Termination proof of /tmp/tmpDk3Qaq/DivMinus2.jar

Termination of the given JBC could be proven:

0 JBC

↳1 JBCToGraph (⇒, 300 ms)

↳2 JBCTerminationGraph

↳3 TerminationGraphToSCCProof (⇒, 10 ms)

↳4 JBCTerminationSCC

↳5 SCCToIDPv1Proof (⇒, 90 ms)

↳6 IDP

↳7 IDPNonInfProof (⇒, 340 ms)

↳8 AND

    ↳9 IDP

        ↳10 IDependencyGraphProof (⇔, 0 ms)

        ↳11 IDP

        ↳12 IDPNonInfProof (⇒, 160 ms)

        ↳13 IDP

        ↳14 IDependencyGraphProof (⇔, 0 ms)

        ↳15 TRUE

    ↳16 IDP

        ↳17 IDependencyGraphProof (⇔, 0 ms)

        ↳18 IDP

        ↳19 IDPNonInfProof (⇒, 140 ms)

        ↳20 IDP

        ↳21 IDependencyGraphProof (⇔, 0 ms)

        ↳22 TRUE

(0) Obligation:

JBC Problem based on JBC Program:
Manifest-Version: 1.0 Created-By: 1.6.0_16 (Sun Microsystems Inc.) Main-Class: DivMinus2

public class DivMinus2 {
  public static int div(int x, int y) {
    int res = 0;
    while (x >= y && y > 0) {
      x = minus(x,y);
      res = res + 1;
    }
    return res;
  }

  public static int minus(int x, int y) {
    while (y != 0) {
      if (y > 0)  {
        y--;
        x--;
      } else  {
        y++;
        x++;
      }
    }
    return x;
  }

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


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:
DivMinus2.main([Ljava/lang/String;)V: Graph of 213 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: DivMinus2.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 31 rules for P and 0 rules for R.

P rules:
1019_0_div_Load(EOS(STATIC_1019), i220, i219, i220, i219) → 1021_0_div_LT(EOS(STATIC_1021), i220, i219, i220, i219, i220)
1021_0_div_LT(EOS(STATIC_1021), i220, i219, i220, i219, i220) → 1025_0_div_LT(EOS(STATIC_1025), i220, i219, i220, i219, i220)
1025_0_div_LT(EOS(STATIC_1025), i220, i219, i220, i219, i220) → 1029_0_div_Load(EOS(STATIC_1029), i220, i219, i220) | >=(i219, i220)
1029_0_div_Load(EOS(STATIC_1029), i220, i219, i220) → 1034_0_div_LE(EOS(STATIC_1034), i220, i219, i220, i220)
1034_0_div_LE(EOS(STATIC_1034), i230, i219, i230, i230) → 1040_0_div_LE(EOS(STATIC_1040), i230, i219, i230, i230)
1040_0_div_LE(EOS(STATIC_1040), i230, i219, i230, i230) → 1046_0_div_Load(EOS(STATIC_1046), i230, i219, i230) | >(i230, 0)
1046_0_div_Load(EOS(STATIC_1046), i230, i219, i230) → 1053_0_div_Load(EOS(STATIC_1053), i230, i230, i219)
1053_0_div_Load(EOS(STATIC_1053), i230, i230, i219) → 1059_0_div_InvokeMethod(EOS(STATIC_1059), i230, i230, i219, i230)
1059_0_div_InvokeMethod(EOS(STATIC_1059), i230, i230, i219, i230) → 1063_0_minus_Load(EOS(STATIC_1063), i230, i230, i219, i230, i219, i230)
1063_0_minus_Load(EOS(STATIC_1063), i230, i230, i219, i230, i219, i230) → 1081_0_minus_Load(EOS(STATIC_1081), i230, i230, i219, i230, i219, i230)
1081_0_minus_Load(EOS(STATIC_1081), i230, i230, i219, i230, i233, i234) → 1083_0_minus_EQ(EOS(STATIC_1083), i230, i230, i219, i230, i233, i234, i234)
1083_0_minus_EQ(EOS(STATIC_1083), i230, i230, i219, i230, i233, i239, i239) → 1085_0_minus_EQ(EOS(STATIC_1085), i230, i230, i219, i230, i233, i239, i239)
1083_0_minus_EQ(EOS(STATIC_1083), i230, i230, i219, i230, i233, matching1, matching2) → 1086_0_minus_EQ(EOS(STATIC_1086), i230, i230, i219, i230, i233, 0, 0) | &&(=(matching1, 0), =(matching2, 0))
1085_0_minus_EQ(EOS(STATIC_1085), i230, i230, i219, i230, i233, i239, i239) → 1087_0_minus_Load(EOS(STATIC_1087), i230, i230, i219, i230, i233, i239) | >(i239, 0)
1087_0_minus_Load(EOS(STATIC_1087), i230, i230, i219, i230, i233, i239) → 1090_0_minus_LE(EOS(STATIC_1090), i230, i230, i219, i230, i233, i239, i239)
1090_0_minus_LE(EOS(STATIC_1090), i230, i230, i219, i230, i233, i239, i239) → 1093_0_minus_Inc(EOS(STATIC_1093), i230, i230, i219, i230, i233, i239) | >(i239, 0)
1093_0_minus_Inc(EOS(STATIC_1093), i230, i230, i219, i230, i233, i239) → 1096_0_minus_Inc(EOS(STATIC_1096), i230, i230, i219, i230, i233, +(i239, -1)) | >(i239, 0)
1096_0_minus_Inc(EOS(STATIC_1096), i230, i230, i219, i230, i233, i240) → 1099_0_minus_JMP(EOS(STATIC_1099), i230, i230, i219, i230, +(i233, -1), i240)
1099_0_minus_JMP(EOS(STATIC_1099), i230, i230, i219, i230, i241, i240) → 1103_0_minus_Load(EOS(STATIC_1103), i230, i230, i219, i230, i241, i240)
1103_0_minus_Load(EOS(STATIC_1103), i230, i230, i219, i230, i241, i240) → 1081_0_minus_Load(EOS(STATIC_1081), i230, i230, i219, i230, i241, i240)
1086_0_minus_EQ(EOS(STATIC_1086), i230, i230, i219, i230, i233, matching1, matching2) → 1089_0_minus_Load(EOS(STATIC_1089), i230, i230, i219, i230, i233) | &&(=(matching1, 0), =(matching2, 0))
1089_0_minus_Load(EOS(STATIC_1089), i230, i230, i219, i230, i233) → 1092_0_minus_Return(EOS(STATIC_1092), i230, i230, i219, i230, i233)
1092_0_minus_Return(EOS(STATIC_1092), i230, i230, i219, i230, i233) → 1095_0_div_Store(EOS(STATIC_1095), i230, i230, i233)
1095_0_div_Store(EOS(STATIC_1095), i230, i230, i233) → 1098_0_div_Load(EOS(STATIC_1098), i230, i233, i230)
1098_0_div_Load(EOS(STATIC_1098), i230, i233, i230) → 1101_0_div_ConstantStackPush(EOS(STATIC_1101), i230, i233, i230)
1101_0_div_ConstantStackPush(EOS(STATIC_1101), i230, i233, i230) → 1104_0_div_IntArithmetic(EOS(STATIC_1104), i230, i233, i230)
1104_0_div_IntArithmetic(EOS(STATIC_1104), i230, i233, i230) → 1106_0_div_Store(EOS(STATIC_1106), i230, i233, i230)
1106_0_div_Store(EOS(STATIC_1106), i230, i233, i230) → 1107_0_div_JMP(EOS(STATIC_1107), i230, i233, i230)
1107_0_div_JMP(EOS(STATIC_1107), i230, i233, i230) → 1110_0_div_Load(EOS(STATIC_1110), i230, i233, i230)
1110_0_div_Load(EOS(STATIC_1110), i230, i233, i230) → 1014_0_div_Load(EOS(STATIC_1014), i230, i233, i230)
1014_0_div_Load(EOS(STATIC_1014), i220, i219, i220) → 1019_0_div_Load(EOS(STATIC_1019), i220, i219, i220, i219)
R rules:

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

P rules:
1083_0_minus_EQ(EOS(STATIC_1083), x0, x0, x1, x0, x2, x3, x3) → 1083_0_minus_EQ(EOS(STATIC_1083), x0, x0, x1, x0, +(x2, -1), +(x3, -1), +(x3, -1)) | >(x3, 0)
1083_0_minus_EQ(EOS(STATIC_1083), x0, x0, x1, x0, x2, 0, 0) → 1083_0_minus_EQ(EOS(STATIC_1083), x0, x0, x2, x0, x2, x0, x0) | &&(>=(x2, x0), >(x0, 0))
R rules:

Filtered ground terms:

1083_0_minus_EQ(x1, x2, x3, x4, x5, x6, x7, x8) → 1083_0_minus_EQ(x2, x3, x4, x5, x6, x7, x8)
EOS(x1) → EOS
Cond_1083_0_minus_EQ1(x1, x2, x3, x4, x5, x6, x7, x8, x9) → Cond_1083_0_minus_EQ1(x1, x3, x4, x5, x6, x7)
Cond_1083_0_minus_EQ(x1, x2, x3, x4, x5, x6, x7, x8, x9) → Cond_1083_0_minus_EQ(x1, x3, x4, x5, x6, x7, x8, x9)

Filtered duplicate args:

1083_0_minus_EQ(x1, x2, x3, x4, x5, x6, x7) → 1083_0_minus_EQ(x3, x4, x5, x7)
Cond_1083_0_minus_EQ(x1, x2, x3, x4, x5, x6, x7, x8) → Cond_1083_0_minus_EQ(x1, x4, x5, x6, x8)
Cond_1083_0_minus_EQ1(x1, x2, x3, x4, x5, x6) → Cond_1083_0_minus_EQ1(x1, x4, x5, x6)

Filtered unneeded arguments:

Cond_1083_0_minus_EQ(x1, x2, x3, x4, x5) → Cond_1083_0_minus_EQ(x1, x3, x4, x5)
1083_0_minus_EQ(x1, x2, x3, x4) → 1083_0_minus_EQ(x2, x3, x4)
Cond_1083_0_minus_EQ1(x1, x2, x3, x4) → Cond_1083_0_minus_EQ1(x1, x3, x4)

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

P rules:
1083_0_minus_EQ(x0, x2, x3) → 1083_0_minus_EQ(x0, +(x2, -1), +(x3, -1)) | >(x3, 0)
1083_0_minus_EQ(x0, x2, 0) → 1083_0_minus_EQ(x0, x2, x0) | &&(>=(x2, x0), >(x0, 0))
R rules:

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

P rules:
1083_0_MINUS_EQ(x0, x2, x3) → COND_1083_0_MINUS_EQ(>(x3, 0), x0, x2, x3)
COND_1083_0_MINUS_EQ(TRUE, x0, x2, x3) → 1083_0_MINUS_EQ(x0, +(x2, -1), +(x3, -1))
1083_0_MINUS_EQ(x0, x2, 0) → COND_1083_0_MINUS_EQ1(&&(>=(x2, x0), >(x0, 0)), x0, x2, 0)
COND_1083_0_MINUS_EQ1(TRUE, x0, x2, 0) → 1083_0_MINUS_EQ(x0, x2, x0)
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:

Integer, Boolean

R is empty.

The integer pair graph contains the following rules and edges:
(0): 1083_0_MINUS_EQ(x0[0], x2[0], x3[0]) → COND_1083_0_MINUS_EQ(x3[0] > 0, x0[0], x2[0], x3[0])
(1): COND_1083_0_MINUS_EQ(TRUE, x0[1], x2[1], x3[1]) → 1083_0_MINUS_EQ(x0[1], x2[1] + -1, x3[1] + -1)
(2): 1083_0_MINUS_EQ(x0[2], x2[2], 0) → COND_1083_0_MINUS_EQ1(x2[2] >= x0[2] && x0[2] > 0, x0[2], x2[2], 0)
(3): COND_1083_0_MINUS_EQ1(TRUE, x0[3], x2[3], 0) → 1083_0_MINUS_EQ(x0[3], x2[3], x0[3])

(0) -> (1), if (x3[0] > 0 ∧x0[0] →^* x0[1]∧x2[0] →^* x2[1]∧x3[0] →^* x3[1])

(1) -> (0), if (x0[1] →^* x0[0]∧x2[1] + -1 →^* x2[0]∧x3[1] + -1 →^* x3[0])

(1) -> (2), if (x0[1] →^* x0[2]∧x2[1] + -1 →^* x2[2]∧x3[1] + -1 →^* 0)

(2) -> (3), if (x2[2] >= x0[2] && x0[2] > 0 ∧x0[2] →^* x0[3]∧x2[2] →^* x2[3])

(3) -> (0), if (x0[3] →^* x0[0]∧x2[3] →^* x2[0]∧x0[3] →^* x3[0])

(3) -> (2), if (x0[3] →^* x0[2]∧x2[3] →^* x2[2]∧x0[3] →^* 0)

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@46192223 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 1083_0_MINUS_EQ(x0, x2, x3) → COND_1083_0_MINUS_EQ(>(x3, 0), x0, x2, x3) the following chains were created:

We consider the chain 1083_0_MINUS_EQ(x0[0], x2[0], x3[0]) → COND_1083_0_MINUS_EQ(>(x3[0], 0), x0[0], x2[0], x3[0]), COND_1083_0_MINUS_EQ(TRUE, x0[1], x2[1], x3[1]) → 1083_0_MINUS_EQ(x0[1], +(x2[1], -1), +(x3[1], -1)) which results in the following constraint:
(1)    (>(x3[0], 0)=TRUE∧x0[0]=x0[1]∧x2[0]=x2[1]∧x3[0]=x3[1] ⇒ 1083_0_MINUS_EQ(x0[0], x2[0], x3[0])≥NonInfC∧1083_0_MINUS_EQ(x0[0], x2[0], x3[0])≥COND_1083_0_MINUS_EQ(>(x3[0], 0), x0[0], x2[0], x3[0])∧(U^Increasing(COND_1083_0_MINUS_EQ(>(x3[0], 0), x0[0], x2[0], x3[0])), ≥))

We simplified constraint (1) using rule (IV) which results in the following new constraint:
(2)    (>(x3[0], 0)=TRUE ⇒ 1083_0_MINUS_EQ(x0[0], x2[0], x3[0])≥NonInfC∧1083_0_MINUS_EQ(x0[0], x2[0], x3[0])≥COND_1083_0_MINUS_EQ(>(x3[0], 0), x0[0], x2[0], x3[0])∧(U^Increasing(COND_1083_0_MINUS_EQ(>(x3[0], 0), x0[0], x2[0], x3[0])), ≥))

We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(3)    (x3[0] + [-1] ≥ 0 ⇒ (U^Increasing(COND_1083_0_MINUS_EQ(>(x3[0], 0), x0[0], x2[0], x3[0])), ≥)∧[(-1)bni_20 + (-1)Bound*bni_20] + [(-1)bni_20]x3[0] + [bni_20]x2[0] ≥ 0∧[(-1)bso_21] ≥ 0)

We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(4)    (x3[0] + [-1] ≥ 0 ⇒ (U^Increasing(COND_1083_0_MINUS_EQ(>(x3[0], 0), x0[0], x2[0], x3[0])), ≥)∧[(-1)bni_20 + (-1)Bound*bni_20] + [(-1)bni_20]x3[0] + [bni_20]x2[0] ≥ 0∧[(-1)bso_21] ≥ 0)

We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(5)    (x3[0] + [-1] ≥ 0 ⇒ (U^Increasing(COND_1083_0_MINUS_EQ(>(x3[0], 0), x0[0], x2[0], x3[0])), ≥)∧[(-1)bni_20 + (-1)Bound*bni_20] + [(-1)bni_20]x3[0] + [bni_20]x2[0] ≥ 0∧[(-1)bso_21] ≥ 0)

We simplified constraint (5) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint:
(6)    (x3[0] + [-1] ≥ 0 ⇒ (U^Increasing(COND_1083_0_MINUS_EQ(>(x3[0], 0), x0[0], x2[0], x3[0])), ≥)∧[bni_20] = 0∧0 = 0∧[(-1)bni_20 + (-1)Bound*bni_20] + [(-1)bni_20]x3[0] ≥ 0∧0 = 0∧0 = 0∧[(-1)bso_21] ≥ 0)

