Termination proof of /tmp/tmpqHvtFt/PastaB11.jar

Termination of the given JBC could be proven:

0 JBC

↳1 JBCToGraph (⇒, 1390 ms)

↳2 JBCTerminationGraph

↳3 TerminationGraphToSCCProof (⇒, 0 ms)

↳4 JBCTerminationSCC

↳5 SCCToIDPv1Proof (⇒, 120 ms)

↳6 IDP

↳7 IDPNonInfProof (⇒, 270 ms)

↳8 AND

    ↳9 IDP

        ↳10 IDependencyGraphProof (⇔, 0 ms)

        ↳11 TRUE

    ↳12 IDP

        ↳13 IDependencyGraphProof (⇔, 0 ms)

        ↳14 TRUE

(0) Obligation:

JBC Problem based on JBC Program:
Manifest-Version: 1.0 Created-By: 1.6.0_16 (Sun Microsystems Inc.) Main-Class: PastaB11

/**
 * Example taken from "A Term Rewriting Approach to the Automated Termination
 * Analysis of Imperative Programs" (http://www.cs.unm.edu/~spf/papers/2009-02.pdf)
 * and converted to Java.
 */

public class PastaB11 {
    public static void main(String[] args) {
        Random.args = args;
        int x = Random.random();
        int y = Random.random();

        while (x + y > 0) {
            if (x > y) {
                x--;
            } else if (x == y) {
                x--;
            } else {
                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:
PastaB11.main([Ljava/lang/String;)V: Graph of 199 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: PastaB11.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 26 rules for P and 0 rules for R.

P rules:
1633_0_main_Load(EOS(STATIC_1633), i127, i855, i127) → 1635_0_main_IntArithmetic(EOS(STATIC_1635), i127, i855, i127, i855)
1635_0_main_IntArithmetic(EOS(STATIC_1635), i127, i855, i127, i855) → 1637_0_main_LE(EOS(STATIC_1637), i127, i855, +(i127, i855))
1637_0_main_LE(EOS(STATIC_1637), i127, i855, i863) → 1639_0_main_LE(EOS(STATIC_1639), i127, i855, i863)
1639_0_main_LE(EOS(STATIC_1639), i127, i855, i863) → 1643_0_main_Load(EOS(STATIC_1643), i127, i855) | >(i863, 0)
1643_0_main_Load(EOS(STATIC_1643), i127, i855) → 1646_0_main_Load(EOS(STATIC_1646), i127, i855, i127)
1646_0_main_Load(EOS(STATIC_1646), i127, i855, i127) → 1647_0_main_LE(EOS(STATIC_1647), i127, i855, i127, i855)
1647_0_main_LE(EOS(STATIC_1647), i127, i855, i127, i855) → 1649_0_main_LE(EOS(STATIC_1649), i127, i855, i127, i855)
1647_0_main_LE(EOS(STATIC_1647), i127, i855, i127, i855) → 1650_0_main_LE(EOS(STATIC_1650), i127, i855, i127, i855)
1649_0_main_LE(EOS(STATIC_1649), i127, i855, i127, i855) → 1651_0_main_Load(EOS(STATIC_1651), i127, i855) | <=(i127, i855)
1651_0_main_Load(EOS(STATIC_1651), i127, i855) → 1655_0_main_Load(EOS(STATIC_1655), i127, i855, i127)
1655_0_main_Load(EOS(STATIC_1655), i127, i855, i127) → 1658_0_main_NE(EOS(STATIC_1658), i127, i855, i127, i855)
1658_0_main_NE(EOS(STATIC_1658), i127, i855, i127, i855) → 1665_0_main_NE(EOS(STATIC_1665), i127, i855, i127, i855)
1658_0_main_NE(EOS(STATIC_1658), i855, i855, i855, i855) → 1666_0_main_NE(EOS(STATIC_1666), i855, i855, i855, i855)
1665_0_main_NE(EOS(STATIC_1665), i127, i855, i127, i855) → 1918_0_main_Inc(EOS(STATIC_1918), i127, i855) | !(=(i127, i855))
1918_0_main_Inc(EOS(STATIC_1918), i127, i855) → 1921_0_main_JMP(EOS(STATIC_1921), i127, +(i855, -1))
1921_0_main_JMP(EOS(STATIC_1921), i127, i1048) → 1926_0_main_Load(EOS(STATIC_1926), i127, i1048)
1926_0_main_Load(EOS(STATIC_1926), i127, i1048) → 1630_0_main_Load(EOS(STATIC_1630), i127, i1048)
1630_0_main_Load(EOS(STATIC_1630), i127, i855) → 1633_0_main_Load(EOS(STATIC_1633), i127, i855, i127)
1666_0_main_NE(EOS(STATIC_1666), i855, i855, i855, i855) → 1919_0_main_Inc(EOS(STATIC_1919), i855, i855)
1919_0_main_Inc(EOS(STATIC_1919), i855, i855) → 1922_0_main_JMP(EOS(STATIC_1922), +(i855, -1), i855)
1922_0_main_JMP(EOS(STATIC_1922), i1049, i855) → 1929_0_main_Load(EOS(STATIC_1929), i1049, i855)
1929_0_main_Load(EOS(STATIC_1929), i1049, i855) → 1630_0_main_Load(EOS(STATIC_1630), i1049, i855)
1650_0_main_LE(EOS(STATIC_1650), i127, i855, i127, i855) → 1653_0_main_Inc(EOS(STATIC_1653), i127, i855) | >(i127, i855)
1653_0_main_Inc(EOS(STATIC_1653), i127, i855) → 1656_0_main_JMP(EOS(STATIC_1656), +(i127, -1), i855)
1656_0_main_JMP(EOS(STATIC_1656), i868, i855) → 1663_0_main_Load(EOS(STATIC_1663), i868, i855)
1663_0_main_Load(EOS(STATIC_1663), i868, i855) → 1630_0_main_Load(EOS(STATIC_1630), i868, i855)
R rules:

