Termination of the given JBC could be proven:

0 JBC
↳1 JBCToGraph (⇒, 580 ms)
↳2 JBCTerminationGraph
↳3 TerminationGraphToSCCProof (⇒, 0 ms)
↳4 JBCTerminationSCC
↳5 SCCToIDPv1Proof (⇒, 100 ms)
↳6 IDP
↳7 IDPNonInfProof (⇒, 1290 ms)
↳8 IDP
↳9 IDependencyGraphProof (⇔, 0 ms)
↳10 TRUE

(0) Obligation:

JBC Problem based on JBC Program:
Manifest-Version: 1.0 Created-By: 1.6.0_16 (Sun Microsystems Inc.) Main-Class: LogAG 
public class LogAG{

  // adapted from Arts&Giesl, 2001

  public static int half(int x) {

    int res = 0;

    while (x > 1) {

      x = x-2;
      res++;

    }

    return res;

  }


  public static int log(int x) {

    int res = 0;

    while (x > 1) {

      x = half(x-2)+1;
      res++;

    }

    return res;

  }


  public static void main(String[] args) {
    Random.args = args;
    int x  = Random.random();
    log(x);
  }
}



public class Random {
  static String[] args;
  static int index = 0;

  public static int random() {
    String string = args[index];
    index++;
    return string.length();
  }
}


(1) JBCToGraph (SOUND transformation)

Constructed TerminationGraph.

(2) Obligation:

Termination Graph based on JBC Program:
LogAG.main([Ljava/lang/String;)V: Graph of 145 nodes with 1 SCC.


(3) TerminationGraphToSCCProof (SOUND transformation)

Splitted TerminationGraph to 1 SCCs.

(4) Obligation:

SCC of termination graph based on JBC Program.
SCC contains nodes from the following methods: LogAG.main([Ljava/lang/String;)V
SCC calls the following helper methods:
Performed SCC analyses: UsedFieldsAnalysis

(5) SCCToIDPv1Proof (SOUND transformation)

Transformed FIGraph SCCs to IDPs. Log:

Generated 35 rules for P and 0 rules for R.


P rules:
1387_0_log_ConstantStackPush(EOS(STATIC_1387), i392, i392) → 1389_0_log_LE(EOS(STATIC_1389), i392, i392, 1)
1389_0_log_LE(EOS(STATIC_1389), i420, i420, matching1) → 1392_0_log_LE(EOS(STATIC_1392), i420, i420, 1) | =(matching1, 1)
1392_0_log_LE(EOS(STATIC_1392), i420, i420, matching1) → 1395_0_log_Load(EOS(STATIC_1395), i420) | &&(>(i420, 1), =(matching1, 1))
1395_0_log_Load(EOS(STATIC_1395), i420) → 1398_0_log_ConstantStackPush(EOS(STATIC_1398), i420)
1398_0_log_ConstantStackPush(EOS(STATIC_1398), i420) → 1401_0_log_IntArithmetic(EOS(STATIC_1401), i420, 2)
1401_0_log_IntArithmetic(EOS(STATIC_1401), i420, matching1) → 1404_0_log_InvokeMethod(EOS(STATIC_1404), -(i420, 2)) | &&(>(i420, 0), =(matching1, 2))
1404_0_log_InvokeMethod(EOS(STATIC_1404), i421) → 1406_0_half_ConstantStackPush(EOS(STATIC_1406), i421, i421)
1406_0_half_ConstantStackPush(EOS(STATIC_1406), i421, i421) → 1408_0_half_Store(EOS(STATIC_1408), i421, i421, 0)
1408_0_half_Store(EOS(STATIC_1408), i421, i421, matching1) → 1409_0_half_Load(EOS(STATIC_1409), i421, i421, 0) | =(matching1, 0)
1409_0_half_Load(EOS(STATIC_1409), i421, i421, matching1) → 1437_0_half_Load(EOS(STATIC_1437), i421, i421, 0) | =(matching1, 0)
1437_0_half_Load(EOS(STATIC_1437), i431, i429, i430) → 1464_0_half_Load(EOS(STATIC_1464), i431, i429, i430)
1464_0_half_Load(EOS(STATIC_1464), i431, i444, i445) → 1491_0_half_Load(EOS(STATIC_1491), i431, i444, i445)
1491_0_half_Load(EOS(STATIC_1491), i431, i457, i458) → 1521_0_half_Load(EOS(STATIC_1521), i431, i457, i458)
1521_0_half_Load(EOS(STATIC_1521), i431, i470, i471) → 1523_0_half_ConstantStackPush(EOS(STATIC_1523), i431, i470, i471, i470)
1523_0_half_ConstantStackPush(EOS(STATIC_1523), i431, i470, i471, i470) → 1524_0_half_LE(EOS(STATIC_1524), i431, i470, i471, i470, 1)
1524_0_half_LE(EOS(STATIC_1524), i431, i477, i471, i477, matching1) → 1526_0_half_LE(EOS(STATIC_1526), i431, i477, i471, i477, 1) | =(matching1, 1)
1524_0_half_LE(EOS(STATIC_1524), i431, i478, i471, i478, matching1) → 1527_0_half_LE(EOS(STATIC_1527), i431, i478, i471, i478, 1) | =(matching1, 1)
1526_0_half_LE(EOS(STATIC_1526), i431, i477, i471, i477, matching1) → 1528_0_half_Load(EOS(STATIC_1528), i431, i471) | &&(<=(i477, 1), =(matching1, 1))
1528_0_half_Load(EOS(STATIC_1528), i431, i471) → 1531_0_half_Return(EOS(STATIC_1531), i431, i471)
1531_0_half_Return(EOS(STATIC_1531), i431, i471) → 1534_0_log_ConstantStackPush(EOS(STATIC_1534), i471)
1534_0_log_ConstantStackPush(EOS(STATIC_1534), i471) → 1537_0_log_IntArithmetic(EOS(STATIC_1537), i471, 1)
1537_0_log_IntArithmetic(EOS(STATIC_1537), i471, matching1) → 1540_0_log_Store(EOS(STATIC_1540), +(i471, 1)) | &&(>=(i471, 0), =(matching1, 1))
1540_0_log_Store(EOS(STATIC_1540), i480) → 1543_0_log_Inc(EOS(STATIC_1543), i480)
1543_0_log_Inc(EOS(STATIC_1543), i480) → 1545_0_log_JMP(EOS(STATIC_1545), i480)
1545_0_log_JMP(EOS(STATIC_1545), i480) → 1550_0_log_Load(EOS(STATIC_1550), i480)
1550_0_log_Load(EOS(STATIC_1550), i480) → 1364_0_log_Load(EOS(STATIC_1364), i480)
1364_0_log_Load(EOS(STATIC_1364), i392) → 1387_0_log_ConstantStackPush(EOS(STATIC_1387), i392, i392)
1527_0_half_LE(EOS(STATIC_1527), i431, i478, i471, i478, matching1) → 1530_0_half_Load(EOS(STATIC_1530), i431, i478, i471) | &&(>(i478, 1), =(matching1, 1))
1530_0_half_Load(EOS(STATIC_1530), i431, i478, i471) → 1533_0_half_ConstantStackPush(EOS(STATIC_1533), i431, i471, i478)
1533_0_half_ConstantStackPush(EOS(STATIC_1533), i431, i471, i478) → 1536_0_half_IntArithmetic(EOS(STATIC_1536), i431, i471, i478, 2)
1536_0_half_IntArithmetic(EOS(STATIC_1536), i431, i471, i478, matching1) → 1539_0_half_Store(EOS(STATIC_1539), i431, i471, -(i478, 2)) | &&(>(i478, 0), =(matching1, 2))
1539_0_half_Store(EOS(STATIC_1539), i431, i471, i479) → 1541_0_half_Inc(EOS(STATIC_1541), i431, i479, i471)
1541_0_half_Inc(EOS(STATIC_1541), i431, i479, i471) → 1544_0_half_JMP(EOS(STATIC_1544), i431, i479, +(i471, 1)) | >=(i471, 0)
1544_0_half_JMP(EOS(STATIC_1544), i431, i479, i481) → 1548_0_half_Load(EOS(STATIC_1548), i431, i479, i481)
1548_0_half_Load(EOS(STATIC_1548), i431, i479, i481) → 1521_0_half_Load(EOS(STATIC_1521), i431, i479, i481)
R rules:

Combined rules. Obtained 2 conditional rules for P and 0 conditional rules for R.


