Termination proof of /tmp/tmpoKxZyX/GCD4.jar

Termination of the given JBC could be proven:

0 JBC

↳1 JBCToGraph (⇒, 670 ms)

↳2 JBCTerminationGraph

↳3 TerminationGraphToSCCProof (⇒, 0 ms)

↳4 JBCTerminationSCC

↳5 SCCToIDPv1Proof (⇒, 90 ms)

↳6 IDP

↳7 IDPNonInfProof (⇒, 260 ms)

↳8 IDP

↳9 IDependencyGraphProof (⇔, 0 ms)

↳10 IDP

↳11 IDPNonInfProof (⇒, 170 ms)

↳12 AND

    ↳13 IDP

        ↳14 IDependencyGraphProof (⇔, 0 ms)

        ↳15 TRUE

    ↳16 IDP

        ↳17 IDependencyGraphProof (⇔, 0 ms)

        ↳18 TRUE

(0) Obligation:

JBC Problem based on JBC Program:
Manifest-Version: 1.0 Created-By: 1.6.0_16 (Sun Microsystems Inc.) Main-Class: GCD4

public class GCD4 {
  public static int mod(int a, int b) {
    while(a>=b && b > 0) {
      a -= b;
    }
    return a;
  }

  public static int gcd(int a, int b) {
    int tmp;
    while(b > 0 && a > 0) {
      tmp = b;
      b = mod(a, b);
      a = tmp;
    }
    return a;
  }

  public static void main(String[] args) {
    Random.args = args;
    int x = Random.random();
    int y = Random.random();
    gcd(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:
GCD4.main([Ljava/lang/String;)V: Graph of 212 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: GCD4.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 33 rules for P and 0 rules for R.

P rules:
983_0_gcd_LE(EOS(STATIC_983), i194, i202, i202) → 986_0_gcd_LE(EOS(STATIC_986), i194, i202, i202)
986_0_gcd_LE(EOS(STATIC_986), i194, i202, i202) → 989_0_gcd_Load(EOS(STATIC_989), i194, i202) | >(i202, 0)
989_0_gcd_Load(EOS(STATIC_989), i194, i202) → 992_0_gcd_LE(EOS(STATIC_992), i194, i202, i194)
992_0_gcd_LE(EOS(STATIC_992), i206, i202, i206) → 997_0_gcd_LE(EOS(STATIC_997), i206, i202, i206)
997_0_gcd_LE(EOS(STATIC_997), i206, i202, i206) → 1002_0_gcd_Load(EOS(STATIC_1002), i206, i202) | >(i206, 0)
1002_0_gcd_Load(EOS(STATIC_1002), i206, i202) → 1008_0_gcd_Store(EOS(STATIC_1008), i206, i202, i202)
1008_0_gcd_Store(EOS(STATIC_1008), i206, i202, i202) → 1011_0_gcd_Load(EOS(STATIC_1011), i206, i202, i202)
1011_0_gcd_Load(EOS(STATIC_1011), i206, i202, i202) → 1015_0_gcd_Load(EOS(STATIC_1015), i202, i202, i206)
1015_0_gcd_Load(EOS(STATIC_1015), i202, i202, i206) → 1018_0_gcd_InvokeMethod(EOS(STATIC_1018), i202, i206, i202)
1018_0_gcd_InvokeMethod(EOS(STATIC_1018), i202, i206, i202) → 1020_0_mod_Load(EOS(STATIC_1020), i202, i206, i202, i206, i202)
1020_0_mod_Load(EOS(STATIC_1020), i202, i206, i202, i206, i202) → 1201_0_mod_Load(EOS(STATIC_1201), i202, i206, i202, i206, i202)
1201_0_mod_Load(EOS(STATIC_1201), i202, i206, i202, i321, i202) → 1203_0_mod_Load(EOS(STATIC_1203), i202, i206, i202, i321, i202, i321)
1203_0_mod_Load(EOS(STATIC_1203), i202, i206, i202, i321, i202, i321) → 1204_0_mod_LT(EOS(STATIC_1204), i202, i206, i202, i321, i202, i321, i202)
1204_0_mod_LT(EOS(STATIC_1204), i202, i206, i202, i321, i202, i321, i202) → 1206_0_mod_LT(EOS(STATIC_1206), i202, i206, i202, i321, i202, i321, i202)
1204_0_mod_LT(EOS(STATIC_1204), i202, i206, i202, i321, i202, i321, i202) → 1207_0_mod_LT(EOS(STATIC_1207), i202, i206, i202, i321, i202, i321, i202)
1206_0_mod_LT(EOS(STATIC_1206), i202, i206, i202, i321, i202, i321, i202) → 1209_0_mod_Load(EOS(STATIC_1209), i202, i206, i202, i321) | <(i321, i202)
1209_0_mod_Load(EOS(STATIC_1209), i202, i206, i202, i321) → 1212_0_mod_Return(EOS(STATIC_1212), i202, i206, i202, i321)
1212_0_mod_Return(EOS(STATIC_1212), i202, i206, i202, i321) → 1214_0_gcd_Store(EOS(STATIC_1214), i202, i321)
1214_0_gcd_Store(EOS(STATIC_1214), i202, i321) → 1217_0_gcd_Load(EOS(STATIC_1217), i321, i202)
1217_0_gcd_Load(EOS(STATIC_1217), i321, i202) → 1220_0_gcd_Store(EOS(STATIC_1220), i321, i202)
1220_0_gcd_Store(EOS(STATIC_1220), i321, i202) → 1223_0_gcd_JMP(EOS(STATIC_1223), i202, i321)
1223_0_gcd_JMP(EOS(STATIC_1223), i202, i321) → 1227_0_gcd_Load(EOS(STATIC_1227), i202, i321)
1227_0_gcd_Load(EOS(STATIC_1227), i202, i321) → 979_0_gcd_Load(EOS(STATIC_979), i202, i321)
979_0_gcd_Load(EOS(STATIC_979), i194, i195) → 983_0_gcd_LE(EOS(STATIC_983), i194, i195, i195)
1207_0_mod_LT(EOS(STATIC_1207), i202, i206, i202, i321, i202, i321, i202) → 1210_0_mod_Load(EOS(STATIC_1210), i202, i206, i202, i321, i202) | >=(i321, i202)
1210_0_mod_Load(EOS(STATIC_1210), i202, i206, i202, i321, i202) → 1213_0_mod_LE(EOS(STATIC_1213), i202, i206, i202, i321, i202, i202)
1213_0_mod_LE(EOS(STATIC_1213), i202, i206, i202, i321, i202, i202) → 1216_0_mod_Load(EOS(STATIC_1216), i202, i206, i202, i321, i202) | >(i202, 0)
1216_0_mod_Load(EOS(STATIC_1216), i202, i206, i202, i321, i202) → 1219_0_mod_Load(EOS(STATIC_1219), i202, i206, i202, i202, i321)
1219_0_mod_Load(EOS(STATIC_1219), i202, i206, i202, i202, i321) → 1222_0_mod_IntArithmetic(EOS(STATIC_1222), i202, i206, i202, i202, i321, i202)
1222_0_mod_IntArithmetic(EOS(STATIC_1222), i202, i206, i202, i202, i321, i202) → 1224_0_mod_Store(EOS(STATIC_1224), i202, i206, i202, i202, -(i321, i202)) | >(i202, 0)
1224_0_mod_Store(EOS(STATIC_1224), i202, i206, i202, i202, i323) → 1229_0_mod_JMP(EOS(STATIC_1229), i202, i206, i202, i323, i202)
1229_0_mod_JMP(EOS(STATIC_1229), i202, i206, i202, i323, i202) → 1371_0_mod_Load(EOS(STATIC_1371), i202, i206, i202, i323, i202)
1371_0_mod_Load(EOS(STATIC_1371), i202, i206, i202, i323, i202) → 1201_0_mod_Load(EOS(STATIC_1201), i202, i206, i202, i323, i202)
R rules:

