → EVAL(-@z(x[2], 1@z), y[2])
(3): COND_EVAL1(TRUE, x[3], y[3]) → EVAL(y[3], y[3])

(0) -> (3), if ((x[0]* x[3])∧(y[0]* y[3])∧(&&(&&(>@z(x[0], 0@z), !(=@z(x[0], 0@z))), >@z(x[0], y[0])) →* TRUE))


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


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


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


(3) -> (0), if ((y[3]* y[0])∧(y[3]* x[0]))


(3) -> (1), if ((y[3]* y[1])∧(y[3]* x[1]))



The set Q consists of the following terms:

Cond_eval(TRUE, x0, x1)
eval(x0, x1)
Cond_eval1(TRUE, 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): EVAL(x[0], y[0]) → COND_EVAL1(&&(&&(>@z(x[0], 0@z), !(=@z(x[0], 0@z))), >@z(x[0], y[0])), x[0], y[0])
(1): EVAL(x[1], y[1]) → COND_EVAL(&&(&&(>@z(x[1], 0@z), !(=@z(x[1], 0@z))), <=@z(x[1], y[1])), x[1], y[1])
(2): COND_EVAL(TRUE, x[2], y[2]) → EVAL(-@z(x[2], 1@z), y[2])
(3): COND_EVAL1(TRUE, x[3], y[3]) → EVAL(y[3], y[3])

(0) -> (3), if ((x[0]* x[3])∧(y[0]* y[3])∧(&&(&&(>@z(x[0], 0@z), !(=@z(x[0], 0@z))), >@z(x[0], y[0])) →* TRUE))


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


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


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


(3) -> (0), if ((y[3]* y[0])∧(y[3]* x[0]))


(3) -> (1), if ((y[3]* y[1])∧(y[3]* x[1]))



The set Q consists of the following terms:

Cond_eval(TRUE, x0, x1)
eval(x0, x1)
Cond_eval1(TRUE, 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 EVAL(x, y) → COND_EVAL1(&&(&&(>@z(x, 0@z), !(=@z(x, 0@z))), >@z(x, y)), x, y) the following chains were created:




For Pair EVAL(x, y) → COND_EVAL(&&(&&(>@z(x, 0@z), !(=@z(x, 0@z))), <=@z(x, y)), x, y) the following chains were created:




For Pair COND_EVAL(TRUE, x, y) → EVAL(-@z(x, 1@z), y) the following chains were created:




For Pair COND_EVAL1(TRUE, x, y) → EVAL(y, y) the following chains were created:




To summarize, we get the following constraints P for the following pairs.



The constraints for P> respective Pbound 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 Pbound.
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(0@z) = 0   
POL(COND_EVAL1(x1, x2, x3)) = (-1)x3 + x2 + (-1)x1   
POL(TRUE) = -1   
POL(&&(x1, x2)) = -1   
POL(COND_EVAL(x1, x2, x3)) = -1 + (-1)x3 + x2 + (-1)x1   
POL(!(x1)) = -1   
POL(FALSE) = -1   
POL(>@z(x1, x2)) = -1   
POL(=@z(x1, x2)) = -1   
POL(EVAL(x1, x2)) = 1 + (-1)x2 + x1   
POL(1@z) = 1   
POL(undefined) = -1   

The following pairs are in P>:

EVAL(x[1], y[1]) → COND_EVAL(&&(&&(>@z(x[1], 0@z), !(=@z(x[1], 0@z))), <=@z(x[1], y[1])), x[1], y[1])

The following pairs are in Pbound:

COND_EVAL1(TRUE, x[3], y[3]) → EVAL(y[3], y[3])

The following pairs are in P:

EVAL(x[0], y[0]) → COND_EVAL1(&&(&&(>@z(x[0], 0@z), !(=@z(x[0], 0@z))), >@z(x[0], y[0])), x[0], y[0])
COND_EVAL(TRUE, x[2], y[2]) → EVAL(-@z(x[2], 1@z), y[2])
COND_EVAL1(TRUE, x[3], y[3]) → EVAL(y[3], y[3])