Combined rules. Obtained 3 conditional rules for P and 0 conditional rules for R.

P rules:
1633_0_main_Load(EOS(STATIC_1633), x0, x1, x0) → 1633_0_main_Load(EOS(STATIC_1633), x0, +(x1, -1), x0) | &&(>(x1, x0), <(0, +(x0, x1)))
1633_0_main_Load(EOS(STATIC_1633), x0, x0, x0) → 1633_0_main_Load(EOS(STATIC_1633), +(x0, -1), x0, +(x0, -1)) | <(0, +(x0, x0))
1633_0_main_Load(EOS(STATIC_1633), x0, x1, x0) → 1633_0_main_Load(EOS(STATIC_1633), +(x0, -1), x1, +(x0, -1)) | &&(<(x1, x0), <(0, +(x0, x1)))
R rules:

Filtered ground terms:

1633_0_main_Load(x1, x2, x3, x4) → 1633_0_main_Load(x2, x3, x4)
EOS(x1) → EOS
Cond_1633_0_main_Load2(x1, x2, x3, x4, x5) → Cond_1633_0_main_Load2(x1, x3, x4, x5)
Cond_1633_0_main_Load1(x1, x2, x3, x4, x5) → Cond_1633_0_main_Load1(x1, x3, x4, x5)
Cond_1633_0_main_Load(x1, x2, x3, x4, x5) → Cond_1633_0_main_Load(x1, x3, x4, x5)

Filtered duplicate args:

1633_0_main_Load(x1, x2, x3) → 1633_0_main_Load(x2, x3)
Cond_1633_0_main_Load(x1, x2, x3, x4) → Cond_1633_0_main_Load(x1, x3, x4)
Cond_1633_0_main_Load1(x1, x2, x3, x4) → Cond_1633_0_main_Load1(x1, x4)
Cond_1633_0_main_Load2(x1, x2, x3, x4) → Cond_1633_0_main_Load2(x1, x3, x4)

Combined rules. Obtained 3 conditional rules for P and 0 conditional rules for R.

P rules:
1633_0_main_Load(x1, x0) → 1633_0_main_Load(+(x1, -1), x0) | &&(>(x1, x0), <(0, +(x0, x1)))
1633_0_main_Load(x0, x0) → 1633_0_main_Load(x0, +(x0, -1)) | <(0, +(x0, x0))
1633_0_main_Load(x1, x0) → 1633_0_main_Load(x1, +(x0, -1)) | &&(<(x1, x0), <(0, +(x0, x1)))
R rules:

Finished conversion. Obtained 6 rules for P and 0 rules for R. System has predefined symbols.

P rules:
1633_0_MAIN_LOAD(x1, x0) → COND_1633_0_MAIN_LOAD(&&(>(x1, x0), <(0, +(x0, x1))), x1, x0)
COND_1633_0_MAIN_LOAD(TRUE, x1, x0) → 1633_0_MAIN_LOAD(+(x1, -1), x0)
1633_0_MAIN_LOAD(x0, x0) → COND_1633_0_MAIN_LOAD1(<(0, +(x0, x0)), x0, x0)
COND_1633_0_MAIN_LOAD1(TRUE, x0, x0) → 1633_0_MAIN_LOAD(x0, +(x0, -1))
1633_0_MAIN_LOAD(x1, x0) → COND_1633_0_MAIN_LOAD2(&&(<(x1, x0), <(0, +(x0, x1))), x1, x0)
COND_1633_0_MAIN_LOAD2(TRUE, x1, x0) → 1633_0_MAIN_LOAD(x1, +(x0, -1))
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): 1633_0_MAIN_LOAD(x1[0], x0[0]) → COND_1633_0_MAIN_LOAD(x1[0] > x0[0] && 0 < x0[0] + x1[0], x1[0], x0[0])
(1): COND_1633_0_MAIN_LOAD(TRUE, x1[1], x0[1]) → 1633_0_MAIN_LOAD(x1[1] + -1, x0[1])
(2): 1633_0_MAIN_LOAD(x0[2], x0[2]) → COND_1633_0_MAIN_LOAD1(0 < x0[2] + x0[2], x0[2], x0[2])
(3): COND_1633_0_MAIN_LOAD1(TRUE, x0[3], x0[3]) → 1633_0_MAIN_LOAD(x0[3], x0[3] + -1)
(4): 1633_0_MAIN_LOAD(x1[4], x0[4]) → COND_1633_0_MAIN_LOAD2(x1[4] < x0[4] && 0 < x0[4] + x1[4], x1[4], x0[4])
(5): COND_1633_0_MAIN_LOAD2(TRUE, x1[5], x0[5]) → 1633_0_MAIN_LOAD(x1[5], x0[5] + -1)