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

P rules:
1204_0_mod_LT(EOS(STATIC_1204), x0, x1, x0, x2, x0, x2, x0) → 1204_0_mod_LT(EOS(STATIC_1204), x2, x0, x2, x0, x2, x0, x2) | &&(&&(>(x2, 0), <(x2, x0)), >(x0, 0))
1204_0_mod_LT(EOS(STATIC_1204), x0, x1, x0, x2, x0, x2, x0) → 1204_0_mod_LT(EOS(STATIC_1204), x0, x1, x0, -(x2, x0), x0, -(x2, x0), x0) | &&(>=(x2, x0), >(x0, 0))
R rules:

Filtered ground terms:

1204_0_mod_LT(x1, x2, x3, x4, x5, x6, x7, x8) → 1204_0_mod_LT(x2, x3, x4, x5, x6, x7, x8)
EOS(x1) → EOS
Cond_1204_0_mod_LT1(x1, x2, x3, x4, x5, x6, x7, x8, x9) → Cond_1204_0_mod_LT1(x1, x3, x4, x5, x6, x7, x8, x9)
Cond_1204_0_mod_LT(x1, x2, x3, x4, x5, x6, x7, x8, x9) → Cond_1204_0_mod_LT(x1, x3, x4, x5, x6, x7, x8, x9)

Filtered duplicate args:

1204_0_mod_LT(x1, x2, x3, x4, x5, x6, x7) → 1204_0_mod_LT(x2, x6, x7)
Cond_1204_0_mod_LT(x1, x2, x3, x4, x5, x6, x7, x8) → Cond_1204_0_mod_LT(x1, x3, x7, x8)
Cond_1204_0_mod_LT1(x1, x2, x3, x4, x5, x6, x7, x8) → Cond_1204_0_mod_LT1(x1, x3, x7, x8)

Filtered unneeded arguments:

Cond_1204_0_mod_LT(x1, x2, x3, x4) → Cond_1204_0_mod_LT(x1, x3, x4)
Cond_1204_0_mod_LT1(x1, x2, x3, x4) → Cond_1204_0_mod_LT1(x1, x3, x4)
1204_0_mod_LT(x1, x2, x3) → 1204_0_mod_LT(x2, x3)

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

P rules:
1204_0_mod_LT(x2, x0) → 1204_0_mod_LT(x0, x2) | &&(&&(>(x2, 0), <(x2, x0)), >(x0, 0))
1204_0_mod_LT(x2, x0) → 1204_0_mod_LT(-(x2, x0), 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:
1204_0_MOD_LT(x2, x0) → COND_1204_0_MOD_LT(&&(&&(>(x2, 0), <(x2, x0)), >(x0, 0)), x2, x0)
COND_1204_0_MOD_LT(TRUE, x2, x0) → 1204_0_MOD_LT(x0, x2)
1204_0_MOD_LT(x2, x0) → COND_1204_0_MOD_LT1(&&(>=(x2, x0), >(x0, 0)), x2, x0)
COND_1204_0_MOD_LT1(TRUE, x2, x0) → 1204_0_MOD_LT(-(x2, x0), 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:

Boolean, Integer

R is empty.

The integer pair graph contains the following rules and edges:
(0): 1204_0_MOD_LT(x2[0], x0[0]) → COND_1204_0_MOD_LT(x2[0] > 0 && x2[0] < x0[0] && x0[0] > 0, x2[0], x0[0])
(1): COND_1204_0_MOD_LT(TRUE, x2[1], x0[1]) → 1204_0_MOD_LT(x0[1], x2[1])
(2): 1204_0_MOD_LT(x2[2], x0[2]) → COND_1204_0_MOD_LT1(x2[2] >= x0[2] && x0[2] > 0, x2[2], x0[2])
(3): COND_1204_0_MOD_LT1(TRUE, x2[3], x0[3]) → 1204_0_MOD_LT(x2[3] - x0[3], x0[3])

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

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

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

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

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

(3) -> (2), if (x2[3] - x0[3] →^* x2[2]∧x0[3] →^* x0[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@55a2a0d8 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 1204_0_MOD_LT(x2, x0) → COND_1204_0_MOD_LT(&&(&&(>(x2, 0), <(x2, x0)), >(x0, 0)), x2, x0) the following chains were created:

We consider the chain 1204_0_MOD_LT(x2[0], x0[0]) → COND_1204_0_MOD_LT(&&(&&(>(x2[0], 0), <(x2[0], x0[0])), >(x0[0], 0)), x2[0], x0[0]), COND_1204_0_MOD_LT(TRUE, x2[1], x0[1]) → 1204_0_MOD_LT(x0[1], x2[1]) which results in the following constraint:
(1)    (&&(&&(>(x2[0], 0), <(x2[0], x0[0])), >(x0[0], 0))=TRUE∧x2[0]=x2[1]∧x0[0]=x0[1] ⇒ 1204_0_MOD_LT(x2[0], x0[0])≥NonInfC∧1204_0_MOD_LT(x2[0], x0[0])≥COND_1204_0_MOD_LT(&&(&&(>(x2[0], 0), <(x2[0], x0[0])), >(x0[0], 0)), x2[0], x0[0])∧(U^Increasing(COND_1204_0_MOD_LT(&&(&&(>(x2[0], 0), <(x2[0], x0[0])), >(x0[0], 0)), x2[0], x0[0])), ≥))