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

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


↳ ITRS
  ↳ ITRStoIDPProof
    ↳ IDP
      ↳ UsableRulesProof
        ↳ 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:

(0): EVAL(x[0], y[0]) → COND_EVAL1(&&(&&(>@z(x[0], 0@z), !(=@z(x[0], 0@z))), >@z(x[0], y[0])), x[0], y[0])
(1): EVAL(x[1], y[1]) → COND_EVAL(&&(&&(>@z(x[1], 0@z), !(=@z(x[1], 0@z))), <=@z(x[1], y[1])), x[1], y[1])
(2): COND_EVAL(TRUE, x[2], y[2]) → EVAL(-@z(x[2], 1@z), y[2])

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


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


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



The set Q consists of the following terms:

Cond_eval(TRUE, x0, x1)
eval(x0, x1)
Cond_eval1(TRUE, x0, x1)


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

↳ ITRS
  ↳ ITRStoIDPProof
    ↳ IDP
      ↳ UsableRulesProof
        ↳ IDP
          ↳ IDPNonInfProof
            ↳ AND
              ↳ IDP
                ↳ IDependencyGraphProof
IDP
                    ↳ IDPNonInfProof
              ↳ IDP

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], 0@z), !(=@z(x[1], 0@z))), <=@z(x[1], y[1])), x[1], y[1])
(2): COND_EVAL(TRUE, x[2], y[2]) → EVAL(-@z(x[2], 1@z), y[2])

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


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



The set Q consists of the following terms:

Cond_eval(TRUE, x0, x1)
eval(x0, x1)
Cond_eval1(TRUE, 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 EVAL(x[1], y[1]) → COND_EVAL(&&(&&(>@z(x[1], 0@z), !(=@z(x[1], 0@z))), <=@z(x[1], y[1])), x[1], y[1]) the following chains were created:




For Pair COND_EVAL(TRUE, x[2], y[2]) → EVAL(-@z(x[2], 1@z), y[2]) the following chains were created:




To summarize, we get the following constraints P for the following pairs.



The constraints for P> respective Pbound 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 Pbound.
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(0@z) = 0   
POL(TRUE) = 2   
POL(&&(x1, x2)) = 0   
POL(COND_EVAL(x1, x2, x3)) = 1 + x3 + x2   
POL(!(x1)) = -1   
POL(FALSE) = -1   
POL(>@z(x1, x2)) = -1   
POL(=@z(x1, x2)) = -1   
POL(EVAL(x1, x2)) = 1 + x2 + x1   
POL(1@z) = 1   
POL(undefined) = -1   

The following pairs are in P>:

COND_EVAL(TRUE, x[2], y[2]) → EVAL(-@z(x[2], 1@z), y[2])

The following pairs are in Pbound:

COND_EVAL(TRUE, x[2], y[2]) → EVAL(-@z(x[2], 1@z), y[2])

The following pairs are in P:

EVAL(x[1], y[1]) → COND_EVAL(&&(&&(>@z(x[1], 0@z), !(=@z(x[1], 0@z))), <=@z(x[1], y[1])), x[1], y[1])

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

&&(FALSE, FALSE)1FALSE1
-@z1
&&(FALSE, TRUE)1FALSE1


↳ ITRS
  ↳ ITRStoIDPProof
    ↳ IDP
      ↳ UsableRulesProof
        ↳ IDP
          ↳ IDPNonInfProof
            ↳ AND
              ↳ IDP
                ↳ IDependencyGraphProof
                  ↳ IDP
                    ↳ IDPNonInfProof
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:

(1): EVAL(x[1], y[1]) → COND_EVAL(&&(&&(>@z(x[1], 0@z), !(=@z(x[1], 0@z))), <=@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)
Cond_eval1(TRUE, x0, x1)


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

↳ ITRS
  ↳ ITRStoIDPProof
    ↳ IDP
      ↳ UsableRulesProof
        ↳ 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:

(0): EVAL(x[0], y[0]) → COND_EVAL1(&&(&&(>@z(x[0], 0@z), !(=@z(x[0], 0@z))), >@z(x[0], y[0])), x[0], y[0])
(2): COND_EVAL(TRUE, x[2], y[2]) → EVAL(-@z(x[2], 1@z), y[2])
(3): COND_EVAL1(TRUE, x[3], y[3]) → EVAL(y[3], y[3])

(0) -> (3), if ((x[0]* x[3])∧(y[0]* y[3])∧(&&(&&(>@z(x[0], 0@z), !(=@z(x[0], 0@z))), >@z(x[0], y[0])) →* TRUE))


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


(3) -> (0), if ((y[3]* y[0])∧(y[3]* x[0]))



The set Q consists of the following terms:

Cond_eval(TRUE, x0, x1)
eval(x0, x1)
Cond_eval1(TRUE, x0, x1)


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

↳ ITRS
  ↳ ITRStoIDPProof
    ↳ IDP
      ↳ UsableRulesProof
        ↳ IDP
          ↳ IDPNonInfProof
            ↳ AND
              ↳ IDP
              ↳ 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:

(3): COND_EVAL1(TRUE, x[3], y[3]) → EVAL(y[3], y[3])
(0): EVAL(x[0], y[0]) → COND_EVAL1(&&(&&(>@z(x[0], 0@z), !(=@z(x[0], 0@z))), >@z(x[0], y[0])), x[0], y[0])

(0) -> (3), if ((x[0]* x[3])∧(y[0]* y[3])∧(&&(&&(>@z(x[0], 0@z), !(=@z(x[0], 0@z))), >@z(x[0], y[0])) →* TRUE))


(3) -> (0), if ((y[3]* y[0])∧(y[3]* x[0]))



The set Q consists of the following terms:

Cond_eval(TRUE, x0, x1)
eval(x0, x1)
Cond_eval1(TRUE, 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_EVAL1(TRUE, x[3], y[3]) → EVAL(y[3], y[3]) the following chains were created:




For Pair EVAL(x[0], y[0]) → COND_EVAL1(&&(&&(>@z(x[0], 0@z), !(=@z(x[0], 0@z))), >@z(x[0], y[0])), x[0], y[0]) the following chains were created:




To summarize, we get the following constraints P for the following pairs.



The constraints for P> respective Pbound 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 Pbound.
Using the following integer polynomial ordering the resulting constraints can be solved
Polynomial interpretation over integers[POLO]:

POL(=@z(x1, x2)) = -1   
POL(0@z) = 0   
POL(COND_EVAL1(x1, x2, x3)) = -1 + (2)x2   
POL(TRUE) = 2   
POL(&&(x1, x2)) = -1   
POL(EVAL(x1, x2)) = -1 + (2)x1   
POL(!(x1)) = -1   
POL(FALSE) = -1   
POL(undefined) = -1   
POL(>@z(x1, x2)) = -1   

The following pairs are in P>:

COND_EVAL1(TRUE, x[3], y[3]) → EVAL(y[3], y[3])

The following pairs are in Pbound:

COND_EVAL1(TRUE, x[3], y[3]) → EVAL(y[3], y[3])

The following pairs are in P:

EVAL(x[0], y[0]) → COND_EVAL1(&&(&&(>@z(x[0], 0@z), !(=@z(x[0], 0@z))), >@z(x[0], y[0])), x[0], y[0])

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

&&(FALSE, FALSE)1FALSE1
&&(FALSE, TRUE)1FALSE1
&&(TRUE, FALSE)1FALSE1


↳ ITRS
  ↳ ITRStoIDPProof
    ↳ IDP
      ↳ UsableRulesProof
        ↳ IDP
          ↳ IDPNonInfProof
            ↳ AND
              ↳ IDP
              ↳ IDP
                ↳ IDependencyGraphProof
                  ↳ 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(x[0], y[0]) → COND_EVAL1(&&(&&(>@z(x[0], 0@z), !(=@z(x[0], 0@z))), >@z(x[0], y[0])), x[0], y[0])


The set Q consists of the following terms:

Cond_eval(TRUE, x0, x1)
eval(x0, x1)
Cond_eval1(TRUE, x0, x1)


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