Termination of the given JBC could be proven:

0 JBC
↳1 JBCToGraph (⇒, 635 ms)
↳2 JBCTerminationGraph
↳3 TerminationGraphToSCCProof (⇒, 0 ms)
↳4 JBCTerminationSCC
↳5 SCCToIDPv1Proof (⇒, 52 ms)
↳6 IDP
↳7 IDPNonInfProof (⇒, 894 ms)
↳8 IDP
↳9 IDependencyGraphProof (⇔, 0 ms)
↳10 IDP
↳11 IDPNonInfProof (⇒, 161 ms)
↳12 IDP
↳13 IDependencyGraphProof (⇔, 0 ms)
↳14 TRUE

(0) Obligation:

JBC Problem based on JBC Program:
public class Log{
  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);
      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:
Log.main([Ljava/lang/String;)V: Graph of 141 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: Log.main([Ljava/lang/String;)V
SCC calls the following helper methods:
Performed SCC analyses:
  • Used field analysis yielded the following read fields:
  • Marker field analysis yielded the following relations that could be markers:

(5) SCCToIDPv1Proof (SOUND transformation)

Transformed FIGraph SCCs to IDPs. Log:

Generated 31 rules for P and 0 rules for R.


P rules:
f704_0_log_ConstantStackPush(EOS(STATIC_704), i123, i123) → f707_0_log_LE(EOS(STATIC_707), i123, i123, 1)
f707_0_log_LE(EOS(STATIC_707), i131, i131, matching1) → f710_0_log_LE(EOS(STATIC_710), i131, i131, 1) | =(matching1, 1)
f710_0_log_LE(EOS(STATIC_710), i131, i131, matching1) → f714_0_log_Load(EOS(STATIC_714), i131) | &&(>(i131, 1), =(matching1, 1))
f714_0_log_Load(EOS(STATIC_714), i131) → f717_0_log_InvokeMethod(EOS(STATIC_717), i131)
f717_0_log_InvokeMethod(EOS(STATIC_717), i131) → f722_0_half_ConstantStackPush(EOS(STATIC_722), i131, i131)
f722_0_half_ConstantStackPush(EOS(STATIC_722), i131, i131) → f728_0_half_Store(EOS(STATIC_728), i131, i131, 0)
f728_0_half_Store(EOS(STATIC_728), i131, i131, matching1) → f730_0_half_Load(EOS(STATIC_730), i131, i131, 0) | =(matching1, 0)
f730_0_half_Load(EOS(STATIC_730), i131, i131, matching1) → f765_0_half_Load(EOS(STATIC_765), i131, i131, 0) | =(matching1, 0)
f765_0_half_Load(EOS(STATIC_765), i131, i135, i136) → f1129_0_half_Load(EOS(STATIC_1129), i131, i135, i136)
f1129_0_half_Load(EOS(STATIC_1129), i131, i191, i192) → f1407_0_half_Load(EOS(STATIC_1407), i131, i191, i192)
f1407_0_half_Load(EOS(STATIC_1407), i131, i244, i245) → f1790_0_half_Load(EOS(STATIC_1790), i131, i244, i245)
f1790_0_half_Load(EOS(STATIC_1790), i131, i297, i298) → f1794_0_half_ConstantStackPush(EOS(STATIC_1794), i131, i297, i298, i297)
f1794_0_half_ConstantStackPush(EOS(STATIC_1794), i131, i297, i298, i297) → f1798_0_half_LE(EOS(STATIC_1798), i131, i297, i298, i297, 1)
f1798_0_half_LE(EOS(STATIC_1798), i131, i304, i298, i304, matching1) → f1801_0_half_LE(EOS(STATIC_1801), i131, i304, i298, i304, 1) | =(matching1, 1)
f1798_0_half_LE(EOS(STATIC_1798), i131, i305, i298, i305, matching1) → f1802_0_half_LE(EOS(STATIC_1802), i131, i305, i298, i305, 1) | =(matching1, 1)
f1801_0_half_LE(EOS(STATIC_1801), i131, i304, i298, i304, matching1) → f1804_0_half_Load(EOS(STATIC_1804), i131, i298) | &&(<=(i304, 1), =(matching1, 1))
f1804_0_half_Load(EOS(STATIC_1804), i131, i298) → f1808_0_half_Return(EOS(STATIC_1808), i131, i298)
f1808_0_half_Return(EOS(STATIC_1808), i131, i298) → f1813_0_log_Store(EOS(STATIC_1813), i298)
f1813_0_log_Store(EOS(STATIC_1813), i298) → f1817_0_log_Inc(EOS(STATIC_1817), i298)
f1817_0_log_Inc(EOS(STATIC_1817), i298) → f1821_0_log_JMP(EOS(STATIC_1821), i298)
f1821_0_log_JMP(EOS(STATIC_1821), i298) → f1831_0_log_Load(EOS(STATIC_1831), i298)
f1831_0_log_Load(EOS(STATIC_1831), i298) → f701_0_log_Load(EOS(STATIC_701), i298)
f701_0_log_Load(EOS(STATIC_701), i123) → f704_0_log_ConstantStackPush(EOS(STATIC_704), i123, i123)
f1802_0_half_LE(EOS(STATIC_1802), i131, i305, i298, i305, matching1) → f1806_0_half_Load(EOS(STATIC_1806), i131, i305, i298) | &&(>(i305, 1), =(matching1, 1))
f1806_0_half_Load(EOS(STATIC_1806), i131, i305, i298) → f1810_0_half_ConstantStackPush(EOS(STATIC_1810), i131, i298, i305)
f1810_0_half_ConstantStackPush(EOS(STATIC_1810), i131, i298, i305) → f1815_0_half_IntArithmetic(EOS(STATIC_1815), i131, i298, i305, 2)
f1815_0_half_IntArithmetic(EOS(STATIC_1815), i131, i298, i305, matching1) → f1819_0_half_Store(EOS(STATIC_1819), i131, i298, -(i305, 2)) | &&(>(i305, 0), =(matching1, 2))
f1819_0_half_Store(EOS(STATIC_1819), i131, i298, i306) → f1823_0_half_Inc(EOS(STATIC_1823), i131, i306, i298)
f1823_0_half_Inc(EOS(STATIC_1823), i131, i306, i298) → f1833_0_half_JMP(EOS(STATIC_1833), i131, i306, +(i298, 1)) | >=(i298, 0)
f1833_0_half_JMP(EOS(STATIC_1833), i131, i306, i310) → f2151_0_half_Load(EOS(STATIC_2151), i131, i306, i310)
f2151_0_half_Load(EOS(STATIC_2151), i131, i306, i310) → f1790_0_half_Load(EOS(STATIC_1790), i131, i306, i310)
R rules:

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


P rules:
f1798_0_half_LE(EOS(STATIC_1798), x0, x1, x2, x1, 1) → f1798_0_half_LE(EOS(STATIC_1798), x2, x2, 0, x2, 1) | &&(<=(x1, 1), >(x2, 1))
f1798_0_half_LE(EOS(STATIC_1798), x0, x1, x2, x1, 1) → f1798_0_half_LE(EOS(STATIC_1798), x0, -(x1, 2), +(x2, 1), -(x1, 2), 1) | &&(>(x1, 1), >(+(x2, 1), 0))
R rules:

Filtered ground terms:



f1798_0_half_LE(x1, x2, x3, x4, x5, x6) → f1798_0_half_LE(x2, x3, x4, x5)
Cond_f1798_0_half_LE(x1, x2, x3, x4, x5, x6, x7) → Cond_f1798_0_half_LE(x1, x3, x4, x5, x6)
Cond_f1798_0_half_LE1(x1, x2, x3, x4, x5, x6, x7) → Cond_f1798_0_half_LE1(x1, x3, x4, x5, x6)
EOS(x1) → EOS