(0) -> (1), if (x1[0] > x0[0] && 0 < x0[0] + x1[0] ∧x1[0] →^* x1[1]∧x0[0] →^* x0[1])

(1) -> (0), if (x1[1] + -1 →^* x1[0]∧x0[1] →^* x0[0])

(1) -> (2), if (x1[1] + -1 →^* x0[2]∧x0[1] →^* x0[2])

(1) -> (4), if (x1[1] + -1 →^* x1[4]∧x0[1] →^* x0[4])

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

(3) -> (0), if (x0[3] →^* x1[0]∧x0[3] + -1 →^* x0[0])

(3) -> (2), if (x0[3] →^* x0[2]∧x0[3] + -1 →^* x0[2])

(3) -> (4), if (x0[3] →^* x1[4]∧x0[3] + -1 →^* x0[4])

(4) -> (5), if (x1[4] < x0[4] && 0 < x0[4] + x1[4] ∧x1[4] →^* x1[5]∧x0[4] →^* x0[5])

(5) -> (0), if (x1[5] →^* x1[0]∧x0[5] + -1 →^* x0[0])

(5) -> (2), if (x1[5] →^* x0[2]∧x0[5] + -1 →^* x0[2])

(5) -> (4), if (x1[5] →^* x1[4]∧x0[5] + -1 →^* x0[4])

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.IdpCand1ShapeHeuristic@4ad3597f Constraint Generator: NonInfConstraintGenerator: PathGenerator: MetricPathGenerator: Max Left Steps: 0 Max Right Steps: 0

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 1633_0_MAIN_LOAD(x1, x0) → COND_1633_0_MAIN_LOAD(&&(>(x1, x0), <(0, +(x0, x1))), x1, x0) the following chains were created:

We consider the chain 1633_0_MAIN_LOAD(x1[0], x0[0]) → COND_1633_0_MAIN_LOAD(&&(>(x1[0], x0[0]), <(0, +(x0[0], x1[0]))), x1[0], x0[0]), COND_1633_0_MAIN_LOAD(TRUE, x1[1], x0[1]) → 1633_0_MAIN_LOAD(+(x1[1], -1), x0[1]) which results in the following constraint:
(1)    (&&(>(x1[0], x0[0]), <(0, +(x0[0], x1[0])))=TRUE∧x1[0]=x1[1]∧x0[0]=x0[1] ⇒ 1633_0_MAIN_LOAD(x1[0], x0[0])≥NonInfC∧1633_0_MAIN_LOAD(x1[0], x0[0])≥COND_1633_0_MAIN_LOAD(&&(>(x1[0], x0[0]), <(0, +(x0[0], x1[0]))), x1[0], x0[0])∧(U^Increasing(COND_1633_0_MAIN_LOAD(&&(>(x1[0], x0[0]), <(0, +(x0[0], x1[0]))), x1[0], x0[0])), ≥))

We simplified constraint (1) using rules (IV), (IDP_BOOLEAN) which results in the following new constraint:
(2)    (>(x1[0], x0[0])=TRUE∧<(0, +(x0[0], x1[0]))=TRUE ⇒ 1633_0_MAIN_LOAD(x1[0], x0[0])≥NonInfC∧1633_0_MAIN_LOAD(x1[0], x0[0])≥COND_1633_0_MAIN_LOAD(&&(>(x1[0], x0[0]), <(0, +(x0[0], x1[0]))), x1[0], x0[0])∧(U^Increasing(COND_1633_0_MAIN_LOAD(&&(>(x1[0], x0[0]), <(0, +(x0[0], x1[0]))), x1[0], x0[0])), ≥))

We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(3)    (x1[0] + [-1] + [-1]x0[0] ≥ 0∧x0[0] + [-1] + x1[0] ≥ 0 ⇒ (U^Increasing(COND_1633_0_MAIN_LOAD(&&(>(x1[0], x0[0]), <(0, +(x0[0], x1[0]))), x1[0], x0[0])), ≥)∧[(-1)Bound*bni_14] + [(2)bni_14]x0[0] + [(2)bni_14]x1[0] ≥ 0∧[(-1)bso_15] ≥ 0)

We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(4)    (x1[0] + [-1] + [-1]x0[0] ≥ 0∧x0[0] + [-1] + x1[0] ≥ 0 ⇒ (U^Increasing(COND_1633_0_MAIN_LOAD(&&(>(x1[0], x0[0]), <(0, +(x0[0], x1[0]))), x1[0], x0[0])), ≥)∧[(-1)Bound*bni_14] + [(2)bni_14]x0[0] + [(2)bni_14]x1[0] ≥ 0∧[(-1)bso_15] ≥ 0)