We simplified constraint (6) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(7)    (x3[0] ≥ 0 ⇒ (U^Increasing(COND_1083_0_MINUS_EQ(>(x3[0], 0), x0[0], x2[0], x3[0])), ≥)∧[bni_20] = 0∧0 = 0∧[(-2)bni_20 + (-1)Bound*bni_20] + [(-1)bni_20]x3[0] ≥ 0∧0 = 0∧0 = 0∧[(-1)bso_21] ≥ 0)

For Pair COND_1083_0_MINUS_EQ(TRUE, x0, x2, x3) → 1083_0_MINUS_EQ(x0, +(x2, -1), +(x3, -1)) the following chains were created:

We consider the chain 1083_0_MINUS_EQ(x0[0], x2[0], x3[0]) → COND_1083_0_MINUS_EQ(>(x3[0], 0), x0[0], x2[0], x3[0]), COND_1083_0_MINUS_EQ(TRUE, x0[1], x2[1], x3[1]) → 1083_0_MINUS_EQ(x0[1], +(x2[1], -1), +(x3[1], -1)), 1083_0_MINUS_EQ(x0[0], x2[0], x3[0]) → COND_1083_0_MINUS_EQ(>(x3[0], 0), x0[0], x2[0], x3[0]) which results in the following constraint:
(8)    (>(x3[0], 0)=TRUE∧x0[0]=x0[1]∧x2[0]=x2[1]∧x3[0]=x3[1]∧x0[1]=x0[0]1∧+(x2[1], -1)=x2[0]1∧+(x3[1], -1)=x3[0]1 ⇒ COND_1083_0_MINUS_EQ(TRUE, x0[1], x2[1], x3[1])≥NonInfC∧COND_1083_0_MINUS_EQ(TRUE, x0[1], x2[1], x3[1])≥1083_0_MINUS_EQ(x0[1], +(x2[1], -1), +(x3[1], -1))∧(U^Increasing(1083_0_MINUS_EQ(x0[1], +(x2[1], -1), +(x3[1], -1))), ≥))

We simplified constraint (8) using rules (III), (IV) which results in the following new constraint:
(9)    (>(x3[0], 0)=TRUE ⇒ COND_1083_0_MINUS_EQ(TRUE, x0[0], x2[0], x3[0])≥NonInfC∧COND_1083_0_MINUS_EQ(TRUE, x0[0], x2[0], x3[0])≥1083_0_MINUS_EQ(x0[0], +(x2[0], -1), +(x3[0], -1))∧(U^Increasing(1083_0_MINUS_EQ(x0[1], +(x2[1], -1), +(x3[1], -1))), ≥))

We simplified constraint (9) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(10)    (x3[0] + [-1] ≥ 0 ⇒ (U^Increasing(1083_0_MINUS_EQ(x0[1], +(x2[1], -1), +(x3[1], -1))), ≥)∧[(-1)bni_22 + (-1)Bound*bni_22] + [(-1)bni_22]x3[0] + [bni_22]x2[0] ≥ 0∧[(-1)bso_23] ≥ 0)

We simplified constraint (10) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(11)    (x3[0] + [-1] ≥ 0 ⇒ (U^Increasing(1083_0_MINUS_EQ(x0[1], +(x2[1], -1), +(x3[1], -1))), ≥)∧[(-1)bni_22 + (-1)Bound*bni_22] + [(-1)bni_22]x3[0] + [bni_22]x2[0] ≥ 0∧[(-1)bso_23] ≥ 0)

We simplified constraint (11) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(12)    (x3[0] + [-1] ≥ 0 ⇒ (U^Increasing(1083_0_MINUS_EQ(x0[1], +(x2[1], -1), +(x3[1], -1))), ≥)∧[(-1)bni_22 + (-1)Bound*bni_22] + [(-1)bni_22]x3[0] + [bni_22]x2[0] ≥ 0∧[(-1)bso_23] ≥ 0)

We simplified constraint (12) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint:
(13)    (x3[0] + [-1] ≥ 0 ⇒ (U^Increasing(1083_0_MINUS_EQ(x0[1], +(x2[1], -1), +(x3[1], -1))), ≥)∧[bni_22] = 0∧0 = 0∧[(-1)bni_22 + (-1)Bound*bni_22] + [(-1)bni_22]x3[0] ≥ 0∧0 = 0∧0 = 0∧[(-1)bso_23] ≥ 0)

We simplified constraint (13) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(14)    (x3[0] ≥ 0 ⇒ (U^Increasing(1083_0_MINUS_EQ(x0[1], +(x2[1], -1), +(x3[1], -1))), ≥)∧[bni_22] = 0∧0 = 0∧[(-2)bni_22 + (-1)Bound*bni_22] + [(-1)bni_22]x3[0] ≥ 0∧0 = 0∧0 = 0∧[(-1)bso_23] ≥ 0)
We consider the chain 1083_0_MINUS_EQ(x0[0], x2[0], x3[0]) → COND_1083_0_MINUS_EQ(>(x3[0], 0), x0[0], x2[0], x3[0]), COND_1083_0_MINUS_EQ(TRUE, x0[1], x2[1], x3[1]) → 1083_0_MINUS_EQ(x0[1], +(x2[1], -1), +(x3[1], -1)), 1083_0_MINUS_EQ(x0[2], x2[2], 0) → COND_1083_0_MINUS_EQ1(&&(>=(x2[2], x0[2]), >(x0[2], 0)), x0[2], x2[2], 0) which results in the following constraint:
(15)    (>(x3[0], 0)=TRUE∧x0[0]=x0[1]∧x2[0]=x2[1]∧x3[0]=x3[1]∧x0[1]=x0[2]∧+(x2[1], -1)=x2[2]∧+(x3[1], -1)=0 ⇒ COND_1083_0_MINUS_EQ(TRUE, x0[1], x2[1], x3[1])≥NonInfC∧COND_1083_0_MINUS_EQ(TRUE, x0[1], x2[1], x3[1])≥1083_0_MINUS_EQ(x0[1], +(x2[1], -1), +(x3[1], -1))∧(U^Increasing(1083_0_MINUS_EQ(x0[1], +(x2[1], -1), +(x3[1], -1))), ≥))

We simplified constraint (15) using rules (III), (IV) which results in the following new constraint:
(16)    (>(x3[0], 0)=TRUE∧+(x3[0], -1)=0 ⇒ COND_1083_0_MINUS_EQ(TRUE, x0[0], x2[0], x3[0])≥NonInfC∧COND_1083_0_MINUS_EQ(TRUE, x0[0], x2[0], x3[0])≥1083_0_MINUS_EQ(x0[0], +(x2[0], -1), +(x3[0], -1))∧(U^Increasing(1083_0_MINUS_EQ(x0[1], +(x2[1], -1), +(x3[1], -1))), ≥))

We simplified constraint (16) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(17)    (x3[0] + [-1] ≥ 0∧x3[0] + [-1] ≥ 0 ⇒ (U^Increasing(1083_0_MINUS_EQ(x0[1], +(x2[1], -1), +(x3[1], -1))), ≥)∧[(-1)bni_22 + (-1)Bound*bni_22] + [(-1)bni_22]x3[0] + [bni_22]x2[0] ≥ 0∧[(-1)bso_23] ≥ 0)

We simplified constraint (17) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(18)    (x3[0] + [-1] ≥ 0∧x3[0] + [-1] ≥ 0 ⇒ (U^Increasing(1083_0_MINUS_EQ(x0[1], +(x2[1], -1), +(x3[1], -1))), ≥)∧[(-1)bni_22 + (-1)Bound*bni_22] + [(-1)bni_22]x3[0] + [bni_22]x2[0] ≥ 0∧[(-1)bso_23] ≥ 0)

We simplified constraint (18) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(19)    (x3[0] + [-1] ≥ 0∧x3[0] + [-1] ≥ 0 ⇒ (U^Increasing(1083_0_MINUS_EQ(x0[1], +(x2[1], -1), +(x3[1], -1))), ≥)∧[(-1)bni_22 + (-1)Bound*bni_22] + [(-1)bni_22]x3[0] + [bni_22]x2[0] ≥ 0∧[(-1)bso_23] ≥ 0)