We simplified constraint (1) using rules (IV), (IDP_BOOLEAN) which results in the following new constraint:
(2)    (>(x0[0], 0)=TRUE∧>(x2[0], 0)=TRUE∧<(x2[0], x0[0])=TRUE ⇒ 1204_0_MOD_LT(x2[0], x0[0])≥NonInfC∧1204_0_MOD_LT(x2[0], x0[0])≥COND_1204_0_MOD_LT(&&(&&(>(x2[0], 0), <(x2[0], x0[0])), >(x0[0], 0)), x2[0], x0[0])∧(U^Increasing(COND_1204_0_MOD_LT(&&(&&(>(x2[0], 0), <(x2[0], x0[0])), >(x0[0], 0)), x2[0], x0[0])), ≥))

We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(3)    (x0[0] + [-1] ≥ 0∧x2[0] + [-1] ≥ 0∧x0[0] + [-1] + [-1]x2[0] ≥ 0 ⇒ (U^Increasing(COND_1204_0_MOD_LT(&&(&&(>(x2[0], 0), <(x2[0], x0[0])), >(x0[0], 0)), x2[0], x0[0])), ≥)∧[(-1)bni_18 + (-1)Bound*bni_18] + [bni_18]x0[0] + [bni_18]x2[0] ≥ 0∧[(-1)bso_19] ≥ 0)

We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(4)    (x0[0] + [-1] ≥ 0∧x2[0] + [-1] ≥ 0∧x0[0] + [-1] + [-1]x2[0] ≥ 0 ⇒ (U^Increasing(COND_1204_0_MOD_LT(&&(&&(>(x2[0], 0), <(x2[0], x0[0])), >(x0[0], 0)), x2[0], x0[0])), ≥)∧[(-1)bni_18 + (-1)Bound*bni_18] + [bni_18]x0[0] + [bni_18]x2[0] ≥ 0∧[(-1)bso_19] ≥ 0)

We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(5)    (x0[0] + [-1] ≥ 0∧x2[0] + [-1] ≥ 0∧x0[0] + [-1] + [-1]x2[0] ≥ 0 ⇒ (U^Increasing(COND_1204_0_MOD_LT(&&(&&(>(x2[0], 0), <(x2[0], x0[0])), >(x0[0], 0)), x2[0], x0[0])), ≥)∧[(-1)bni_18 + (-1)Bound*bni_18] + [bni_18]x0[0] + [bni_18]x2[0] ≥ 0∧[(-1)bso_19] ≥ 0)

We simplified constraint (5) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(6)    (x0[0] ≥ 0∧x2[0] + [-1] ≥ 0∧x0[0] + [-1]x2[0] ≥ 0 ⇒ (U^Increasing(COND_1204_0_MOD_LT(&&(&&(>(x2[0], 0), <(x2[0], x0[0])), >(x0[0], 0)), x2[0], x0[0])), ≥)∧[(-1)Bound*bni_18] + [bni_18]x0[0] + [bni_18]x2[0] ≥ 0∧[(-1)bso_19] ≥ 0)

We simplified constraint (6) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(7)    (x2[0] + x0[0] ≥ 0∧x2[0] + [-1] ≥ 0∧x0[0] ≥ 0 ⇒ (U^Increasing(COND_1204_0_MOD_LT(&&(&&(>(x2[0], 0), <(x2[0], x0[0])), >(x0[0], 0)), x2[0], x0[0])), ≥)∧[(-1)Bound*bni_18] + [(2)bni_18]x2[0] + [bni_18]x0[0] ≥ 0∧[(-1)bso_19] ≥ 0)

We simplified constraint (7) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(8)    ([1] + x2[0] + x0[0] ≥ 0∧x2[0] ≥ 0∧x0[0] ≥ 0 ⇒ (U^Increasing(COND_1204_0_MOD_LT(&&(&&(>(x2[0], 0), <(x2[0], x0[0])), >(x0[0], 0)), x2[0], x0[0])), ≥)∧[(2)bni_18 + (-1)Bound*bni_18] + [(2)bni_18]x2[0] + [bni_18]x0[0] ≥ 0∧[(-1)bso_19] ≥ 0)

For Pair COND_1204_0_MOD_LT(TRUE, x2, x0) → 1204_0_MOD_LT(x0, x2) the following chains were created:

We consider the chain COND_1204_0_MOD_LT(TRUE, x2[1], x0[1]) → 1204_0_MOD_LT(x0[1], x2[1]), 1204_0_MOD_LT(x2[0], x0[0]) → COND_1204_0_MOD_LT(&&(&&(>(x2[0], 0), <(x2[0], x0[0])), >(x0[0], 0)), x2[0], x0[0]) which results in the following constraint:
(9)    (x0[1]=x2[0]∧x2[1]=x0[0] ⇒ COND_1204_0_MOD_LT(TRUE, x2[1], x0[1])≥NonInfC∧COND_1204_0_MOD_LT(TRUE, x2[1], x0[1])≥1204_0_MOD_LT(x0[1], x2[1])∧(U^Increasing(1204_0_MOD_LT(x0[1], x2[1])), ≥))

We simplified constraint (9) using rule (IV) which results in the following new constraint:
(10)    (COND_1204_0_MOD_LT(TRUE, x2[1], x0[1])≥NonInfC∧COND_1204_0_MOD_LT(TRUE, x2[1], x0[1])≥1204_0_MOD_LT(x0[1], x2[1])∧(U^Increasing(1204_0_MOD_LT(x0[1], x2[1])), ≥))

We simplified constraint (10) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(11)    ((U^Increasing(1204_0_MOD_LT(x0[1], x2[1])), ≥)∧[bni_20] = 0∧[(-1)bso_21] ≥ 0)

We simplified constraint (11) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(12)    ((U^Increasing(1204_0_MOD_LT(x0[1], x2[1])), ≥)∧[bni_20] = 0∧[(-1)bso_21] ≥ 0)

We simplified constraint (12) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(13)    ((U^Increasing(1204_0_MOD_LT(x0[1], x2[1])), ≥)∧[bni_20] = 0∧[(-1)bso_21] ≥ 0)

We simplified constraint (13) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint:
(14)    ((U^Increasing(1204_0_MOD_LT(x0[1], x2[1])), ≥)∧[bni_20] = 0∧0 = 0∧0 = 0∧[(-1)bso_21] ≥ 0)
We consider the chain COND_1204_0_MOD_LT(TRUE, x2[1], x0[1]) → 1204_0_MOD_LT(x0[1], x2[1]), 1204_0_MOD_LT(x2[2], x0[2]) → COND_1204_0_MOD_LT1(&&(>=(x2[2], x0[2]), >(x0[2], 0)), x2[2], x0[2]) which results in the following constraint:
(15)    (x0[1]=x2[2]∧x2[1]=x0[2] ⇒ COND_1204_0_MOD_LT(TRUE, x2[1], x0[1])≥NonInfC∧COND_1204_0_MOD_LT(TRUE, x2[1], x0[1])≥1204_0_MOD_LT(x0[1], x2[1])∧(U^Increasing(1204_0_MOD_LT(x0[1], x2[1])), ≥))