We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(5)    (x1[0] + [-1] + [-1]x0[0] ≥ 0∧x0[0] + [-1] + x1[0] ≥ 0 ⇒ (U^Increasing(COND_1633_0_MAIN_LOAD(&&(>(x1[0], x0[0]), <(0, +(x0[0], x1[0]))), x1[0], x0[0])), ≥)∧[(-1)Bound*bni_14] + [(2)bni_14]x0[0] + [(2)bni_14]x1[0] ≥ 0∧[(-1)bso_15] ≥ 0)

We simplified constraint (5) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(6)    (x1[0] ≥ 0∧[2]x0[0] + x1[0] ≥ 0 ⇒ (U^Increasing(COND_1633_0_MAIN_LOAD(&&(>(x1[0], x0[0]), <(0, +(x0[0], x1[0]))), x1[0], x0[0])), ≥)∧[(-1)Bound*bni_14 + (2)bni_14] + [(4)bni_14]x0[0] + [(2)bni_14]x1[0] ≥ 0∧[(-1)bso_15] ≥ 0)

We simplified constraint (6) using rule (IDP_SMT_SPLIT) which results in the following new constraints:
(7)    (x1[0] ≥ 0∧[2]x0[0] + x1[0] ≥ 0∧x0[0] ≥ 0 ⇒ (U^Increasing(COND_1633_0_MAIN_LOAD(&&(>(x1[0], x0[0]), <(0, +(x0[0], x1[0]))), x1[0], x0[0])), ≥)∧[(-1)Bound*bni_14 + (2)bni_14] + [(4)bni_14]x0[0] + [(2)bni_14]x1[0] ≥ 0∧[(-1)bso_15] ≥ 0)

(8)    (x1[0] ≥ 0∧[-2]x0[0] + x1[0] ≥ 0∧x0[0] ≥ 0 ⇒ (U^Increasing(COND_1633_0_MAIN_LOAD(&&(>(x1[0], x0[0]), <(0, +(x0[0], x1[0]))), x1[0], x0[0])), ≥)∧[(-1)Bound*bni_14 + (2)bni_14] + [(-4)bni_14]x0[0] + [(2)bni_14]x1[0] ≥ 0∧[(-1)bso_15] ≥ 0)

We simplified constraint (8) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(9)    ([2]x0[0] + x1[0] ≥ 0∧x1[0] ≥ 0∧x0[0] ≥ 0 ⇒ (U^Increasing(COND_1633_0_MAIN_LOAD(&&(>(x1[0], x0[0]), <(0, +(x0[0], x1[0]))), x1[0], x0[0])), ≥)∧[(-1)Bound*bni_14 + (2)bni_14] + [(2)bni_14]x1[0] ≥ 0∧[(-1)bso_15] ≥ 0)

For Pair COND_1633_0_MAIN_LOAD(TRUE, x1, x0) → 1633_0_MAIN_LOAD(+(x1, -1), x0) the following chains were created:

We consider the chain COND_1633_0_MAIN_LOAD(TRUE, x1[1], x0[1]) → 1633_0_MAIN_LOAD(+(x1[1], -1), x0[1]) which results in the following constraint:
(10)    (COND_1633_0_MAIN_LOAD(TRUE, x1[1], x0[1])≥NonInfC∧COND_1633_0_MAIN_LOAD(TRUE, x1[1], x0[1])≥1633_0_MAIN_LOAD(+(x1[1], -1), x0[1])∧(U^Increasing(1633_0_MAIN_LOAD(+(x1[1], -1), x0[1])), ≥))

We simplified constraint (10) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(11)    ((U^Increasing(1633_0_MAIN_LOAD(+(x1[1], -1), x0[1])), ≥)∧[bni_16] = 0∧[2 + (-1)bso_17] ≥ 0)

We simplified constraint (11) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(12)    ((U^Increasing(1633_0_MAIN_LOAD(+(x1[1], -1), x0[1])), ≥)∧[bni_16] = 0∧[2 + (-1)bso_17] ≥ 0)

We simplified constraint (12) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(13)    ((U^Increasing(1633_0_MAIN_LOAD(+(x1[1], -1), x0[1])), ≥)∧[bni_16] = 0∧[2 + (-1)bso_17] ≥ 0)

We simplified constraint (13) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint:
(14)    ((U^Increasing(1633_0_MAIN_LOAD(+(x1[1], -1), x0[1])), ≥)∧[bni_16] = 0∧0 = 0∧0 = 0∧[2 + (-1)bso_17] ≥ 0)

For Pair 1633_0_MAIN_LOAD(x0, x0) → COND_1633_0_MAIN_LOAD1(<(0, +(x0, x0)), x0, x0) the following chains were created:

We consider the chain 1633_0_MAIN_LOAD(x0[2], x0[2]) → COND_1633_0_MAIN_LOAD1(<(0, +(x0[2], x0[2])), x0[2], x0[2]), COND_1633_0_MAIN_LOAD1(TRUE, x0[3], x0[3]) → 1633_0_MAIN_LOAD(x0[3], +(x0[3], -1)) which results in the following constraint:
(15)    (<(0, +(x0[2], x0[2]))=TRUE∧x0[2]=x0[3] ⇒ 1633_0_MAIN_LOAD(x0[2], x0[2])≥NonInfC∧1633_0_MAIN_LOAD(x0[2], x0[2])≥COND_1633_0_MAIN_LOAD1(<(0, +(x0[2], x0[2])), x0[2], x0[2])∧(U^Increasing(COND_1633_0_MAIN_LOAD1(<(0, +(x0[2], x0[2])), x0[2], x0[2])), ≥))

We simplified constraint (15) using rule (IV) which results in the following new constraint:
(16)    (<(0, +(x0[2], x0[2]))=TRUE ⇒ 1633_0_MAIN_LOAD(x0[2], x0[2])≥NonInfC∧1633_0_MAIN_LOAD(x0[2], x0[2])≥COND_1633_0_MAIN_LOAD1(<(0, +(x0[2], x0[2])), x0[2], x0[2])∧(U^Increasing(COND_1633_0_MAIN_LOAD1(<(0, +(x0[2], x0[2])), x0[2], x0[2])), ≥))

We simplified constraint (16) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(17)    ([2]x0[2] + [-1] ≥ 0 ⇒ (U^Increasing(COND_1633_0_MAIN_LOAD1(<(0, +(x0[2], x0[2])), x0[2], x0[2])), ≥)∧[(-1)Bound*bni_18] + [(4)bni_18]x0[2] ≥ 0∧[1 + (-1)bso_19] ≥ 0)

We simplified constraint (17) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(18)    ([2]x0[2] + [-1] ≥ 0 ⇒ (U^Increasing(COND_1633_0_MAIN_LOAD1(<(0, +(x0[2], x0[2])), x0[2], x0[2])), ≥)∧[(-1)Bound*bni_18] + [(4)bni_18]x0[2] ≥ 0∧[1 + (-1)bso_19] ≥ 0)

We simplified constraint (18) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(19)    ([2]x0[2] + [-1] ≥ 0 ⇒ (U^Increasing(COND_1633_0_MAIN_LOAD1(<(0, +(x0[2], x0[2])), x0[2], x0[2])), ≥)∧[(-1)Bound*bni_18] + [(4)bni_18]x0[2] ≥ 0∧[1 + (-1)bso_19] ≥ 0)

For Pair COND_1633_0_MAIN_LOAD1(TRUE, x0, x0) → 1633_0_MAIN_LOAD(x0, +(x0, -1)) the following chains were created:

We consider the chain COND_1633_0_MAIN_LOAD1(TRUE, x0[3], x0[3]) → 1633_0_MAIN_LOAD(x0[3], +(x0[3], -1)) which results in the following constraint:
(20)    (COND_1633_0_MAIN_LOAD1(TRUE, x0[3], x0[3])≥NonInfC∧COND_1633_0_MAIN_LOAD1(TRUE, x0[3], x0[3])≥1633_0_MAIN_LOAD(x0[3], +(x0[3], -1))∧(U^Increasing(1633_0_MAIN_LOAD(x0[3], +(x0[3], -1))), ≥))

We simplified constraint (20) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(21)    ((U^Increasing(1633_0_MAIN_LOAD(x0[3], +(x0[3], -1))), ≥)∧[bni_20] = 0∧[1 + (-1)bso_21] ≥ 0)

We simplified constraint (21) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(22)    ((U^Increasing(1633_0_MAIN_LOAD(x0[3], +(x0[3], -1))), ≥)∧[bni_20] = 0∧[1 + (-1)bso_21] ≥ 0)

We simplified constraint (22) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(23)    ((U^Increasing(1633_0_MAIN_LOAD(x0[3], +(x0[3], -1))), ≥)∧[bni_20] = 0∧[1 + (-1)bso_21] ≥ 0)

We simplified constraint (23) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint:
(24)    ((U^Increasing(1633_0_MAIN_LOAD(x0[3], +(x0[3], -1))), ≥)∧[bni_20] = 0∧0 = 0∧[1 + (-1)bso_21] ≥ 0)

For Pair 1633_0_MAIN_LOAD(x1, x0) → COND_1633_0_MAIN_LOAD2(&&(<(x1, x0), <(0, +(x0, x1))), x1, x0) the following chains were created:

We consider the chain 1633_0_MAIN_LOAD(x1[4], x0[4]) → COND_1633_0_MAIN_LOAD2(&&(<(x1[4], x0[4]), <(0, +(x0[4], x1[4]))), x1[4], x0[4]), COND_1633_0_MAIN_LOAD2(TRUE, x1[5], x0[5]) → 1633_0_MAIN_LOAD(x1[5], +(x0[5], -1)) which results in the following constraint:
(25)    (&&(<(x1[4], x0[4]), <(0, +(x0[4], x1[4])))=TRUE∧x1[4]=x1[5]∧x0[4]=x0[5] ⇒ 1633_0_MAIN_LOAD(x1[4], x0[4])≥NonInfC∧1633_0_MAIN_LOAD(x1[4], x0[4])≥COND_1633_0_MAIN_LOAD2(&&(<(x1[4], x0[4]), <(0, +(x0[4], x1[4]))), x1[4], x0[4])∧(U^Increasing(COND_1633_0_MAIN_LOAD2(&&(<(x1[4], x0[4]), <(0, +(x0[4], x1[4]))), x1[4], x0[4])), ≥))