We simplified constraint (19) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint:
(20)    (x3[0] + [-1] ≥ 0∧x3[0] + [-1] ≥ 0 ⇒ (U^Increasing(1083_0_MINUS_EQ(x0[1], +(x2[1], -1), +(x3[1], -1))), ≥)∧[bni_22] = 0∧0 = 0∧[(-1)bni_22 + (-1)Bound*bni_22] + [(-1)bni_22]x3[0] ≥ 0∧0 = 0∧0 = 0∧[(-1)bso_23] ≥ 0)

We simplified constraint (20) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(21)    (x3[0] ≥ 0∧x3[0] ≥ 0 ⇒ (U^Increasing(1083_0_MINUS_EQ(x0[1], +(x2[1], -1), +(x3[1], -1))), ≥)∧[bni_22] = 0∧0 = 0∧[(-2)bni_22 + (-1)Bound*bni_22] + [(-1)bni_22]x3[0] ≥ 0∧0 = 0∧0 = 0∧[(-1)bso_23] ≥ 0)

For Pair 1083_0_MINUS_EQ(x0, x2, 0) → COND_1083_0_MINUS_EQ1(&&(>=(x2, x0), >(x0, 0)), x0, x2, 0) the following chains were created:

We consider the chain 1083_0_MINUS_EQ(x0[2], x2[2], 0) → COND_1083_0_MINUS_EQ1(&&(>=(x2[2], x0[2]), >(x0[2], 0)), x0[2], x2[2], 0), COND_1083_0_MINUS_EQ1(TRUE, x0[3], x2[3], 0) → 1083_0_MINUS_EQ(x0[3], x2[3], x0[3]) which results in the following constraint:
(22)    (&&(>=(x2[2], x0[2]), >(x0[2], 0))=TRUE∧x0[2]=x0[3]∧x2[2]=x2[3] ⇒ 1083_0_MINUS_EQ(x0[2], x2[2], 0)≥NonInfC∧1083_0_MINUS_EQ(x0[2], x2[2], 0)≥COND_1083_0_MINUS_EQ1(&&(>=(x2[2], x0[2]), >(x0[2], 0)), x0[2], x2[2], 0)∧(U^Increasing(COND_1083_0_MINUS_EQ1(&&(>=(x2[2], x0[2]), >(x0[2], 0)), x0[2], x2[2], 0)), ≥))

We simplified constraint (22) using rules (IV), (IDP_BOOLEAN) which results in the following new constraint:
(23)    (>=(x2[2], x0[2])=TRUE∧>(x0[2], 0)=TRUE ⇒ 1083_0_MINUS_EQ(x0[2], x2[2], 0)≥NonInfC∧1083_0_MINUS_EQ(x0[2], x2[2], 0)≥COND_1083_0_MINUS_EQ1(&&(>=(x2[2], x0[2]), >(x0[2], 0)), x0[2], x2[2], 0)∧(U^Increasing(COND_1083_0_MINUS_EQ1(&&(>=(x2[2], x0[2]), >(x0[2], 0)), x0[2], x2[2], 0)), ≥))

We simplified constraint (23) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(24)    (x2[2] + [-1]x0[2] ≥ 0∧x0[2] + [-1] ≥ 0 ⇒ (U^Increasing(COND_1083_0_MINUS_EQ1(&&(>=(x2[2], x0[2]), >(x0[2], 0)), x0[2], x2[2], 0)), ≥)∧[(-1)bni_24 + (-1)Bound*bni_24] + [bni_24]x2[2] ≥ 0∧[-1 + (-1)bso_25] + x0[2] ≥ 0)

We simplified constraint (24) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(25)    (x2[2] + [-1]x0[2] ≥ 0∧x0[2] + [-1] ≥ 0 ⇒ (U^Increasing(COND_1083_0_MINUS_EQ1(&&(>=(x2[2], x0[2]), >(x0[2], 0)), x0[2], x2[2], 0)), ≥)∧[(-1)bni_24 + (-1)Bound*bni_24] + [bni_24]x2[2] ≥ 0∧[-1 + (-1)bso_25] + x0[2] ≥ 0)

We simplified constraint (25) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(26)    (x2[2] + [-1]x0[2] ≥ 0∧x0[2] + [-1] ≥ 0 ⇒ (U^Increasing(COND_1083_0_MINUS_EQ1(&&(>=(x2[2], x0[2]), >(x0[2], 0)), x0[2], x2[2], 0)), ≥)∧[(-1)bni_24 + (-1)Bound*bni_24] + [bni_24]x2[2] ≥ 0∧[-1 + (-1)bso_25] + x0[2] ≥ 0)

We simplified constraint (26) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(27)    (x2[2] ≥ 0∧x0[2] + [-1] ≥ 0 ⇒ (U^Increasing(COND_1083_0_MINUS_EQ1(&&(>=(x2[2], x0[2]), >(x0[2], 0)), x0[2], x2[2], 0)), ≥)∧[(-1)bni_24 + (-1)Bound*bni_24] + [bni_24]x0[2] + [bni_24]x2[2] ≥ 0∧[-1 + (-1)bso_25] + x0[2] ≥ 0)

We simplified constraint (27) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(28)    (x2[2] ≥ 0∧x0[2] ≥ 0 ⇒ (U^Increasing(COND_1083_0_MINUS_EQ1(&&(>=(x2[2], x0[2]), >(x0[2], 0)), x0[2], x2[2], 0)), ≥)∧[(-1)Bound*bni_24] + [bni_24]x0[2] + [bni_24]x2[2] ≥ 0∧[(-1)bso_25] + x0[2] ≥ 0)

For Pair COND_1083_0_MINUS_EQ1(TRUE, x0, x2, 0) → 1083_0_MINUS_EQ(x0, x2, x0) the following chains were created:

We consider the chain COND_1083_0_MINUS_EQ1(TRUE, x0[3], x2[3], 0) → 1083_0_MINUS_EQ(x0[3], x2[3], x0[3]), 1083_0_MINUS_EQ(x0[0], x2[0], x3[0]) → COND_1083_0_MINUS_EQ(>(x3[0], 0), x0[0], x2[0], x3[0]) which results in the following constraint:
(29)    (x0[3]=x0[0]∧x2[3]=x2[0]∧x0[3]=x3[0] ⇒ COND_1083_0_MINUS_EQ1(TRUE, x0[3], x2[3], 0)≥NonInfC∧COND_1083_0_MINUS_EQ1(TRUE, x0[3], x2[3], 0)≥1083_0_MINUS_EQ(x0[3], x2[3], x0[3])∧(U^Increasing(1083_0_MINUS_EQ(x0[3], x2[3], x0[3])), ≥))

We simplified constraint (29) using rule (IV) which results in the following new constraint:
(30)    (COND_1083_0_MINUS_EQ1(TRUE, x0[3], x2[3], 0)≥NonInfC∧COND_1083_0_MINUS_EQ1(TRUE, x0[3], x2[3], 0)≥1083_0_MINUS_EQ(x0[3], x2[3], x0[3])∧(U^Increasing(1083_0_MINUS_EQ(x0[3], x2[3], x0[3])), ≥))

We simplified constraint (30) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(31)    ((U^Increasing(1083_0_MINUS_EQ(x0[3], x2[3], x0[3])), ≥)∧[bni_26] = 0∧[1 + (-1)bso_27] ≥ 0)

We simplified constraint (31) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(32)    ((U^Increasing(1083_0_MINUS_EQ(x0[3], x2[3], x0[3])), ≥)∧[bni_26] = 0∧[1 + (-1)bso_27] ≥ 0)

We simplified constraint (32) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(33)    ((U^Increasing(1083_0_MINUS_EQ(x0[3], x2[3], x0[3])), ≥)∧[bni_26] = 0∧[1 + (-1)bso_27] ≥ 0)