P rules:
1524_0_half_LE(EOS(STATIC_1524), x0, x1, x2, x1, 1) → 1524_0_half_LE(EOS(STATIC_1524), -(+(x2, 1), 2), -(+(x2, 1), 2), 0, -(+(x2, 1), 2), 1) | &&(>(x2, 0), <=(x1, 1))
1524_0_half_LE(EOS(STATIC_1524), x0, x1, x2, x1, 1) → 1524_0_half_LE(EOS(STATIC_1524), x0, -(x1, 2), +(x2, 1), -(x1, 2), 1) | &&(>(+(x2, 1), 0), >(x1, 1))
R rules:

Filtered ground terms:



1524_0_half_LE(x1, x2, x3, x4, x5, x6) → 1524_0_half_LE(x2, x3, x4, x5)
EOS(x1) → EOS
Cond_1524_0_half_LE1(x1, x2, x3, x4, x5, x6, x7) → Cond_1524_0_half_LE1(x1, x3, x4, x5, x6)
Cond_1524_0_half_LE(x1, x2, x3, x4, x5, x6, x7) → Cond_1524_0_half_LE(x1, x3, x4, x5, x6)

Filtered duplicate args:



1524_0_half_LE(x1, x2, x3, x4) → 1524_0_half_LE(x1, x3, x4)
Cond_1524_0_half_LE(x1, x2, x3, x4, x5) → Cond_1524_0_half_LE(x1, x2, x4, x5)
Cond_1524_0_half_LE1(x1, x2, x3, x4, x5) → Cond_1524_0_half_LE1(x1, x2, x4, x5)

Filtered unneeded arguments:



Cond_1524_0_half_LE(x1, x2, x3, x4) → Cond_1524_0_half_LE(x1, x3)
Cond_1524_0_half_LE1(x1, x2, x3, x4) → Cond_1524_0_half_LE1(x1, x3, x4)
1524_0_half_LE(x1, x2, x3) → 1524_0_half_LE(x2, x3)

Combined rules. Obtained 2 conditional rules for P and 0 conditional rules for R.


P rules:
1524_0_half_LE(x2, x1) → 1524_0_half_LE(0, -(+(x2, 1), 2)) | &&(>(x2, 0), <=(x1, 1))
1524_0_half_LE(x2, x1) → 1524_0_half_LE(+(x2, 1), -(x1, 2)) | &&(>(x2, -1), >(x1, 1))
R rules:

Finished conversion. Obtained 4 rules for P and 0 rules for R. System has predefined symbols.


P rules:
1524_0_HALF_LE(x2, x1) → COND_1524_0_HALF_LE(&&(>(x2, 0), <=(x1, 1)), x2, x1)
COND_1524_0_HALF_LE(TRUE, x2, x1) → 1524_0_HALF_LE(0, -(+(x2, 1), 2))
1524_0_HALF_LE(x2, x1) → COND_1524_0_HALF_LE1(&&(>(x2, -1), >(x1, 1)), x2, x1)
COND_1524_0_HALF_LE1(TRUE, x2, x1) → 1524_0_HALF_LE(+(x2, 1), -(x1, 2))
R rules:

(6) Obligation:

IDP problem:
The following function symbols are pre-defined:
!=~Neq: (Integer, Integer) -> Boolean
*~Mul: (Integer, Integer) -> Integer
>=~Ge: (Integer, Integer) -> Boolean
-1~UnaryMinus: (Integer) -> Integer
|~Bwor: (Integer, Integer) -> Integer
/~Div: (Integer, Integer) -> Integer
=~Eq: (Integer, Integer) -> Boolean
~Bwxor: (Integer, Integer) -> Integer
||~Lor: (Boolean, Boolean) -> Boolean
!~Lnot: (Boolean) -> Boolean
<~Lt: (Integer, Integer) -> Boolean
-~Sub: (Integer, Integer) -> Integer
<=~Le: (Integer, Integer) -> Boolean
>~Gt: (Integer, Integer) -> Boolean
~~Bwnot: (Integer) -> Integer
%~Mod: (Integer, Integer) -> Integer
&~Bwand: (Integer, Integer) -> Integer
+~Add: (Integer, Integer) -> Integer
&&~Land: (Boolean, Boolean) -> Boolean


The following domains are used:

Boolean, Integer


R is empty.

The integer pair graph contains the following rules and edges:
(0): 1524_0_HALF_LE(x2[0], x1[0]) → COND_1524_0_HALF_LE(x2[0] > 0 && x1[0] <= 1, x2[0], x1[0])
(1): COND_1524_0_HALF_LE(TRUE, x2[1], x1[1]) → 1524_0_HALF_LE(0, x2[1] + 1 - 2)
(2): 1524_0_HALF_LE(x2[2], x1[2]) → COND_1524_0_HALF_LE1(x2[2] > -1 && x1[2] > 1, x2[2], x1[2])
(3): COND_1524_0_HALF_LE1(TRUE, x2[3], x1[3]) → 1524_0_HALF_LE(x2[3] + 1, x1[3] - 2)

(0) -> (1), if (x2[0] > 0 && x1[0] <= 1 ∧x2[0] →* x2[1]∧x1[0] →* x1[1])


(1) -> (0), if (0 →* x2[0]∧x2[1] + 1 - 2 →* x1[0])


(1) -> (2), if (0 →* x2[2]∧x2[1] + 1 - 2 →* x1[2])


(2) -> (3), if (x2[2] > -1 && x1[2] > 1 ∧x2[2] →* x2[3]∧x1[2] →* x1[3])


(3) -> (0), if (x2[3] + 1 →* x2[0]∧x1[3] - 2 →* x1[0])


(3) -> (2), if (x2[3] + 1 →* x2[2]∧x1[3] - 2 →* x1[2])



The set Q is empty.

(7) IDPNonInfProof (SOUND transformation)

Used the following options for this NonInfProof:
IDPGPoloSolver: Range: [(-1,2)] IsNat: false Interpretation Shape Heuristic: aprove.DPFramework.IDPProblem.Processors.nonInf.poly.IdpDefaultShapeHeuristic@73ab28e1 Constraint Generator: NonInfConstraintGenerator: PathGenerator: MetricPathGenerator: Max Left Steps: 3 Max Right Steps: 2

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 1524_0_HALF_LE(x2, x1) → COND_1524_0_HALF_LE(&&(>(x2, 0), <=(x1, 1)), x2, x1) the following chains were created:
  • We consider the chain 1524_0_HALF_LE(x2[0], x1[0]) → COND_1524_0_HALF_LE(&&(>(x2[0], 0), <=(x1[0], 1)), x2[0], x1[0]), COND_1524_0_HALF_LE(TRUE, x2[1], x1[1]) → 1524_0_HALF_LE(0, -(+(x2[1], 1), 2)) which results in the following constraint:

    (1)    (&&(>(x2[0], 0), <=(x1[0], 1))=TRUE∧x2[0]=x2[1]∧x1[0]=x1[1] ⇒ 1524_0_HALF_LE(x2[0], x1[0])≥NonInfC∧1524_0_HALF_LE(x2[0], x1[0])≥COND_1524_0_HALF_LE(&&(>(x2[0], 0), <=(x1[0], 1)), x2[0], x1[0])∧(UIncreasing(COND_1524_0_HALF_LE(&&(>(x2[0], 0), <=(x1[0], 1)), x2[0], x1[0])), ≥))



    We simplified constraint (1) using rules (IV), (IDP_BOOLEAN) which results in the following new constraint:

    (2)    (>(x2[0], 0)=TRUE∧<=(x1[0], 1)=TRUE ⇒ 1524_0_HALF_LE(x2[0], x1[0])≥NonInfC∧1524_0_HALF_LE(x2[0], x1[0])≥COND_1524_0_HALF_LE(&&(>(x2[0], 0), <=(x1[0], 1)), x2[0], x1[0])∧(UIncreasing(COND_1524_0_HALF_LE(&&(>(x2[0], 0), <=(x1[0], 1)), x2[0], x1[0])), ≥))



    We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint:

    (3)    (x2[0] + [-1] ≥ 0∧[1] + [-1]x1[0] ≥ 0 ⇒ (UIncreasing(COND_1524_0_HALF_LE(&&(>(x2[0], 0), <=(x1[0], 1)), x2[0], x1[0])), ≥)∧[(-1)bni_17 + (-1)Bound*bni_17] + [bni_17]x1[0] + [bni_17]x2[0] ≥ 0∧[(-1)bso_18] ≥ 0)



    We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:

    (4)    (x2[0] + [-1] ≥ 0∧[1] + [-1]x1[0] ≥ 0 ⇒ (UIncreasing(COND_1524_0_HALF_LE(&&(>(x2[0], 0), <=(x1[0], 1)), x2[0], x1[0])), ≥)∧[(-1)bni_17 + (-1)Bound*bni_17] + [bni_17]x1[0] + [bni_17]x2[0] ≥ 0∧[(-1)bso_18] ≥ 0)



    We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:

    (5)    (x2[0] + [-1] ≥ 0∧[1] + [-1]x1[0] ≥ 0 ⇒ (UIncreasing(COND_1524_0_HALF_LE(&&(>(x2[0], 0), <=(x1[0], 1)), x2[0], x1[0])), ≥)∧[(-1)bni_17 + (-1)Bound*bni_17] + [bni_17]x1[0] + [bni_17]x2[0] ≥ 0∧[(-1)bso_18] ≥ 0)



