Termination proof of /tmp/tmppVoiXs/LogAG.jar

Termination of the given JBC could be proven:

0 JBC

↳1 JBCToGraph (⇒, 606 ms)

↳2 JBCTerminationGraph

↳3 TerminationGraphToSCCProof (⇒, 0 ms)

↳4 JBCTerminationSCC

↳5 SCCToIDPv1Proof (⇒, 28 ms)

↳6 IDP

↳7 IDPNonInfProof (⇒, 1131 ms)

↳8 IDP

↳9 IDependencyGraphProof (⇔, 0 ms)

↳10 TRUE

(0) Obligation:

JBC Problem based on JBC Program:

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:

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 35 rules for P and 0 rules for R.

P rules:
f1741_0_log_ConstantStackPush(EOS(STATIC_1741), i371, i371) → f1744_0_log_LE(EOS(STATIC_1744), i371, i371, 1)
f1744_0_log_LE(EOS(STATIC_1744), i387, i387, matching1) → f1747_0_log_LE(EOS(STATIC_1747), i387, i387, 1) | =(matching1, 1)
f1747_0_log_LE(EOS(STATIC_1747), i387, i387, matching1) → f1752_0_log_Load(EOS(STATIC_1752), i387) | &&(>(i387, 1), =(matching1, 1))
f1752_0_log_Load(EOS(STATIC_1752), i387) → f1756_0_log_ConstantStackPush(EOS(STATIC_1756), i387)
f1756_0_log_ConstantStackPush(EOS(STATIC_1756), i387) → f1760_0_log_IntArithmetic(EOS(STATIC_1760), i387, 2)
f1760_0_log_IntArithmetic(EOS(STATIC_1760), i387, matching1) → f1764_0_log_InvokeMethod(EOS(STATIC_1764), -(i387, 2)) | &&(>(i387, 0), =(matching1, 2))
f1764_0_log_InvokeMethod(EOS(STATIC_1764), i389) → f1768_0_half_ConstantStackPush(EOS(STATIC_1768), i389, i389)
f1768_0_half_ConstantStackPush(EOS(STATIC_1768), i389, i389) → f1770_0_half_Store(EOS(STATIC_1770), i389, i389, 0)
f1770_0_half_Store(EOS(STATIC_1770), i389, i389, matching1) → f1772_0_half_Load(EOS(STATIC_1772), i389, i389, 0) | =(matching1, 0)
f1772_0_half_Load(EOS(STATIC_1772), i389, i389, matching1) → f1824_0_half_Load(EOS(STATIC_1824), i389, i389, 0) | =(matching1, 0)
f1824_0_half_Load(EOS(STATIC_1824), i398, i396, i397) → f1877_0_half_Load(EOS(STATIC_1877), i398, i396, i397)
f1877_0_half_Load(EOS(STATIC_1877), i398, i412, i413) → f1932_0_half_Load(EOS(STATIC_1932), i398, i412, i413)
f1932_0_half_Load(EOS(STATIC_1932), i398, i425, i426) → f1988_0_half_Load(EOS(STATIC_1988), i398, i425, i426)
f1988_0_half_Load(EOS(STATIC_1988), i398, i440, i441) → f1992_0_half_ConstantStackPush(EOS(STATIC_1992), i398, i440, i441, i440)
f1992_0_half_ConstantStackPush(EOS(STATIC_1992), i398, i440, i441, i440) → f1995_0_half_LE(EOS(STATIC_1995), i398, i440, i441, i440, 1)
f1995_0_half_LE(EOS(STATIC_1995), i398, i447, i441, i447, matching1) → f1998_0_half_LE(EOS(STATIC_1998), i398, i447, i441, i447, 1) | =(matching1, 1)
f1995_0_half_LE(EOS(STATIC_1995), i398, i448, i441, i448, matching1) → f1999_0_half_LE(EOS(STATIC_1999), i398, i448, i441, i448, 1) | =(matching1, 1)
f1998_0_half_LE(EOS(STATIC_1998), i398, i447, i441, i447, matching1) → f2002_0_half_Load(EOS(STATIC_2002), i398, i441) | &&(<=(i447, 1), =(matching1, 1))
f2002_0_half_Load(EOS(STATIC_2002), i398, i441) → f2006_0_half_Return(EOS(STATIC_2006), i398, i441)
f2006_0_half_Return(EOS(STATIC_2006), i398, i441) → f2010_0_log_ConstantStackPush(EOS(STATIC_2010), i441)
f2010_0_log_ConstantStackPush(EOS(STATIC_2010), i441) → f2014_0_log_IntArithmetic(EOS(STATIC_2014), i441, 1)
f2014_0_log_IntArithmetic(EOS(STATIC_2014), i441, matching1) → f2018_0_log_Store(EOS(STATIC_2018), +(i441, 1)) | &&(>=(i441, 0), =(matching1, 1))
f2018_0_log_Store(EOS(STATIC_2018), i451) → f2022_0_log_Inc(EOS(STATIC_2022), i451)
f2022_0_log_Inc(EOS(STATIC_2022), i451) → f2026_0_log_JMP(EOS(STATIC_2026), i451)
f2026_0_log_JMP(EOS(STATIC_2026), i451) → f2043_0_log_Load(EOS(STATIC_2043), i451)
f2043_0_log_Load(EOS(STATIC_2043), i451) → f1671_0_log_Load(EOS(STATIC_1671), i451)
f1671_0_log_Load(EOS(STATIC_1671), i371) → f1741_0_log_ConstantStackPush(EOS(STATIC_1741), i371, i371)
f1999_0_half_LE(EOS(STATIC_1999), i398, i448, i441, i448, matching1) → f2004_0_half_Load(EOS(STATIC_2004), i398, i448, i441) | &&(>(i448, 1), =(matching1, 1))
f2004_0_half_Load(EOS(STATIC_2004), i398, i448, i441) → f2008_0_half_ConstantStackPush(EOS(STATIC_2008), i398, i441, i448)
f2008_0_half_ConstantStackPush(EOS(STATIC_2008), i398, i441, i448) → f2012_0_half_IntArithmetic(EOS(STATIC_2012), i398, i441, i448, 2)
f2012_0_half_IntArithmetic(EOS(STATIC_2012), i398, i441, i448, matching1) → f2016_0_half_Store(EOS(STATIC_2016), i398, i441, -(i448, 2)) | &&(>(i448, 0), =(matching1, 2))
f2016_0_half_Store(EOS(STATIC_2016), i398, i441, i450) → f2020_0_half_Inc(EOS(STATIC_2020), i398, i450, i441)
f2020_0_half_Inc(EOS(STATIC_2020), i398, i450, i441) → f2024_0_half_JMP(EOS(STATIC_2024), i398, i450, +(i441, 1)) | >=(i441, 0)
f2024_0_half_JMP(EOS(STATIC_2024), i398, i450, i452) → f2035_0_half_Load(EOS(STATIC_2035), i398, i450, i452)
f2035_0_half_Load(EOS(STATIC_2035), i398, i450, i452) → f1988_0_half_Load(EOS(STATIC_1988), i398, i450, i452)
R rules:

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

P rules:
f1995_0_half_LE(EOS(STATIC_1995), x0, x1, x2, x1, 1) → f1995_0_half_LE(EOS(STATIC_1995), -(x2, 1), -(x2, 1), 0, -(x2, 1), 1) | &&(>(x2, 0), <=(x1, 1))
f1995_0_half_LE(EOS(STATIC_1995), x0, x1, x2, x1, 1) → f1995_0_half_LE(EOS(STATIC_1995), x0, -(x1, 2), +(x2, 1), -(x1, 2), 1) | &&(>(x1, 1), >(+(x2, 1), 0))
R rules:

Filtered ground terms:

f1995_0_half_LE(x1, x2, x3, x4, x5, x6) → f1995_0_half_LE(x2, x3, x4, x5)
Cond_f1995_0_half_LE(x1, x2, x3, x4, x5, x6, x7) → Cond_f1995_0_half_LE(x1, x3, x4, x5, x6)
Cond_f1995_0_half_LE1(x1, x2, x3, x4, x5, x6, x7) → Cond_f1995_0_half_LE1(x1, x3, x4, x5, x6)
EOS(x1) → EOS

Filtered unneeded arguments:

f1995_0_half_LE(x1, x2, x3, x4) → f1995_0_half_LE(x2, x3, x4)
Cond_f1995_0_half_LE(x1, x2, x3, x4, x5) → Cond_f1995_0_half_LE(x1, x3, x4, x5)
Cond_f1995_0_half_LE1(x1, x2, x3, x4, x5) → Cond_f1995_0_half_LE1(x1, x3, x4, x5)

Filtered duplicate args:

f1995_0_half_LE(x1, x2, x3) → f1995_0_half_LE(x2, x3)
Cond_f1995_0_half_LE(x1, x2, x3, x4) → Cond_f1995_0_half_LE(x1, x3, x4)
Cond_f1995_0_half_LE1(x1, x2, x3, x4) → Cond_f1995_0_half_LE1(x1, x3, x4)

Filtered unneeded arguments:

Cond_f1995_0_half_LE(x1, x2, x3) → Cond_f1995_0_half_LE(x1, x2)

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

P rules:
F1995_0_HALF_LE(x2, x1) → F1995_0_HALF_LE(0, -(x2, 1)) | &&(<=(x1, 1), >(x2, 0))
F1995_0_HALF_LE(x2, x1) → F1995_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:
F1995_0_HALF_LE'(x2, x1) → COND_F1995_0_HALF_LE(&&(<=(x1, 1), >(x2, 0)), x2, x1)
COND_F1995_0_HALF_LE(TRUE, x2, x1) → F1995_0_HALF_LE'(0, -(x2, 1))
F1995_0_HALF_LE'(x2, x1) → COND_F1995_0_HALF_LE1(&&(>(x1, 1), >(x2, -1)), x2, x1)
COND_F1995_0_HALF_LE1(TRUE, x2, x1) → F1995_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): F1995_0_HALF_LE'(x2[0], x1[0]) → COND_F1995_0_HALF_LE(x1[0] <= 1 && x2[0] > 0, x2[0], x1[0])
(1): COND_F1995_0_HALF_LE(TRUE, x2[1], x1[1]) → F1995_0_HALF_LE'(0, x2[1] - 1)
(2): F1995_0_HALF_LE'(x2[2], x1[2]) → COND_F1995_0_HALF_LE1(x1[2] > 1 && x2[2] > -1, x2[2], x1[2])
(3): COND_F1995_0_HALF_LE1(TRUE, x2[3], x1[3]) → F1995_0_HALF_LE'(x2[3] + 1, x1[3] - 2)

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

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

(1) -> (2), if (0 →^* x2[2]∧x2[1] - 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@71660c8c 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 F1995_0_HALF_LE'(x2, x1) → COND_F1995_0_HALF_LE(&&(<=(x1, 1), >(x2, 0)), x2, x1) the following chains were created:

We consider the chain F1995_0_HALF_LE'(x2[0], x1[0]) → COND_F1995_0_HALF_LE(&&(<=(x1[0], 1), >(x2[0], 0)), x2[0], x1[0]), COND_F1995_0_HALF_LE(TRUE, x2[1], x1[1]) → F1995_0_HALF_LE'(0, -(x2[1], 1)) which results in the following constraint:
(1)    (&&(<=(x1[0], 1), >(x2[0], 0))=TRUE∧x2[0]=x2[1]∧x1[0]=x1[1] ⇒ F1995_0_HALF_LE'(x2[0], x1[0])≥NonInfC∧F1995_0_HALF_LE'(x2[0], x1[0])≥COND_F1995_0_HALF_LE(&&(<=(x1[0], 1), >(x2[0], 0)), x2[0], x1[0])∧(U^Increasing(COND_F1995_0_HALF_LE(&&(<=(x1[0], 1), >(x2[0], 0)), 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], 0)=TRUE ⇒ F1995_0_HALF_LE'(x2[0], x1[0])≥NonInfC∧F1995_0_HALF_LE'(x2[0], x1[0])≥COND_F1995_0_HALF_LE(&&(<=(x1[0], 1), >(x2[0], 0)), x2[0], x1[0])∧(U^Increasing(COND_F1995_0_HALF_LE(&&(<=(x1[0], 1), >(x2[0], 0)), 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] + [-1] ≥ 0 ⇒ (U^Increasing(COND_F1995_0_HALF_LE(&&(<=(x1[0], 1), >(x2[0], 0)), 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] + [-1] ≥ 0 ⇒ (U^Increasing(COND_F1995_0_HALF_LE(&&(<=(x1[0], 1), >(x2[0], 0)), 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] + [-1] ≥ 0 ⇒ (U^Increasing(COND_F1995_0_HALF_LE(&&(<=(x1[0], 1), >(x2[0], 0)), 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 ⇒ (U^Increasing(COND_F1995_0_HALF_LE(&&(<=(x1[0], 1), >(x2[0], 0)), 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)    ([1] + [-1]x1[0] ≥ 0∧x2[0] ≥ 0∧x1[0] ≥ 0 ⇒ (U^Increasing(COND_F1995_0_HALF_LE(&&(<=(x1[0], 1), >(x2[0], 0)), x2[0], x1[0])), ≥)∧[(-1)Bound*bni_17] + [bni_17]x1[0] + [bni_17]x2[0] ≥ 0∧[(-1)bso_18] ≥ 0)

(8)    ([1] + x1[0] ≥ 0∧x2[0] ≥ 0∧x1[0] ≥ 0 ⇒ (U^Increasing(COND_F1995_0_HALF_LE(&&(<=(x1[0], 1), >(x2[0], 0)), 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_F1995_0_HALF_LE(TRUE, x2, x1) → F1995_0_HALF_LE'(0, -(x2, 1)) the following chains were created:

We consider the chain F1995_0_HALF_LE'(x2[0], x1[0]) → COND_F1995_0_HALF_LE(&&(<=(x1[0], 1), >(x2[0], 0)), x2[0], x1[0]), COND_F1995_0_HALF_LE(TRUE, x2[1], x1[1]) → F1995_0_HALF_LE'(0, -(x2[1], 1)), F1995_0_HALF_LE'(x2[0], x1[0]) → COND_F1995_0_HALF_LE(&&(<=(x1[0], 1), >(x2[0], 0)), x2[0], x1[0]), COND_F1995_0_HALF_LE(TRUE, x2[1], x1[1]) → F1995_0_HALF_LE'(0, -(x2[1], 1)), F1995_0_HALF_LE'(x2[0], x1[0]) → COND_F1995_0_HALF_LE(&&(<=(x1[0], 1), >(x2[0], 0)), x2[0], x1[0]), COND_F1995_0_HALF_LE(TRUE, x2[1], x1[1]) → F1995_0_HALF_LE'(0, -(x2[1], 1)) which results in the following constraint:
(9) (&&(<=(x1[0], 1), >(x2[0], 0))=TRUE∧x2[0]=x2[1]∧x1[0]=x1[1]∧0=x2[0]1∧-(x2[1], 1)=x1[0]1∧&&(<=(x1[0]1, 1), >(x2[0]1, 0))=TRUE∧x2[0]1=x2[1]1∧x1[0]1=x1[1]1∧0=x2[0]2∧-(x2[1]1, 1)=x1[0]2∧&&(<=(x1[0]2, 1), >(x2[0]2, 0))=TRUE∧x2[0]2=x2[1]2∧x1[0]2=x1[1]2 ⇒ COND_F1995_0_HALF_LE(TRUE, x2[1]1, x1[1]1)≥NonInfC∧COND_F1995_0_HALF_LE(TRUE, x2[1]1, x1[1]1)≥F1995_0_HALF_LE'(0, -(x2[1]1, 1))∧(U^Increasing(F1995_0_HALF_LE'(0, -(x2[1]1, 1))), ≥))