We simplified constraint (25) using rules (IV), (IDP_BOOLEAN) which results in the following new constraint:
(26)    (<(x1[4], x0[4])=TRUE∧<(0, +(x0[4], x1[4]))=TRUE ⇒ 1633_0_MAIN_LOAD(x1[4], x0[4])≥NonInfC∧1633_0_MAIN_LOAD(x1[4], x0[4])≥COND_1633_0_MAIN_LOAD2(&&(<(x1[4], x0[4]), <(0, +(x0[4], x1[4]))), x1[4], x0[4])∧(U^Increasing(COND_1633_0_MAIN_LOAD2(&&(<(x1[4], x0[4]), <(0, +(x0[4], x1[4]))), x1[4], x0[4])), ≥))

We simplified constraint (26) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(27)    (x0[4] + [-1] + [-1]x1[4] ≥ 0∧x0[4] + [-1] + x1[4] ≥ 0 ⇒ (U^Increasing(COND_1633_0_MAIN_LOAD2(&&(<(x1[4], x0[4]), <(0, +(x0[4], x1[4]))), x1[4], x0[4])), ≥)∧[(-1)Bound*bni_22] + [(2)bni_22]x0[4] + [(2)bni_22]x1[4] ≥ 0∧[1 + (-1)bso_23] ≥ 0)

We simplified constraint (27) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(28)    (x0[4] + [-1] + [-1]x1[4] ≥ 0∧x0[4] + [-1] + x1[4] ≥ 0 ⇒ (U^Increasing(COND_1633_0_MAIN_LOAD2(&&(<(x1[4], x0[4]), <(0, +(x0[4], x1[4]))), x1[4], x0[4])), ≥)∧[(-1)Bound*bni_22] + [(2)bni_22]x0[4] + [(2)bni_22]x1[4] ≥ 0∧[1 + (-1)bso_23] ≥ 0)

We simplified constraint (28) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(29)    (x0[4] + [-1] + [-1]x1[4] ≥ 0∧x0[4] + [-1] + x1[4] ≥ 0 ⇒ (U^Increasing(COND_1633_0_MAIN_LOAD2(&&(<(x1[4], x0[4]), <(0, +(x0[4], x1[4]))), x1[4], x0[4])), ≥)∧[(-1)Bound*bni_22] + [(2)bni_22]x0[4] + [(2)bni_22]x1[4] ≥ 0∧[1 + (-1)bso_23] ≥ 0)

We simplified constraint (29) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(30)    (x0[4] ≥ 0∧[2]x1[4] + x0[4] ≥ 0 ⇒ (U^Increasing(COND_1633_0_MAIN_LOAD2(&&(<(x1[4], x0[4]), <(0, +(x0[4], x1[4]))), x1[4], x0[4])), ≥)∧[(-1)Bound*bni_22 + (2)bni_22] + [(4)bni_22]x1[4] + [(2)bni_22]x0[4] ≥ 0∧[1 + (-1)bso_23] ≥ 0)

We simplified constraint (30) using rule (IDP_SMT_SPLIT) which results in the following new constraints:
(31)    (x0[4] ≥ 0∧[2]x1[4] + x0[4] ≥ 0∧x1[4] ≥ 0 ⇒ (U^Increasing(COND_1633_0_MAIN_LOAD2(&&(<(x1[4], x0[4]), <(0, +(x0[4], x1[4]))), x1[4], x0[4])), ≥)∧[(-1)Bound*bni_22 + (2)bni_22] + [(4)bni_22]x1[4] + [(2)bni_22]x0[4] ≥ 0∧[1 + (-1)bso_23] ≥ 0)

(32)    (x0[4] ≥ 0∧[-2]x1[4] + x0[4] ≥ 0∧x1[4] ≥ 0 ⇒ (U^Increasing(COND_1633_0_MAIN_LOAD2(&&(<(x1[4], x0[4]), <(0, +(x0[4], x1[4]))), x1[4], x0[4])), ≥)∧[(-1)Bound*bni_22 + (2)bni_22] + [(-4)bni_22]x1[4] + [(2)bni_22]x0[4] ≥ 0∧[1 + (-1)bso_23] ≥ 0)

We simplified constraint (32) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(33)    ([2]x1[4] + x0[4] ≥ 0∧x0[4] ≥ 0∧x1[4] ≥ 0 ⇒ (U^Increasing(COND_1633_0_MAIN_LOAD2(&&(<(x1[4], x0[4]), <(0, +(x0[4], x1[4]))), x1[4], x0[4])), ≥)∧[(-1)Bound*bni_22 + (2)bni_22] + [(2)bni_22]x0[4] ≥ 0∧[1 + (-1)bso_23] ≥ 0)

For Pair COND_1633_0_MAIN_LOAD2(TRUE, x1, x0) → 1633_0_MAIN_LOAD(x1, +(x0, -1)) the following chains were created:

We consider the chain COND_1633_0_MAIN_LOAD2(TRUE, x1[5], x0[5]) → 1633_0_MAIN_LOAD(x1[5], +(x0[5], -1)) which results in the following constraint:
(34)    (COND_1633_0_MAIN_LOAD2(TRUE, x1[5], x0[5])≥NonInfC∧COND_1633_0_MAIN_LOAD2(TRUE, x1[5], x0[5])≥1633_0_MAIN_LOAD(x1[5], +(x0[5], -1))∧(U^Increasing(1633_0_MAIN_LOAD(x1[5], +(x0[5], -1))), ≥))

We simplified constraint (34) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(35)    ((U^Increasing(1633_0_MAIN_LOAD(x1[5], +(x0[5], -1))), ≥)∧[bni_24] = 0∧[1 + (-1)bso_25] ≥ 0)

We simplified constraint (35) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(36)    ((U^Increasing(1633_0_MAIN_LOAD(x1[5], +(x0[5], -1))), ≥)∧[bni_24] = 0∧[1 + (-1)bso_25] ≥ 0)

We simplified constraint (36) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(37)    ((U^Increasing(1633_0_MAIN_LOAD(x1[5], +(x0[5], -1))), ≥)∧[bni_24] = 0∧[1 + (-1)bso_25] ≥ 0)

We simplified constraint (37) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint:
(38)    ((U^Increasing(1633_0_MAIN_LOAD(x1[5], +(x0[5], -1))), ≥)∧[bni_24] = 0∧0 = 0∧0 = 0∧[1 + (-1)bso_25] ≥ 0)

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

1633_0_MAIN_LOAD(x1, x0) → COND_1633_0_MAIN_LOAD(&&(>(x1, x0), <(0, +(x0, x1))), x1, x0)
- (x1[0] ≥ 0∧[2]x0[0] + x1[0] ≥ 0∧x0[0] ≥ 0 ⇒ (U^Increasing(COND_1633_0_MAIN_LOAD(&&(>(x1[0], x0[0]), <(0, +(x0[0], x1[0]))), x1[0], x0[0])), ≥)∧[(-1)Bound*bni_14 + (2)bni_14] + [(4)bni_14]x0[0] + [(2)bni_14]x1[0] ≥ 0∧[(-1)bso_15] ≥ 0)
- ([2]x0[0] + x1[0] ≥ 0∧x1[0] ≥ 0∧x0[0] ≥ 0 ⇒ (U^Increasing(COND_1633_0_MAIN_LOAD(&&(>(x1[0], x0[0]), <(0, +(x0[0], x1[0]))), x1[0], x0[0])), ≥)∧[(-1)Bound*bni_14 + (2)bni_14] + [(2)bni_14]x1[0] ≥ 0∧[(-1)bso_15] ≥ 0)
COND_1633_0_MAIN_LOAD(TRUE, x1, x0) → 1633_0_MAIN_LOAD(+(x1, -1), x0)
- ((U^Increasing(1633_0_MAIN_LOAD(+(x1[1], -1), x0[1])), ≥)∧[bni_16] = 0∧0 = 0∧0 = 0∧[2 + (-1)bso_17] ≥ 0)
1633_0_MAIN_LOAD(x0, x0) → COND_1633_0_MAIN_LOAD1(<(0, +(x0, x0)), x0, x0)
- ([2]x0[2] + [-1] ≥ 0 ⇒ (U^Increasing(COND_1633_0_MAIN_LOAD1(<(0, +(x0[2], x0[2])), x0[2], x0[2])), ≥)∧[(-1)Bound*bni_18] + [(4)bni_18]x0[2] ≥ 0∧[1 + (-1)bso_19] ≥ 0)
COND_1633_0_MAIN_LOAD1(TRUE, x0, x0) → 1633_0_MAIN_LOAD(x0, +(x0, -1))
- ((U^Increasing(1633_0_MAIN_LOAD(x0[3], +(x0[3], -1))), ≥)∧[bni_20] = 0∧0 = 0∧[1 + (-1)bso_21] ≥ 0)
1633_0_MAIN_LOAD(x1, x0) → COND_1633_0_MAIN_LOAD2(&&(<(x1, x0), <(0, +(x0, x1))), x1, x0)
- (x0[4] ≥ 0∧[2]x1[4] + x0[4] ≥ 0∧x1[4] ≥ 0 ⇒ (U^Increasing(COND_1633_0_MAIN_LOAD2(&&(<(x1[4], x0[4]), <(0, +(x0[4], x1[4]))), x1[4], x0[4])), ≥)∧[(-1)Bound*bni_22 + (2)bni_22] + [(4)bni_22]x1[4] + [(2)bni_22]x0[4] ≥ 0∧[1 + (-1)bso_23] ≥ 0)
- ([2]x1[4] + x0[4] ≥ 0∧x0[4] ≥ 0∧x1[4] ≥ 0 ⇒ (U^Increasing(COND_1633_0_MAIN_LOAD2(&&(<(x1[4], x0[4]), <(0, +(x0[4], x1[4]))), x1[4], x0[4])), ≥)∧[(-1)Bound*bni_22 + (2)bni_22] + [(2)bni_22]x0[4] ≥ 0∧[1 + (-1)bso_23] ≥ 0)
COND_1633_0_MAIN_LOAD2(TRUE, x1, x0) → 1633_0_MAIN_LOAD(x1, +(x0, -1))
- ((U^Increasing(1633_0_MAIN_LOAD(x1[5], +(x0[5], -1))), ≥)∧[bni_24] = 0∧0 = 0∧0 = 0∧[1 + (-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(1633_0_MAIN_LOAD(x₁, x₂)) = [2]x₂ + [2]x₁
POL(COND_1633_0_MAIN_LOAD(x₁, x₂, x₃)) = [2]x₃ + [2]x₂
POL(&&(x₁, x₂)) = [-1]
POL(>(x₁, x₂)) = [-1]
POL(<(x₁, x₂)) = [-1]
POL(0) = 0
POL(+(x₁, x₂)) = x₁ + x₂
POL(-1) = [-1]
POL(COND_1633_0_MAIN_LOAD1(x₁, x₂, x₃)) = [-1] + [2]x₃ + [2]x₂
POL(COND_1633_0_MAIN_LOAD2(x₁, x₂, x₃)) = [-1] + [2]x₃ + [2]x₂