We simplified constraint (15) using rule (IV) which results in the following new constraint:
(16)    (COND_1204_0_MOD_LT(TRUE, x2[1], x0[1])≥NonInfC∧COND_1204_0_MOD_LT(TRUE, x2[1], x0[1])≥1204_0_MOD_LT(x0[1], x2[1])∧(U^Increasing(1204_0_MOD_LT(x0[1], x2[1])), ≥))

We simplified constraint (16) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(17)    ((U^Increasing(1204_0_MOD_LT(x0[1], x2[1])), ≥)∧[bni_20] = 0∧[(-1)bso_21] ≥ 0)

We simplified constraint (17) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(18)    ((U^Increasing(1204_0_MOD_LT(x0[1], x2[1])), ≥)∧[bni_20] = 0∧[(-1)bso_21] ≥ 0)

We simplified constraint (18) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(19)    ((U^Increasing(1204_0_MOD_LT(x0[1], x2[1])), ≥)∧[bni_20] = 0∧[(-1)bso_21] ≥ 0)

We simplified constraint (19) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint:
(20)    ((U^Increasing(1204_0_MOD_LT(x0[1], x2[1])), ≥)∧[bni_20] = 0∧0 = 0∧0 = 0∧[(-1)bso_21] ≥ 0)

For Pair 1204_0_MOD_LT(x2, x0) → COND_1204_0_MOD_LT1(&&(>=(x2, x0), >(x0, 0)), x2, x0) the following chains were created:

We consider the chain 1204_0_MOD_LT(x2[2], x0[2]) → COND_1204_0_MOD_LT1(&&(>=(x2[2], x0[2]), >(x0[2], 0)), x2[2], x0[2]), COND_1204_0_MOD_LT1(TRUE, x2[3], x0[3]) → 1204_0_MOD_LT(-(x2[3], x0[3]), x0[3]) which results in the following constraint:
(21)    (&&(>=(x2[2], x0[2]), >(x0[2], 0))=TRUE∧x2[2]=x2[3]∧x0[2]=x0[3] ⇒ 1204_0_MOD_LT(x2[2], x0[2])≥NonInfC∧1204_0_MOD_LT(x2[2], x0[2])≥COND_1204_0_MOD_LT1(&&(>=(x2[2], x0[2]), >(x0[2], 0)), x2[2], x0[2])∧(U^Increasing(COND_1204_0_MOD_LT1(&&(>=(x2[2], x0[2]), >(x0[2], 0)), x2[2], x0[2])), ≥))

We simplified constraint (21) using rules (IV), (IDP_BOOLEAN) which results in the following new constraint:
(22)    (>=(x2[2], x0[2])=TRUE∧>(x0[2], 0)=TRUE ⇒ 1204_0_MOD_LT(x2[2], x0[2])≥NonInfC∧1204_0_MOD_LT(x2[2], x0[2])≥COND_1204_0_MOD_LT1(&&(>=(x2[2], x0[2]), >(x0[2], 0)), x2[2], x0[2])∧(U^Increasing(COND_1204_0_MOD_LT1(&&(>=(x2[2], x0[2]), >(x0[2], 0)), x2[2], x0[2])), ≥))

We simplified constraint (22) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(23)    (x2[2] + [-1]x0[2] ≥ 0∧x0[2] + [-1] ≥ 0 ⇒ (U^Increasing(COND_1204_0_MOD_LT1(&&(>=(x2[2], x0[2]), >(x0[2], 0)), x2[2], x0[2])), ≥)∧[(-1)bni_22 + (-1)Bound*bni_22] + [bni_22]x0[2] + [bni_22]x2[2] ≥ 0∧[(-1)bso_23] + x0[2] ≥ 0)

We simplified constraint (23) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(24)    (x2[2] + [-1]x0[2] ≥ 0∧x0[2] + [-1] ≥ 0 ⇒ (U^Increasing(COND_1204_0_MOD_LT1(&&(>=(x2[2], x0[2]), >(x0[2], 0)), x2[2], x0[2])), ≥)∧[(-1)bni_22 + (-1)Bound*bni_22] + [bni_22]x0[2] + [bni_22]x2[2] ≥ 0∧[(-1)bso_23] + x0[2] ≥ 0)

We simplified constraint (24) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(25)    (x2[2] + [-1]x0[2] ≥ 0∧x0[2] + [-1] ≥ 0 ⇒ (U^Increasing(COND_1204_0_MOD_LT1(&&(>=(x2[2], x0[2]), >(x0[2], 0)), x2[2], x0[2])), ≥)∧[(-1)bni_22 + (-1)Bound*bni_22] + [bni_22]x0[2] + [bni_22]x2[2] ≥ 0∧[(-1)bso_23] + x0[2] ≥ 0)

We simplified constraint (25) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(26)    (x2[2] ≥ 0∧x0[2] + [-1] ≥ 0 ⇒ (U^Increasing(COND_1204_0_MOD_LT1(&&(>=(x2[2], x0[2]), >(x0[2], 0)), x2[2], x0[2])), ≥)∧[(-1)bni_22 + (-1)Bound*bni_22] + [(2)bni_22]x0[2] + [bni_22]x2[2] ≥ 0∧[(-1)bso_23] + 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] ≥ 0 ⇒ (U^Increasing(COND_1204_0_MOD_LT1(&&(>=(x2[2], x0[2]), >(x0[2], 0)), x2[2], x0[2])), ≥)∧[(-1)Bound*bni_22 + bni_22] + [(2)bni_22]x0[2] + [bni_22]x2[2] ≥ 0∧[1 + (-1)bso_23] + x0[2] ≥ 0)

For Pair COND_1204_0_MOD_LT1(TRUE, x2, x0) → 1204_0_MOD_LT(-(x2, x0), x0) the following chains were created:

We consider the chain 1204_0_MOD_LT(x2[2], x0[2]) → COND_1204_0_MOD_LT1(&&(>=(x2[2], x0[2]), >(x0[2], 0)), x2[2], x0[2]), COND_1204_0_MOD_LT1(TRUE, x2[3], x0[3]) → 1204_0_MOD_LT(-(x2[3], x0[3]), x0[3]), 1204_0_MOD_LT(x2[0], x0[0]) → COND_1204_0_MOD_LT(&&(&&(>(x2[0], 0), <(x2[0], x0[0])), >(x0[0], 0)), x2[0], x0[0]) which results in the following constraint:
(28)    (&&(>=(x2[2], x0[2]), >(x0[2], 0))=TRUE∧x2[2]=x2[3]∧x0[2]=x0[3]∧-(x2[3], x0[3])=x2[0]∧x0[3]=x0[0] ⇒ COND_1204_0_MOD_LT1(TRUE, x2[3], x0[3])≥NonInfC∧COND_1204_0_MOD_LT1(TRUE, x2[3], x0[3])≥1204_0_MOD_LT(-(x2[3], x0[3]), x0[3])∧(U^Increasing(1204_0_MOD_LT(-(x2[3], x0[3]), x0[3])), ≥))

We simplified constraint (28) using rules (III), (IV), (IDP_BOOLEAN) which results in the following new constraint:
(29)    (>=(x2[2], x0[2])=TRUE∧>(x0[2], 0)=TRUE ⇒ COND_1204_0_MOD_LT1(TRUE, x2[2], x0[2])≥NonInfC∧COND_1204_0_MOD_LT1(TRUE, x2[2], x0[2])≥1204_0_MOD_LT(-(x2[2], x0[2]), x0[2])∧(U^Increasing(1204_0_MOD_LT(-(x2[3], x0[3]), x0[3])), ≥))

We simplified constraint (29) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(30)    (x2[2] + [-1]x0[2] ≥ 0∧x0[2] + [-1] ≥ 0 ⇒ (U^Increasing(1204_0_MOD_LT(-(x2[3], x0[3]), x0[3])), ≥)∧[(-1)bni_24 + (-1)Bound*bni_24] + [bni_24]x2[2] ≥ 0∧[(-1)bso_25] ≥ 0)

We simplified constraint (30) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(31)    (x2[2] + [-1]x0[2] ≥ 0∧x0[2] + [-1] ≥ 0 ⇒ (U^Increasing(1204_0_MOD_LT(-(x2[3], x0[3]), x0[3])), ≥)∧[(-1)bni_24 + (-1)Bound*bni_24] + [bni_24]x2[2] ≥ 0∧[(-1)bso_25] ≥ 0)

We simplified constraint (31) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(32)    (x2[2] + [-1]x0[2] ≥ 0∧x0[2] + [-1] ≥ 0 ⇒ (U^Increasing(1204_0_MOD_LT(-(x2[3], x0[3]), x0[3])), ≥)∧[(-1)bni_24 + (-1)Bound*bni_24] + [bni_24]x2[2] ≥ 0∧[(-1)bso_25] ≥ 0)

We simplified constraint (32) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(33)    (x2[2] ≥ 0∧x0[2] + [-1] ≥ 0 ⇒ (U^Increasing(1204_0_MOD_LT(-(x2[3], x0[3]), x0[3])), ≥)∧[(-1)bni_24 + (-1)Bound*bni_24] + [bni_24]x0[2] + [bni_24]x2[2] ≥ 0∧[(-1)bso_25] ≥ 0)

We simplified constraint (33) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(34)    (x2[2] ≥ 0∧x0[2] ≥ 0 ⇒ (U^Increasing(1204_0_MOD_LT(-(x2[3], x0[3]), x0[3])), ≥)∧[(-1)Bound*bni_24] + [bni_24]x0[2] + [bni_24]x2[2] ≥ 0∧[(-1)bso_25] ≥ 0)
We consider the chain 1204_0_MOD_LT(x2[2], x0[2]) → COND_1204_0_MOD_LT1(&&(>=(x2[2], x0[2]), >(x0[2], 0)), x2[2], x0[2]), COND_1204_0_MOD_LT1(TRUE, x2[3], x0[3]) → 1204_0_MOD_LT(-(x2[3], x0[3]), x0[3]), 1204_0_MOD_LT(x2[2], x0[2]) → COND_1204_0_MOD_LT1(&&(>=(x2[2], x0[2]), >(x0[2], 0)), x2[2], x0[2]) which results in the following constraint:
(35)    (&&(>=(x2[2], x0[2]), >(x0[2], 0))=TRUE∧x2[2]=x2[3]∧x0[2]=x0[3]∧-(x2[3], x0[3])=x2[2]1∧x0[3]=x0[2]1 ⇒ COND_1204_0_MOD_LT1(TRUE, x2[3], x0[3])≥NonInfC∧COND_1204_0_MOD_LT1(TRUE, x2[3], x0[3])≥1204_0_MOD_LT(-(x2[3], x0[3]), x0[3])∧(U^Increasing(1204_0_MOD_LT(-(x2[3], x0[3]), x0[3])), ≥))

We simplified constraint (35) using rules (III), (IV), (IDP_BOOLEAN) which results in the following new constraint:
(36)    (>=(x2[2], x0[2])=TRUE∧>(x0[2], 0)=TRUE ⇒ COND_1204_0_MOD_LT1(TRUE, x2[2], x0[2])≥NonInfC∧COND_1204_0_MOD_LT1(TRUE, x2[2], x0[2])≥1204_0_MOD_LT(-(x2[2], x0[2]), x0[2])∧(U^Increasing(1204_0_MOD_LT(-(x2[3], x0[3]), x0[3])), ≥))

We simplified constraint (36) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(37)    (x2[2] + [-1]x0[2] ≥ 0∧x0[2] + [-1] ≥ 0 ⇒ (U^Increasing(1204_0_MOD_LT(-(x2[3], x0[3]), x0[3])), ≥)∧[(-1)bni_24 + (-1)Bound*bni_24] + [bni_24]x2[2] ≥ 0∧[(-1)bso_25] ≥ 0)

We simplified constraint (37) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(38)    (x2[2] + [-1]x0[2] ≥ 0∧x0[2] + [-1] ≥ 0 ⇒ (U^Increasing(1204_0_MOD_LT(-(x2[3], x0[3]), x0[3])), ≥)∧[(-1)bni_24 + (-1)Bound*bni_24] + [bni_24]x2[2] ≥ 0∧[(-1)bso_25] ≥ 0)

We simplified constraint (38) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(39)    (x2[2] + [-1]x0[2] ≥ 0∧x0[2] + [-1] ≥ 0 ⇒ (U^Increasing(1204_0_MOD_LT(-(x2[3], x0[3]), x0[3])), ≥)∧[(-1)bni_24 + (-1)Bound*bni_24] + [bni_24]x2[2] ≥ 0∧[(-1)bso_25] ≥ 0)

We simplified constraint (39) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(40)    (x2[2] ≥ 0∧x0[2] + [-1] ≥ 0 ⇒ (U^Increasing(1204_0_MOD_LT(-(x2[3], x0[3]), x0[3])), ≥)∧[(-1)bni_24 + (-1)Bound*bni_24] + [bni_24]x0[2] + [bni_24]x2[2] ≥ 0∧[(-1)bso_25] ≥ 0)

We simplified constraint (40) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(41)    (x2[2] ≥ 0∧x0[2] ≥ 0 ⇒ (U^Increasing(1204_0_MOD_LT(-(x2[3], x0[3]), x0[3])), ≥)∧[(-1)Bound*bni_24] + [bni_24]x0[2] + [bni_24]x2[2] ≥ 0∧[(-1)bso_25] ≥ 0)

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

1204_0_MOD_LT(x2, x0) → COND_1204_0_MOD_LT(&&(&&(>(x2, 0), <(x2, x0)), >(x0, 0)), x2, x0)
- ([1] + x2[0] + x0[0] ≥ 0∧x2[0] ≥ 0∧x0[0] ≥ 0 ⇒ (U^Increasing(COND_1204_0_MOD_LT(&&(&&(>(x2[0], 0), <(x2[0], x0[0])), >(x0[0], 0)), x2[0], x0[0])), ≥)∧[(2)bni_18 + (-1)Bound*bni_18] + [(2)bni_18]x2[0] + [bni_18]x0[0] ≥ 0∧[(-1)bso_19] ≥ 0)
COND_1204_0_MOD_LT(TRUE, x2, x0) → 1204_0_MOD_LT(x0, x2)
- ((U^Increasing(1204_0_MOD_LT(x0[1], x2[1])), ≥)∧[bni_20] = 0∧0 = 0∧0 = 0∧[(-1)bso_21] ≥ 0)
- ((U^Increasing(1204_0_MOD_LT(x0[1], x2[1])), ≥)∧[bni_20] = 0∧0 = 0∧0 = 0∧[(-1)bso_21] ≥ 0)
1204_0_MOD_LT(x2, x0) → COND_1204_0_MOD_LT1(&&(>=(x2, x0), >(x0, 0)), x2, x0)
- (x2[2] ≥ 0∧x0[2] ≥ 0 ⇒ (U^Increasing(COND_1204_0_MOD_LT1(&&(>=(x2[2], x0[2]), >(x0[2], 0)), x2[2], x0[2])), ≥)∧[(-1)Bound*bni_22 + bni_22] + [(2)bni_22]x0[2] + [bni_22]x2[2] ≥ 0∧[1 + (-1)bso_23] + x0[2] ≥ 0)
COND_1204_0_MOD_LT1(TRUE, x2, x0) → 1204_0_MOD_LT(-(x2, x0), x0)
- (x2[2] ≥ 0∧x0[2] ≥ 0 ⇒ (U^Increasing(1204_0_MOD_LT(-(x2[3], x0[3]), x0[3])), ≥)∧[(-1)Bound*bni_24] + [bni_24]x0[2] + [bni_24]x2[2] ≥ 0∧[(-1)bso_25] ≥ 0)
- (x2[2] ≥ 0∧x0[2] ≥ 0 ⇒ (U^Increasing(1204_0_MOD_LT(-(x2[3], x0[3]), x0[3])), ≥)∧[(-1)Bound*bni_24] + [bni_24]x0[2] + [bni_24]x2[2] ≥ 0∧[(-1)bso_25] ≥ 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(1204_0_MOD_LT(x₁, x₂)) = [-1] + x₂ + x₁
POL(COND_1204_0_MOD_LT(x₁, x₂, x₃)) = [-1] + x₃ + x₂ + [-1]x₁
POL(&&(x₁, x₂)) = 0
POL(>(x₁, x₂)) = [-1]
POL(0) = 0
POL(<(x₁, x₂)) = [-1]
POL(COND_1204_0_MOD_LT1(x₁, x₂, x₃)) = [-1] + x₂ + [-1]x₁
POL(>=(x₁, x₂)) = [-1]
POL(-(x₁, x₂)) = x₁ + [-1]x₂

The following pairs are in P_>:

1204_0_MOD_LT(x2[2], x0[2]) → COND_1204_0_MOD_LT1(&&(>=(x2[2], x0[2]), >(x0[2], 0)), x2[2], x0[2])

The following pairs are in P_bound:

1204_0_MOD_LT(x2[0], x0[0]) → COND_1204_0_MOD_LT(&&(&&(>(x2[0], 0), <(x2[0], x0[0])), >(x0[0], 0)), x2[0], x0[0])
1204_0_MOD_LT(x2[2], x0[2]) → COND_1204_0_MOD_LT1(&&(>=(x2[2], x0[2]), >(x0[2], 0)), x2[2], x0[2])
COND_1204_0_MOD_LT1(TRUE, x2[3], x0[3]) → 1204_0_MOD_LT(-(x2[3], x0[3]), x0[3])

The following pairs are in P_≥:

1204_0_MOD_LT(x2[0], x0[0]) → COND_1204_0_MOD_LT(&&(&&(>(x2[0], 0), <(x2[0], x0[0])), >(x0[0], 0)), x2[0], x0[0])
COND_1204_0_MOD_LT(TRUE, x2[1], x0[1]) → 1204_0_MOD_LT(x0[1], x2[1])
COND_1204_0_MOD_LT1(TRUE, x2[3], x0[3]) → 1204_0_MOD_LT(-(x2[3], x0[3]), x0[3])

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) 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): 1204_0_MOD_LT(x2[0], x0[0]) → COND_1204_0_MOD_LT(x2[0] > 0 && x2[0] < x0[0] && x0[0] > 0, x2[0], x0[0])
(1): COND_1204_0_MOD_LT(TRUE, x2[1], x0[1]) → 1204_0_MOD_LT(x0[1], x2[1])
(3): COND_1204_0_MOD_LT1(TRUE, x2[3], x0[3]) → 1204_0_MOD_LT(x2[3] - x0[3], x0[3])

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

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

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

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_1204_0_MOD_LT(TRUE, x2[1], x0[1]) → 1204_0_MOD_LT(x0[1], x2[1])
(0): 1204_0_MOD_LT(x2[0], x0[0]) → COND_1204_0_MOD_LT(x2[0] > 0 && x2[0] < x0[0] && x0[0] > 0, x2[0], x0[0])

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

(0) -> (1), if (x2[0] > 0 && x2[0] < x0[0] && x0[0] > 0 ∧x2[0] →^* x2[1]∧x0[0] →^* x0[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@55a2a0d8 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_1204_0_MOD_LT(TRUE, x2[1], x0[1]) → 1204_0_MOD_LT(x0[1], x2[1]) the following chains were created:

We consider the chain 1204_0_MOD_LT(x2[0], x0[0]) → COND_1204_0_MOD_LT(&&(&&(>(x2[0], 0), <(x2[0], x0[0])), >(x0[0], 0)), x2[0], x0[0]), COND_1204_0_MOD_LT(TRUE, x2[1], x0[1]) → 1204_0_MOD_LT(x0[1], x2[1]), 1204_0_MOD_LT(x2[0], x0[0]) → COND_1204_0_MOD_LT(&&(&&(>(x2[0], 0), <(x2[0], x0[0])), >(x0[0], 0)), x2[0], x0[0]) which results in the following constraint:
(1)    (&&(&&(>(x2[0], 0), <(x2[0], x0[0])), >(x0[0], 0))=TRUE∧x2[0]=x2[1]∧x0[0]=x0[1]∧x0[1]=x2[0]1∧x2[1]=x0[0]1 ⇒ COND_1204_0_MOD_LT(TRUE, x2[1], x0[1])≥NonInfC∧COND_1204_0_MOD_LT(TRUE, x2[1], x0[1])≥1204_0_MOD_LT(x0[1], x2[1])∧(U^Increasing(1204_0_MOD_LT(x0[1], x2[1])), ≥))

We simplified constraint (1) using rules (III), (IV), (IDP_BOOLEAN) which results in the following new constraint:
(2)    (>(x0[0], 0)=TRUE∧>(x2[0], 0)=TRUE∧<(x2[0], x0[0])=TRUE ⇒ COND_1204_0_MOD_LT(TRUE, x2[0], x0[0])≥NonInfC∧COND_1204_0_MOD_LT(TRUE, x2[0], x0[0])≥1204_0_MOD_LT(x0[0], x2[0])∧(U^Increasing(1204_0_MOD_LT(x0[1], x2[1])), ≥))

We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(3)    (x0[0] + [-1] ≥ 0∧x2[0] + [-1] ≥ 0∧x0[0] + [-1] + [-1]x2[0] ≥ 0 ⇒ (U^Increasing(1204_0_MOD_LT(x0[1], x2[1])), ≥)∧[bni_13 + (-1)Bound*bni_13] + [(-1)bni_13]x0[0] + [bni_13]x2[0] ≥ 0∧[2 + (-1)bso_14] ≥ 0)

We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(4)    (x0[0] + [-1] ≥ 0∧x2[0] + [-1] ≥ 0∧x0[0] + [-1] + [-1]x2[0] ≥ 0 ⇒ (U^Increasing(1204_0_MOD_LT(x0[1], x2[1])), ≥)∧[bni_13 + (-1)Bound*bni_13] + [(-1)bni_13]x0[0] + [bni_13]x2[0] ≥ 0∧[2 + (-1)bso_14] ≥ 0)

We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(5)    (x0[0] + [-1] ≥ 0∧x2[0] + [-1] ≥ 0∧x0[0] + [-1] + [-1]x2[0] ≥ 0 ⇒ (U^Increasing(1204_0_MOD_LT(x0[1], x2[1])), ≥)∧[bni_13 + (-1)Bound*bni_13] + [(-1)bni_13]x0[0] + [bni_13]x2[0] ≥ 0∧[2 + (-1)bso_14] ≥ 0)

We simplified constraint (5) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(6)    (x0[0] ≥ 0∧x2[0] + [-1] ≥ 0∧x0[0] + [-1]x2[0] ≥ 0 ⇒ (U^Increasing(1204_0_MOD_LT(x0[1], x2[1])), ≥)∧[(-1)Bound*bni_13] + [(-1)bni_13]x0[0] + [bni_13]x2[0] ≥ 0∧[2 + (-1)bso_14] ≥ 0)

We simplified constraint (6) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(7)    (x2[0] + x0[0] ≥ 0∧x2[0] + [-1] ≥ 0∧x0[0] ≥ 0 ⇒ (U^Increasing(1204_0_MOD_LT(x0[1], x2[1])), ≥)∧[(-1)Bound*bni_13] + [(-1)bni_13]x0[0] ≥ 0∧[2 + (-1)bso_14] ≥ 0)

We simplified constraint (7) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(8)    ([1] + x2[0] + x0[0] ≥ 0∧x2[0] ≥ 0∧x0[0] ≥ 0 ⇒ (U^Increasing(1204_0_MOD_LT(x0[1], x2[1])), ≥)∧[(-1)Bound*bni_13] + [(-1)bni_13]x0[0] ≥ 0∧[2 + (-1)bso_14] ≥ 0)

For Pair 1204_0_MOD_LT(x2[0], x0[0]) → COND_1204_0_MOD_LT(&&(&&(>(x2[0], 0), <(x2[0], x0[0])), >(x0[0], 0)), x2[0], x0[0]) the following chains were created:

We consider the chain COND_1204_0_MOD_LT(TRUE, x2[1], x0[1]) → 1204_0_MOD_LT(x0[1], x2[1]), 1204_0_MOD_LT(x2[0], x0[0]) → COND_1204_0_MOD_LT(&&(&&(>(x2[0], 0), <(x2[0], x0[0])), >(x0[0], 0)), x2[0], x0[0]), COND_1204_0_MOD_LT(TRUE, x2[1], x0[1]) → 1204_0_MOD_LT(x0[1], x2[1]) which results in the following constraint:
(9)    (x0[1]=x2[0]∧x2[1]=x0[0]∧&&(&&(>(x2[0], 0), <(x2[0], x0[0])), >(x0[0], 0))=TRUE∧x2[0]=x2[1]1∧x0[0]=x0[1]1 ⇒ 1204_0_MOD_LT(x2[0], x0[0])≥NonInfC∧1204_0_MOD_LT(x2[0], x0[0])≥COND_1204_0_MOD_LT(&&(&&(>(x2[0], 0), <(x2[0], x0[0])), >(x0[0], 0)), x2[0], x0[0])∧(U^Increasing(COND_1204_0_MOD_LT(&&(&&(>(x2[0], 0), <(x2[0], x0[0])), >(x0[0], 0)), x2[0], x0[0])), ≥))

We simplified constraint (9) using rules (III), (IV), (IDP_BOOLEAN) which results in the following new constraint:
(10)    (>(x0[0], 0)=TRUE∧>(x2[0], 0)=TRUE∧<(x2[0], x0[0])=TRUE ⇒ 1204_0_MOD_LT(x2[0], x0[0])≥NonInfC∧1204_0_MOD_LT(x2[0], x0[0])≥COND_1204_0_MOD_LT(&&(&&(>(x2[0], 0), <(x2[0], x0[0])), >(x0[0], 0)), x2[0], x0[0])∧(U^Increasing(COND_1204_0_MOD_LT(&&(&&(>(x2[0], 0), <(x2[0], x0[0])), >(x0[0], 0)), x2[0], x0[0])), ≥))

We simplified constraint (10) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(11)    (x0[0] + [-1] ≥ 0∧x2[0] + [-1] ≥ 0∧x0[0] + [-1] + [-1]x2[0] ≥ 0 ⇒ (U^Increasing(COND_1204_0_MOD_LT(&&(&&(>(x2[0], 0), <(x2[0], x0[0])), >(x0[0], 0)), x2[0], x0[0])), ≥)∧[(-1)bni_15 + (-1)Bound*bni_15] + [bni_15]x0[0] + [(-1)bni_15]x2[0] ≥ 0∧[-2 + (-1)bso_16] + [2]x0[0] + [-2]x2[0] ≥ 0)

We simplified constraint (11) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(12)    (x0[0] + [-1] ≥ 0∧x2[0] + [-1] ≥ 0∧x0[0] + [-1] + [-1]x2[0] ≥ 0 ⇒ (U^Increasing(COND_1204_0_MOD_LT(&&(&&(>(x2[0], 0), <(x2[0], x0[0])), >(x0[0], 0)), x2[0], x0[0])), ≥)∧[(-1)bni_15 + (-1)Bound*bni_15] + [bni_15]x0[0] + [(-1)bni_15]x2[0] ≥ 0∧[-2 + (-1)bso_16] + [2]x0[0] + [-2]x2[0] ≥ 0)

We simplified constraint (12) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(13)    (x0[0] + [-1] ≥ 0∧x2[0] + [-1] ≥ 0∧x0[0] + [-1] + [-1]x2[0] ≥ 0 ⇒ (U^Increasing(COND_1204_0_MOD_LT(&&(&&(>(x2[0], 0), <(x2[0], x0[0])), >(x0[0], 0)), x2[0], x0[0])), ≥)∧[(-1)bni_15 + (-1)Bound*bni_15] + [bni_15]x0[0] + [(-1)bni_15]x2[0] ≥ 0∧[-2 + (-1)bso_16] + [2]x0[0] + [-2]x2[0] ≥ 0)

We simplified constraint (13) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(14)    (x0[0] ≥ 0∧x2[0] + [-1] ≥ 0∧x0[0] + [-1]x2[0] ≥ 0 ⇒ (U^Increasing(COND_1204_0_MOD_LT(&&(&&(>(x2[0], 0), <(x2[0], x0[0])), >(x0[0], 0)), x2[0], x0[0])), ≥)∧[(-1)Bound*bni_15] + [bni_15]x0[0] + [(-1)bni_15]x2[0] ≥ 0∧[(-1)bso_16] + [2]x0[0] + [-2]x2[0] ≥ 0)

We simplified constraint (14) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(15)    (x2[0] + x0[0] ≥ 0∧x2[0] + [-1] ≥ 0∧x0[0] ≥ 0 ⇒ (U^Increasing(COND_1204_0_MOD_LT(&&(&&(>(x2[0], 0), <(x2[0], x0[0])), >(x0[0], 0)), x2[0], x0[0])), ≥)∧[(-1)Bound*bni_15] + [bni_15]x0[0] ≥ 0∧[(-1)bso_16] + [2]x0[0] ≥ 0)

We simplified constraint (15) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(16)    ([1] + x2[0] + x0[0] ≥ 0∧x2[0] ≥ 0∧x0[0] ≥ 0 ⇒ (U^Increasing(COND_1204_0_MOD_LT(&&(&&(>(x2[0], 0), <(x2[0], x0[0])), >(x0[0], 0)), x2[0], x0[0])), ≥)∧[(-1)Bound*bni_15] + [bni_15]x0[0] ≥ 0∧[(-1)bso_16] + [2]x0[0] ≥ 0)

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

COND_1204_0_MOD_LT(TRUE, x2[1], x0[1]) → 1204_0_MOD_LT(x0[1], x2[1])
- ([1] + x2[0] + x0[0] ≥ 0∧x2[0] ≥ 0∧x0[0] ≥ 0 ⇒ (U^Increasing(1204_0_MOD_LT(x0[1], x2[1])), ≥)∧[(-1)Bound*bni_13] + [(-1)bni_13]x0[0] ≥ 0∧[2 + (-1)bso_14] ≥ 0)
1204_0_MOD_LT(x2[0], x0[0]) → COND_1204_0_MOD_LT(&&(&&(>(x2[0], 0), <(x2[0], x0[0])), >(x0[0], 0)), x2[0], x0[0])
- ([1] + x2[0] + x0[0] ≥ 0∧x2[0] ≥ 0∧x0[0] ≥ 0 ⇒ (U^Increasing(COND_1204_0_MOD_LT(&&(&&(>(x2[0], 0), <(x2[0], x0[0])), >(x0[0], 0)), x2[0], x0[0])), ≥)∧[(-1)Bound*bni_15] + [bni_15]x0[0] ≥ 0∧[(-1)bso_16] + [2]x0[0] ≥ 0)

POL(TRUE) = 0
POL(FALSE) = 0
POL(COND_1204_0_MOD_LT(x₁, x₂, x₃)) = [1] + [-1]x₃ + x₂ + [-1]x₁
POL(1204_0_MOD_LT(x₁, x₂)) = [-1] + x₂ + [-1]x₁
POL(&&(x₁, x₂)) = 0
POL(>(x₁, x₂)) = [-1]
POL(0) = 0
POL(<(x₁, x₂)) = [-1]

The following pairs are in P_>:

COND_1204_0_MOD_LT(TRUE, x2[1], x0[1]) → 1204_0_MOD_LT(x0[1], x2[1])

The following pairs are in P_bound:

1204_0_MOD_LT(x2[0], x0[0]) → COND_1204_0_MOD_LT(&&(&&(>(x2[0], 0), <(x2[0], x0[0])), >(x0[0], 0)), x2[0], x0[0])

The following pairs are in P_≥:

1204_0_MOD_LT(x2[0], x0[0]) → COND_1204_0_MOD_LT(&&(&&(>(x2[0], 0), <(x2[0], x0[0])), >(x0[0], 0)), x2[0], x0[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

(12) Complex Obligation (AND)

(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:

Boolean, Integer

(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:
none

R is empty.

The integer pair graph contains the following rules and edges:
(1): COND_1204_0_MOD_LT(TRUE, x2[1], x0[1]) → 1204_0_MOD_LT(x0[1], x2[1])

The set Q is empty.

(17) 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) Obligation:

(9) IDependencyGraphProof (EQUIVALENT transformation)

(10) Obligation:

(11) IDPNonInfProof (SOUND transformation)

(12) Complex Obligation (AND)

(13) Obligation:

(14) IDependencyGraphProof (EQUIVALENT transformation)

(15) TRUE

(16) Obligation:

(17) IDependencyGraphProof (EQUIVALENT transformation)

(18) TRUE