    We simplified constraint (5) using rule (IDP_SMT_SPLIT) which results in the following new constraint:

    (6)    (x2[0] ≥ 0∧[1] + [-1]x1[0] ≥ 0 ⇒ (UIncreasing(COND_1524_0_HALF_LE(&&(>(x2[0], 0), <=(x1[0], 1)), x2[0], x1[0])), ≥)∧[(-1)Bound*bni_17] + [bni_17]x1[0] + [bni_17]x2[0] ≥ 0∧[(-1)bso_18] ≥ 0)



    We simplified constraint (6) using rule (IDP_SMT_SPLIT) which results in the following new constraints:

    (7)    (x2[0] ≥ 0∧[1] + [-1]x1[0] ≥ 0∧x1[0] ≥ 0 ⇒ (UIncreasing(COND_1524_0_HALF_LE(&&(>(x2[0], 0), <=(x1[0], 1)), x2[0], x1[0])), ≥)∧[(-1)Bound*bni_17] + [bni_17]x1[0] + [bni_17]x2[0] ≥ 0∧[(-1)bso_18] ≥ 0)


    (8)    (x2[0] ≥ 0∧[1] + x1[0] ≥ 0∧x1[0] ≥ 0 ⇒ (UIncreasing(COND_1524_0_HALF_LE(&&(>(x2[0], 0), <=(x1[0], 1)), x2[0], x1[0])), ≥)∧[(-1)Bound*bni_17] + [(-1)bni_17]x1[0] + [bni_17]x2[0] ≥ 0∧[(-1)bso_18] ≥ 0)







For Pair COND_1524_0_HALF_LE(TRUE, x2, x1) → 1524_0_HALF_LE(0, -(+(x2, 1), 2)) the following chains were created:
  • We consider the chain 1524_0_HALF_LE(x2[0], x1[0]) → COND_1524_0_HALF_LE(&&(>(x2[0], 0), <=(x1[0], 1)), x2[0], x1[0]), COND_1524_0_HALF_LE(TRUE, x2[1], x1[1]) → 1524_0_HALF_LE(0, -(+(x2[1], 1), 2)), 1524_0_HALF_LE(x2[0], x1[0]) → COND_1524_0_HALF_LE(&&(>(x2[0], 0), <=(x1[0], 1)), x2[0], x1[0]), COND_1524_0_HALF_LE(TRUE, x2[1], x1[1]) → 1524_0_HALF_LE(0, -(+(x2[1], 1), 2)), 1524_0_HALF_LE(x2[0], x1[0]) → COND_1524_0_HALF_LE(&&(>(x2[0], 0), <=(x1[0], 1)), x2[0], x1[0]), COND_1524_0_HALF_LE(TRUE, x2[1], x1[1]) → 1524_0_HALF_LE(0, -(+(x2[1], 1), 2)) which results in the following constraint:

    (9)    (&&(>(x2[0], 0), <=(x1[0], 1))=TRUE∧x2[0]=x2[1]∧x1[0]=x1[1]∧0=x2[0]1∧-(+(x2[1], 1), 2)=x1[0]1∧&&(>(x2[0]1, 0), <=(x1[0]1, 1))=TRUE∧x2[0]1=x2[1]1∧x1[0]1=x1[1]1∧0=x2[0]2∧-(+(x2[1]1, 1), 2)=x1[0]2∧&&(>(x2[0]2, 0), <=(x1[0]2, 1))=TRUE∧x2[0]2=x2[1]2∧x1[0]2=x1[1]2 ⇒ COND_1524_0_HALF_LE(TRUE, x2[1]1, x1[1]1)≥NonInfC∧COND_1524_0_HALF_LE(TRUE, x2[1]1, x1[1]1)≥1524_0_HALF_LE(0, -(+(x2[1]1, 1), 2))∧(UIncreasing(1524_0_HALF_LE(0, -(+(x2[1]1, 1), 2))), ≥))



    We solved constraint (9) using rules (I), (II), (III), (IV), (IDP_CONSTANT_FOLD), (IDP_BOOLEAN).
  • We consider the chain 1524_0_HALF_LE(x2[0], x1[0]) → COND_1524_0_HALF_LE(&&(>(x2[0], 0), <=(x1[0], 1)), x2[0], x1[0]), COND_1524_0_HALF_LE(TRUE, x2[1], x1[1]) → 1524_0_HALF_LE(0, -(+(x2[1], 1), 2)), 1524_0_HALF_LE(x2[0], x1[0]) → COND_1524_0_HALF_LE(&&(>(x2[0], 0), <=(x1[0], 1)), x2[0], x1[0]), COND_1524_0_HALF_LE(TRUE, x2[1], x1[1]) → 1524_0_HALF_LE(0, -(+(x2[1], 1), 2)), 1524_0_HALF_LE(x2[2], x1[2]) → COND_1524_0_HALF_LE1(&&(>(x2[2], -1), >(x1[2], 1)), x2[2], x1[2]), COND_1524_0_HALF_LE1(TRUE, x2[3], x1[3]) → 1524_0_HALF_LE(+(x2[3], 1), -(x1[3], 2)) which results in the following constraint:

    (10)    (&&(>(x2[0], 0), <=(x1[0], 1))=TRUE∧x2[0]=x2[1]∧x1[0]=x1[1]∧0=x2[0]1∧-(+(x2[1], 1), 2)=x1[0]1∧&&(>(x2[0]1, 0), <=(x1[0]1, 1))=TRUE∧x2[0]1=x2[1]1∧x1[0]1=x1[1]1∧0=x2[2]∧-(+(x2[1]1, 1), 2)=x1[2]∧&&(>(x2[2], -1), >(x1[2], 1))=TRUE∧x2[2]=x2[3]∧x1[2]=x1[3] ⇒ COND_1524_0_HALF_LE(TRUE, x2[1]1, x1[1]1)≥NonInfC∧COND_1524_0_HALF_LE(TRUE, x2[1]1, x1[1]1)≥1524_0_HALF_LE(0, -(+(x2[1]1, 1), 2))∧(UIncreasing(1524_0_HALF_LE(0, -(+(x2[1]1, 1), 2))), ≥))



    We solved constraint (10) using rules (I), (II), (III), (IV), (IDP_CONSTANT_FOLD), (IDP_BOOLEAN).
  • We consider the chain 1524_0_HALF_LE(x2[2], x1[2]) → COND_1524_0_HALF_LE1(&&(>(x2[2], -1), >(x1[2], 1)), x2[2], x1[2]), COND_1524_0_HALF_LE1(TRUE, x2[3], x1[3]) → 1524_0_HALF_LE(+(x2[3], 1), -(x1[3], 2)), 1524_0_HALF_LE(x2[0], x1[0]) → COND_1524_0_HALF_LE(&&(>(x2[0], 0), <=(x1[0], 1)), x2[0], x1[0]), COND_1524_0_HALF_LE(TRUE, x2[1], x1[1]) → 1524_0_HALF_LE(0, -(+(x2[1], 1), 2)), 1524_0_HALF_LE(x2[0], x1[0]) → COND_1524_0_HALF_LE(&&(>(x2[0], 0), <=(x1[0], 1)), x2[0], x1[0]), COND_1524_0_HALF_LE(TRUE, x2[1], x1[1]) → 1524_0_HALF_LE(0, -(+(x2[1], 1), 2)) which results in the following constraint:

    (11)    (&&(>(x2[2], -1), >(x1[2], 1))=TRUE∧x2[2]=x2[3]∧x1[2]=x1[3]∧+(x2[3], 1)=x2[0]∧-(x1[3], 2)=x1[0]∧&&(>(x2[0], 0), <=(x1[0], 1))=TRUE∧x2[0]=x2[1]∧x1[0]=x1[1]∧0=x2[0]1∧-(+(x2[1], 1), 2)=x1[0]1∧&&(>(x2[0]1, 0), <=(x1[0]1, 1))=TRUE∧x2[0]1=x2[1]1∧x1[0]1=x1[1]1 ⇒ COND_1524_0_HALF_LE(TRUE, x2[1], x1[1])≥NonInfC∧COND_1524_0_HALF_LE(TRUE, x2[1], x1[1])≥1524_0_HALF_LE(0, -(+(x2[1], 1), 2))∧(UIncreasing(1524_0_HALF_LE(0, -(+(x2[1], 1), 2))), ≥))



    We solved constraint (11) using rules (I), (II), (III), (IV), (IDP_CONSTANT_FOLD), (IDP_BOOLEAN).
  • We consider the chain 1524_0_HALF_LE(x2[2], x1[2]) → COND_1524_0_HALF_LE1(&&(>(x2[2], -1), >(x1[2], 1)), x2[2], x1[2]), COND_1524_0_HALF_LE1(TRUE, x2[3], x1[3]) → 1524_0_HALF_LE(+(x2[3], 1), -(x1[3], 2)), 1524_0_HALF_LE(x2[0], x1[0]) → COND_1524_0_HALF_LE(&&(>(x2[0], 0), <=(x1[0], 1)), x2[0], x1[0]), COND_1524_0_HALF_LE(TRUE, x2[1], x1[1]) → 1524_0_HALF_LE(0, -(+(x2[1], 1), 2)), 1524_0_HALF_LE(x2[2], x1[2]) → COND_1524_0_HALF_LE1(&&(>(x2[2], -1), >(x1[2], 1)), x2[2], x1[2]), COND_1524_0_HALF_LE1(TRUE, x2[3], x1[3]) → 1524_0_HALF_LE(+(x2[3], 1), -(x1[3], 2)) which results in the following constraint:

    (12)    (&&(>(x2[2], -1), >(x1[2], 1))=TRUE∧x2[2]=x2[3]∧x1[2]=x1[3]∧+(x2[3], 1)=x2[0]∧-(x1[3], 2)=x1[0]∧&&(>(x2[0], 0), <=(x1[0], 1))=TRUE∧x2[0]=x2[1]∧x1[0]=x1[1]∧0=x2[2]1∧-(+(x2[1], 1), 2)=x1[2]1∧&&(>(x2[2]1, -1), >(x1[2]1, 1))=TRUE∧x2[2]1=x2[3]1∧x1[2]1=x1[3]1 ⇒ COND_1524_0_HALF_LE(TRUE, x2[1], x1[1])≥NonInfC∧COND_1524_0_HALF_LE(TRUE, x2[1], x1[1])≥1524_0_HALF_LE(0, -(+(x2[1], 1), 2))∧(UIncreasing(1524_0_HALF_LE(0, -(+(x2[1], 1), 2))), ≥))



    We simplified constraint (12) using rules (III), (IV), (IDP_CONSTANT_FOLD), (DELETE_TRIVIAL_REDUCESTO), (IDP_BOOLEAN) which results in the following new constraint:

    (13)    (>(x2[2], -1)=TRUE∧>(x1[2], 1)=TRUE∧>(+(x2[2], 1), 0)=TRUE∧<=(-(x1[2], 2), 1)=TRUE∧>(-(+(+(x2[2], 1), 1), 2), 1)=TRUE ⇒ COND_1524_0_HALF_LE(TRUE, +(x2[2], 1), -(x1[2], 2))≥NonInfC∧COND_1524_0_HALF_LE(TRUE, +(x2[2], 1), -(x1[2], 2))≥1524_0_HALF_LE(0, -(+(+(x2[2], 1), 1), 2))∧(UIncreasing(1524_0_HALF_LE(0, -(+(x2[1], 1), 2))), ≥))



    We simplified constraint (13) using rule (POLY_CONSTRAINTS) which results in the following new constraint:

    (14)    (x2[2] ≥ 0∧x1[2] + [-2] ≥ 0∧x2[2] ≥ 0∧[3] + [-1]x1[2] ≥ 0∧x2[2] + [-2] ≥ 0 ⇒ (UIncreasing(1524_0_HALF_LE(0, -(+(x2[1], 1), 2))), ≥)∧[(-2)bni_19 + (-1)Bound*bni_19] + [bni_19]x1[2] + [bni_19]x2[2] ≥ 0∧[-1 + (-1)bso_20] + x1[2] ≥ 0)



    We simplified constraint (14) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:

    (15)    (x2[2] ≥ 0∧x1[2] + [-2] ≥ 0∧x2[2] ≥ 0∧[3] + [-1]x1[2] ≥ 0∧x2[2] + [-2] ≥ 0 ⇒ (UIncreasing(1524_0_HALF_LE(0, -(+(x2[1], 1), 2))), ≥)∧[(-2)bni_19 + (-1)Bound*bni_19] + [bni_19]x1[2] + [bni_19]x2[2] ≥ 0∧[-1 + (-1)bso_20] + x1[2] ≥ 0)



    We simplified constraint (15) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:

    (16)    (x2[2] ≥ 0∧x1[2] + [-2] ≥ 0∧x2[2] ≥ 0∧[3] + [-1]x1[2] ≥ 0∧x2[2] + [-2] ≥ 0 ⇒ (UIncreasing(1524_0_HALF_LE(0, -(+(x2[1], 1), 2))), ≥)∧[(-2)bni_19 + (-1)Bound*bni_19] + [bni_19]x1[2] + [bni_19]x2[2] ≥ 0∧[-1 + (-1)bso_20] + x1[2] ≥ 0)



    We simplified constraint (16) using rule (IDP_SMT_SPLIT) which results in the following new constraint:

    (17)    ([2] + x2[2] ≥ 0∧x1[2] + [-2] ≥ 0∧[2] + x2[2] ≥ 0∧[3] + [-1]x1[2] ≥ 0∧x2[2] ≥ 0 ⇒ (UIncreasing(1524_0_HALF_LE(0, -(+(x2[1], 1), 2))), ≥)∧[(-1)Bound*bni_19] + [bni_19]x1[2] + [bni_19]x2[2] ≥ 0∧[-1 + (-1)bso_20] + x1[2] ≥ 0)



    We simplified constraint (17) using rule (IDP_SMT_SPLIT) which results in the following new constraint:

    (18)    ([2] + x2[2] ≥ 0∧x1[2] ≥ 0∧[2] + x2[2] ≥ 0∧[1] + [-1]x1[2] ≥ 0∧x2[2] ≥ 0 ⇒ (UIncreasing(1524_0_HALF_LE(0, -(+(x2[1], 1), 2))), ≥)∧[(-1)Bound*bni_19 + (2)bni_19] + [bni_19]x1[2] + [bni_19]x2[2] ≥ 0∧[1 + (-1)bso_20] + x1[2] ≥ 0)







For Pair 1524_0_HALF_LE(x2, x1) → COND_1524_0_HALF_LE1(&&(>(x2, -1), >(x1, 1)), x2, x1) the following chains were created:
  • We consider the chain 1524_0_HALF_LE(x2[2], x1[2]) → COND_1524_0_HALF_LE1(&&(>(x2[2], -1), >(x1[2], 1)), x2[2], x1[2]), COND_1524_0_HALF_LE1(TRUE, x2[3], x1[3]) → 1524_0_HALF_LE(+(x2[3], 1), -(x1[3], 2)) which results in the following constraint:

    (19)    (&&(>(x2[2], -1), >(x1[2], 1))=TRUE∧x2[2]=x2[3]∧x1[2]=x1[3] ⇒ 1524_0_HALF_LE(x2[2], x1[2])≥NonInfC∧1524_0_HALF_LE(x2[2], x1[2])≥COND_1524_0_HALF_LE1(&&(>(x2[2], -1), >(x1[2], 1)), x2[2], x1[2])∧(UIncreasing(COND_1524_0_HALF_LE1(&&(>(x2[2], -1), >(x1[2], 1)), x2[2], x1[2])), ≥))



    We simplified constraint (19) using rules (IV), (IDP_BOOLEAN) which results in the following new constraint:

    (20)    (>(x2[2], -1)=TRUE∧>(x1[2], 1)=TRUE ⇒ 1524_0_HALF_LE(x2[2], x1[2])≥NonInfC∧1524_0_HALF_LE(x2[2], x1[2])≥COND_1524_0_HALF_LE1(&&(>(x2[2], -1), >(x1[2], 1)), x2[2], x1[2])∧(UIncreasing(COND_1524_0_HALF_LE1(&&(>(x2[2], -1), >(x1[2], 1)), x2[2], x1[2])), ≥))



    We simplified constraint (20) using rule (POLY_CONSTRAINTS) which results in the following new constraint:

    (21)    (x2[2] ≥ 0∧x1[2] + [-2] ≥ 0 ⇒ (UIncreasing(COND_1524_0_HALF_LE1(&&(>(x2[2], -1), >(x1[2], 1)), x2[2], x1[2])), ≥)∧[(-1)bni_21 + (-1)Bound*bni_21] + [bni_21]x1[2] + [bni_21]x2[2] ≥ 0∧[(-1)bso_22] ≥ 0)



    We simplified constraint (21) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:

    (22)    (x2[2] ≥ 0∧x1[2] + [-2] ≥ 0 ⇒ (UIncreasing(COND_1524_0_HALF_LE1(&&(>(x2[2], -1), >(x1[2], 1)), x2[2], x1[2])), ≥)∧[(-1)bni_21 + (-1)Bound*bni_21] + [bni_21]x1[2] + [bni_21]x2[2] ≥ 0∧[(-1)bso_22] ≥ 0)



    We simplified constraint (22) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:

    (23)    (x2[2] ≥ 0∧x1[2] + [-2] ≥ 0 ⇒ (UIncreasing(COND_1524_0_HALF_LE1(&&(>(x2[2], -1), >(x1[2], 1)), x2[2], x1[2])), ≥)∧[(-1)bni_21 + (-1)Bound*bni_21] + [bni_21]x1[2] + [bni_21]x2[2] ≥ 0∧[(-1)bso_22] ≥ 0)



    We simplified constraint (23) using rule (IDP_SMT_SPLIT) which results in the following new constraint:

    (24)    (x2[2] ≥ 0∧x1[2] ≥ 0 ⇒ (UIncreasing(COND_1524_0_HALF_LE1(&&(>(x2[2], -1), >(x1[2], 1)), x2[2], x1[2])), ≥)∧[bni_21 + (-1)Bound*bni_21] + [bni_21]x1[2] + [bni_21]x2[2] ≥ 0∧[(-1)bso_22] ≥ 0)







For Pair COND_1524_0_HALF_LE1(TRUE, x2, x1) → 1524_0_HALF_LE(+(x2, 1), -(x1, 2)) the following chains were created:
  • We consider the chain 1524_0_HALF_LE(x2[0], x1[0]) → COND_1524_0_HALF_LE(&&(>(x2[0], 0), <=(x1[0], 1)), x2[0], x1[0]), COND_1524_0_HALF_LE(TRUE, x2[1], x1[1]) → 1524_0_HALF_LE(0, -(+(x2[1], 1), 2)), 1524_0_HALF_LE(x2[2], x1[2]) → COND_1524_0_HALF_LE1(&&(>(x2[2], -1), >(x1[2], 1)), x2[2], x1[2]), COND_1524_0_HALF_LE1(TRUE, x2[3], x1[3]) → 1524_0_HALF_LE(+(x2[3], 1), -(x1[3], 2)), 1524_0_HALF_LE(x2[0], x1[0]) → COND_1524_0_HALF_LE(&&(>(x2[0], 0), <=(x1[0], 1)), x2[0], x1[0]), COND_1524_0_HALF_LE(TRUE, x2[1], x1[1]) → 1524_0_HALF_LE(0, -(+(x2[1], 1), 2)) which results in the following constraint:

    (25)    (&&(>(x2[0], 0), <=(x1[0], 1))=TRUE∧x2[0]=x2[1]∧x1[0]=x1[1]∧0=x2[2]∧-(+(x2[1], 1), 2)=x1[2]∧&&(>(x2[2], -1), >(x1[2], 1))=TRUE∧x2[2]=x2[3]∧x1[2]=x1[3]∧+(x2[3], 1)=x2[0]1∧-(x1[3], 2)=x1[0]1∧&&(>(x2[0]1, 0), <=(x1[0]1, 1))=TRUE∧x2[0]1=x2[1]1∧x1[0]1=x1[1]1 ⇒ COND_1524_0_HALF_LE1(TRUE, x2[3], x1[3])≥NonInfC∧COND_1524_0_HALF_LE1(TRUE, x2[3], x1[3])≥1524_0_HALF_LE(+(x2[3], 1), -(x1[3], 2))∧(UIncreasing(1524_0_HALF_LE(+(x2[3], 1), -(x1[3], 2))), ≥))



    We simplified constraint (25) using rules (III), (IV), (IDP_CONSTANT_FOLD), (DELETE_TRIVIAL_REDUCESTO), (IDP_BOOLEAN) which results in the following new constraint:

    (26)    (>(x2[0], 0)=TRUE∧<=(x1[0], 1)=TRUE∧>(-(+(x2[0], 1), 2), 1)=TRUE∧<=(-(-(+(x2[0], 1), 2), 2), 1)=TRUE ⇒ COND_1524_0_HALF_LE1(TRUE, 0, -(+(x2[0], 1), 2))≥NonInfC∧COND_1524_0_HALF_LE1(TRUE, 0, -(+(x2[0], 1), 2))≥1524_0_HALF_LE(+(0, 1), -(-(+(x2[0], 1), 2), 2))∧(UIncreasing(1524_0_HALF_LE(+(x2[3], 1), -(x1[3], 2))), ≥))



    We simplified constraint (26) using rule (POLY_CONSTRAINTS) which results in the following new constraint:

    (27)    (x2[0] + [-1] ≥ 0∧[1] + [-1]x1[0] ≥ 0∧x2[0] + [-3] ≥ 0∧[4] + [-1]x2[0] ≥ 0 ⇒ (UIncreasing(1524_0_HALF_LE(+(x2[3], 1), -(x1[3], 2))), ≥)∧[(-2)bni_23 + (-1)Bound*bni_23] + [bni_23]x2[0] ≥ 0∧[1 + (-1)bso_24] ≥ 0)



    We simplified constraint (27) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:

    (28)    (x2[0] + [-1] ≥ 0∧[1] + [-1]x1[0] ≥ 0∧x2[0] + [-3] ≥ 0∧[4] + [-1]x2[0] ≥ 0 ⇒ (UIncreasing(1524_0_HALF_LE(+(x2[3], 1), -(x1[3], 2))), ≥)∧[(-2)bni_23 + (-1)Bound*bni_23] + [bni_23]x2[0] ≥ 0∧[1 + (-1)bso_24] ≥ 0)



    We simplified constraint (28) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:

    (29)    (x2[0] + [-1] ≥ 0∧[1] + [-1]x1[0] ≥ 0∧x2[0] + [-3] ≥ 0∧[4] + [-1]x2[0] ≥ 0 ⇒ (UIncreasing(1524_0_HALF_LE(+(x2[3], 1), -(x1[3], 2))), ≥)∧[(-2)bni_23 + (-1)Bound*bni_23] + [bni_23]x2[0] ≥ 0∧[1 + (-1)bso_24] ≥ 0)



    We simplified constraint (29) using rule (IDP_SMT_SPLIT) which results in the following new constraint:

    (30)    (x2[0] ≥ 0∧[1] + [-1]x1[0] ≥ 0∧[-2] + x2[0] ≥ 0∧[3] + [-1]x2[0] ≥ 0 ⇒ (UIncreasing(1524_0_HALF_LE(+(x2[3], 1), -(x1[3], 2))), ≥)∧[(-1)bni_23 + (-1)Bound*bni_23] + [bni_23]x2[0] ≥ 0∧[1 + (-1)bso_24] ≥ 0)



    We simplified constraint (30) using rule (IDP_SMT_SPLIT) which results in the following new constraint:

    (31)    ([2] + x2[0] ≥ 0∧[1] + [-1]x1[0] ≥ 0∧x2[0] ≥ 0∧[1] + [-1]x2[0] ≥ 0 ⇒ (UIncreasing(1524_0_HALF_LE(+(x2[3], 1), -(x1[3], 2))), ≥)∧[bni_23 + (-1)Bound*bni_23] + [bni_23]x2[0] ≥ 0∧[1 + (-1)bso_24] ≥ 0)



    We simplified constraint (31) using rule (IDP_SMT_SPLIT) which results in the following new constraint:

    (32)    ([2] + x2[0] ≥ 0∧x2[0] ≥ 0∧[1] + [-1]x2[0] ≥ 0 ⇒ (UIncreasing(1524_0_HALF_LE(+(x2[3], 1), -(x1[3], 2))), ≥)∧[bni_23 + (-1)Bound*bni_23] + [bni_23]x2[0] ≥ 0∧[1 + (-1)bso_24] ≥ 0)



  • We consider the chain 1524_0_HALF_LE(x2[0], x1[0]) → COND_1524_0_HALF_LE(&&(>(x2[0], 0), <=(x1[0], 1)), x2[0], x1[0]), COND_1524_0_HALF_LE(TRUE, x2[1], x1[1]) → 1524_0_HALF_LE(0, -(+(x2[1], 1), 2)), 1524_0_HALF_LE(x2[2], x1[2]) → COND_1524_0_HALF_LE1(&&(>(x2[2], -1), >(x1[2], 1)), x2[2], x1[2]), COND_1524_0_HALF_LE1(TRUE, x2[3], x1[3]) → 1524_0_HALF_LE(+(x2[3], 1), -(x1[3], 2)), 1524_0_HALF_LE(x2[2], x1[2]) → COND_1524_0_HALF_LE1(&&(>(x2[2], -1), >(x1[2], 1)), x2[2], x1[2]), COND_1524_0_HALF_LE1(TRUE, x2[3], x1[3]) → 1524_0_HALF_LE(+(x2[3], 1), -(x1[3], 2)) which results in the following constraint:

    (33)    (&&(>(x2[0], 0), <=(x1[0], 1))=TRUE∧x2[0]=x2[1]∧x1[0]=x1[1]∧0=x2[2]∧-(+(x2[1], 1), 2)=x1[2]∧&&(>(x2[2], -1), >(x1[2], 1))=TRUE∧x2[2]=x2[3]∧x1[2]=x1[3]∧+(x2[3], 1)=x2[2]1∧-(x1[3], 2)=x1[2]1∧&&(>(x2[2]1, -1), >(x1[2]1, 1))=TRUE∧x2[2]1=x2[3]1∧x1[2]1=x1[3]1 ⇒ COND_1524_0_HALF_LE1(TRUE, x2[3], x1[3])≥NonInfC∧COND_1524_0_HALF_LE1(TRUE, x2[3], x1[3])≥1524_0_HALF_LE(+(x2[3], 1), -(x1[3], 2))∧(UIncreasing(1524_0_HALF_LE(+(x2[3], 1), -(x1[3], 2))), ≥))