We simplified constraint (33) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint:
(34)    ((U^Increasing(1083_0_MINUS_EQ(x0[3], x2[3], x0[3])), ≥)∧[bni_26] = 0∧0 = 0∧0 = 0∧[1 + (-1)bso_27] ≥ 0)
We consider the chain COND_1083_0_MINUS_EQ1(TRUE, x0[3], x2[3], 0) → 1083_0_MINUS_EQ(x0[3], x2[3], x0[3]), 1083_0_MINUS_EQ(x0[2], x2[2], 0) → COND_1083_0_MINUS_EQ1(&&(>=(x2[2], x0[2]), >(x0[2], 0)), x0[2], x2[2], 0) which results in the following constraint:
(35)    (x0[3]=x0[2]∧x2[3]=x2[2]∧x0[3]=0 ⇒ COND_1083_0_MINUS_EQ1(TRUE, x0[3], x2[3], 0)≥NonInfC∧COND_1083_0_MINUS_EQ1(TRUE, x0[3], x2[3], 0)≥1083_0_MINUS_EQ(x0[3], x2[3], x0[3])∧(U^Increasing(1083_0_MINUS_EQ(x0[3], x2[3], x0[3])), ≥))

We simplified constraint (35) using rules (III), (IV) which results in the following new constraint:
(36)    (COND_1083_0_MINUS_EQ1(TRUE, 0, x2[3], 0)≥NonInfC∧COND_1083_0_MINUS_EQ1(TRUE, 0, x2[3], 0)≥1083_0_MINUS_EQ(0, x2[3], 0)∧(U^Increasing(1083_0_MINUS_EQ(x0[3], x2[3], x0[3])), ≥))

We simplified constraint (36) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(37)    ((U^Increasing(1083_0_MINUS_EQ(x0[3], x2[3], x0[3])), ≥)∧[bni_26] = 0∧[1 + (-1)bso_27] ≥ 0)

We simplified constraint (37) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(38)    ((U^Increasing(1083_0_MINUS_EQ(x0[3], x2[3], x0[3])), ≥)∧[bni_26] = 0∧[1 + (-1)bso_27] ≥ 0)

We simplified constraint (38) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(39)    ((U^Increasing(1083_0_MINUS_EQ(x0[3], x2[3], x0[3])), ≥)∧[bni_26] = 0∧[1 + (-1)bso_27] ≥ 0)

We simplified constraint (39) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint:
(40)    ((U^Increasing(1083_0_MINUS_EQ(x0[3], x2[3], x0[3])), ≥)∧[bni_26] = 0∧0 = 0∧[1 + (-1)bso_27] ≥ 0)

To summarize, we get the following constraints P_≥ for the following pairs.

1083_0_MINUS_EQ(x0, x2, x3) → COND_1083_0_MINUS_EQ(>(x3, 0), x0, x2, x3)
- (x3[0] ≥ 0 ⇒ (U^Increasing(COND_1083_0_MINUS_EQ(>(x3[0], 0), x0[0], x2[0], x3[0])), ≥)∧[bni_20] = 0∧0 = 0∧[(-2)bni_20 + (-1)Bound*bni_20] + [(-1)bni_20]x3[0] ≥ 0∧0 = 0∧0 = 0∧[(-1)bso_21] ≥ 0)
COND_1083_0_MINUS_EQ(TRUE, x0, x2, x3) → 1083_0_MINUS_EQ(x0, +(x2, -1), +(x3, -1))
- (x3[0] ≥ 0 ⇒ (U^Increasing(1083_0_MINUS_EQ(x0[1], +(x2[1], -1), +(x3[1], -1))), ≥)∧[bni_22] = 0∧0 = 0∧[(-2)bni_22 + (-1)Bound*bni_22] + [(-1)bni_22]x3[0] ≥ 0∧0 = 0∧0 = 0∧[(-1)bso_23] ≥ 0)
- (x3[0] ≥ 0∧x3[0] ≥ 0 ⇒ (U^Increasing(1083_0_MINUS_EQ(x0[1], +(x2[1], -1), +(x3[1], -1))), ≥)∧[bni_22] = 0∧0 = 0∧[(-2)bni_22 + (-1)Bound*bni_22] + [(-1)bni_22]x3[0] ≥ 0∧0 = 0∧0 = 0∧[(-1)bso_23] ≥ 0)
1083_0_MINUS_EQ(x0, x2, 0) → COND_1083_0_MINUS_EQ1(&&(>=(x2, x0), >(x0, 0)), x0, x2, 0)
- (x2[2] ≥ 0∧x0[2] ≥ 0 ⇒ (U^Increasing(COND_1083_0_MINUS_EQ1(&&(>=(x2[2], x0[2]), >(x0[2], 0)), x0[2], x2[2], 0)), ≥)∧[(-1)Bound*bni_24] + [bni_24]x0[2] + [bni_24]x2[2] ≥ 0∧[(-1)bso_25] + x0[2] ≥ 0)
COND_1083_0_MINUS_EQ1(TRUE, x0, x2, 0) → 1083_0_MINUS_EQ(x0, x2, x0)
- ((U^Increasing(1083_0_MINUS_EQ(x0[3], x2[3], x0[3])), ≥)∧[bni_26] = 0∧0 = 0∧0 = 0∧[1 + (-1)bso_27] ≥ 0)
- ((U^Increasing(1083_0_MINUS_EQ(x0[3], x2[3], x0[3])), ≥)∧[bni_26] = 0∧0 = 0∧[1 + (-1)bso_27] ≥ 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) = 0
POL(1083_0_MINUS_EQ(x₁, x₂, x₃)) = [-1] + [-1]x₃ + x₂
POL(COND_1083_0_MINUS_EQ(x₁, x₂, x₃, x₄)) = [-1] + [-1]x₄ + x₃
POL(>(x₁, x₂)) = [-1]
POL(0) = 0
POL(+(x₁, x₂)) = x₁ + x₂
POL(-1) = [-1]
POL(COND_1083_0_MINUS_EQ1(x₁, x₂, x₃, x₄)) = [-1]x₄ + x₃ + [-1]x₂ + [-1]x₁
POL(&&(x₁, x₂)) = 0
POL(>=(x₁, x₂)) = [-1]

The following pairs are in P_>:

COND_1083_0_MINUS_EQ1(TRUE, x0[3], x2[3], 0) → 1083_0_MINUS_EQ(x0[3], x2[3], x0[3])

The following pairs are in P_bound:

1083_0_MINUS_EQ(x0[2], x2[2], 0) → COND_1083_0_MINUS_EQ1(&&(>=(x2[2], x0[2]), >(x0[2], 0)), x0[2], x2[2], 0)

The following pairs are in P_≥:

1083_0_MINUS_EQ(x0[0], x2[0], x3[0]) → COND_1083_0_MINUS_EQ(>(x3[0], 0), x0[0], x2[0], x3[0])
COND_1083_0_MINUS_EQ(TRUE, x0[1], x2[1], x3[1]) → 1083_0_MINUS_EQ(x0[1], +(x2[1], -1), +(x3[1], -1))
1083_0_MINUS_EQ(x0[2], x2[2], 0) → COND_1083_0_MINUS_EQ1(&&(>=(x2[2], x0[2]), >(x0[2], 0)), x0[2], x2[2], 0)

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

(8) Complex Obligation (AND)

(9) 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:

Integer, Boolean

(1) -> (0), if (x0[1] →^* x0[0]∧x2[1] + -1 →^* x2[0]∧x3[1] + -1 →^* x3[0])

(0) -> (1), if (x3[0] > 0 ∧x0[0] →^* x0[1]∧x2[0] →^* x2[1]∧x3[0] →^* x3[1])

(1) -> (2), if (x0[1] →^* x0[2]∧x2[1] + -1 →^* x2[2]∧x3[1] + -1 →^* 0)

The set Q is empty.

(10) IDependencyGraphProof (EQUIVALENT transformation)

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

(11) 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:

Integer

R is empty.

The integer pair graph contains the following rules and edges:
(1): COND_1083_0_MINUS_EQ(TRUE, x0[1], x2[1], x3[1]) → 1083_0_MINUS_EQ(x0[1], x2[1] + -1, x3[1] + -1)
(0): 1083_0_MINUS_EQ(x0[0], x2[0], x3[0]) → COND_1083_0_MINUS_EQ(x3[0] > 0, x0[0], x2[0], x3[0])

(1) -> (0), if (x0[1] →^* x0[0]∧x2[1] + -1 →^* x2[0]∧x3[1] + -1 →^* x3[0])

(0) -> (1), if (x3[0] > 0 ∧x0[0] →^* x0[1]∧x2[0] →^* x2[1]∧x3[0] →^* x3[1])

The set Q is empty.