Filtered unneeded arguments:



f1798_0_half_LE(x1, x2, x3, x4) → f1798_0_half_LE(x2, x3, x4)
Cond_f1798_0_half_LE(x1, x2, x3, x4, x5) → Cond_f1798_0_half_LE(x1, x3, x4, x5)
Cond_f1798_0_half_LE1(x1, x2, x3, x4, x5) → Cond_f1798_0_half_LE1(x1, x3, x4, x5)

Filtered duplicate args:



f1798_0_half_LE(x1, x2, x3) → f1798_0_half_LE(x2, x3)
Cond_f1798_0_half_LE(x1, x2, x3, x4) → Cond_f1798_0_half_LE(x1, x3, x4)
Cond_f1798_0_half_LE1(x1, x2, x3, x4) → Cond_f1798_0_half_LE1(x1, x3, x4)

Filtered unneeded arguments:



Cond_f1798_0_half_LE(x1, x2, x3) → Cond_f1798_0_half_LE(x1, x2)

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


P rules:
F1798_0_HALF_LE(x2, x1) → F1798_0_HALF_LE(0, x2) | &&(<=(x1, 1), >(x2, 1))
F1798_0_HALF_LE(x2, x1) → F1798_0_HALF_LE(+(x2, 1), -(x1, 2)) | &&(>(x1, 1), >(x2, -1))
R rules:

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


P rules:
F1798_0_HALF_LE'(x2, x1) → COND_F1798_0_HALF_LE(&&(<=(x1, 1), >(x2, 1)), x2, x1)
COND_F1798_0_HALF_LE(TRUE, x2, x1) → F1798_0_HALF_LE'(0, x2)
F1798_0_HALF_LE'(x2, x1) → COND_F1798_0_HALF_LE1(&&(>(x1, 1), >(x2, -1)), x2, x1)
COND_F1798_0_HALF_LE1(TRUE, x2, x1) → F1798_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): F1798_0_HALF_LE'(x2[0], x1[0]) → COND_F1798_0_HALF_LE(x1[0] <= 1 && x2[0] > 1, x2[0], x1[0])
(1): COND_F1798_0_HALF_LE(TRUE, x2[1], x1[1]) → F1798_0_HALF_LE'(0, x2[1])
(2): F1798_0_HALF_LE'(x2[2], x1[2]) → COND_F1798_0_HALF_LE1(x1[2] > 1 && x2[2] > -1, x2[2], x1[2])
(3): COND_F1798_0_HALF_LE1(TRUE, x2[3], x1[3]) → F1798_0_HALF_LE'(x2[3] + 1, x1[3] - 2)

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


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


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


(2) -> (3), if (x1[2] > 1 && x2[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@4f77450a 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 F1798_0_HALF_LE'(x2, x1) → COND_F1798_0_HALF_LE(&&(<=(x1, 1), >(x2, 1)), x2, x1) the following chains were created:
  • We consider the chain F1798_0_HALF_LE'(x2[0], x1[0]) → COND_F1798_0_HALF_LE(&&(<=(x1[0], 1), >(x2[0], 1)), x2[0], x1[0]), COND_F1798_0_HALF_LE(TRUE, x2[1], x1[1]) → F1798_0_HALF_LE'(0, x2[1]) which results in the following constraint:

    (1)    (&&(<=(x1[0], 1), >(x2[0], 1))=TRUE∧x2[0]=x2[1]∧x1[0]=x1[1] ⇒ F1798_0_HALF_LE'(x2[0], x1[0])≥NonInfC∧F1798_0_HALF_LE'(x2[0], x1[0])≥COND_F1798_0_HALF_LE(&&(<=(x1[0], 1), >(x2[0], 1)), x2[0], x1[0])∧(UIncreasing(COND_F1798_0_HALF_LE(&&(<=(x1[0], 1), >(x2[0], 1)), x2[0], x1[0])), ≥))



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

    (2)    (<=(x1[0], 1)=TRUE∧>(x2[0], 1)=TRUE ⇒ F1798_0_HALF_LE'(x2[0], x1[0])≥NonInfC∧F1798_0_HALF_LE'(x2[0], x1[0])≥COND_F1798_0_HALF_LE(&&(<=(x1[0], 1), >(x2[0], 1)), x2[0], x1[0])∧(UIncreasing(COND_F1798_0_HALF_LE(&&(<=(x1[0], 1), >(x2[0], 1)), x2[0], x1[0])), ≥))



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

    (3)    ([1] + [-1]x1[0] ≥ 0∧x2[0] + [-2] ≥ 0 ⇒ (UIncreasing(COND_F1798_0_HALF_LE(&&(<=(x1[0], 1), >(x2[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)    ([1] + [-1]x1[0] ≥ 0∧x2[0] + [-2] ≥ 0 ⇒ (UIncreasing(COND_F1798_0_HALF_LE(&&(<=(x1[0], 1), >(x2[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)    ([1] + [-1]x1[0] ≥ 0∧x2[0] + [-2] ≥ 0 ⇒ (UIncreasing(COND_F1798_0_HALF_LE(&&(<=(x1[0], 1), >(x2[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)    ([1] + [-1]x1[0] ≥ 0∧x2[0] ≥ 0 ⇒ (UIncreasing(COND_F1798_0_HALF_LE(&&(<=(x1[0], 1), >(x2[0], 1)), x2[0], x1[0])), ≥)∧[bni_17 + (-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)    ([1] + x1[0] ≥ 0∧x2[0] ≥ 0∧x1[0] ≥ 0 ⇒ (UIncreasing(COND_F1798_0_HALF_LE(&&(<=(x1[0], 1), >(x2[0], 1)), x2[0], x1[0])), ≥)∧[bni_17 + (-1)Bound*bni_17] + [(-1)bni_17]x1[0] + [bni_17]x2[0] ≥ 0∧[(-1)bso_18] ≥ 0)


    (8)    ([1] + [-1]x1[0] ≥ 0∧x2[0] ≥ 0∧x1[0] ≥ 0 ⇒ (UIncreasing(COND_F1798_0_HALF_LE(&&(<=(x1[0], 1), >(x2[0], 1)), x2[0], x1[0])), ≥)∧[bni_17 + (-1)Bound*bni_17] + [bni_17]x1[0] + [bni_17]x2[0] ≥ 0∧[(-1)bso_18] ≥ 0)







For Pair COND_F1798_0_HALF_LE(TRUE, x2, x1) → F1798_0_HALF_LE'(0, x2) the following chains were created:
  • We consider the chain F1798_0_HALF_LE'(x2[0], x1[0]) → COND_F1798_0_HALF_LE(&&(<=(x1[0], 1), >(x2[0], 1)), x2[0], x1[0]), COND_F1798_0_HALF_LE(TRUE, x2[1], x1[1]) → F1798_0_HALF_LE'(0, x2[1]), F1798_0_HALF_LE'(x2[0], x1[0]) → COND_F1798_0_HALF_LE(&&(<=(x1[0], 1), >(x2[0], 1)), x2[0], x1[0]), COND_F1798_0_HALF_LE(TRUE, x2[1], x1[1]) → F1798_0_HALF_LE'(0, x2[1]), F1798_0_HALF_LE'(x2[0], x1[0]) → COND_F1798_0_HALF_LE(&&(<=(x1[0], 1), >(x2[0], 1)), x2[0], x1[0]), COND_F1798_0_HALF_LE(TRUE, x2[1], x1[1]) → F1798_0_HALF_LE'(0, x2[1]) which results in the following constraint:

    (9)    (&&(<=(x1[0], 1), >(x2[0], 1))=TRUE∧x2[0]=x2[1]∧x1[0]=x1[1]∧0=x2[0]1∧x2[1]=x1[0]1∧&&(<=(x1[0]1, 1), >(x2[0]1, 1))=TRUE∧x2[0]1=x2[1]1∧x1[0]1=x1[1]1∧0=x2[0]2∧x2[1]1=x1[0]2∧&&(<=(x1[0]2, 1), >(x2[0]2, 1))=TRUE∧x2[0]2=x2[1]2∧x1[0]2=x1[1]2 ⇒ COND_F1798_0_HALF_LE(TRUE, x2[1]1, x1[1]1)≥NonInfC∧COND_F1798_0_HALF_LE(TRUE, x2[1]1, x1[1]1)≥F1798_0_HALF_LE'(0, x2[1]1)∧(UIncreasing(F1798_0_HALF_LE'(0, x2[1]1)), ≥))