    We simplified constraint (33) using rules (III), (IV), (IDP_CONSTANT_FOLD), (DELETE_TRIVIAL_REDUCESTO), (IDP_BOOLEAN) which results in the following new constraint:

    (34)    (>(x2[0], 0)=TRUE∧<=(x1[0], 1)=TRUE∧>(-(+(x2[0], 1), 2), 1)=TRUE∧>(-(-(+(x2[0], 1), 2), 2), 1)=TRUE ⇒ COND_1524_0_HALF_LE1(TRUE, 0, -(+(x2[0], 1), 2))≥NonInfC∧COND_1524_0_HALF_LE1(TRUE, 0, -(+(x2[0], 1), 2))≥1524_0_HALF_LE(+(0, 1), -(-(+(x2[0], 1), 2), 2))∧(UIncreasing(1524_0_HALF_LE(+(x2[3], 1), -(x1[3], 2))), ≥))



    We simplified constraint (34) using rule (POLY_CONSTRAINTS) which results in the following new constraint:

    (35)    (x2[0] + [-1] ≥ 0∧[1] + [-1]x1[0] ≥ 0∧x2[0] + [-3] ≥ 0∧x2[0] + [-5] ≥ 0 ⇒ (UIncreasing(1524_0_HALF_LE(+(x2[3], 1), -(x1[3], 2))), ≥)∧[(-2)bni_23 + (-1)Bound*bni_23] + [bni_23]x2[0] ≥ 0∧[1 + (-1)bso_24] ≥ 0)



    We simplified constraint (35) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:

    (36)    (x2[0] + [-1] ≥ 0∧[1] + [-1]x1[0] ≥ 0∧x2[0] + [-3] ≥ 0∧x2[0] + [-5] ≥ 0 ⇒ (UIncreasing(1524_0_HALF_LE(+(x2[3], 1), -(x1[3], 2))), ≥)∧[(-2)bni_23 + (-1)Bound*bni_23] + [bni_23]x2[0] ≥ 0∧[1 + (-1)bso_24] ≥ 0)



    We simplified constraint (36) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:

    (37)    (x2[0] + [-1] ≥ 0∧[1] + [-1]x1[0] ≥ 0∧x2[0] + [-3] ≥ 0∧x2[0] + [-5] ≥ 0 ⇒ (UIncreasing(1524_0_HALF_LE(+(x2[3], 1), -(x1[3], 2))), ≥)∧[(-2)bni_23 + (-1)Bound*bni_23] + [bni_23]x2[0] ≥ 0∧[1 + (-1)bso_24] ≥ 0)



    We simplified constraint (37) using rule (IDP_SMT_SPLIT) which results in the following new constraint:

    (38)    (x2[0] ≥ 0∧[1] + [-1]x1[0] ≥ 0∧[-2] + x2[0] ≥ 0∧[-4] + x2[0] ≥ 0 ⇒ (UIncreasing(1524_0_HALF_LE(+(x2[3], 1), -(x1[3], 2))), ≥)∧[(-1)bni_23 + (-1)Bound*bni_23] + [bni_23]x2[0] ≥ 0∧[1 + (-1)bso_24] ≥ 0)



    We simplified constraint (38) using rule (IDP_SMT_SPLIT) which results in the following new constraint:

    (39)    ([2] + x2[0] ≥ 0∧[1] + [-1]x1[0] ≥ 0∧x2[0] ≥ 0∧[-2] + x2[0] ≥ 0 ⇒ (UIncreasing(1524_0_HALF_LE(+(x2[3], 1), -(x1[3], 2))), ≥)∧[bni_23 + (-1)Bound*bni_23] + [bni_23]x2[0] ≥ 0∧[1 + (-1)bso_24] ≥ 0)



    We simplified constraint (39) using rule (IDP_SMT_SPLIT) which results in the following new constraint:

    (40)    ([4] + x2[0] ≥ 0∧[1] + [-1]x1[0] ≥ 0∧[2] + x2[0] ≥ 0∧x2[0] ≥ 0 ⇒ (UIncreasing(1524_0_HALF_LE(+(x2[3], 1), -(x1[3], 2))), ≥)∧[(3)bni_23 + (-1)Bound*bni_23] + [bni_23]x2[0] ≥ 0∧[1 + (-1)bso_24] ≥ 0)



    We simplified constraint (40) using rule (IDP_SMT_SPLIT) which results in the following new constraint:

    (41)    ([4] + x2[0] ≥ 0∧[2] + x2[0] ≥ 0∧x2[0] ≥ 0 ⇒ (UIncreasing(1524_0_HALF_LE(+(x2[3], 1), -(x1[3], 2))), ≥)∧[(3)bni_23 + (-1)Bound*bni_23] + [bni_23]x2[0] ≥ 0∧[1 + (-1)bso_24] ≥ 0)



  • We consider the chain 1524_0_HALF_LE(x2[2], x1[2]) → COND_1524_0_HALF_LE1(&&(>(x2[2], -1), >(x1[2], 1)), x2[2], x1[2]), COND_1524_0_HALF_LE1(TRUE, x2[3], x1[3]) → 1524_0_HALF_LE(+(x2[3], 1), -(x1[3], 2)), 1524_0_HALF_LE(x2[2], x1[2]) → COND_1524_0_HALF_LE1(&&(>(x2[2], -1), >(x1[2], 1)), x2[2], x1[2]), COND_1524_0_HALF_LE1(TRUE, x2[3], x1[3]) → 1524_0_HALF_LE(+(x2[3], 1), -(x1[3], 2)), 1524_0_HALF_LE(x2[0], x1[0]) → COND_1524_0_HALF_LE(&&(>(x2[0], 0), <=(x1[0], 1)), x2[0], x1[0]), COND_1524_0_HALF_LE(TRUE, x2[1], x1[1]) → 1524_0_HALF_LE(0, -(+(x2[1], 1), 2)) which results in the following constraint:

    (42)    (&&(>(x2[2], -1), >(x1[2], 1))=TRUE∧x2[2]=x2[3]∧x1[2]=x1[3]∧+(x2[3], 1)=x2[2]1∧-(x1[3], 2)=x1[2]1∧&&(>(x2[2]1, -1), >(x1[2]1, 1))=TRUE∧x2[2]1=x2[3]1∧x1[2]1=x1[3]1∧+(x2[3]1, 1)=x2[0]∧-(x1[3]1, 2)=x1[0]∧&&(>(x2[0], 0), <=(x1[0], 1))=TRUE∧x2[0]=x2[1]∧x1[0]=x1[1] ⇒ COND_1524_0_HALF_LE1(TRUE, x2[3]1, x1[3]1)≥NonInfC∧COND_1524_0_HALF_LE1(TRUE, x2[3]1, x1[3]1)≥1524_0_HALF_LE(+(x2[3]1, 1), -(x1[3]1, 2))∧(UIncreasing(1524_0_HALF_LE(+(x2[3]1, 1), -(x1[3]1, 2))), ≥))



    We simplified constraint (42) using rules (III), (IV), (IDP_BOOLEAN) which results in the following new constraint:

    (43)    (>(x2[2], -1)=TRUE∧>(x1[2], 1)=TRUE∧>(+(x2[2], 1), -1)=TRUE∧>(-(x1[2], 2), 1)=TRUE∧>(+(+(x2[2], 1), 1), 0)=TRUE∧<=(-(-(x1[2], 2), 2), 1)=TRUE ⇒ COND_1524_0_HALF_LE1(TRUE, +(x2[2], 1), -(x1[2], 2))≥NonInfC∧COND_1524_0_HALF_LE1(TRUE, +(x2[2], 1), -(x1[2], 2))≥1524_0_HALF_LE(+(+(x2[2], 1), 1), -(-(x1[2], 2), 2))∧(UIncreasing(1524_0_HALF_LE(+(x2[3]1, 1), -(x1[3]1, 2))), ≥))



    We simplified constraint (43) using rule (POLY_CONSTRAINTS) which results in the following new constraint:

    (44)    (x2[2] ≥ 0∧x1[2] + [-2] ≥ 0∧x2[2] + [1] ≥ 0∧x1[2] + [-4] ≥ 0∧x2[2] + [1] ≥ 0∧[5] + [-1]x1[2] ≥ 0 ⇒ (UIncreasing(1524_0_HALF_LE(+(x2[3]1, 1), -(x1[3]1, 2))), ≥)∧[(-2)bni_23 + (-1)Bound*bni_23] + [bni_23]x1[2] + [bni_23]x2[2] ≥ 0∧[1 + (-1)bso_24] ≥ 0)



    We simplified constraint (44) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:

    (45)    (x2[2] ≥ 0∧x1[2] + [-2] ≥ 0∧x2[2] + [1] ≥ 0∧x1[2] + [-4] ≥ 0∧x2[2] + [1] ≥ 0∧[5] + [-1]x1[2] ≥ 0 ⇒ (UIncreasing(1524_0_HALF_LE(+(x2[3]1, 1), -(x1[3]1, 2))), ≥)∧[(-2)bni_23 + (-1)Bound*bni_23] + [bni_23]x1[2] + [bni_23]x2[2] ≥ 0∧[1 + (-1)bso_24] ≥ 0)



    We simplified constraint (45) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:

    (46)    (x2[2] ≥ 0∧x1[2] + [-2] ≥ 0∧x2[2] + [1] ≥ 0∧x1[2] + [-4] ≥ 0∧x2[2] + [1] ≥ 0∧[5] + [-1]x1[2] ≥ 0 ⇒ (UIncreasing(1524_0_HALF_LE(+(x2[3]1, 1), -(x1[3]1, 2))), ≥)∧[(-2)bni_23 + (-1)Bound*bni_23] + [bni_23]x1[2] + [bni_23]x2[2] ≥ 0∧[1 + (-1)bso_24] ≥ 0)



    We simplified constraint (46) using rule (IDP_SMT_SPLIT) which results in the following new constraint:

    (47)    (x2[2] ≥ 0∧x1[2] ≥ 0∧x2[2] + [1] ≥ 0∧[-2] + x1[2] ≥ 0∧x2[2] + [1] ≥ 0∧[3] + [-1]x1[2] ≥ 0 ⇒ (UIncreasing(1524_0_HALF_LE(+(x2[3]1, 1), -(x1[3]1, 2))), ≥)∧[(-1)Bound*bni_23] + [bni_23]x1[2] + [bni_23]x2[2] ≥ 0∧[1 + (-1)bso_24] ≥ 0)



    We simplified constraint (47) using rule (IDP_SMT_SPLIT) which results in the following new constraint:

    (48)    (x2[2] ≥ 0∧[2] + x1[2] ≥ 0∧x2[2] + [1] ≥ 0∧x1[2] ≥ 0∧x2[2] + [1] ≥ 0∧[1] + [-1]x1[2] ≥ 0 ⇒ (UIncreasing(1524_0_HALF_LE(+(x2[3]1, 1), -(x1[3]1, 2))), ≥)∧[(2)bni_23 + (-1)Bound*bni_23] + [bni_23]x1[2] + [bni_23]x2[2] ≥ 0∧[1 + (-1)bso_24] ≥ 0)



  • We consider the chain 1524_0_HALF_LE(x2[2], x1[2]) → COND_1524_0_HALF_LE1(&&(>(x2[2], -1), >(x1[2], 1)), x2[2], x1[2]), COND_1524_0_HALF_LE1(TRUE, x2[3], x1[3]) → 1524_0_HALF_LE(+(x2[3], 1), -(x1[3], 2)), 1524_0_HALF_LE(x2[2], x1[2]) → COND_1524_0_HALF_LE1(&&(>(x2[2], -1), >(x1[2], 1)), x2[2], x1[2]), COND_1524_0_HALF_LE1(TRUE, x2[3], x1[3]) → 1524_0_HALF_LE(+(x2[3], 1), -(x1[3], 2)), 1524_0_HALF_LE(x2[2], x1[2]) → COND_1524_0_HALF_LE1(&&(>(x2[2], -1), >(x1[2], 1)), x2[2], x1[2]), COND_1524_0_HALF_LE1(TRUE, x2[3], x1[3]) → 1524_0_HALF_LE(+(x2[3], 1), -(x1[3], 2)) which results in the following constraint:

    (49)    (&&(>(x2[2], -1), >(x1[2], 1))=TRUE∧x2[2]=x2[3]∧x1[2]=x1[3]∧+(x2[3], 1)=x2[2]1∧-(x1[3], 2)=x1[2]1∧&&(>(x2[2]1, -1), >(x1[2]1, 1))=TRUE∧x2[2]1=x2[3]1∧x1[2]1=x1[3]1∧+(x2[3]1, 1)=x2[2]2∧-(x1[3]1, 2)=x1[2]2∧&&(>(x2[2]2, -1), >(x1[2]2, 1))=TRUE∧x2[2]2=x2[3]2∧x1[2]2=x1[3]2 ⇒ COND_1524_0_HALF_LE1(TRUE, x2[3]1, x1[3]1)≥NonInfC∧COND_1524_0_HALF_LE1(TRUE, x2[3]1, x1[3]1)≥1524_0_HALF_LE(+(x2[3]1, 1), -(x1[3]1, 2))∧(UIncreasing(1524_0_HALF_LE(+(x2[3]1, 1), -(x1[3]1, 2))), ≥))



    We simplified constraint (49) using rules (III), (IV), (IDP_BOOLEAN) which results in the following new constraint:

    (50)    (>(x2[2], -1)=TRUE∧>(x1[2], 1)=TRUE∧>(+(x2[2], 1), -1)=TRUE∧>(-(x1[2], 2), 1)=TRUE∧>(+(+(x2[2], 1), 1), -1)=TRUE∧>(-(-(x1[2], 2), 2), 1)=TRUE ⇒ COND_1524_0_HALF_LE1(TRUE, +(x2[2], 1), -(x1[2], 2))≥NonInfC∧COND_1524_0_HALF_LE1(TRUE, +(x2[2], 1), -(x1[2], 2))≥1524_0_HALF_LE(+(+(x2[2], 1), 1), -(-(x1[2], 2), 2))∧(UIncreasing(1524_0_HALF_LE(+(x2[3]1, 1), -(x1[3]1, 2))), ≥))



    We simplified constraint (50) using rule (POLY_CONSTRAINTS) which results in the following new constraint:

    (51)    (x2[2] ≥ 0∧x1[2] + [-2] ≥ 0∧x2[2] + [1] ≥ 0∧x1[2] + [-4] ≥ 0∧x2[2] + [2] ≥ 0∧x1[2] + [-6] ≥ 0 ⇒ (UIncreasing(1524_0_HALF_LE(+(x2[3]1, 1), -(x1[3]1, 2))), ≥)∧[(-2)bni_23 + (-1)Bound*bni_23] + [bni_23]x1[2] + [bni_23]x2[2] ≥ 0∧[1 + (-1)bso_24] ≥ 0)



    We simplified constraint (51) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:

    (52)    (x2[2] ≥ 0∧x1[2] + [-2] ≥ 0∧x2[2] + [1] ≥ 0∧x1[2] + [-4] ≥ 0∧x2[2] + [2] ≥ 0∧x1[2] + [-6] ≥ 0 ⇒ (UIncreasing(1524_0_HALF_LE(+(x2[3]1, 1), -(x1[3]1, 2))), ≥)∧[(-2)bni_23 + (-1)Bound*bni_23] + [bni_23]x1[2] + [bni_23]x2[2] ≥ 0∧[1 + (-1)bso_24] ≥ 0)



    We simplified constraint (52) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:

    (53)    (x2[2] ≥ 0∧x1[2] + [-2] ≥ 0∧x2[2] + [1] ≥ 0∧x1[2] + [-4] ≥ 0∧x2[2] + [2] ≥ 0∧x1[2] + [-6] ≥ 0 ⇒ (UIncreasing(1524_0_HALF_LE(+(x2[3]1, 1), -(x1[3]1, 2))), ≥)∧[(-2)bni_23 + (-1)Bound*bni_23] + [bni_23]x1[2] + [bni_23]x2[2] ≥ 0∧[1 + (-1)bso_24] ≥ 0)



    We simplified constraint (53) using rule (IDP_SMT_SPLIT) which results in the following new constraint:

    (54)    (x2[2] ≥ 0∧x1[2] ≥ 0∧x2[2] + [1] ≥ 0∧[-2] + x1[2] ≥ 0∧x2[2] + [2] ≥ 0∧[-4] + x1[2] ≥ 0 ⇒ (UIncreasing(1524_0_HALF_LE(+(x2[3]1, 1), -(x1[3]1, 2))), ≥)∧[(-1)Bound*bni_23] + [bni_23]x1[2] + [bni_23]x2[2] ≥ 0∧[1 + (-1)bso_24] ≥ 0)



    We simplified constraint (54) using rule (IDP_SMT_SPLIT) which results in the following new constraint:

    (55)    (x2[2] ≥ 0∧[2] + x1[2] ≥ 0∧x2[2] + [1] ≥ 0∧x1[2] ≥ 0∧x2[2] + [2] ≥ 0∧[-2] + x1[2] ≥ 0 ⇒ (UIncreasing(1524_0_HALF_LE(+(x2[3]1, 1), -(x1[3]1, 2))), ≥)∧[(2)bni_23 + (-1)Bound*bni_23] + [bni_23]x1[2] + [bni_23]x2[2] ≥ 0∧[1 + (-1)bso_24] ≥ 0)



    We simplified constraint (55) using rule (IDP_SMT_SPLIT) which results in the following new constraint:

    (56)    (x2[2] ≥ 0∧[4] + x1[2] ≥ 0∧x2[2] + [1] ≥ 0∧[2] + x1[2] ≥ 0∧x2[2] + [2] ≥ 0∧x1[2] ≥ 0 ⇒ (UIncreasing(1524_0_HALF_LE(+(x2[3]1, 1), -(x1[3]1, 2))), ≥)∧[(4)bni_23 + (-1)Bound*bni_23] + [bni_23]x1[2] + [bni_23]x2[2] ≥ 0∧[1 + (-1)bso_24] ≥ 0)







To summarize, we get the following constraints P≥ for the following pairs.
  • 1524_0_HALF_LE(x2, x1) → COND_1524_0_HALF_LE(&&(>(x2, 0), <=(x1, 1)), x2, x1)
    • (x2[0] ≥ 0∧[1] + [-1]x1[0] ≥ 0∧x1[0] ≥ 0 ⇒ (UIncreasing(COND_1524_0_HALF_LE(&&(>(x2[0], 0), <=(x1[0], 1)), x2[0], x1[0])), ≥)∧[(-1)Bound*bni_17] + [bni_17]x1[0] + [bni_17]x2[0] ≥ 0∧[(-1)bso_18] ≥ 0)
    • (x2[0] ≥ 0∧[1] + x1[0] ≥ 0∧x1[0] ≥ 0 ⇒ (UIncreasing(COND_1524_0_HALF_LE(&&(>(x2[0], 0), <=(x1[0], 1)), x2[0], x1[0])), ≥)∧[(-1)Bound*bni_17] + [(-1)bni_17]x1[0] + [bni_17]x2[0] ≥ 0∧[(-1)bso_18] ≥ 0)

  • COND_1524_0_HALF_LE(TRUE, x2, x1) → 1524_0_HALF_LE(0, -(+(x2, 1), 2))
    • ([2] + x2[2] ≥ 0∧x1[2] ≥ 0∧[2] + x2[2] ≥ 0∧[1] + [-1]x1[2] ≥ 0∧x2[2] ≥ 0 ⇒ (UIncreasing(1524_0_HALF_LE(0, -(+(x2[1], 1), 2))), ≥)∧[(-1)Bound*bni_19 + (2)bni_19] + [bni_19]x1[2] + [bni_19]x2[2] ≥ 0∧[1 + (-1)bso_20] + x1[2] ≥ 0)

  • 1524_0_HALF_LE(x2, x1) → COND_1524_0_HALF_LE1(&&(>(x2, -1), >(x1, 1)), x2, x1)
    • (x2[2] ≥ 0∧x1[2] ≥ 0 ⇒ (UIncreasing(COND_1524_0_HALF_LE1(&&(>(x2[2], -1), >(x1[2], 1)), x2[2], x1[2])), ≥)∧[bni_21 + (-1)Bound*bni_21] + [bni_21]x1[2] + [bni_21]x2[2] ≥ 0∧[(-1)bso_22] ≥ 0)

  • COND_1524_0_HALF_LE1(TRUE, x2, x1) → 1524_0_HALF_LE(+(x2, 1), -(x1, 2))
    • ([2] + x2[0] ≥ 0∧x2[0] ≥ 0∧[1] + [-1]x2[0] ≥ 0 ⇒ (UIncreasing(1524_0_HALF_LE(+(x2[3], 1), -(x1[3], 2))), ≥)∧[bni_23 + (-1)Bound*bni_23] + [bni_23]x2[0] ≥ 0∧[1 + (-1)bso_24] ≥ 0)
    • ([4] + x2[0] ≥ 0∧[2] + x2[0] ≥ 0∧x2[0] ≥ 0 ⇒ (UIncreasing(1524_0_HALF_LE(+(x2[3], 1), -(x1[3], 2))), ≥)∧[(3)bni_23 + (-1)Bound*bni_23] + [bni_23]x2[0] ≥ 0∧[1 + (-1)bso_24] ≥ 0)
    • (x2[2] ≥ 0∧[2] + x1[2] ≥ 0∧x2[2] + [1] ≥ 0∧x1[2] ≥ 0∧x2[2] + [1] ≥ 0∧[1] + [-1]x1[2] ≥ 0 ⇒ (UIncreasing(1524_0_HALF_LE(+(x2[3]1, 1), -(x1[3]1, 2))), ≥)∧[(2)bni_23 + (-1)Bound*bni_23] + [bni_23]x1[2] + [bni_23]x2[2] ≥ 0∧[1 + (-1)bso_24] ≥ 0)
    • (x2[2] ≥ 0∧[4] + x1[2] ≥ 0∧x2[2] + [1] ≥ 0∧[2] + x1[2] ≥ 0∧x2[2] + [2] ≥ 0∧x1[2] ≥ 0 ⇒ (UIncreasing(1524_0_HALF_LE(+(x2[3]1, 1), -(x1[3]1, 2))), ≥)∧[(4)bni_23 + (-1)Bound*bni_23] + [bni_23]x1[2] + [bni_23]x2[2] ≥ 0∧[1 + (-1)bso_24] ≥ 0)




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(TRUE) = 0   
POL(FALSE) = 0   
POL(1524_0_HALF_LE(x1, x2)) = [-1] + x2 + x1   
POL(COND_1524_0_HALF_LE(x1, x2, x3)) = [-1] + x3 + x2 + [-1]x1   
POL(&&(x1, x2)) = 0   
POL(>(x1, x2)) = [-1]   
POL(0) = 0   
POL(<=(x1, x2)) = [-1]   
POL(1) = [1]   
POL(-(x1, x2)) = x1 + [-1]x2   
POL(+(x1, x2)) = x1 + x2   
POL(2) = [2]   
POL(COND_1524_0_HALF_LE1(x1, x2, x3)) = [-1] + x3 + x2 + [-1]x1   
POL(-1) = [-1]   

The following pairs are in P>:

COND_1524_0_HALF_LE(TRUE, x2[1], x1[1]) → 1524_0_HALF_LE(0, -(+(x2[1], 1), 2))
COND_1524_0_HALF_LE1(TRUE, x2[3], x1[3]) → 1524_0_HALF_LE(+(x2[3], 1), -(x1[3], 2))

The following pairs are in Pbound:

COND_1524_0_HALF_LE(TRUE, x2[1], x1[1]) → 1524_0_HALF_LE(0, -(+(x2[1], 1), 2))
1524_0_HALF_LE(x2[2], x1[2]) → COND_1524_0_HALF_LE1(&&(>(x2[2], -1), >(x1[2], 1)), x2[2], x1[2])
COND_1524_0_HALF_LE1(TRUE, x2[3], x1[3]) → 1524_0_HALF_LE(+(x2[3], 1), -(x1[3], 2))

The following pairs are in P≥:

1524_0_HALF_LE(x2[0], x1[0]) → COND_1524_0_HALF_LE(&&(>(x2[0], 0), <=(x1[0], 1)), x2[0], x1[0])
1524_0_HALF_LE(x2[2], x1[2]) → COND_1524_0_HALF_LE1(&&(>(x2[2], -1), >(x1[2], 1)), x2[2], x1[2])

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

TRUE1 → &&(TRUE, TRUE)1
&&(TRUE, FALSE)1 ↔ FALSE1
&&(FALSE, TRUE)1 ↔ FALSE1
&&(FALSE, FALSE)1 ↔ FALSE1

(8) Obligation:

IDP problem:
The following function symbols are pre-defined:
!=~Neq: (Integer, Integer) -> Boolean
*~Mul: (Integer, Integer) -> Integer
>=~Ge: (Integer, Integer) -> Boolean
-1~UnaryMinus: (Integer) -> Integer
|~Bwor: (Integer, Integer) -> Integer
/~Div: (Integer, Integer) -> Integer
=~Eq: (Integer, Integer) -> Boolean
~Bwxor: (Integer, Integer) -> Integer
||~Lor: (Boolean, Boolean) -> Boolean
!~Lnot: (Boolean) -> Boolean
<~Lt: (Integer, Integer) -> Boolean
-~Sub: (Integer, Integer) -> Integer
<=~Le: (Integer, Integer) -> Boolean
>~Gt: (Integer, Integer) -> Boolean
~~Bwnot: (Integer) -> Integer
%~Mod: (Integer, Integer) -> Integer
&~Bwand: (Integer, Integer) -> Integer
+~Add: (Integer, Integer) -> Integer
&&~Land: (Boolean, Boolean) -> Boolean


The following domains are used:

Boolean, Integer


R is empty.

The integer pair graph contains the following rules and edges:
(0): 1524_0_HALF_LE(x2[0], x1[0]) → COND_1524_0_HALF_LE(x2[0] > 0 && x1[0] <= 1, x2[0], x1[0])
(2): 1524_0_HALF_LE(x2[2], x1[2]) → COND_1524_0_HALF_LE1(x2[2] > -1 && x1[2] > 1, x2[2], x1[2])


The set Q is empty.

(9) IDependencyGraphProof (EQUIVALENT transformation)

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

(10) TRUE