(12) 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@46192223 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_1083_0_MINUS_EQ(TRUE, x0[1], x2[1], x3[1]) → 1083_0_MINUS_EQ(x0[1], +(x2[1], -1), +(x3[1], -1)) the following chains were created:

We consider the chain 1083_0_MINUS_EQ(x0[0], x2[0], x3[0]) → COND_1083_0_MINUS_EQ(>(x3[0], 0), x0[0], x2[0], x3[0]), COND_1083_0_MINUS_EQ(TRUE, x0[1], x2[1], x3[1]) → 1083_0_MINUS_EQ(x0[1], +(x2[1], -1), +(x3[1], -1)), 1083_0_MINUS_EQ(x0[0], x2[0], x3[0]) → COND_1083_0_MINUS_EQ(>(x3[0], 0), x0[0], x2[0], x3[0]) which results in the following constraint:
(1)    (>(x3[0], 0)=TRUE∧x0[0]=x0[1]∧x2[0]=x2[1]∧x3[0]=x3[1]∧x0[1]=x0[0]1∧+(x2[1], -1)=x2[0]1∧+(x3[1], -1)=x3[0]1 ⇒ COND_1083_0_MINUS_EQ(TRUE, x0[1], x2[1], x3[1])≥NonInfC∧COND_1083_0_MINUS_EQ(TRUE, x0[1], x2[1], x3[1])≥1083_0_MINUS_EQ(x0[1], +(x2[1], -1), +(x3[1], -1))∧(U^Increasing(1083_0_MINUS_EQ(x0[1], +(x2[1], -1), +(x3[1], -1))), ≥))

We simplified constraint (1) using rules (III), (IV) which results in the following new constraint:
(2)    (>(x3[0], 0)=TRUE ⇒ COND_1083_0_MINUS_EQ(TRUE, x0[0], x2[0], x3[0])≥NonInfC∧COND_1083_0_MINUS_EQ(TRUE, x0[0], x2[0], x3[0])≥1083_0_MINUS_EQ(x0[0], +(x2[0], -1), +(x3[0], -1))∧(U^Increasing(1083_0_MINUS_EQ(x0[1], +(x2[1], -1), +(x3[1], -1))), ≥))

We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(3)    (x3[0] + [-1] ≥ 0 ⇒ (U^Increasing(1083_0_MINUS_EQ(x0[1], +(x2[1], -1), +(x3[1], -1))), ≥)∧[(-1)bni_13 + (-1)Bound*bni_13] + [bni_13]x3[0] ≥ 0∧[1 + (-1)bso_14] ≥ 0)

We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(4)    (x3[0] + [-1] ≥ 0 ⇒ (U^Increasing(1083_0_MINUS_EQ(x0[1], +(x2[1], -1), +(x3[1], -1))), ≥)∧[(-1)bni_13 + (-1)Bound*bni_13] + [bni_13]x3[0] ≥ 0∧[1 + (-1)bso_14] ≥ 0)

We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(5)    (x3[0] + [-1] ≥ 0 ⇒ (U^Increasing(1083_0_MINUS_EQ(x0[1], +(x2[1], -1), +(x3[1], -1))), ≥)∧[(-1)bni_13 + (-1)Bound*bni_13] + [bni_13]x3[0] ≥ 0∧[1 + (-1)bso_14] ≥ 0)

We simplified constraint (5) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint:
(6)    (x3[0] + [-1] ≥ 0 ⇒ (U^Increasing(1083_0_MINUS_EQ(x0[1], +(x2[1], -1), +(x3[1], -1))), ≥)∧0 = 0∧0 = 0∧[(-1)bni_13 + (-1)Bound*bni_13] + [bni_13]x3[0] ≥ 0∧0 = 0∧0 = 0∧[1 + (-1)bso_14] ≥ 0)

We simplified constraint (6) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(7)    (x3[0] ≥ 0 ⇒ (U^Increasing(1083_0_MINUS_EQ(x0[1], +(x2[1], -1), +(x3[1], -1))), ≥)∧0 = 0∧0 = 0∧[(-1)Bound*bni_13] + [bni_13]x3[0] ≥ 0∧0 = 0∧0 = 0∧[1 + (-1)bso_14] ≥ 0)

For Pair 1083_0_MINUS_EQ(x0[0], x2[0], x3[0]) → COND_1083_0_MINUS_EQ(>(x3[0], 0), x0[0], x2[0], x3[0]) the following chains were created:

We consider the chain 1083_0_MINUS_EQ(x0[0], x2[0], x3[0]) → COND_1083_0_MINUS_EQ(>(x3[0], 0), x0[0], x2[0], x3[0]), COND_1083_0_MINUS_EQ(TRUE, x0[1], x2[1], x3[1]) → 1083_0_MINUS_EQ(x0[1], +(x2[1], -1), +(x3[1], -1)) which results in the following constraint:
(8)    (>(x3[0], 0)=TRUE∧x0[0]=x0[1]∧x2[0]=x2[1]∧x3[0]=x3[1] ⇒ 1083_0_MINUS_EQ(x0[0], x2[0], x3[0])≥NonInfC∧1083_0_MINUS_EQ(x0[0], x2[0], x3[0])≥COND_1083_0_MINUS_EQ(>(x3[0], 0), x0[0], x2[0], x3[0])∧(U^Increasing(COND_1083_0_MINUS_EQ(>(x3[0], 0), x0[0], x2[0], x3[0])), ≥))

We simplified constraint (8) using rule (IV) which results in the following new constraint:
(9)    (>(x3[0], 0)=TRUE ⇒ 1083_0_MINUS_EQ(x0[0], x2[0], x3[0])≥NonInfC∧1083_0_MINUS_EQ(x0[0], x2[0], x3[0])≥COND_1083_0_MINUS_EQ(>(x3[0], 0), x0[0], x2[0], x3[0])∧(U^Increasing(COND_1083_0_MINUS_EQ(>(x3[0], 0), x0[0], x2[0], x3[0])), ≥))

We simplified constraint (9) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(10)    (x3[0] + [-1] ≥ 0 ⇒ (U^Increasing(COND_1083_0_MINUS_EQ(>(x3[0], 0), x0[0], x2[0], x3[0])), ≥)∧[(-1)bni_15 + (-1)Bound*bni_15] + [bni_15]x3[0] ≥ 0∧[(-1)bso_16] ≥ 0)

We simplified constraint (10) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(11)    (x3[0] + [-1] ≥ 0 ⇒ (U^Increasing(COND_1083_0_MINUS_EQ(>(x3[0], 0), x0[0], x2[0], x3[0])), ≥)∧[(-1)bni_15 + (-1)Bound*bni_15] + [bni_15]x3[0] ≥ 0∧[(-1)bso_16] ≥ 0)

We simplified constraint (11) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(12)    (x3[0] + [-1] ≥ 0 ⇒ (U^Increasing(COND_1083_0_MINUS_EQ(>(x3[0], 0), x0[0], x2[0], x3[0])), ≥)∧[(-1)bni_15 + (-1)Bound*bni_15] + [bni_15]x3[0] ≥ 0∧[(-1)bso_16] ≥ 0)

We simplified constraint (12) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint:
(13)    (x3[0] + [-1] ≥ 0 ⇒ (U^Increasing(COND_1083_0_MINUS_EQ(>(x3[0], 0), x0[0], x2[0], x3[0])), ≥)∧0 = 0∧0 = 0∧[(-1)bni_15 + (-1)Bound*bni_15] + [bni_15]x3[0] ≥ 0∧0 = 0∧0 = 0∧[(-1)bso_16] ≥ 0)

We simplified constraint (13) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(14)    (x3[0] ≥ 0 ⇒ (U^Increasing(COND_1083_0_MINUS_EQ(>(x3[0], 0), x0[0], x2[0], x3[0])), ≥)∧0 = 0∧0 = 0∧[(-1)Bound*bni_15] + [bni_15]x3[0] ≥ 0∧0 = 0∧0 = 0∧[(-1)bso_16] ≥ 0)

To summarize, we get the following constraints P_≥ for the following pairs.

