/export/starexec/sandbox2/solver/bin/starexec_run_standard /export/starexec/sandbox2/benchmark/theBenchmark.xml /export/starexec/sandbox2/output/output_files


--------------------------------------------------------------------------------


NO
proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml
# AProVE Commit ID: c69e44bd14796315568835c1ffa2502984884775 mhark 20210624 unpublished


Termination w.r.t. Q of the given QTRS could be disproven:

(0) QTRS
(1) DependencyPairsProof [EQUIVALENT, 0 ms]
(2) QDP
(3) DependencyGraphProof [EQUIVALENT, 0 ms]
(4) AND
    (5) QDP
        (6) UsableRulesProof [EQUIVALENT, 0 ms]
        (7) QDP
        (8) QDPSizeChangeProof [EQUIVALENT, 0 ms]
        (9) YES
    (10) QDP
        (11) UsableRulesProof [EQUIVALENT, 0 ms]
        (12) QDP
        (13) QDPSizeChangeProof [EQUIVALENT, 0 ms]
        (14) YES
    (15) QDP
        (16) NonLoopProof [COMPLETE, 9295 ms]
        (17) NO


----------------------------------------

(0)
Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:

   notZero(pos(s(x))) -> true
   notZero(neg(s(x))) -> true
   notZero(neg(0)) -> false
   notZero(pos(0)) -> false
   greaterZero(pos(s(x))) -> true
   greaterZero(pos(0)) -> false
   greaterZero(neg(x)) -> false
   and(false, false) -> false
   and(false, true) -> false
   and(true, false) -> false
   and(true, true) -> true
   minusT(0, y) -> neg(y)
   minusT(x, 0) -> pos(x)
   minusT(s(x), s(y)) -> minusT(x, y)
   plusNat(0, y) -> y
   plusNat(s(x), y) -> plusNat(x, s(y))
   negate(pos(x)) -> neg(x)
   negate(neg(x)) -> pos(x)
   minus(pos(x), pos(y)) -> minusT(x, y)
   minus(neg(x), neg(y)) -> negate(minusT(x, y))
   minus(pos(x), neg(y)) -> pos(plusNat(x, y))
   minus(neg(x), pos(y)) -> neg(plusNat(x, y))
   while(true, i, y) -> while(and(notZero(y), greaterZero(i)), minus(i, y), y)

Q is empty.

----------------------------------------

(1) DependencyPairsProof (EQUIVALENT)
Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem.
----------------------------------------

(2)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   MINUST(s(x), s(y)) -> MINUST(x, y)
   PLUSNAT(s(x), y) -> PLUSNAT(x, s(y))
   MINUS(pos(x), pos(y)) -> MINUST(x, y)
   MINUS(neg(x), neg(y)) -> NEGATE(minusT(x, y))
   MINUS(neg(x), neg(y)) -> MINUST(x, y)
   MINUS(pos(x), neg(y)) -> PLUSNAT(x, y)
   MINUS(neg(x), pos(y)) -> PLUSNAT(x, y)
   WHILE(true, i, y) -> WHILE(and(notZero(y), greaterZero(i)), minus(i, y), y)
   WHILE(true, i, y) -> AND(notZero(y), greaterZero(i))
   WHILE(true, i, y) -> NOTZERO(y)
   WHILE(true, i, y) -> GREATERZERO(i)
   WHILE(true, i, y) -> MINUS(i, y)

The TRS R consists of the following rules:

   notZero(pos(s(x))) -> true
   notZero(neg(s(x))) -> true
   notZero(neg(0)) -> false
   notZero(pos(0)) -> false
   greaterZero(pos(s(x))) -> true
   greaterZero(pos(0)) -> false
   greaterZero(neg(x)) -> false
   and(false, false) -> false
   and(false, true) -> false
   and(true, false) -> false
   and(true, true) -> true
   minusT(0, y) -> neg(y)
   minusT(x, 0) -> pos(x)
   minusT(s(x), s(y)) -> minusT(x, y)
   plusNat(0, y) -> y
   plusNat(s(x), y) -> plusNat(x, s(y))
   negate(pos(x)) -> neg(x)
   negate(neg(x)) -> pos(x)
   minus(pos(x), pos(y)) -> minusT(x, y)
   minus(neg(x), neg(y)) -> negate(minusT(x, y))
   minus(pos(x), neg(y)) -> pos(plusNat(x, y))
   minus(neg(x), pos(y)) -> neg(plusNat(x, y))
   while(true, i, y) -> while(and(notZero(y), greaterZero(i)), minus(i, y), y)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(3) DependencyGraphProof (EQUIVALENT)
The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 3 SCCs with 9 less nodes.
----------------------------------------

(4)
Complex Obligation (AND)

----------------------------------------

(5)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   PLUSNAT(s(x), y) -> PLUSNAT(x, s(y))

The TRS R consists of the following rules:

   notZero(pos(s(x))) -> true
   notZero(neg(s(x))) -> true
   notZero(neg(0)) -> false
   notZero(pos(0)) -> false
   greaterZero(pos(s(x))) -> true
   greaterZero(pos(0)) -> false
   greaterZero(neg(x)) -> false
   and(false, false) -> false
   and(false, true) -> false
   and(true, false) -> false
   and(true, true) -> true
   minusT(0, y) -> neg(y)
   minusT(x, 0) -> pos(x)
   minusT(s(x), s(y)) -> minusT(x, y)
   plusNat(0, y) -> y
   plusNat(s(x), y) -> plusNat(x, s(y))
   negate(pos(x)) -> neg(x)
   negate(neg(x)) -> pos(x)
   minus(pos(x), pos(y)) -> minusT(x, y)
   minus(neg(x), neg(y)) -> negate(minusT(x, y))
   minus(pos(x), neg(y)) -> pos(plusNat(x, y))
   minus(neg(x), pos(y)) -> neg(plusNat(x, y))
   while(true, i, y) -> while(and(notZero(y), greaterZero(i)), minus(i, y), y)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(6) UsableRulesProof (EQUIVALENT)
We can use the usable rules and reduction pair processor [LPAR04] with the Ce-compatible extension of the polynomial order that maps every function symbol to the sum of its arguments. Then, we can delete all non-usable rules [FROCOS05] from R.
----------------------------------------

(7)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   PLUSNAT(s(x), y) -> PLUSNAT(x, s(y))

R is empty.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(8) QDPSizeChangeProof (EQUIVALENT)
By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] we have proven that there are no infinite chains for this DP problem. 

From the DPs we obtained the following set of size-change graphs:
*PLUSNAT(s(x), y) -> PLUSNAT(x, s(y))
The graph contains the following edges 1 > 1


----------------------------------------

(9)
YES

----------------------------------------

(10)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   MINUST(s(x), s(y)) -> MINUST(x, y)

The TRS R consists of the following rules:

   notZero(pos(s(x))) -> true
   notZero(neg(s(x))) -> true
   notZero(neg(0)) -> false
   notZero(pos(0)) -> false
   greaterZero(pos(s(x))) -> true
   greaterZero(pos(0)) -> false
   greaterZero(neg(x)) -> false
   and(false, false) -> false
   and(false, true) -> false
   and(true, false) -> false
   and(true, true) -> true
   minusT(0, y) -> neg(y)
   minusT(x, 0) -> pos(x)
   minusT(s(x), s(y)) -> minusT(x, y)
   plusNat(0, y) -> y
   plusNat(s(x), y) -> plusNat(x, s(y))
   negate(pos(x)) -> neg(x)
   negate(neg(x)) -> pos(x)
   minus(pos(x), pos(y)) -> minusT(x, y)
   minus(neg(x), neg(y)) -> negate(minusT(x, y))
   minus(pos(x), neg(y)) -> pos(plusNat(x, y))
   minus(neg(x), pos(y)) -> neg(plusNat(x, y))
   while(true, i, y) -> while(and(notZero(y), greaterZero(i)), minus(i, y), y)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(11) UsableRulesProof (EQUIVALENT)
We can use the usable rules and reduction pair processor [LPAR04] with the Ce-compatible extension of the polynomial order that maps every function symbol to the sum of its arguments. Then, we can delete all non-usable rules [FROCOS05] from R.
----------------------------------------

(12)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   MINUST(s(x), s(y)) -> MINUST(x, y)

R is empty.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(13) QDPSizeChangeProof (EQUIVALENT)
By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] we have proven that there are no infinite chains for this DP problem. 

From the DPs we obtained the following set of size-change graphs:
*MINUST(s(x), s(y)) -> MINUST(x, y)
The graph contains the following edges 1 > 1, 2 > 2


----------------------------------------

(14)
YES

----------------------------------------

(15)
Obligation:
Q DP problem:
The TRS P consists of the following rules:

   WHILE(true, i, y) -> WHILE(and(notZero(y), greaterZero(i)), minus(i, y), y)

The TRS R consists of the following rules:

   notZero(pos(s(x))) -> true
   notZero(neg(s(x))) -> true
   notZero(neg(0)) -> false
   notZero(pos(0)) -> false
   greaterZero(pos(s(x))) -> true
   greaterZero(pos(0)) -> false
   greaterZero(neg(x)) -> false
   and(false, false) -> false
   and(false, true) -> false
   and(true, false) -> false
   and(true, true) -> true
   minusT(0, y) -> neg(y)
   minusT(x, 0) -> pos(x)
   minusT(s(x), s(y)) -> minusT(x, y)
   plusNat(0, y) -> y
   plusNat(s(x), y) -> plusNat(x, s(y))
   negate(pos(x)) -> neg(x)
   negate(neg(x)) -> pos(x)
   minus(pos(x), pos(y)) -> minusT(x, y)
   minus(neg(x), neg(y)) -> negate(minusT(x, y))
   minus(pos(x), neg(y)) -> pos(plusNat(x, y))
   minus(neg(x), pos(y)) -> neg(plusNat(x, y))
   while(true, i, y) -> while(and(notZero(y), greaterZero(i)), minus(i, y), y)

Q is empty.
We have to consider all minimal (P,Q,R)-chains.
----------------------------------------

(16) NonLoopProof (COMPLETE)
By Theorem 8 [NONLOOP] we deduce infiniteness of the QDP.
We apply the theorem with m = 1, b = 1, 
σ' = [ ], and μ' = [ ] on the rule
WHILE(true, pos(s(s(zr1))), neg(s(0)))[zr1 / s(zr1)]^n[zr1 / 0] -> WHILE(true, pos(s(s(s(zr1)))), neg(s(0)))[zr1 / s(zr1)]^n[zr1 / 0]
This rule is correct for the QDP as the following derivation shows:

WHILE(true, pos(s(s(zr1))), neg(s(0)))[zr1 / s(zr1)]^n[zr1 / 0] -> WHILE(true, pos(s(s(s(zr1)))), neg(s(0)))[zr1 / s(zr1)]^n[zr1 / 0]
    by Equivalency by Simplifying Mu with mu1: [zl1 / 0] mu2: [zr1 / 0]
    intermediate steps: Equiv Smu (lhs) - Equiv DR (rhs) - Equiv DR (lhs) - Equiv DR (rhs) - Equiv DR (lhs) - Instantiate mu - Instantiate mu - Equiv DR (lhs) - Equiv DR (rhs) - Equiv DR (lhs) - Equiv IPS (rhs) - Equiv IPS (lhs)
    WHILE(true, pos(s(s(zl1))), neg(s(x0)))[zr1 / s(zr1), zl1 / s(zl1)]^n[zr1 / x0, zl1 / 0] -> WHILE(true, pos(s(s(s(zr1)))), neg(s(x0)))[zr1 / s(zr1), zl1 / s(zl1)]^n[zr1 / x0, zl1 / 0]
        by Narrowing at position: [1,0]
        intermediate steps: Equiv IPS (rhs) - Equiv IPS (lhs) - Equiv IPS (rhs) - Equiv IPS (lhs) - Instantiation - Equiv DR (rhs) - Equiv DR (lhs) - Instantiation - Equiv DR (rhs) - Equiv DR (lhs) - Equiv IPS (rhs) - Equiv IPS (lhs)
        WHILE(true, pos(s(s(zs1))), neg(s(x0)))[zt1 / s(zt1), zs1 / s(zs1)]^n[zt1 / x0, zs1 / y1] -> WHILE(true, pos(plusNat(y1, s(s(s(zt1))))), neg(s(x0)))[zt1 / s(zt1), zs1 / s(zs1)]^n[zt1 / x0, zs1 / y1]
            by Narrowing at position: [1,0]
            intermediate steps: Equiv IPS (rhs) - Equiv IPS (lhs) - Instantiate mu - Equiv IPS (rhs) - Equiv IPS (lhs) - Instantiate Sigma - Instantiation - Instantiation
            WHILE(true, pos(s(y0)), neg(s(x1)))[ ]^n[ ] -> WHILE(true, pos(plusNat(y0, s(s(x1)))), neg(s(x1)))[ ]^n[ ]
                by Rewrite t with the rewrite sequence <Pos,Rule>: [([0],and(true, true) -> true), ([1],minus(pos(x), neg(y)) -> pos(plusNat(x, y))), ([1,0],plusNat(s(x), y) -> plusNat(x, s(y)))]
                WHILE(true, pos(s(y0)), neg(s(x1)))[ ]^n[ ] -> WHILE(and(true, true), minus(pos(s(y0)), neg(s(x1))), neg(s(x1)))[ ]^n[ ]
                    by Narrowing at position: [0,1]
                    intermediate steps: Instantiation - Instantiation
                    WHILE(true, x0, neg(s(y0)))[ ]^n[ ] -> WHILE(and(true, greaterZero(x0)), minus(x0, neg(s(y0))), neg(s(y0)))[ ]^n[ ]
                        by Narrowing at position: [0,0]
                        intermediate steps: Instantiation - Instantiation
                        WHILE(true, i, y)[ ]^n[ ] -> WHILE(and(notZero(y), greaterZero(i)), minus(i, y), y)[ ]^n[ ]
                            by Rule from TRS P

                        intermediate steps: Instantiation
                        notZero(neg(s(x)))[ ]^n[ ] -> true[ ]^n[ ]
                            by Rule from TRS R

                    intermediate steps: Instantiation
                    greaterZero(pos(s(x)))[ ]^n[ ] -> true[ ]^n[ ]
                        by Rule from TRS R

            intermediate steps: Equiv IPS (rhs) - Equiv IPS (lhs) - Equiv IPS (rhs) - Equiv IPS (lhs) - Equiv Smu (rhs) - Instantiation - Instantiation - Equiv DR (rhs) - Equiv DR (lhs) - Instantiation - Equiv DR (rhs) - Equiv DR (lhs)
            plusNat(s(x), y)[x / s(x)]^n[ ] -> plusNat(x, s(y))[y / s(y)]^n[ ]
                by PatternCreation I with delta: [ ], theta: [y / s(y)], sigma: [x / s(x)]
                plusNat(s(x), y)[ ]^n[ ] -> plusNat(x, s(y))[ ]^n[ ]
                    by Rule from TRS R

        intermediate steps: Equiv IPS (rhs) - Equiv IPS (lhs) - Instantiate mu - Equiv IPS (rhs) - Equiv IPS (lhs) - Instantiate Sigma - Instantiation - Instantiation
        plusNat(0, y)[ ]^n[ ] -> y[ ]^n[ ]
            by Rule from TRS R
----------------------------------------

(17)
NO