The following pairs are in P_>:

COND_1633_0_MAIN_LOAD(TRUE, x1[1], x0[1]) → 1633_0_MAIN_LOAD(+(x1[1], -1), x0[1])
1633_0_MAIN_LOAD(x0[2], x0[2]) → COND_1633_0_MAIN_LOAD1(<(0, +(x0[2], x0[2])), x0[2], x0[2])
COND_1633_0_MAIN_LOAD1(TRUE, x0[3], x0[3]) → 1633_0_MAIN_LOAD(x0[3], +(x0[3], -1))
1633_0_MAIN_LOAD(x1[4], x0[4]) → COND_1633_0_MAIN_LOAD2(&&(<(x1[4], x0[4]), <(0, +(x0[4], x1[4]))), x1[4], x0[4])
COND_1633_0_MAIN_LOAD2(TRUE, x1[5], x0[5]) → 1633_0_MAIN_LOAD(x1[5], +(x0[5], -1))

The following pairs are in P_bound:

1633_0_MAIN_LOAD(x1[0], x0[0]) → COND_1633_0_MAIN_LOAD(&&(>(x1[0], x0[0]), <(0, +(x0[0], x1[0]))), x1[0], x0[0])
1633_0_MAIN_LOAD(x0[2], x0[2]) → COND_1633_0_MAIN_LOAD1(<(0, +(x0[2], x0[2])), x0[2], x0[2])
1633_0_MAIN_LOAD(x1[4], x0[4]) → COND_1633_0_MAIN_LOAD2(&&(<(x1[4], x0[4]), <(0, +(x0[4], x1[4]))), x1[4], x0[4])

The following pairs are in P_≥:

1633_0_MAIN_LOAD(x1[0], x0[0]) → COND_1633_0_MAIN_LOAD(&&(>(x1[0], x0[0]), <(0, +(x0[0], x1[0]))), x1[0], x0[0])

There are no usable rules.

(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:

Boolean, Integer

(10) IDependencyGraphProof (EQUIVALENT transformation)

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

(11) TRUE

(12) Obligation:

IDP problem:
The following function symbols are pre-defined:

!=	~	Neq: (Integer, Integer) -> Boolean
*	~	Mul: (Integer, Integer) -> Integer
>=	~	Ge: (Integer, Integer) -> Boolean
-1	~	UnaryMinus: (Integer) -> Integer
\|	~	Bwor: (Integer, Integer) -> Integer
/	~	Div: (Integer, Integer) -> Integer
=	~	Eq: (Integer, Integer) -> Boolean
	~	Bwxor: (Integer, Integer) -> Integer
\|\|	~	Lor: (Boolean, Boolean) -> Boolean
!	~	Lnot: (Boolean) -> Boolean
<	~	Lt: (Integer, Integer) -> Boolean
-	~	Sub: (Integer, Integer) -> Integer
<=	~	Le: (Integer, Integer) -> Boolean
>	~	Gt: (Integer, Integer) -> Boolean
~	~	Bwnot: (Integer) -> Integer
%	~	Mod: (Integer, Integer) -> Integer
&	~	Bwand: (Integer, Integer) -> Integer
+	~	Add: (Integer, Integer) -> Integer
&&	~	Land: (Boolean, Boolean) -> Boolean

The following domains are used:

Integer

R is empty.

The integer pair graph contains the following rules and edges:
(1): COND_1633_0_MAIN_LOAD(TRUE, x1[1], x0[1]) → 1633_0_MAIN_LOAD(x1[1] + -1, x0[1])
(3): COND_1633_0_MAIN_LOAD1(TRUE, x0[3], x0[3]) → 1633_0_MAIN_LOAD(x0[3], x0[3] + -1)
(5): COND_1633_0_MAIN_LOAD2(TRUE, x1[5], x0[5]) → 1633_0_MAIN_LOAD(x1[5], x0[5] + -1)

The set Q is empty.

(13) IDependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 3 less nodes.

(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) TRUE

(12) Obligation:

(13) IDependencyGraphProof (EQUIVALENT transformation)

(14) TRUE