COND_1083_0_MINUS_EQ(TRUE, x0[1], x2[1], x3[1]) → 1083_0_MINUS_EQ(x0[1], +(x2[1], -1), +(x3[1], -1))
- (x3[0] ≥ 0 ⇒ (U^Increasing(1083_0_MINUS_EQ(x0[1], +(x2[1], -1), +(x3[1], -1))), ≥)∧0 = 0∧0 = 0∧[(-1)Bound*bni_13] + [bni_13]x3[0] ≥ 0∧0 = 0∧0 = 0∧[1 + (-1)bso_14] ≥ 0)
1083_0_MINUS_EQ(x0[0], x2[0], x3[0]) → COND_1083_0_MINUS_EQ(>(x3[0], 0), x0[0], x2[0], x3[0])
- (x3[0] ≥ 0 ⇒ (U^Increasing(COND_1083_0_MINUS_EQ(>(x3[0], 0), x0[0], x2[0], x3[0])), ≥)∧0 = 0∧0 = 0∧[(-1)Bound*bni_15] + [bni_15]x3[0] ≥ 0∧0 = 0∧0 = 0∧[(-1)bso_16] ≥ 0)

POL(TRUE) = 0
POL(FALSE) = 0
POL(COND_1083_0_MINUS_EQ(x₁, x₂, x₃, x₄)) = [-1] + x₄
POL(1083_0_MINUS_EQ(x₁, x₂, x₃)) = [-1] + x₃
POL(+(x₁, x₂)) = x₁ + x₂
POL(-1) = [-1]
POL(>(x₁, x₂)) = [-1]
POL(0) = 0

The following pairs are in P_>:

COND_1083_0_MINUS_EQ(TRUE, x0[1], x2[1], x3[1]) → 1083_0_MINUS_EQ(x0[1], +(x2[1], -1), +(x3[1], -1))

The following pairs are in P_bound:

COND_1083_0_MINUS_EQ(TRUE, x0[1], x2[1], x3[1]) → 1083_0_MINUS_EQ(x0[1], +(x2[1], -1), +(x3[1], -1))
1083_0_MINUS_EQ(x0[0], x2[0], x3[0]) → COND_1083_0_MINUS_EQ(>(x3[0], 0), x0[0], x2[0], x3[0])

The following pairs are in P_≥:

1083_0_MINUS_EQ(x0[0], x2[0], x3[0]) → COND_1083_0_MINUS_EQ(>(x3[0], 0), x0[0], x2[0], x3[0])

There are no usable rules.

(13) 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:

Integer

R is empty.

The integer pair graph contains the following rules and edges:
(0): 1083_0_MINUS_EQ(x0[0], x2[0], x3[0]) → COND_1083_0_MINUS_EQ(x3[0] > 0, x0[0], x2[0], x3[0])

The set Q is empty.

(14) IDependencyGraphProof (EQUIVALENT transformation)

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

(15) TRUE

(16) 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:

Integer

R is empty.

The integer pair graph contains the following rules and edges:
(0): 1083_0_MINUS_EQ(x0[0], x2[0], x3[0]) → COND_1083_0_MINUS_EQ(x3[0] > 0, x0[0], x2[0], x3[0])
(1): COND_1083_0_MINUS_EQ(TRUE, x0[1], x2[1], x3[1]) → 1083_0_MINUS_EQ(x0[1], x2[1] + -1, x3[1] + -1)
(3): COND_1083_0_MINUS_EQ1(TRUE, x0[3], x2[3], 0) → 1083_0_MINUS_EQ(x0[3], x2[3], x0[3])

(1) -> (0), if (x0[1] →^* x0[0]∧x2[1] + -1 →^* x2[0]∧x3[1] + -1 →^* x3[0])

(3) -> (0), if (x0[3] →^* x0[0]∧x2[3] →^* x2[0]∧x0[3] →^* x3[0])

(0) -> (1), if (x3[0] > 0 ∧x0[0] →^* x0[1]∧x2[0] →^* x2[1]∧x3[0] →^* x3[1])

The set Q is empty.

(17) IDependencyGraphProof (EQUIVALENT transformation)

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

(18) 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:

Integer

(1) -> (0), if (x0[1] →^* x0[0]∧x2[1] + -1 →^* x2[0]∧x3[1] + -1 →^* x3[0])

(0) -> (1), if (x3[0] > 0 ∧x0[0] →^* x0[1]∧x2[0] →^* x2[1]∧x3[0] →^* x3[1])

The set Q is empty.

(19) 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@46192223 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_1083_0_MINUS_EQ(TRUE, x0[1], x2[1], x3[1]) → 1083_0_MINUS_EQ(x0[1], +(x2[1], -1), +(x3[1], -1)) the following chains were created:

We consider the chain 1083_0_MINUS_EQ(x0[0], x2[0], x3[0]) → COND_1083_0_MINUS_EQ(>(x3[0], 0), x0[0], x2[0], x3[0]), COND_1083_0_MINUS_EQ(TRUE, x0[1], x2[1], x3[1]) → 1083_0_MINUS_EQ(x0[1], +(x2[1], -1), +(x3[1], -1)), 1083_0_MINUS_EQ(x0[0], x2[0], x3[0]) → COND_1083_0_MINUS_EQ(>(x3[0], 0), x0[0], x2[0], x3[0]) which results in the following constraint:
(1)    (>(x3[0], 0)=TRUE∧x0[0]=x0[1]∧x2[0]=x2[1]∧x3[0]=x3[1]∧x0[1]=x0[0]1∧+(x2[1], -1)=x2[0]1∧+(x3[1], -1)=x3[0]1 ⇒ COND_1083_0_MINUS_EQ(TRUE, x0[1], x2[1], x3[1])≥NonInfC∧COND_1083_0_MINUS_EQ(TRUE, x0[1], x2[1], x3[1])≥1083_0_MINUS_EQ(x0[1], +(x2[1], -1), +(x3[1], -1))∧(U^Increasing(1083_0_MINUS_EQ(x0[1], +(x2[1], -1), +(x3[1], -1))), ≥))

We simplified constraint (1) using rules (III), (IV) which results in the following new constraint:
(2)    (>(x3[0], 0)=TRUE ⇒ COND_1083_0_MINUS_EQ(TRUE, x0[0], x2[0], x3[0])≥NonInfC∧COND_1083_0_MINUS_EQ(TRUE, x0[0], x2[0], x3[0])≥1083_0_MINUS_EQ(x0[0], +(x2[0], -1), +(x3[0], -1))∧(U^Increasing(1083_0_MINUS_EQ(x0[1], +(x2[1], -1), +(x3[1], -1))), ≥))

We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(3)    (x3[0] + [-1] ≥ 0 ⇒ (U^Increasing(1083_0_MINUS_EQ(x0[1], +(x2[1], -1), +(x3[1], -1))), ≥)∧[(-1)bni_13 + (-1)Bound*bni_13] + [bni_13]x3[0] ≥ 0∧[1 + (-1)bso_14] ≥ 0)

We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(4)    (x3[0] + [-1] ≥ 0 ⇒ (U^Increasing(1083_0_MINUS_EQ(x0[1], +(x2[1], -1), +(x3[1], -1))), ≥)∧[(-1)bni_13 + (-1)Bound*bni_13] + [bni_13]x3[0] ≥ 0∧[1 + (-1)bso_14] ≥ 0)

We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(5)    (x3[0] + [-1] ≥ 0 ⇒ (U^Increasing(1083_0_MINUS_EQ(x0[1], +(x2[1], -1), +(x3[1], -1))), ≥)∧[(-1)bni_13 + (-1)Bound*bni_13] + [bni_13]x3[0] ≥ 0∧[1 + (-1)bso_14] ≥ 0)

We simplified constraint (5) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint:
(6)    (x3[0] + [-1] ≥ 0 ⇒ (U^Increasing(1083_0_MINUS_EQ(x0[1], +(x2[1], -1), +(x3[1], -1))), ≥)∧0 = 0∧0 = 0∧[(-1)bni_13 + (-1)Bound*bni_13] + [bni_13]x3[0] ≥ 0∧0 = 0∧0 = 0∧[1 + (-1)bso_14] ≥ 0)

We simplified constraint (6) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(7)    (x3[0] ≥ 0 ⇒ (U^Increasing(1083_0_MINUS_EQ(x0[1], +(x2[1], -1), +(x3[1], -1))), ≥)∧0 = 0∧0 = 0∧[(-1)Bound*bni_13] + [bni_13]x3[0] ≥ 0∧0 = 0∧0 = 0∧[1 + (-1)bso_14] ≥ 0)

For Pair 1083_0_MINUS_EQ(x0[0], x2[0], x3[0]) → COND_1083_0_MINUS_EQ(>(x3[0], 0), x0[0], x2[0], x3[0]) the following chains were created:

We consider the chain 1083_0_MINUS_EQ(x0[0], x2[0], x3[0]) → COND_1083_0_MINUS_EQ(>(x3[0], 0), x0[0], x2[0], x3[0]), COND_1083_0_MINUS_EQ(TRUE, x0[1], x2[1], x3[1]) → 1083_0_MINUS_EQ(x0[1], +(x2[1], -1), +(x3[1], -1)) which results in the following constraint:
(8)    (>(x3[0], 0)=TRUE∧x0[0]=x0[1]∧x2[0]=x2[1]∧x3[0]=x3[1] ⇒ 1083_0_MINUS_EQ(x0[0], x2[0], x3[0])≥NonInfC∧1083_0_MINUS_EQ(x0[0], x2[0], x3[0])≥COND_1083_0_MINUS_EQ(>(x3[0], 0), x0[0], x2[0], x3[0])∧(U^Increasing(COND_1083_0_MINUS_EQ(>(x3[0], 0), x0[0], x2[0], x3[0])), ≥))

We simplified constraint (8) using rule (IV) which results in the following new constraint:
(9)    (>(x3[0], 0)=TRUE ⇒ 1083_0_MINUS_EQ(x0[0], x2[0], x3[0])≥NonInfC∧1083_0_MINUS_EQ(x0[0], x2[0], x3[0])≥COND_1083_0_MINUS_EQ(>(x3[0], 0), x0[0], x2[0], x3[0])∧(U^Increasing(COND_1083_0_MINUS_EQ(>(x3[0], 0), x0[0], x2[0], x3[0])), ≥))

We simplified constraint (9) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(10)    (x3[0] + [-1] ≥ 0 ⇒ (U^Increasing(COND_1083_0_MINUS_EQ(>(x3[0], 0), x0[0], x2[0], x3[0])), ≥)∧[(-1)bni_15 + (-1)Bound*bni_15] + [bni_15]x3[0] ≥ 0∧[(-1)bso_16] ≥ 0)

We simplified constraint (10) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(11)    (x3[0] + [-1] ≥ 0 ⇒ (U^Increasing(COND_1083_0_MINUS_EQ(>(x3[0], 0), x0[0], x2[0], x3[0])), ≥)∧[(-1)bni_15 + (-1)Bound*bni_15] + [bni_15]x3[0] ≥ 0∧[(-1)bso_16] ≥ 0)

We simplified constraint (11) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(12)    (x3[0] + [-1] ≥ 0 ⇒ (U^Increasing(COND_1083_0_MINUS_EQ(>(x3[0], 0), x0[0], x2[0], x3[0])), ≥)∧[(-1)bni_15 + (-1)Bound*bni_15] + [bni_15]x3[0] ≥ 0∧[(-1)bso_16] ≥ 0)

We simplified constraint (12) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint:
(13)    (x3[0] + [-1] ≥ 0 ⇒ (U^Increasing(COND_1083_0_MINUS_EQ(>(x3[0], 0), x0[0], x2[0], x3[0])), ≥)∧0 = 0∧0 = 0∧[(-1)bni_15 + (-1)Bound*bni_15] + [bni_15]x3[0] ≥ 0∧0 = 0∧0 = 0∧[(-1)bso_16] ≥ 0)

We simplified constraint (13) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(14)    (x3[0] ≥ 0 ⇒ (U^Increasing(COND_1083_0_MINUS_EQ(>(x3[0], 0), x0[0], x2[0], x3[0])), ≥)∧0 = 0∧0 = 0∧[(-1)Bound*bni_15] + [bni_15]x3[0] ≥ 0∧0 = 0∧0 = 0∧[(-1)bso_16] ≥ 0)

To summarize, we get the following constraints P_≥ for the following pairs.

COND_1083_0_MINUS_EQ(TRUE, x0[1], x2[1], x3[1]) → 1083_0_MINUS_EQ(x0[1], +(x2[1], -1), +(x3[1], -1))
- (x3[0] ≥ 0 ⇒ (U^Increasing(1083_0_MINUS_EQ(x0[1], +(x2[1], -1), +(x3[1], -1))), ≥)∧0 = 0∧0 = 0∧[(-1)Bound*bni_13] + [bni_13]x3[0] ≥ 0∧0 = 0∧0 = 0∧[1 + (-1)bso_14] ≥ 0)
1083_0_MINUS_EQ(x0[0], x2[0], x3[0]) → COND_1083_0_MINUS_EQ(>(x3[0], 0), x0[0], x2[0], x3[0])
- (x3[0] ≥ 0 ⇒ (U^Increasing(COND_1083_0_MINUS_EQ(>(x3[0], 0), x0[0], x2[0], x3[0])), ≥)∧0 = 0∧0 = 0∧[(-1)Bound*bni_15] + [bni_15]x3[0] ≥ 0∧0 = 0∧0 = 0∧[(-1)bso_16] ≥ 0)

POL(TRUE) = 0
POL(FALSE) = 0
POL(COND_1083_0_MINUS_EQ(x₁, x₂, x₃, x₄)) = [-1] + x₄
POL(1083_0_MINUS_EQ(x₁, x₂, x₃)) = [-1] + x₃
POL(+(x₁, x₂)) = x₁ + x₂
POL(-1) = [-1]
POL(>(x₁, x₂)) = [-1]
POL(0) = 0

The following pairs are in P_>:

COND_1083_0_MINUS_EQ(TRUE, x0[1], x2[1], x3[1]) → 1083_0_MINUS_EQ(x0[1], +(x2[1], -1), +(x3[1], -1))

The following pairs are in P_bound:

COND_1083_0_MINUS_EQ(TRUE, x0[1], x2[1], x3[1]) → 1083_0_MINUS_EQ(x0[1], +(x2[1], -1), +(x3[1], -1))
1083_0_MINUS_EQ(x0[0], x2[0], x3[0]) → COND_1083_0_MINUS_EQ(>(x3[0], 0), x0[0], x2[0], x3[0])

The following pairs are in P_≥:

1083_0_MINUS_EQ(x0[0], x2[0], x3[0]) → COND_1083_0_MINUS_EQ(>(x3[0], 0), x0[0], x2[0], x3[0])

There are no usable rules.

(20) 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:

Integer

R is empty.

The integer pair graph contains the following rules and edges:
(0): 1083_0_MINUS_EQ(x0[0], x2[0], x3[0]) → COND_1083_0_MINUS_EQ(x3[0] > 0, x0[0], x2[0], x3[0])

The set Q is empty.

(21) IDependencyGraphProof (EQUIVALENT transformation)

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

(0) Obligation:

(1) JBCToGraph (SOUND transformation)

(2) Obligation:

(3) TerminationGraphToSCCProof (SOUND transformation)

(4) Obligation:

(5) SCCToIDPv1Proof (SOUND transformation)

(6) Obligation:

(7) IDPNonInfProof (SOUND transformation)

(8) Complex Obligation (AND)

(9) Obligation:

(10) IDependencyGraphProof (EQUIVALENT transformation)

(11) Obligation:

(12) IDPNonInfProof (SOUND transformation)

(13) Obligation:

(14) IDependencyGraphProof (EQUIVALENT transformation)

(15) TRUE

(16) Obligation:

(17) IDependencyGraphProof (EQUIVALENT transformation)

(18) Obligation:

(19) IDPNonInfProof (SOUND transformation)

(20) Obligation:

(21) IDependencyGraphProof (EQUIVALENT transformation)

(22) TRUE