We solved constraint (9) using rules (I), (II), (III), (IV), (IDP_CONSTANT_FOLD), (IDP_BOOLEAN).
We consider the chain F1995_0_HALF_LE'(x2[0], x1[0]) → COND_F1995_0_HALF_LE(&&(<=(x1[0], 1), >(x2[0], 0)), x2[0], x1[0]), COND_F1995_0_HALF_LE(TRUE, x2[1], x1[1]) → F1995_0_HALF_LE'(0, -(x2[1], 1)), F1995_0_HALF_LE'(x2[0], x1[0]) → COND_F1995_0_HALF_LE(&&(<=(x1[0], 1), >(x2[0], 0)), x2[0], x1[0]), COND_F1995_0_HALF_LE(TRUE, x2[1], x1[1]) → F1995_0_HALF_LE'(0, -(x2[1], 1)), F1995_0_HALF_LE'(x2[2], x1[2]) → COND_F1995_0_HALF_LE1(&&(>(x1[2], 1), >(x2[2], -1)), x2[2], x1[2]), COND_F1995_0_HALF_LE1(TRUE, x2[3], x1[3]) → F1995_0_HALF_LE'(+(x2[3], 1), -(x1[3], 2)) which results in the following constraint:
(10) (&&(<=(x1[0], 1), >(x2[0], 0))=TRUE∧x2[0]=x2[1]∧x1[0]=x1[1]∧0=x2[0]1∧-(x2[1], 1)=x1[0]1∧&&(<=(x1[0]1, 1), >(x2[0]1, 0))=TRUE∧x2[0]1=x2[1]1∧x1[0]1=x1[1]1∧0=x2[2]∧-(x2[1]1, 1)=x1[2]∧&&(>(x1[2], 1), >(x2[2], -1))=TRUE∧x2[2]=x2[3]∧x1[2]=x1[3] ⇒ COND_F1995_0_HALF_LE(TRUE, x2[1]1, x1[1]1)≥NonInfC∧COND_F1995_0_HALF_LE(TRUE, x2[1]1, x1[1]1)≥F1995_0_HALF_LE'(0, -(x2[1]1, 1))∧(U^Increasing(F1995_0_HALF_LE'(0, -(x2[1]1, 1))), ≥))

We solved constraint (10) using rules (I), (II), (III), (IV), (IDP_CONSTANT_FOLD), (IDP_BOOLEAN).
We consider the chain F1995_0_HALF_LE'(x2[2], x1[2]) → COND_F1995_0_HALF_LE1(&&(>(x1[2], 1), >(x2[2], -1)), x2[2], x1[2]), COND_F1995_0_HALF_LE1(TRUE, x2[3], x1[3]) → F1995_0_HALF_LE'(+(x2[3], 1), -(x1[3], 2)), F1995_0_HALF_LE'(x2[0], x1[0]) → COND_F1995_0_HALF_LE(&&(<=(x1[0], 1), >(x2[0], 0)), x2[0], x1[0]), COND_F1995_0_HALF_LE(TRUE, x2[1], x1[1]) → F1995_0_HALF_LE'(0, -(x2[1], 1)), F1995_0_HALF_LE'(x2[0], x1[0]) → COND_F1995_0_HALF_LE(&&(<=(x1[0], 1), >(x2[0], 0)), x2[0], x1[0]), COND_F1995_0_HALF_LE(TRUE, x2[1], x1[1]) → F1995_0_HALF_LE'(0, -(x2[1], 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], 0))=TRUE∧x2[0]=x2[1]∧x1[0]=x1[1]∧0=x2[0]1∧-(x2[1], 1)=x1[0]1∧&&(<=(x1[0]1, 1), >(x2[0]1, 0))=TRUE∧x2[0]1=x2[1]1∧x1[0]1=x1[1]1 ⇒ COND_F1995_0_HALF_LE(TRUE, x2[1], x1[1])≥NonInfC∧COND_F1995_0_HALF_LE(TRUE, x2[1], x1[1])≥F1995_0_HALF_LE'(0, -(x2[1], 1))∧(U^Increasing(F1995_0_HALF_LE'(0, -(x2[1], 1))), ≥))

We solved constraint (11) using rules (I), (II), (III), (IV), (IDP_CONSTANT_FOLD), (IDP_BOOLEAN).
We consider the chain F1995_0_HALF_LE'(x2[2], x1[2]) → COND_F1995_0_HALF_LE1(&&(>(x1[2], 1), >(x2[2], -1)), x2[2], x1[2]), COND_F1995_0_HALF_LE1(TRUE, x2[3], x1[3]) → F1995_0_HALF_LE'(+(x2[3], 1), -(x1[3], 2)), F1995_0_HALF_LE'(x2[0], x1[0]) → COND_F1995_0_HALF_LE(&&(<=(x1[0], 1), >(x2[0], 0)), x2[0], x1[0]), COND_F1995_0_HALF_LE(TRUE, x2[1], x1[1]) → F1995_0_HALF_LE'(0, -(x2[1], 1)), F1995_0_HALF_LE'(x2[2], x1[2]) → COND_F1995_0_HALF_LE1(&&(>(x1[2], 1), >(x2[2], -1)), x2[2], x1[2]), COND_F1995_0_HALF_LE1(TRUE, x2[3], x1[3]) → F1995_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], 0))=TRUE∧x2[0]=x2[1]∧x1[0]=x1[1]∧0=x2[2]1∧-(x2[1], 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_F1995_0_HALF_LE(TRUE, x2[1], x1[1])≥NonInfC∧COND_F1995_0_HALF_LE(TRUE, x2[1], x1[1])≥F1995_0_HALF_LE'(0, -(x2[1], 1))∧(U^Increasing(F1995_0_HALF_LE'(0, -(x2[1], 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), 0)=TRUE∧>(-(+(x2[2], 1), 1), 1)=TRUE ⇒ COND_F1995_0_HALF_LE(TRUE, +(x2[2], 1), -(x1[2], 2))≥NonInfC∧COND_F1995_0_HALF_LE(TRUE, +(x2[2], 1), -(x1[2], 2))≥F1995_0_HALF_LE'(0, -(+(x2[2], 1), 1))∧(U^Increasing(F1995_0_HALF_LE'(0, -(x2[1], 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] ≥ 0∧x2[2] + [-2] ≥ 0 ⇒ (U^Increasing(F1995_0_HALF_LE'(0, -(x2[1], 1))), ≥)∧[(-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)    (x1[2] + [-2] ≥ 0∧x2[2] ≥ 0∧[3] + [-1]x1[2] ≥ 0∧x2[2] ≥ 0∧x2[2] + [-2] ≥ 0 ⇒ (U^Increasing(F1995_0_HALF_LE'(0, -(x2[1], 1))), ≥)∧[(-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)    (x1[2] + [-2] ≥ 0∧x2[2] ≥ 0∧[3] + [-1]x1[2] ≥ 0∧x2[2] ≥ 0∧x2[2] + [-2] ≥ 0 ⇒ (U^Increasing(F1995_0_HALF_LE'(0, -(x2[1], 1))), ≥)∧[(-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)    (x1[2] ≥ 0∧x2[2] ≥ 0∧[1] + [-1]x1[2] ≥ 0∧x2[2] ≥ 0∧x2[2] + [-2] ≥ 0 ⇒ (U^Increasing(F1995_0_HALF_LE'(0, -(x2[1], 1))), ≥)∧[(-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)    (x1[2] ≥ 0∧[2] + x2[2] ≥ 0∧[1] + [-1]x1[2] ≥ 0∧[2] + x2[2] ≥ 0∧x2[2] ≥ 0 ⇒ (U^Increasing(F1995_0_HALF_LE'(0, -(x2[1], 1))), ≥)∧[(-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 F1995_0_HALF_LE'(x2, x1) → COND_F1995_0_HALF_LE1(&&(>(x1, 1), >(x2, -1)), x2, x1) the following chains were created:

We consider the chain F1995_0_HALF_LE'(x2[2], x1[2]) → COND_F1995_0_HALF_LE1(&&(>(x1[2], 1), >(x2[2], -1)), x2[2], x1[2]), COND_F1995_0_HALF_LE1(TRUE, x2[3], x1[3]) → F1995_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] ⇒ F1995_0_HALF_LE'(x2[2], x1[2])≥NonInfC∧F1995_0_HALF_LE'(x2[2], x1[2])≥COND_F1995_0_HALF_LE1(&&(>(x1[2], 1), >(x2[2], -1)), x2[2], x1[2])∧(U^Increasing(COND_F1995_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 ⇒ F1995_0_HALF_LE'(x2[2], x1[2])≥NonInfC∧F1995_0_HALF_LE'(x2[2], x1[2])≥COND_F1995_0_HALF_LE1(&&(>(x1[2], 1), >(x2[2], -1)), x2[2], x1[2])∧(U^Increasing(COND_F1995_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 ⇒ (U^Increasing(COND_F1995_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 ⇒ (U^Increasing(COND_F1995_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 ⇒ (U^Increasing(COND_F1995_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 ⇒ (U^Increasing(COND_F1995_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_F1995_0_HALF_LE1(TRUE, x2, x1) → F1995_0_HALF_LE'(+(x2, 1), -(x1, 2)) the following chains were created:

We consider the chain F1995_0_HALF_LE'(x2[0], x1[0]) → COND_F1995_0_HALF_LE(&&(<=(x1[0], 1), >(x2[0], 0)), x2[0], x1[0]), COND_F1995_0_HALF_LE(TRUE, x2[1], x1[1]) → F1995_0_HALF_LE'(0, -(x2[1], 1)), F1995_0_HALF_LE'(x2[2], x1[2]) → COND_F1995_0_HALF_LE1(&&(>(x1[2], 1), >(x2[2], -1)), x2[2], x1[2]), COND_F1995_0_HALF_LE1(TRUE, x2[3], x1[3]) → F1995_0_HALF_LE'(+(x2[3], 1), -(x1[3], 2)), F1995_0_HALF_LE'(x2[0], x1[0]) → COND_F1995_0_HALF_LE(&&(<=(x1[0], 1), >(x2[0], 0)), x2[0], x1[0]), COND_F1995_0_HALF_LE(TRUE, x2[1], x1[1]) → F1995_0_HALF_LE'(0, -(x2[1], 1)) which results in the following constraint:
(25)    (&&(<=(x1[0], 1), >(x2[0], 0))=TRUE∧x2[0]=x2[1]∧x1[0]=x1[1]∧0=x2[2]∧-(x2[1], 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, 0))=TRUE∧x2[0]1=x2[1]1∧x1[0]1=x1[1]1 ⇒ COND_F1995_0_HALF_LE1(TRUE, x2[3], x1[3])≥NonInfC∧COND_F1995_0_HALF_LE1(TRUE, x2[3], x1[3])≥F1995_0_HALF_LE'(+(x2[3], 1), -(x1[3], 2))∧(U^Increasing(F1995_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)    (<=(x1[0], 1)=TRUE∧>(x2[0], 0)=TRUE∧>(-(x2[0], 1), 1)=TRUE∧<=(-(-(x2[0], 1), 2), 1)=TRUE ⇒ COND_F1995_0_HALF_LE1(TRUE, 0, -(x2[0], 1))≥NonInfC∧COND_F1995_0_HALF_LE1(TRUE, 0, -(x2[0], 1))≥F1995_0_HALF_LE'(+(0, 1), -(-(x2[0], 1), 2))∧(U^Increasing(F1995_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)    ([1] + [-1]x1[0] ≥ 0∧x2[0] + [-1] ≥ 0∧x2[0] + [-3] ≥ 0∧[4] + [-1]x2[0] ≥ 0 ⇒ (U^Increasing(F1995_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)    ([1] + [-1]x1[0] ≥ 0∧x2[0] + [-1] ≥ 0∧x2[0] + [-3] ≥ 0∧[4] + [-1]x2[0] ≥ 0 ⇒ (U^Increasing(F1995_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)    ([1] + [-1]x1[0] ≥ 0∧x2[0] + [-1] ≥ 0∧x2[0] + [-3] ≥ 0∧[4] + [-1]x2[0] ≥ 0 ⇒ (U^Increasing(F1995_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)    ([1] + [-1]x1[0] ≥ 0∧x2[0] ≥ 0∧[-2] + x2[0] ≥ 0∧[3] + [-1]x2[0] ≥ 0 ⇒ (U^Increasing(F1995_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∧[2] + x2[0] ≥ 0∧x2[0] ≥ 0∧[1] + [-1]x2[0] ≥ 0 ⇒ (U^Increasing(F1995_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 ⇒ (U^Increasing(F1995_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 F1995_0_HALF_LE'(x2[0], x1[0]) → COND_F1995_0_HALF_LE(&&(<=(x1[0], 1), >(x2[0], 0)), x2[0], x1[0]), COND_F1995_0_HALF_LE(TRUE, x2[1], x1[1]) → F1995_0_HALF_LE'(0, -(x2[1], 1)), F1995_0_HALF_LE'(x2[2], x1[2]) → COND_F1995_0_HALF_LE1(&&(>(x1[2], 1), >(x2[2], -1)), x2[2], x1[2]), COND_F1995_0_HALF_LE1(TRUE, x2[3], x1[3]) → F1995_0_HALF_LE'(+(x2[3], 1), -(x1[3], 2)), F1995_0_HALF_LE'(x2[2], x1[2]) → COND_F1995_0_HALF_LE1(&&(>(x1[2], 1), >(x2[2], -1)), x2[2], x1[2]), COND_F1995_0_HALF_LE1(TRUE, x2[3], x1[3]) → F1995_0_HALF_LE'(+(x2[3], 1), -(x1[3], 2)) which results in the following constraint:
(33)    (&&(<=(x1[0], 1), >(x2[0], 0))=TRUE∧x2[0]=x2[1]∧x1[0]=x1[1]∧0=x2[2]∧-(x2[1], 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_F1995_0_HALF_LE1(TRUE, x2[3], x1[3])≥NonInfC∧COND_F1995_0_HALF_LE1(TRUE, x2[3], x1[3])≥F1995_0_HALF_LE'(+(x2[3], 1), -(x1[3], 2))∧(U^Increasing(F1995_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)    (<=(x1[0], 1)=TRUE∧>(x2[0], 0)=TRUE∧>(-(x2[0], 1), 1)=TRUE∧>(-(-(x2[0], 1), 2), 1)=TRUE ⇒ COND_F1995_0_HALF_LE1(TRUE, 0, -(x2[0], 1))≥NonInfC∧COND_F1995_0_HALF_LE1(TRUE, 0, -(x2[0], 1))≥F1995_0_HALF_LE'(+(0, 1), -(-(x2[0], 1), 2))∧(U^Increasing(F1995_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)    ([1] + [-1]x1[0] ≥ 0∧x2[0] + [-1] ≥ 0∧x2[0] + [-3] ≥ 0∧x2[0] + [-5] ≥ 0 ⇒ (U^Increasing(F1995_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)    ([1] + [-1]x1[0] ≥ 0∧x2[0] + [-1] ≥ 0∧x2[0] + [-3] ≥ 0∧x2[0] + [-5] ≥ 0 ⇒ (U^Increasing(F1995_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)    ([1] + [-1]x1[0] ≥ 0∧x2[0] + [-1] ≥ 0∧x2[0] + [-3] ≥ 0∧x2[0] + [-5] ≥ 0 ⇒ (U^Increasing(F1995_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)    ([1] + [-1]x1[0] ≥ 0∧x2[0] ≥ 0∧[-2] + x2[0] ≥ 0∧[-4] + x2[0] ≥ 0 ⇒ (U^Increasing(F1995_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)    ([1] + [-1]x1[0] ≥ 0∧[2] + x2[0] ≥ 0∧x2[0] ≥ 0∧[-2] + x2[0] ≥ 0 ⇒ (U^Increasing(F1995_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)    ([1] + [-1]x1[0] ≥ 0∧[4] + x2[0] ≥ 0∧[2] + x2[0] ≥ 0∧x2[0] ≥ 0 ⇒ (U^Increasing(F1995_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 ⇒ (U^Increasing(F1995_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 F1995_0_HALF_LE'(x2[2], x1[2]) → COND_F1995_0_HALF_LE1(&&(>(x1[2], 1), >(x2[2], -1)), x2[2], x1[2]), COND_F1995_0_HALF_LE1(TRUE, x2[3], x1[3]) → F1995_0_HALF_LE'(+(x2[3], 1), -(x1[3], 2)), F1995_0_HALF_LE'(x2[2], x1[2]) → COND_F1995_0_HALF_LE1(&&(>(x1[2], 1), >(x2[2], -1)), x2[2], x1[2]), COND_F1995_0_HALF_LE1(TRUE, x2[3], x1[3]) → F1995_0_HALF_LE'(+(x2[3], 1), -(x1[3], 2)), F1995_0_HALF_LE'(x2[0], x1[0]) → COND_F1995_0_HALF_LE(&&(<=(x1[0], 1), >(x2[0], 0)), x2[0], x1[0]), COND_F1995_0_HALF_LE(TRUE, x2[1], x1[1]) → F1995_0_HALF_LE'(0, -(x2[1], 1)) which results in the following constraint:
(42)    (&&(>(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], 0))=TRUE∧x2[0]=x2[1]∧x1[0]=x1[1] ⇒ COND_F1995_0_HALF_LE1(TRUE, x2[3]1, x1[3]1)≥NonInfC∧COND_F1995_0_HALF_LE1(TRUE, x2[3]1, x1[3]1)≥F1995_0_HALF_LE'(+(x2[3]1, 1), -(x1[3]1, 2))∧(U^Increasing(F1995_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)    (>(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), 0)=TRUE ⇒ COND_F1995_0_HALF_LE1(TRUE, +(x2[2], 1), -(x1[2], 2))≥NonInfC∧COND_F1995_0_HALF_LE1(TRUE, +(x2[2], 1), -(x1[2], 2))≥F1995_0_HALF_LE'(+(+(x2[2], 1), 1), -(-(x1[2], 2), 2))∧(U^Increasing(F1995_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)    (x1[2] + [-2] ≥ 0∧x2[2] ≥ 0∧x1[2] + [-4] ≥ 0∧x2[2] + [1] ≥ 0∧[5] + [-1]x1[2] ≥ 0∧x2[2] + [1] ≥ 0 ⇒ (U^Increasing(F1995_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)    (x1[2] + [-2] ≥ 0∧x2[2] ≥ 0∧x1[2] + [-4] ≥ 0∧x2[2] + [1] ≥ 0∧[5] + [-1]x1[2] ≥ 0∧x2[2] + [1] ≥ 0 ⇒ (U^Increasing(F1995_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)    (x1[2] + [-2] ≥ 0∧x2[2] ≥ 0∧x1[2] + [-4] ≥ 0∧x2[2] + [1] ≥ 0∧[5] + [-1]x1[2] ≥ 0∧x2[2] + [1] ≥ 0 ⇒ (U^Increasing(F1995_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)    (x1[2] ≥ 0∧x2[2] ≥ 0∧[-2] + x1[2] ≥ 0∧x2[2] + [1] ≥ 0∧[3] + [-1]x1[2] ≥ 0∧x2[2] + [1] ≥ 0 ⇒ (U^Increasing(F1995_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)    ([2] + x1[2] ≥ 0∧x2[2] ≥ 0∧x1[2] ≥ 0∧x2[2] + [1] ≥ 0∧[1] + [-1]x1[2] ≥ 0∧x2[2] + [1] ≥ 0 ⇒ (U^Increasing(F1995_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 F1995_0_HALF_LE'(x2[2], x1[2]) → COND_F1995_0_HALF_LE1(&&(>(x1[2], 1), >(x2[2], -1)), x2[2], x1[2]), COND_F1995_0_HALF_LE1(TRUE, x2[3], x1[3]) → F1995_0_HALF_LE'(+(x2[3], 1), -(x1[3], 2)), F1995_0_HALF_LE'(x2[2], x1[2]) → COND_F1995_0_HALF_LE1(&&(>(x1[2], 1), >(x2[2], -1)), x2[2], x1[2]), COND_F1995_0_HALF_LE1(TRUE, x2[3], x1[3]) → F1995_0_HALF_LE'(+(x2[3], 1), -(x1[3], 2)), F1995_0_HALF_LE'(x2[2], x1[2]) → COND_F1995_0_HALF_LE1(&&(>(x1[2], 1), >(x2[2], -1)), x2[2], x1[2]), COND_F1995_0_HALF_LE1(TRUE, x2[3], x1[3]) → F1995_0_HALF_LE'(+(x2[3], 1), -(x1[3], 2)) which results in the following constraint:
(49)    (&&(>(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_F1995_0_HALF_LE1(TRUE, x2[3]1, x1[3]1)≥NonInfC∧COND_F1995_0_HALF_LE1(TRUE, x2[3]1, x1[3]1)≥F1995_0_HALF_LE'(+(x2[3]1, 1), -(x1[3]1, 2))∧(U^Increasing(F1995_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)    (>(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_F1995_0_HALF_LE1(TRUE, +(x2[2], 1), -(x1[2], 2))≥NonInfC∧COND_F1995_0_HALF_LE1(TRUE, +(x2[2], 1), -(x1[2], 2))≥F1995_0_HALF_LE'(+(+(x2[2], 1), 1), -(-(x1[2], 2), 2))∧(U^Increasing(F1995_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)    (x1[2] + [-2] ≥ 0∧x2[2] ≥ 0∧x1[2] + [-4] ≥ 0∧x2[2] + [1] ≥ 0∧x1[2] + [-6] ≥ 0∧x2[2] + [2] ≥ 0 ⇒ (U^Increasing(F1995_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)    (x1[2] + [-2] ≥ 0∧x2[2] ≥ 0∧x1[2] + [-4] ≥ 0∧x2[2] + [1] ≥ 0∧x1[2] + [-6] ≥ 0∧x2[2] + [2] ≥ 0 ⇒ (U^Increasing(F1995_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)    (x1[2] + [-2] ≥ 0∧x2[2] ≥ 0∧x1[2] + [-4] ≥ 0∧x2[2] + [1] ≥ 0∧x1[2] + [-6] ≥ 0∧x2[2] + [2] ≥ 0 ⇒ (U^Increasing(F1995_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)    (x1[2] ≥ 0∧x2[2] ≥ 0∧[-2] + x1[2] ≥ 0∧x2[2] + [1] ≥ 0∧[-4] + x1[2] ≥ 0∧x2[2] + [2] ≥ 0 ⇒ (U^Increasing(F1995_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)    ([2] + x1[2] ≥ 0∧x2[2] ≥ 0∧x1[2] ≥ 0∧x2[2] + [1] ≥ 0∧[-2] + x1[2] ≥ 0∧x2[2] + [2] ≥ 0 ⇒ (U^Increasing(F1995_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)    ([4] + x1[2] ≥ 0∧x2[2] ≥ 0∧[2] + x1[2] ≥ 0∧x2[2] + [1] ≥ 0∧x1[2] ≥ 0∧x2[2] + [2] ≥ 0 ⇒ (U^Increasing(F1995_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.