    We solved constraint (9) using rules (I), (II), (III), (IV), (IDP_CONSTANT_FOLD), (IDP_BOOLEAN).
  • We consider the chain F1798_0_HALF_LE'(x2[0], x1[0]) → COND_F1798_0_HALF_LE(&&(<=(x1[0], 1), >(x2[0], 1)), x2[0], x1[0]), COND_F1798_0_HALF_LE(TRUE, x2[1], x1[1]) → F1798_0_HALF_LE'(0, x2[1]), F1798_0_HALF_LE'(x2[0], x1[0]) → COND_F1798_0_HALF_LE(&&(<=(x1[0], 1), >(x2[0], 1)), x2[0], x1[0]), COND_F1798_0_HALF_LE(TRUE, x2[1], x1[1]) → F1798_0_HALF_LE'(0, x2[1]), F1798_0_HALF_LE'(x2[2], x1[2]) → COND_F1798_0_HALF_LE1(&&(>(x1[2], 1), >(x2[2], -1)), x2[2], x1[2]), COND_F1798_0_HALF_LE1(TRUE, x2[3], x1[3]) → F1798_0_HALF_LE'(+(x2[3], 1), -(x1[3], 2)) which results in the following constraint:

    (10)    (&&(<=(x1[0], 1), >(x2[0], 1))=TRUE∧x2[0]=x2[1]∧x1[0]=x1[1]∧0=x2[0]1∧x2[1]=x1[0]1∧&&(<=(x1[0]1, 1), >(x2[0]1, 1))=TRUE∧x2[0]1=x2[1]1∧x1[0]1=x1[1]1∧0=x2[2]∧x2[1]1=x1[2]∧&&(>(x1[2], 1), >(x2[2], -1))=TRUE∧x2[2]=x2[3]∧x1[2]=x1[3] ⇒ COND_F1798_0_HALF_LE(TRUE, x2[1]1, x1[1]1)≥NonInfC∧COND_F1798_0_HALF_LE(TRUE, x2[1]1, x1[1]1)≥F1798_0_HALF_LE'(0, x2[1]1)∧(UIncreasing(F1798_0_HALF_LE'(0, x2[1]1)), ≥))



    We solved constraint (10) using rules (I), (II), (III), (IV), (IDP_CONSTANT_FOLD), (IDP_BOOLEAN).
  • We consider the chain F1798_0_HALF_LE'(x2[2], x1[2]) → COND_F1798_0_HALF_LE1(&&(>(x1[2], 1), >(x2[2], -1)), x2[2], x1[2]), COND_F1798_0_HALF_LE1(TRUE, x2[3], x1[3]) → F1798_0_HALF_LE'(+(x2[3], 1), -(x1[3], 2)), F1798_0_HALF_LE'(x2[0], x1[0]) → COND_F1798_0_HALF_LE(&&(<=(x1[0], 1), >(x2[0], 1)), x2[0], x1[0]), COND_F1798_0_HALF_LE(TRUE, x2[1], x1[1]) → F1798_0_HALF_LE'(0, x2[1]), F1798_0_HALF_LE'(x2[0], x1[0]) → COND_F1798_0_HALF_LE(&&(<=(x1[0], 1), >(x2[0], 1)), x2[0], x1[0]), COND_F1798_0_HALF_LE(TRUE, x2[1], x1[1]) → F1798_0_HALF_LE'(0, x2[1]) which results in the following constraint:

    (11)    (&&(>(x1[2], 1), >(x2[2], -1))=TRUE∧x2[2]=x2[3]∧x1[2]=x1[3]∧+(x2[3], 1)=x2[0]∧-(x1[3], 2)=x1[0]∧&&(<=(x1[0], 1), >(x2[0], 1))=TRUE∧x2[0]=x2[1]∧x1[0]=x1[1]∧0=x2[0]1∧x2[1]=x1[0]1∧&&(<=(x1[0]1, 1), >(x2[0]1, 1))=TRUE∧x2[0]1=x2[1]1∧x1[0]1=x1[1]1 ⇒ COND_F1798_0_HALF_LE(TRUE, x2[1], x1[1])≥NonInfC∧COND_F1798_0_HALF_LE(TRUE, x2[1], x1[1])≥F1798_0_HALF_LE'(0, x2[1])∧(UIncreasing(F1798_0_HALF_LE'(0, x2[1])), ≥))



    We solved constraint (11) using rules (I), (II), (III), (IV), (IDP_CONSTANT_FOLD), (IDP_BOOLEAN).
  • We consider the chain F1798_0_HALF_LE'(x2[2], x1[2]) → COND_F1798_0_HALF_LE1(&&(>(x1[2], 1), >(x2[2], -1)), x2[2], x1[2]), COND_F1798_0_HALF_LE1(TRUE, x2[3], x1[3]) → F1798_0_HALF_LE'(+(x2[3], 1), -(x1[3], 2)), F1798_0_HALF_LE'(x2[0], x1[0]) → COND_F1798_0_HALF_LE(&&(<=(x1[0], 1), >(x2[0], 1)), x2[0], x1[0]), COND_F1798_0_HALF_LE(TRUE, x2[1], x1[1]) → F1798_0_HALF_LE'(0, x2[1]), F1798_0_HALF_LE'(x2[2], x1[2]) → COND_F1798_0_HALF_LE1(&&(>(x1[2], 1), >(x2[2], -1)), x2[2], x1[2]), COND_F1798_0_HALF_LE1(TRUE, x2[3], x1[3]) → F1798_0_HALF_LE'(+(x2[3], 1), -(x1[3], 2)) which results in the following constraint:

    (12)    (&&(>(x1[2], 1), >(x2[2], -1))=TRUE∧x2[2]=x2[3]∧x1[2]=x1[3]∧+(x2[3], 1)=x2[0]∧-(x1[3], 2)=x1[0]∧&&(<=(x1[0], 1), >(x2[0], 1))=TRUE∧x2[0]=x2[1]∧x1[0]=x1[1]∧0=x2[2]1∧x2[1]=x1[2]1∧&&(>(x1[2]1, 1), >(x2[2]1, -1))=TRUE∧x2[2]1=x2[3]1∧x1[2]1=x1[3]1 ⇒ COND_F1798_0_HALF_LE(TRUE, x2[1], x1[1])≥NonInfC∧COND_F1798_0_HALF_LE(TRUE, x2[1], x1[1])≥F1798_0_HALF_LE'(0, x2[1])∧(UIncreasing(F1798_0_HALF_LE'(0, x2[1])), ≥))



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

    (13)    (>(x1[2], 1)=TRUE∧>(x2[2], -1)=TRUE∧<=(-(x1[2], 2), 1)=TRUE∧>(+(x2[2], 1), 1)=TRUE ⇒ COND_F1798_0_HALF_LE(TRUE, +(x2[2], 1), -(x1[2], 2))≥NonInfC∧COND_F1798_0_HALF_LE(TRUE, +(x2[2], 1), -(x1[2], 2))≥F1798_0_HALF_LE'(0, +(x2[2], 1))∧(UIncreasing(F1798_0_HALF_LE'(0, x2[1])), ≥))



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

    (14)    (x1[2] + [-2] ≥ 0∧x2[2] ≥ 0∧[3] + [-1]x1[2] ≥ 0∧x2[2] + [-1] ≥ 0 ⇒ (UIncreasing(F1798_0_HALF_LE'(0, x2[1])), ≥)∧[(-2)bni_19 + (-1)Bound*bni_19] + [bni_19]x1[2] + [bni_19]x2[2] ≥ 0∧[-2 + (-1)bso_20] + x1[2] ≥ 0)



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

    (15)    (x1[2] + [-2] ≥ 0∧x2[2] ≥ 0∧[3] + [-1]x1[2] ≥ 0∧x2[2] + [-1] ≥ 0 ⇒ (UIncreasing(F1798_0_HALF_LE'(0, x2[1])), ≥)∧[(-2)bni_19 + (-1)Bound*bni_19] + [bni_19]x1[2] + [bni_19]x2[2] ≥ 0∧[-2 + (-1)bso_20] + x1[2] ≥ 0)



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

    (16)    (x1[2] + [-2] ≥ 0∧x2[2] ≥ 0∧[3] + [-1]x1[2] ≥ 0∧x2[2] + [-1] ≥ 0 ⇒ (UIncreasing(F1798_0_HALF_LE'(0, x2[1])), ≥)∧[(-2)bni_19 + (-1)Bound*bni_19] + [bni_19]x1[2] + [bni_19]x2[2] ≥ 0∧[-2 + (-1)bso_20] + x1[2] ≥ 0)



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

    (17)    (x1[2] ≥ 0∧x2[2] ≥ 0∧[1] + [-1]x1[2] ≥ 0∧x2[2] + [-1] ≥ 0 ⇒ (UIncreasing(F1798_0_HALF_LE'(0, x2[1])), ≥)∧[(-1)Bound*bni_19] + [bni_19]x1[2] + [bni_19]x2[2] ≥ 0∧[(-1)bso_20] + x1[2] ≥ 0)



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

    (18)    (x1[2] ≥ 0∧[1] + x2[2] ≥ 0∧[1] + [-1]x1[2] ≥ 0∧x2[2] ≥ 0 ⇒ (UIncreasing(F1798_0_HALF_LE'(0, x2[1])), ≥)∧[(-1)Bound*bni_19 + bni_19] + [bni_19]x1[2] + [bni_19]x2[2] ≥ 0∧[(-1)bso_20] + x1[2] ≥ 0)







For Pair F1798_0_HALF_LE'(x2, x1) → COND_F1798_0_HALF_LE1(&&(>(x1, 1), >(x2, -1)), x2, x1) the following chains were created:
  • We consider the chain F1798_0_HALF_LE'(x2[2], x1[2]) → COND_F1798_0_HALF_LE1(&&(>(x1[2], 1), >(x2[2], -1)), x2[2], x1[2]), COND_F1798_0_HALF_LE1(TRUE, x2[3], x1[3]) → F1798_0_HALF_LE'(+(x2[3], 1), -(x1[3], 2)) which results in the following constraint:

    (19)    (&&(>(x1[2], 1), >(x2[2], -1))=TRUE∧x2[2]=x2[3]∧x1[2]=x1[3] ⇒ F1798_0_HALF_LE'(x2[2], x1[2])≥NonInfC∧F1798_0_HALF_LE'(x2[2], x1[2])≥COND_F1798_0_HALF_LE1(&&(>(x1[2], 1), >(x2[2], -1)), x2[2], x1[2])∧(UIncreasing(COND_F1798_0_HALF_LE1(&&(>(x1[2], 1), >(x2[2], -1)), x2[2], x1[2])), ≥))



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

    (20)    (>(x1[2], 1)=TRUE∧>(x2[2], -1)=TRUE ⇒ F1798_0_HALF_LE'(x2[2], x1[2])≥NonInfC∧F1798_0_HALF_LE'(x2[2], x1[2])≥COND_F1798_0_HALF_LE1(&&(>(x1[2], 1), >(x2[2], -1)), x2[2], x1[2])∧(UIncreasing(COND_F1798_0_HALF_LE1(&&(>(x1[2], 1), >(x2[2], -1)), x2[2], x1[2])), ≥))



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

    (21)    (x1[2] + [-2] ≥ 0∧x2[2] ≥ 0 ⇒ (UIncreasing(COND_F1798_0_HALF_LE1(&&(>(x1[2], 1), >(x2[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)    (x1[2] + [-2] ≥ 0∧x2[2] ≥ 0 ⇒ (UIncreasing(COND_F1798_0_HALF_LE1(&&(>(x1[2], 1), >(x2[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)    (x1[2] + [-2] ≥ 0∧x2[2] ≥ 0 ⇒ (UIncreasing(COND_F1798_0_HALF_LE1(&&(>(x1[2], 1), >(x2[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)    (x1[2] ≥ 0∧x2[2] ≥ 0 ⇒ (UIncreasing(COND_F1798_0_HALF_LE1(&&(>(x1[2], 1), >(x2[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_F1798_0_HALF_LE1(TRUE, x2, x1) → F1798_0_HALF_LE'(+(x2, 1), -(x1, 2)) the following chains were created:
  • We consider the chain F1798_0_HALF_LE'(x2[0], x1[0]) → COND_F1798_0_HALF_LE(&&(<=(x1[0], 1), >(x2[0], 1)), x2[0], x1[0]), COND_F1798_0_HALF_LE(TRUE, x2[1], x1[1]) → F1798_0_HALF_LE'(0, x2[1]), F1798_0_HALF_LE'(x2[2], x1[2]) → COND_F1798_0_HALF_LE1(&&(>(x1[2], 1), >(x2[2], -1)), x2[2], x1[2]), COND_F1798_0_HALF_LE1(TRUE, x2[3], x1[3]) → F1798_0_HALF_LE'(+(x2[3], 1), -(x1[3], 2)), F1798_0_HALF_LE'(x2[0], x1[0]) → COND_F1798_0_HALF_LE(&&(<=(x1[0], 1), >(x2[0], 1)), x2[0], x1[0]), COND_F1798_0_HALF_LE(TRUE, x2[1], x1[1]) → F1798_0_HALF_LE'(0, x2[1]) which results in the following constraint:

    (25)    (&&(<=(x1[0], 1), >(x2[0], 1))=TRUE∧x2[0]=x2[1]∧x1[0]=x1[1]∧0=x2[2]∧x2[1]=x1[2]∧&&(>(x1[2], 1), >(x2[2], -1))=TRUE∧x2[2]=x2[3]∧x1[2]=x1[3]∧+(x2[3], 1)=x2[0]1∧-(x1[3], 2)=x1[0]1∧&&(<=(x1[0]1, 1), >(x2[0]1, 1))=TRUE∧x2[0]1=x2[1]1∧x1[0]1=x1[1]1 ⇒ COND_F1798_0_HALF_LE1(TRUE, x2[3], x1[3])≥NonInfC∧COND_F1798_0_HALF_LE1(TRUE, x2[3], x1[3])≥F1798_0_HALF_LE'(+(x2[3], 1), -(x1[3], 2))∧(UIncreasing(F1798_0_HALF_LE'(+(x2[3], 1), -(x1[3], 2))), ≥))



    We solved constraint (25) using rules (I), (II), (III), (IV), (IDP_CONSTANT_FOLD), (DELETE_TRIVIAL_REDUCESTO), (IDP_BOOLEAN).
  • We consider the chain F1798_0_HALF_LE'(x2[0], x1[0]) → COND_F1798_0_HALF_LE(&&(<=(x1[0], 1), >(x2[0], 1)), x2[0], x1[0]), COND_F1798_0_HALF_LE(TRUE, x2[1], x1[1]) → F1798_0_HALF_LE'(0, x2[1]), F1798_0_HALF_LE'(x2[2], x1[2]) → COND_F1798_0_HALF_LE1(&&(>(x1[2], 1), >(x2[2], -1)), x2[2], x1[2]), COND_F1798_0_HALF_LE1(TRUE, x2[3], x1[3]) → F1798_0_HALF_LE'(+(x2[3], 1), -(x1[3], 2)), F1798_0_HALF_LE'(x2[2], x1[2]) → COND_F1798_0_HALF_LE1(&&(>(x1[2], 1), >(x2[2], -1)), x2[2], x1[2]), COND_F1798_0_HALF_LE1(TRUE, x2[3], x1[3]) → F1798_0_HALF_LE'(+(x2[3], 1), -(x1[3], 2)) which results in the following constraint:

    (26)    (&&(<=(x1[0], 1), >(x2[0], 1))=TRUE∧x2[0]=x2[1]∧x1[0]=x1[1]∧0=x2[2]∧x2[1]=x1[2]∧&&(>(x1[2], 1), >(x2[2], -1))=TRUE∧x2[2]=x2[3]∧x1[2]=x1[3]∧+(x2[3], 1)=x2[2]1∧-(x1[3], 2)=x1[2]1∧&&(>(x1[2]1, 1), >(x2[2]1, -1))=TRUE∧x2[2]1=x2[3]1∧x1[2]1=x1[3]1 ⇒ COND_F1798_0_HALF_LE1(TRUE, x2[3], x1[3])≥NonInfC∧COND_F1798_0_HALF_LE1(TRUE, x2[3], x1[3])≥F1798_0_HALF_LE'(+(x2[3], 1), -(x1[3], 2))∧(UIncreasing(F1798_0_HALF_LE'(+(x2[3], 1), -(x1[3], 2))), ≥))



    We simplified constraint (26) using rules (III), (IV), (IDP_CONSTANT_FOLD), (DELETE_TRIVIAL_REDUCESTO), (IDP_BOOLEAN) which results in the following new constraint:

    (27)    (<=(x1[0], 1)=TRUE∧>(x2[0], 1)=TRUE∧>(-(x2[0], 2), 1)=TRUE ⇒ COND_F1798_0_HALF_LE1(TRUE, 0, x2[0])≥NonInfC∧COND_F1798_0_HALF_LE1(TRUE, 0, x2[0])≥F1798_0_HALF_LE'(+(0, 1), -(x2[0], 2))∧(UIncreasing(F1798_0_HALF_LE'(+(x2[3], 1), -(x1[3], 2))), ≥))



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

    (28)    ([1] + [-1]x1[0] ≥ 0∧x2[0] + [-2] ≥ 0∧x2[0] + [-4] ≥ 0 ⇒ (UIncreasing(F1798_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 (28) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:

    (29)    ([1] + [-1]x1[0] ≥ 0∧x2[0] + [-2] ≥ 0∧x2[0] + [-4] ≥ 0 ⇒ (UIncreasing(F1798_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 (29) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:

    (30)    ([1] + [-1]x1[0] ≥ 0∧x2[0] + [-2] ≥ 0∧x2[0] + [-4] ≥ 0 ⇒ (UIncreasing(F1798_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)    ([1] + [-1]x1[0] ≥ 0∧x2[0] ≥ 0∧[-2] + x2[0] ≥ 0 ⇒ (UIncreasing(F1798_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)    ([1] + [-1]x1[0] ≥ 0∧[2] + x2[0] ≥ 0∧x2[0] ≥ 0 ⇒ (UIncreasing(F1798_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 (32) using rule (IDP_SMT_SPLIT) which results in the following new constraint:

    (33)    ([2] + x2[0] ≥ 0∧x2[0] ≥ 0 ⇒ (UIncreasing(F1798_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 F1798_0_HALF_LE'(x2[2], x1[2]) → COND_F1798_0_HALF_LE1(&&(>(x1[2], 1), >(x2[2], -1)), x2[2], x1[2]), COND_F1798_0_HALF_LE1(TRUE, x2[3], x1[3]) → F1798_0_HALF_LE'(+(x2[3], 1), -(x1[3], 2)), F1798_0_HALF_LE'(x2[2], x1[2]) → COND_F1798_0_HALF_LE1(&&(>(x1[2], 1), >(x2[2], -1)), x2[2], x1[2]), COND_F1798_0_HALF_LE1(TRUE, x2[3], x1[3]) → F1798_0_HALF_LE'(+(x2[3], 1), -(x1[3], 2)), F1798_0_HALF_LE'(x2[0], x1[0]) → COND_F1798_0_HALF_LE(&&(<=(x1[0], 1), >(x2[0], 1)), x2[0], x1[0]), COND_F1798_0_HALF_LE(TRUE, x2[1], x1[1]) → F1798_0_HALF_LE'(0, x2[1]) which results in the following constraint:

    (34)    (&&(>(x1[2], 1), >(x2[2], -1))=TRUE∧x2[2]=x2[3]∧x1[2]=x1[3]∧+(x2[3], 1)=x2[2]1∧-(x1[3], 2)=x1[2]1∧&&(>(x1[2]1, 1), >(x2[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]∧&&(<=(x1[0], 1), >(x2[0], 1))=TRUE∧x2[0]=x2[1]∧x1[0]=x1[1] ⇒ COND_F1798_0_HALF_LE1(TRUE, x2[3]1, x1[3]1)≥NonInfC∧COND_F1798_0_HALF_LE1(TRUE, x2[3]1, x1[3]1)≥F1798_0_HALF_LE'(+(x2[3]1, 1), -(x1[3]1, 2))∧(UIncreasing(F1798_0_HALF_LE'(+(x2[3]1, 1), -(x1[3]1, 2))), ≥))



    We simplified constraint (34) using rules (III), (IV), (IDP_BOOLEAN) which results in the following new constraint:

    (35)    (>(x1[2], 1)=TRUE∧>(x2[2], -1)=TRUE∧>(-(x1[2], 2), 1)=TRUE∧>(+(x2[2], 1), -1)=TRUE∧<=(-(-(x1[2], 2), 2), 1)=TRUE∧>(+(+(x2[2], 1), 1), 1)=TRUE ⇒ COND_F1798_0_HALF_LE1(TRUE, +(x2[2], 1), -(x1[2], 2))≥NonInfC∧COND_F1798_0_HALF_LE1(TRUE, +(x2[2], 1), -(x1[2], 2))≥F1798_0_HALF_LE'(+(+(x2[2], 1), 1), -(-(x1[2], 2), 2))∧(UIncreasing(F1798_0_HALF_LE'(+(x2[3]1, 1), -(x1[3]1, 2))), ≥))



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

    (36)    (x1[2] + [-2] ≥ 0∧x2[2] ≥ 0∧x1[2] + [-4] ≥ 0∧x2[2] + [1] ≥ 0∧[5] + [-1]x1[2] ≥ 0∧x2[2] ≥ 0 ⇒ (UIncreasing(F1798_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 (36) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:

    (37)    (x1[2] + [-2] ≥ 0∧x2[2] ≥ 0∧x1[2] + [-4] ≥ 0∧x2[2] + [1] ≥ 0∧[5] + [-1]x1[2] ≥ 0∧x2[2] ≥ 0 ⇒ (UIncreasing(F1798_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 (37) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:

    (38)    (x1[2] + [-2] ≥ 0∧x2[2] ≥ 0∧x1[2] + [-4] ≥ 0∧x2[2] + [1] ≥ 0∧[5] + [-1]x1[2] ≥ 0∧x2[2] ≥ 0 ⇒ (UIncreasing(F1798_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 (38) using rule (IDP_SMT_SPLIT) which results in the following new constraint:

    (39)    (x1[2] ≥ 0∧x2[2] ≥ 0∧[-2] + x1[2] ≥ 0∧x2[2] + [1] ≥ 0∧[3] + [-1]x1[2] ≥ 0∧x2[2] ≥ 0 ⇒ (UIncreasing(F1798_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 (39) using rule (IDP_SMT_SPLIT) which results in the following new constraint:

    (40)    ([2] + x1[2] ≥ 0∧x2[2] ≥ 0∧x1[2] ≥ 0∧x2[2] + [1] ≥ 0∧[1] + [-1]x1[2] ≥ 0∧x2[2] ≥ 0 ⇒ (UIncreasing(F1798_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 F1798_0_HALF_LE'(x2[2], x1[2]) → COND_F1798_0_HALF_LE1(&&(>(x1[2], 1), >(x2[2], -1)), x2[2], x1[2]), COND_F1798_0_HALF_LE1(TRUE, x2[3], x1[3]) → F1798_0_HALF_LE'(+(x2[3], 1), -(x1[3], 2)), F1798_0_HALF_LE'(x2[2], x1[2]) → COND_F1798_0_HALF_LE1(&&(>(x1[2], 1), >(x2[2], -1)), x2[2], x1[2]), COND_F1798_0_HALF_LE1(TRUE, x2[3], x1[3]) → F1798_0_HALF_LE'(+(x2[3], 1), -(x1[3], 2)), F1798_0_HALF_LE'(x2[2], x1[2]) → COND_F1798_0_HALF_LE1(&&(>(x1[2], 1), >(x2[2], -1)), x2[2], x1[2]), COND_F1798_0_HALF_LE1(TRUE, x2[3], x1[3]) → F1798_0_HALF_LE'(+(x2[3], 1), -(x1[3], 2)) which results in the following constraint:

    (41)    (&&(>(x1[2], 1), >(x2[2], -1))=TRUE∧x2[2]=x2[3]∧x1[2]=x1[3]∧+(x2[3], 1)=x2[2]1∧-(x1[3], 2)=x1[2]1∧&&(>(x1[2]1, 1), >(x2[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∧&&(>(x1[2]2, 1), >(x2[2]2, -1))=TRUE∧x2[2]2=x2[3]2∧x1[2]2=x1[3]2 ⇒ COND_F1798_0_HALF_LE1(TRUE, x2[3]1, x1[3]1)≥NonInfC∧COND_F1798_0_HALF_LE1(TRUE, x2[3]1, x1[3]1)≥F1798_0_HALF_LE'(+(x2[3]1, 1), -(x1[3]1, 2))∧(UIncreasing(F1798_0_HALF_LE'(+(x2[3]1, 1), -(x1[3]1, 2))), ≥))



    We simplified constraint (41) using rules (III), (IV), (IDP_BOOLEAN) which results in the following new constraint:

    (42)    (>(x1[2], 1)=TRUE∧>(x2[2], -1)=TRUE∧>(-(x1[2], 2), 1)=TRUE∧>(+(x2[2], 1), -1)=TRUE∧>(-(-(x1[2], 2), 2), 1)=TRUE∧>(+(+(x2[2], 1), 1), -1)=TRUE ⇒ COND_F1798_0_HALF_LE1(TRUE, +(x2[2], 1), -(x1[2], 2))≥NonInfC∧COND_F1798_0_HALF_LE1(TRUE, +(x2[2], 1), -(x1[2], 2))≥F1798_0_HALF_LE'(+(+(x2[2], 1), 1), -(-(x1[2], 2), 2))∧(UIncreasing(F1798_0_HALF_LE'(+(x2[3]1, 1), -(x1[3]1, 2))), ≥))



    We simplified constraint (42) using rule (POLY_CONSTRAINTS) which results in the following new constraint:

    (43)    (x1[2] + [-2] ≥ 0∧x2[2] ≥ 0∧x1[2] + [-4] ≥ 0∧x2[2] + [1] ≥ 0∧x1[2] + [-6] ≥ 0∧x2[2] + [2] ≥ 0 ⇒ (UIncreasing(F1798_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 (43) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:

    (44)    (x1[2] + [-2] ≥ 0∧x2[2] ≥ 0∧x1[2] + [-4] ≥ 0∧x2[2] + [1] ≥ 0∧x1[2] + [-6] ≥ 0∧x2[2] + [2] ≥ 0 ⇒ (UIncreasing(F1798_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 (POLY_REMOVE_MIN_MAX) which results in the following new constraint:

    (45)    (x1[2] + [-2] ≥ 0∧x2[2] ≥ 0∧x1[2] + [-4] ≥ 0∧x2[2] + [1] ≥ 0∧x1[2] + [-6] ≥ 0∧x2[2] + [2] ≥ 0 ⇒ (UIncreasing(F1798_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 (IDP_SMT_SPLIT) which results in the following new constraint:

    (46)    (x1[2] ≥ 0∧x2[2] ≥ 0∧[-2] + x1[2] ≥ 0∧x2[2] + [1] ≥ 0∧[-4] + x1[2] ≥ 0∧x2[2] + [2] ≥ 0 ⇒ (UIncreasing(F1798_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 (46) using rule (IDP_SMT_SPLIT) which results in the following new constraint:

    (47)    ([2] + x1[2] ≥ 0∧x2[2] ≥ 0∧x1[2] ≥ 0∧x2[2] + [1] ≥ 0∧[-2] + x1[2] ≥ 0∧x2[2] + [2] ≥ 0 ⇒ (UIncreasing(F1798_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 (47) using rule (IDP_SMT_SPLIT) which results in the following new constraint:

    (48)    ([4] + x1[2] ≥ 0∧x2[2] ≥ 0∧[2] + x1[2] ≥ 0∧x2[2] + [1] ≥ 0∧x1[2] ≥ 0∧x2[2] + [2] ≥ 0 ⇒ (UIncreasing(F1798_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.
  • F1798_0_HALF_LE'(x2, x1) → COND_F1798_0_HALF_LE(&&(<=(x1, 1), >(x2, 1)), x2, x1)
    • ([1] + x1[0] ≥ 0∧x2[0] ≥ 0∧x1[0] ≥ 0 ⇒ (UIncreasing(COND_F1798_0_HALF_LE(&&(<=(x1[0], 1), >(x2[0], 1)), x2[0], x1[0])), ≥)∧[bni_17 + (-1)Bound*bni_17] + [(-1)bni_17]x1[0] + [bni_17]x2[0] ≥ 0∧[(-1)bso_18] ≥ 0)
    • ([1] + [-1]x1[0] ≥ 0∧x2[0] ≥ 0∧x1[0] ≥ 0 ⇒ (UIncreasing(COND_F1798_0_HALF_LE(&&(<=(x1[0], 1), >(x2[0], 1)), x2[0], x1[0])), ≥)∧[bni_17 + (-1)Bound*bni_17] + [bni_17]x1[0] + [bni_17]x2[0] ≥ 0∧[(-1)bso_18] ≥ 0)

  • COND_F1798_0_HALF_LE(TRUE, x2, x1) → F1798_0_HALF_LE'(0, x2)
    • (x1[2] ≥ 0∧[1] + x2[2] ≥ 0∧[1] + [-1]x1[2] ≥ 0∧x2[2] ≥ 0 ⇒ (UIncreasing(F1798_0_HALF_LE'(0, x2[1])), ≥)∧[(-1)Bound*bni_19 + bni_19] + [bni_19]x1[2] + [bni_19]x2[2] ≥ 0∧[(-1)bso_20] + x1[2] ≥ 0)

  • F1798_0_HALF_LE'(x2, x1) → COND_F1798_0_HALF_LE1(&&(>(x1, 1), >(x2, -1)), x2, x1)
    • (x1[2] ≥ 0∧x2[2] ≥ 0 ⇒ (UIncreasing(COND_F1798_0_HALF_LE1(&&(>(x1[2], 1), >(x2[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_F1798_0_HALF_LE1(TRUE, x2, x1) → F1798_0_HALF_LE'(+(x2, 1), -(x1, 2))
    • ([2] + x2[0] ≥ 0∧x2[0] ≥ 0 ⇒ (UIncreasing(F1798_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)
    • ([2] + x1[2] ≥ 0∧x2[2] ≥ 0∧x1[2] ≥ 0∧x2[2] + [1] ≥ 0∧[1] + [-1]x1[2] ≥ 0∧x2[2] ≥ 0 ⇒ (UIncreasing(F1798_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)
    • ([4] + x1[2] ≥ 0∧x2[2] ≥ 0∧[2] + x1[2] ≥ 0∧x2[2] + [1] ≥ 0∧x1[2] ≥ 0∧x2[2] + [2] ≥ 0 ⇒ (UIncreasing(F1798_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) = [1]   
POL(F1798_0_HALF_LE'(x1, x2)) = [-1] + x2 + x1   
POL(COND_F1798_0_HALF_LE(x1, x2, x3)) = [-1] + x3 + x2   
POL(&&(x1, x2)) = [-1]   
POL(<=(x1, x2)) = [-1]   
POL(1) = [1]   
POL(>(x1, x2)) = [-1]   
POL(0) = 0   
POL(COND_F1798_0_HALF_LE1(x1, x2, x3)) = [-1] + x3 + x2   
POL(-1) = [-1]   
POL(+(x1, x2)) = x1 + x2   
POL(-(x1, x2)) = x1 + [-1]x2   
POL(2) = [2]   

The following pairs are in P>:

COND_F1798_0_HALF_LE1(TRUE, x2[3], x1[3]) → F1798_0_HALF_LE'(+(x2[3], 1), -(x1[3], 2))

The following pairs are in Pbound:

COND_F1798_0_HALF_LE(TRUE, x2[1], x1[1]) → F1798_0_HALF_LE'(0, x2[1])
F1798_0_HALF_LE'(x2[2], x1[2]) → COND_F1798_0_HALF_LE1(&&(>(x1[2], 1), >(x2[2], -1)), x2[2], x1[2])
COND_F1798_0_HALF_LE1(TRUE, x2[3], x1[3]) → F1798_0_HALF_LE'(+(x2[3], 1), -(x1[3], 2))

The following pairs are in P≥:

F1798_0_HALF_LE'(x2[0], x1[0]) → COND_F1798_0_HALF_LE(&&(<=(x1[0], 1), >(x2[0], 1)), x2[0], x1[0])
COND_F1798_0_HALF_LE(TRUE, x2[1], x1[1]) → F1798_0_HALF_LE'(0, x2[1])
F1798_0_HALF_LE'(x2[2], x1[2]) → COND_F1798_0_HALF_LE1(&&(>(x1[2], 1), >(x2[2], -1)), x2[2], x1[2])

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

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

(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): F1798_0_HALF_LE'(x2[0], x1[0]) → COND_F1798_0_HALF_LE(x1[0] <= 1 && x2[0] > 1, x2[0], x1[0])
(1): COND_F1798_0_HALF_LE(TRUE, x2[1], x1[1]) → F1798_0_HALF_LE'(0, x2[1])
(2): F1798_0_HALF_LE'(x2[2], x1[2]) → COND_F1798_0_HALF_LE1(x1[2] > 1 && x2[2] > -1, x2[2], x1[2])

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


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


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



The set Q is empty.

(9) IDependencyGraphProof (EQUIVALENT transformation)

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

(10) 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:
(1): COND_F1798_0_HALF_LE(TRUE, x2[1], x1[1]) → F1798_0_HALF_LE'(0, x2[1])
(0): F1798_0_HALF_LE'(x2[0], x1[0]) → COND_F1798_0_HALF_LE(x1[0] <= 1 && x2[0] > 1, x2[0], x1[0])

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


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



The set Q is empty.

(11) 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@1f55bc48 Constraint Generator: NonInfConstraintGenerator: PathGenerator: MetricPathGenerator: Max Left Steps: 1 Max Right Steps: 1

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_F1798_0_HALF_LE(TRUE, x2[1], x1[1]) → F1798_0_HALF_LE'(0, x2[1]) the following chains were created:
  • We consider the chain F1798_0_HALF_LE'(x2[0], x1[0]) → COND_F1798_0_HALF_LE(&&(<=(x1[0], 1), >(x2[0], 1)), x2[0], x1[0]), COND_F1798_0_HALF_LE(TRUE, x2[1], x1[1]) → F1798_0_HALF_LE'(0, x2[1]), F1798_0_HALF_LE'(x2[0], x1[0]) → COND_F1798_0_HALF_LE(&&(<=(x1[0], 1), >(x2[0], 1)), x2[0], x1[0]) which results in the following constraint:

    (1)    (&&(<=(x1[0], 1), >(x2[0], 1))=TRUE∧x2[0]=x2[1]∧x1[0]=x1[1]∧0=x2[0]1∧x2[1]=x1[0]1 ⇒ COND_F1798_0_HALF_LE(TRUE, x2[1], x1[1])≥NonInfC∧COND_F1798_0_HALF_LE(TRUE, x2[1], x1[1])≥F1798_0_HALF_LE'(0, x2[1])∧(UIncreasing(F1798_0_HALF_LE'(0, x2[1])), ≥))



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

    (2)    (<=(x1[0], 1)=TRUE∧>(x2[0], 1)=TRUE ⇒ COND_F1798_0_HALF_LE(TRUE, x2[0], x1[0])≥NonInfC∧COND_F1798_0_HALF_LE(TRUE, x2[0], x1[0])≥F1798_0_HALF_LE'(0, x2[0])∧(UIncreasing(F1798_0_HALF_LE'(0, x2[1])), ≥))



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

    (3)    ([1] + [-1]x1[0] ≥ 0∧x2[0] + [-2] ≥ 0 ⇒ (UIncreasing(F1798_0_HALF_LE'(0, x2[1])), ≥)∧[(-1)bni_13 + (-1)Bound*bni_13] + [(-1)bni_13]x2[0] ≥ 0∧[(-1)bso_14] ≥ 0)



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

    (4)    ([1] + [-1]x1[0] ≥ 0∧x2[0] + [-2] ≥ 0 ⇒ (UIncreasing(F1798_0_HALF_LE'(0, x2[1])), ≥)∧[(-1)bni_13 + (-1)Bound*bni_13] + [(-1)bni_13]x2[0] ≥ 0∧[(-1)bso_14] ≥ 0)



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

    (5)    ([1] + [-1]x1[0] ≥ 0∧x2[0] + [-2] ≥ 0 ⇒ (UIncreasing(F1798_0_HALF_LE'(0, x2[1])), ≥)∧[(-1)bni_13 + (-1)Bound*bni_13] + [(-1)bni_13]x2[0] ≥ 0∧[(-1)bso_14] ≥ 0)



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

    (6)    ([1] + [-1]x1[0] ≥ 0∧x2[0] ≥ 0 ⇒ (UIncreasing(F1798_0_HALF_LE'(0, x2[1])), ≥)∧[(-3)bni_13 + (-1)Bound*bni_13] + [(-1)bni_13]x2[0] ≥ 0∧[(-1)bso_14] ≥ 0)



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

    (7)    ([1] + [-1]x1[0] ≥ 0∧x2[0] ≥ 0∧x1[0] ≥ 0 ⇒ (UIncreasing(F1798_0_HALF_LE'(0, x2[1])), ≥)∧[(-3)bni_13 + (-1)Bound*bni_13] + [(-1)bni_13]x2[0] ≥ 0∧[(-1)bso_14] ≥ 0)


    (8)    ([1] + x1[0] ≥ 0∧x2[0] ≥ 0∧x1[0] ≥ 0 ⇒ (UIncreasing(F1798_0_HALF_LE'(0, x2[1])), ≥)∧[(-3)bni_13 + (-1)Bound*bni_13] + [(-1)bni_13]x2[0] ≥ 0∧[(-1)bso_14] ≥ 0)







For Pair F1798_0_HALF_LE'(x2[0], x1[0]) → COND_F1798_0_HALF_LE(&&(<=(x1[0], 1), >(x2[0], 1)), x2[0], x1[0]) the following chains were created:
  • We consider the chain F1798_0_HALF_LE'(x2[0], x1[0]) → COND_F1798_0_HALF_LE(&&(<=(x1[0], 1), >(x2[0], 1)), x2[0], x1[0]), COND_F1798_0_HALF_LE(TRUE, x2[1], x1[1]) → F1798_0_HALF_LE'(0, x2[1]) which results in the following constraint:

    (9)    (&&(<=(x1[0], 1), >(x2[0], 1))=TRUE∧x2[0]=x2[1]∧x1[0]=x1[1] ⇒ F1798_0_HALF_LE'(x2[0], x1[0])≥NonInfC∧F1798_0_HALF_LE'(x2[0], x1[0])≥COND_F1798_0_HALF_LE(&&(<=(x1[0], 1), >(x2[0], 1)), x2[0], x1[0])∧(UIncreasing(COND_F1798_0_HALF_LE(&&(<=(x1[0], 1), >(x2[0], 1)), x2[0], x1[0])), ≥))



    We simplified constraint (9) using rules (IV), (IDP_BOOLEAN) which results in the following new constraint:

    (10)    (<=(x1[0], 1)=TRUE∧>(x2[0], 1)=TRUE ⇒ F1798_0_HALF_LE'(x2[0], x1[0])≥NonInfC∧F1798_0_HALF_LE'(x2[0], x1[0])≥COND_F1798_0_HALF_LE(&&(<=(x1[0], 1), >(x2[0], 1)), x2[0], x1[0])∧(UIncreasing(COND_F1798_0_HALF_LE(&&(<=(x1[0], 1), >(x2[0], 1)), x2[0], x1[0])), ≥))



    We simplified constraint (10) using rule (POLY_CONSTRAINTS) which results in the following new constraint:

    (11)    ([1] + [-1]x1[0] ≥ 0∧x2[0] + [-2] ≥ 0 ⇒ (UIncreasing(COND_F1798_0_HALF_LE(&&(<=(x1[0], 1), >(x2[0], 1)), x2[0], x1[0])), ≥)∧[(-1)bni_15 + (-1)Bound*bni_15] + [(-1)bni_15]x1[0] + [bni_15]x2[0] ≥ 0∧[(-1)bso_16] + [-1]x1[0] + [2]x2[0] ≥ 0)



    We simplified constraint (11) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:

    (12)    ([1] + [-1]x1[0] ≥ 0∧x2[0] + [-2] ≥ 0 ⇒ (UIncreasing(COND_F1798_0_HALF_LE(&&(<=(x1[0], 1), >(x2[0], 1)), x2[0], x1[0])), ≥)∧[(-1)bni_15 + (-1)Bound*bni_15] + [(-1)bni_15]x1[0] + [bni_15]x2[0] ≥ 0∧[(-1)bso_16] + [-1]x1[0] + [2]x2[0] ≥ 0)



    We simplified constraint (12) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:

    (13)    ([1] + [-1]x1[0] ≥ 0∧x2[0] + [-2] ≥ 0 ⇒ (UIncreasing(COND_F1798_0_HALF_LE(&&(<=(x1[0], 1), >(x2[0], 1)), x2[0], x1[0])), ≥)∧[(-1)bni_15 + (-1)Bound*bni_15] + [(-1)bni_15]x1[0] + [bni_15]x2[0] ≥ 0∧[(-1)bso_16] + [-1]x1[0] + [2]x2[0] ≥ 0)



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

    (14)    ([1] + [-1]x1[0] ≥ 0∧x2[0] ≥ 0 ⇒ (UIncreasing(COND_F1798_0_HALF_LE(&&(<=(x1[0], 1), >(x2[0], 1)), x2[0], x1[0])), ≥)∧[bni_15 + (-1)Bound*bni_15] + [(-1)bni_15]x1[0] + [bni_15]x2[0] ≥ 0∧[4 + (-1)bso_16] + [-1]x1[0] + [2]x2[0] ≥ 0)



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

    (15)    ([1] + [-1]x1[0] ≥ 0∧x2[0] ≥ 0∧x1[0] ≥ 0 ⇒ (UIncreasing(COND_F1798_0_HALF_LE(&&(<=(x1[0], 1), >(x2[0], 1)), x2[0], x1[0])), ≥)∧[bni_15 + (-1)Bound*bni_15] + [(-1)bni_15]x1[0] + [bni_15]x2[0] ≥ 0∧[4 + (-1)bso_16] + [-1]x1[0] + [2]x2[0] ≥ 0)


    (16)    ([1] + x1[0] ≥ 0∧x2[0] ≥ 0∧x1[0] ≥ 0 ⇒ (UIncreasing(COND_F1798_0_HALF_LE(&&(<=(x1[0], 1), >(x2[0], 1)), x2[0], x1[0])), ≥)∧[bni_15 + (-1)Bound*bni_15] + [bni_15]x1[0] + [bni_15]x2[0] ≥ 0∧[4 + (-1)bso_16] + x1[0] + [2]x2[0] ≥ 0)







To summarize, we get the following constraints P≥ for the following pairs.
  • COND_F1798_0_HALF_LE(TRUE, x2[1], x1[1]) → F1798_0_HALF_LE'(0, x2[1])
    • ([1] + [-1]x1[0] ≥ 0∧x2[0] ≥ 0∧x1[0] ≥ 0 ⇒ (UIncreasing(F1798_0_HALF_LE'(0, x2[1])), ≥)∧[(-3)bni_13 + (-1)Bound*bni_13] + [(-1)bni_13]x2[0] ≥ 0∧[(-1)bso_14] ≥ 0)
    • ([1] + x1[0] ≥ 0∧x2[0] ≥ 0∧x1[0] ≥ 0 ⇒ (UIncreasing(F1798_0_HALF_LE'(0, x2[1])), ≥)∧[(-3)bni_13 + (-1)Bound*bni_13] + [(-1)bni_13]x2[0] ≥ 0∧[(-1)bso_14] ≥ 0)

  • F1798_0_HALF_LE'(x2[0], x1[0]) → COND_F1798_0_HALF_LE(&&(<=(x1[0], 1), >(x2[0], 1)), x2[0], x1[0])
    • ([1] + [-1]x1[0] ≥ 0∧x2[0] ≥ 0∧x1[0] ≥ 0 ⇒ (UIncreasing(COND_F1798_0_HALF_LE(&&(<=(x1[0], 1), >(x2[0], 1)), x2[0], x1[0])), ≥)∧[bni_15 + (-1)Bound*bni_15] + [(-1)bni_15]x1[0] + [bni_15]x2[0] ≥ 0∧[4 + (-1)bso_16] + [-1]x1[0] + [2]x2[0] ≥ 0)
    • ([1] + x1[0] ≥ 0∧x2[0] ≥ 0∧x1[0] ≥ 0 ⇒ (UIncreasing(COND_F1798_0_HALF_LE(&&(<=(x1[0], 1), >(x2[0], 1)), x2[0], x1[0])), ≥)∧[bni_15 + (-1)Bound*bni_15] + [bni_15]x1[0] + [bni_15]x2[0] ≥ 0∧[4 + (-1)bso_16] + x1[0] + [2]x2[0] ≥ 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(COND_F1798_0_HALF_LE(x1, x2, x3)) = [-1] + [-1]x2 + [-1]x1   
POL(F1798_0_HALF_LE'(x1, x2)) = [-1] + [-1]x2 + x1   
POL(0) = 0   
POL(&&(x1, x2)) = 0   
POL(<=(x1, x2)) = [-1]   
POL(1) = [1]   
POL(>(x1, x2)) = [-1]   

The following pairs are in P>:

F1798_0_HALF_LE'(x2[0], x1[0]) → COND_F1798_0_HALF_LE(&&(<=(x1[0], 1), >(x2[0], 1)), x2[0], x1[0])

The following pairs are in Pbound:

F1798_0_HALF_LE'(x2[0], x1[0]) → COND_F1798_0_HALF_LE(&&(<=(x1[0], 1), >(x2[0], 1)), x2[0], x1[0])

The following pairs are in P≥:

COND_F1798_0_HALF_LE(TRUE, x2[1], x1[1]) → F1798_0_HALF_LE'(0, x2[1])

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

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

(12) 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:
none


R is empty.

The integer pair graph contains the following rules and edges:
(1): COND_F1798_0_HALF_LE(TRUE, x2[1], x1[1]) → F1798_0_HALF_LE'(0, x2[1])


The set Q is empty.

(13) IDependencyGraphProof (EQUIVALENT transformation)

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

(14) TRUE