F1995_0_HALF_LE'(x2, x1) → COND_F1995_0_HALF_LE(&&(<=(x1, 1), >(x2, 0)), x2, x1)
- ([1] + [-1]x1[0] ≥ 0∧x2[0] ≥ 0∧x1[0] ≥ 0 ⇒ (U^Increasing(COND_F1995_0_HALF_LE(&&(<=(x1[0], 1), >(x2[0], 0)), x2[0], x1[0])), ≥)∧[(-1)Bound*bni_17] + [bni_17]x1[0] + [bni_17]x2[0] ≥ 0∧[(-1)bso_18] ≥ 0)
- ([1] + x1[0] ≥ 0∧x2[0] ≥ 0∧x1[0] ≥ 0 ⇒ (U^Increasing(COND_F1995_0_HALF_LE(&&(<=(x1[0], 1), >(x2[0], 0)), x2[0], x1[0])), ≥)∧[(-1)Bound*bni_17] + [(-1)bni_17]x1[0] + [bni_17]x2[0] ≥ 0∧[(-1)bso_18] ≥ 0)
COND_F1995_0_HALF_LE(TRUE, x2, x1) → F1995_0_HALF_LE'(0, -(x2, 1))
- (x1[2] ≥ 0∧[2] + x2[2] ≥ 0∧[1] + [-1]x1[2] ≥ 0∧[2] + x2[2] ≥ 0∧x2[2] ≥ 0 ⇒ (U^Increasing(F1995_0_HALF_LE'(0, -(x2[1], 1))), ≥)∧[(-1)Bound*bni_19 + (2)bni_19] + [bni_19]x1[2] + [bni_19]x2[2] ≥ 0∧[1 + (-1)bso_20] + x1[2] ≥ 0)
F1995_0_HALF_LE'(x2, x1) → COND_F1995_0_HALF_LE1(&&(>(x1, 1), >(x2, -1)), x2, x1)
- (x1[2] ≥ 0∧x2[2] ≥ 0 ⇒ (U^Increasing(COND_F1995_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_F1995_0_HALF_LE1(TRUE, x2, x1) → F1995_0_HALF_LE'(+(x2, 1), -(x1, 2))
- ([2] + x2[0] ≥ 0∧x2[0] ≥ 0∧[1] + [-1]x2[0] ≥ 0 ⇒ (U^Increasing(F1995_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 ⇒ (U^Increasing(F1995_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] + [1] ≥ 0 ⇒ (U^Increasing(F1995_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 ⇒ (U^Increasing(F1995_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 P_bound are constructed from P_≥ where we just replace every occurence of "t ≥ s" in P_≥ by "t > s" respective "t ≥ c". Here c stands for the fresh constant used for P_bound.
Using the following integer polynomial ordering the resulting constraints can be solved
Polynomial interpretation over integers[POLO]:

POL(TRUE) = 0
POL(FALSE) = [3]
POL(F1995_0_HALF_LE'(x₁, x₂)) = [-1] + x₂ + x₁
POL(COND_F1995_0_HALF_LE(x₁, x₂, x₃)) = [-1] + x₃ + x₂ + [-1]x₁
POL(&&(x₁, x₂)) = 0
POL(<=(x₁, x₂)) = [-1]
POL(1) = [1]
POL(>(x₁, x₂)) = [-1]
POL(0) = 0
POL(-(x₁, x₂)) = x₁ + [-1]x₂
POL(COND_F1995_0_HALF_LE1(x₁, x₂, x₃)) = [-1] + x₃ + x₂ + [-1]x₁
POL(-1) = [-1]
POL(+(x₁, x₂)) = x₁ + x₂
POL(2) = [2]

The following pairs are in P_>:

COND_F1995_0_HALF_LE(TRUE, x2[1], x1[1]) → F1995_0_HALF_LE'(0, -(x2[1], 1))
COND_F1995_0_HALF_LE1(TRUE, x2[3], x1[3]) → F1995_0_HALF_LE'(+(x2[3], 1), -(x1[3], 2))

The following pairs are in P_bound:

COND_F1995_0_HALF_LE(TRUE, x2[1], x1[1]) → F1995_0_HALF_LE'(0, -(x2[1], 1))
F1995_0_HALF_LE'(x2[2], x1[2]) → COND_F1995_0_HALF_LE1(&&(>(x1[2], 1), >(x2[2], -1)), x2[2], x1[2])
COND_F1995_0_HALF_LE1(TRUE, x2[3], x1[3]) → F1995_0_HALF_LE'(+(x2[3], 1), -(x1[3], 2))

The following pairs are in P_≥:

F1995_0_HALF_LE'(x2[0], x1[0]) → COND_F1995_0_HALF_LE(&&(<=(x1[0], 1), >(x2[0], 0)), x2[0], x1[0])
F1995_0_HALF_LE'(x2[2], x1[2]) → COND_F1995_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: