Termination proof of /tmp/tmp0segHD/PastaB10.jar

Termination of the given JBC could be proven:

0 JBC

↳1 JBCToGraph (⇒, 510 ms)

↳2 JBCTerminationGraph

↳3 TerminationGraphToSCCProof (⇒, 0 ms)

↳4 JBCTerminationSCC

↳5 SCCToIDPv1Proof (⇒, 100 ms)

↳6 IDP

↳7 IDPNonInfProof (⇒, 150 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: PastaB10

/**
 * 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 PastaB10 {
    public static void main(String[] args) {
        Random.args = args;
        int x = Random.random();
        int y = Random.random();

        while (x + y > 0) {
            if (x > 0) {
                x--;
            } else if (y > 0) {
                y--;
            } else {
                continue;
            }            
        }
    }
}


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:
PastaB10.main([Ljava/lang/String;)V: Graph of 187 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: PastaB10.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 22 rules for P and 0 rules for R.

P rules:
304_0_main_Load(EOS(STATIC_304), i18, i47, i18) → 306_0_main_IntArithmetic(EOS(STATIC_306), i18, i47, i18, i47)
306_0_main_IntArithmetic(EOS(STATIC_306), i18, i47, i18, i47) → 319_0_main_LE(EOS(STATIC_319), i18, i47, +(i18, i47)) | &&(>=(i18, 0), >=(i47, 0))
319_0_main_LE(EOS(STATIC_319), i18, i47, i56) → 328_0_main_LE(EOS(STATIC_328), i18, i47, i56)
328_0_main_LE(EOS(STATIC_328), i18, i47, i56) → 336_0_main_Load(EOS(STATIC_336), i18, i47) | >(i56, 0)
336_0_main_Load(EOS(STATIC_336), i18, i47) → 347_0_main_LE(EOS(STATIC_347), i18, i47, i18)
347_0_main_LE(EOS(STATIC_347), matching1, i47, matching2) → 356_0_main_LE(EOS(STATIC_356), 0, i47, 0) | &&(=(matching1, 0), =(matching2, 0))
347_0_main_LE(EOS(STATIC_347), i60, i47, i60) → 358_0_main_LE(EOS(STATIC_358), i60, i47, i60)
356_0_main_LE(EOS(STATIC_356), matching1, i47, matching2) → 367_0_main_Load(EOS(STATIC_367), 0, i47) | &&(&&(<=(0, 0), =(matching1, 0)), =(matching2, 0))
367_0_main_Load(EOS(STATIC_367), matching1, i47) → 378_0_main_LE(EOS(STATIC_378), 0, i47, i47) | =(matching1, 0)
378_0_main_LE(EOS(STATIC_378), matching1, matching2, matching3) → 393_0_main_LE(EOS(STATIC_393), 0, 0, 0) | &&(&&(=(matching1, 0), =(matching2, 0)), =(matching3, 0))
378_0_main_LE(EOS(STATIC_378), matching1, i67, i67) → 394_0_main_LE(EOS(STATIC_394), 0, i67, i67) | =(matching1, 0)
393_0_main_LE(EOS(STATIC_393), matching1, matching2, matching3) → 428_0_main_Load(EOS(STATIC_428), 0, 0) | &&(&&(&&(<=(0, 0), =(matching1, 0)), =(matching2, 0)), =(matching3, 0))
428_0_main_Load(EOS(STATIC_428), matching1, matching2) → 297_0_main_Load(EOS(STATIC_297), 0, 0) | &&(=(matching1, 0), =(matching2, 0))
297_0_main_Load(EOS(STATIC_297), i18, i47) → 304_0_main_Load(EOS(STATIC_304), i18, i47, i18)
394_0_main_LE(EOS(STATIC_394), matching1, i67, i67) → 430_0_main_Inc(EOS(STATIC_430), 0, i67) | &&(>(i67, 0), =(matching1, 0))
430_0_main_Inc(EOS(STATIC_430), matching1, i67) → 993_0_main_JMP(EOS(STATIC_993), 0, +(i67, -1)) | &&(>(i67, 0), =(matching1, 0))
993_0_main_JMP(EOS(STATIC_993), matching1, i218) → 998_0_main_Load(EOS(STATIC_998), 0, i218) | =(matching1, 0)
998_0_main_Load(EOS(STATIC_998), matching1, i218) → 297_0_main_Load(EOS(STATIC_297), 0, i218) | =(matching1, 0)
358_0_main_LE(EOS(STATIC_358), i60, i47, i60) → 369_0_main_Inc(EOS(STATIC_369), i60, i47) | >(i60, 0)
369_0_main_Inc(EOS(STATIC_369), i60, i47) → 380_0_main_JMP(EOS(STATIC_380), +(i60, -1), i47) | >(i60, 0)
380_0_main_JMP(EOS(STATIC_380), i64, i47) → 415_0_main_Load(EOS(STATIC_415), i64, i47)
415_0_main_Load(EOS(STATIC_415), i64, i47) → 297_0_main_Load(EOS(STATIC_297), i64, i47)
R rules:

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

P rules:
304_0_main_Load(EOS(STATIC_304), x0, x1, x0) → 304_0_main_Load(EOS(STATIC_304), 0, 0, 0) | FALSE
304_0_main_Load(EOS(STATIC_304), 0, x1, 0) → 304_0_main_Load(EOS(STATIC_304), 0, +(x1, -1), 0) | >(x1, 0)
304_0_main_Load(EOS(STATIC_304), x0, x1, x0) → 304_0_main_Load(EOS(STATIC_304), +(x0, -1), x1, +(x0, -1)) | &&(&&(>(+(x1, 1), 0), >(x0, 0)), <(0, +(x0, x1)))
R rules:

Filtered ground terms:

304_0_main_Load(x1, x2, x3, x4) → 304_0_main_Load(x2, x3, x4)
EOS(x1) → EOS
Cond_304_0_main_Load1(x1, x2, x3, x4, x5) → Cond_304_0_main_Load1(x1, x3, x4, x5)
Cond_304_0_main_Load(x1, x2, x3, x4, x5) → Cond_304_0_main_Load(x1, x4)

Filtered duplicate args:

304_0_main_Load(x1, x2, x3) → 304_0_main_Load(x2, x3)
Cond_304_0_main_Load1(x1, x2, x3, x4) → Cond_304_0_main_Load1(x1, x3, x4)

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

P rules:
304_0_main_Load(x1, 0) → 304_0_main_Load(+(x1, -1), 0) | >(x1, 0)
304_0_main_Load(x1, x0) → 304_0_main_Load(x1, +(x0, -1)) | &&(&&(>(x1, -1), >(x0, 0)), <(0, +(x0, x1)))
R rules:

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

P rules:
304_0_MAIN_LOAD(x1, 0) → COND_304_0_MAIN_LOAD(>(x1, 0), x1, 0)
COND_304_0_MAIN_LOAD(TRUE, x1, 0) → 304_0_MAIN_LOAD(+(x1, -1), 0)
304_0_MAIN_LOAD(x1, x0) → COND_304_0_MAIN_LOAD1(&&(&&(>(x1, -1), >(x0, 0)), <(0, +(x0, x1))), x1, x0)
COND_304_0_MAIN_LOAD1(TRUE, x1, x0) → 304_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:

Integer, Boolean

R is empty.

The integer pair graph contains the following rules and edges:
(0): 304_0_MAIN_LOAD(x1[0], 0) → COND_304_0_MAIN_LOAD(x1[0] > 0, x1[0], 0)
(1): COND_304_0_MAIN_LOAD(TRUE, x1[1], 0) → 304_0_MAIN_LOAD(x1[1] + -1, 0)
(2): 304_0_MAIN_LOAD(x1[2], x0[2]) → COND_304_0_MAIN_LOAD1(x1[2] > -1 && x0[2] > 0 && 0 < x0[2] + x1[2], x1[2], x0[2])
(3): COND_304_0_MAIN_LOAD1(TRUE, x1[3], x0[3]) → 304_0_MAIN_LOAD(x1[3], x0[3] + -1)

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

(1) -> (0), if x1[1] + -1 →^* x1[0]

(1) -> (2), if (x1[1] + -1 →^* x1[2]∧0 →^* x0[2])

(2) -> (3), if (x1[2] > -1 && x0[2] > 0 && 0 < x0[2] + x1[2] ∧x1[2] →^* x1[3]∧x0[2] →^* x0[3])

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

(3) -> (2), if (x1[3] →^* x1[2]∧x0[3] + -1 →^* 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.IdpCand1ShapeHeuristic@30250179 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 304_0_MAIN_LOAD(x1, 0) → COND_304_0_MAIN_LOAD(>(x1, 0), x1, 0) the following chains were created:

We consider the chain 304_0_MAIN_LOAD(x1[0], 0) → COND_304_0_MAIN_LOAD(>(x1[0], 0), x1[0], 0), COND_304_0_MAIN_LOAD(TRUE, x1[1], 0) → 304_0_MAIN_LOAD(+(x1[1], -1), 0) which results in the following constraint:
(1)    (>(x1[0], 0)=TRUE∧x1[0]=x1[1] ⇒ 304_0_MAIN_LOAD(x1[0], 0)≥NonInfC∧304_0_MAIN_LOAD(x1[0], 0)≥COND_304_0_MAIN_LOAD(>(x1[0], 0), x1[0], 0)∧(U^Increasing(COND_304_0_MAIN_LOAD(>(x1[0], 0), x1[0], 0)), ≥))

We simplified constraint (1) using rule (IV) which results in the following new constraint:
(2)    (>(x1[0], 0)=TRUE ⇒ 304_0_MAIN_LOAD(x1[0], 0)≥NonInfC∧304_0_MAIN_LOAD(x1[0], 0)≥COND_304_0_MAIN_LOAD(>(x1[0], 0), x1[0], 0)∧(U^Increasing(COND_304_0_MAIN_LOAD(>(x1[0], 0), x1[0], 0)), ≥))

We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(3)    (x1[0] + [-1] ≥ 0 ⇒ (U^Increasing(COND_304_0_MAIN_LOAD(>(x1[0], 0), x1[0], 0)), ≥)∧[bni_12 + (-1)Bound*bni_12] + [bni_12]x1[0] ≥ 0∧[(-1)bso_13] ≥ 0)

We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(4)    (x1[0] + [-1] ≥ 0 ⇒ (U^Increasing(COND_304_0_MAIN_LOAD(>(x1[0], 0), x1[0], 0)), ≥)∧[bni_12 + (-1)Bound*bni_12] + [bni_12]x1[0] ≥ 0∧[(-1)bso_13] ≥ 0)

We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(5)    (x1[0] + [-1] ≥ 0 ⇒ (U^Increasing(COND_304_0_MAIN_LOAD(>(x1[0], 0), x1[0], 0)), ≥)∧[bni_12 + (-1)Bound*bni_12] + [bni_12]x1[0] ≥ 0∧[(-1)bso_13] ≥ 0)

We simplified constraint (5) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(6)    (x1[0] ≥ 0 ⇒ (U^Increasing(COND_304_0_MAIN_LOAD(>(x1[0], 0), x1[0], 0)), ≥)∧[(2)bni_12 + (-1)Bound*bni_12] + [bni_12]x1[0] ≥ 0∧[(-1)bso_13] ≥ 0)

For Pair COND_304_0_MAIN_LOAD(TRUE, x1, 0) → 304_0_MAIN_LOAD(+(x1, -1), 0) the following chains were created:

We consider the chain COND_304_0_MAIN_LOAD(TRUE, x1[1], 0) → 304_0_MAIN_LOAD(+(x1[1], -1), 0) which results in the following constraint:
(7)    (COND_304_0_MAIN_LOAD(TRUE, x1[1], 0)≥NonInfC∧COND_304_0_MAIN_LOAD(TRUE, x1[1], 0)≥304_0_MAIN_LOAD(+(x1[1], -1), 0)∧(U^Increasing(304_0_MAIN_LOAD(+(x1[1], -1), 0)), ≥))

We simplified constraint (7) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(8)    ((U^Increasing(304_0_MAIN_LOAD(+(x1[1], -1), 0)), ≥)∧[bni_14] = 0∧[1 + (-1)bso_15] ≥ 0)

We simplified constraint (8) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(9)    ((U^Increasing(304_0_MAIN_LOAD(+(x1[1], -1), 0)), ≥)∧[bni_14] = 0∧[1 + (-1)bso_15] ≥ 0)

We simplified constraint (9) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(10)    ((U^Increasing(304_0_MAIN_LOAD(+(x1[1], -1), 0)), ≥)∧[bni_14] = 0∧[1 + (-1)bso_15] ≥ 0)

We simplified constraint (10) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint:
(11)    ((U^Increasing(304_0_MAIN_LOAD(+(x1[1], -1), 0)), ≥)∧[bni_14] = 0∧0 = 0∧[1 + (-1)bso_15] ≥ 0)

For Pair 304_0_MAIN_LOAD(x1, x0) → COND_304_0_MAIN_LOAD1(&&(&&(>(x1, -1), >(x0, 0)), <(0, +(x0, x1))), x1, x0) the following chains were created:

We consider the chain 304_0_MAIN_LOAD(x1[2], x0[2]) → COND_304_0_MAIN_LOAD1(&&(&&(>(x1[2], -1), >(x0[2], 0)), <(0, +(x0[2], x1[2]))), x1[2], x0[2]), COND_304_0_MAIN_LOAD1(TRUE, x1[3], x0[3]) → 304_0_MAIN_LOAD(x1[3], +(x0[3], -1)) which results in the following constraint:
(12)    (&&(&&(>(x1[2], -1), >(x0[2], 0)), <(0, +(x0[2], x1[2])))=TRUE∧x1[2]=x1[3]∧x0[2]=x0[3] ⇒ 304_0_MAIN_LOAD(x1[2], x0[2])≥NonInfC∧304_0_MAIN_LOAD(x1[2], x0[2])≥COND_304_0_MAIN_LOAD1(&&(&&(>(x1[2], -1), >(x0[2], 0)), <(0, +(x0[2], x1[2]))), x1[2], x0[2])∧(U^Increasing(COND_304_0_MAIN_LOAD1(&&(&&(>(x1[2], -1), >(x0[2], 0)), <(0, +(x0[2], x1[2]))), x1[2], x0[2])), ≥))

We simplified constraint (12) using rules (IV), (IDP_BOOLEAN) which results in the following new constraint:
(13)    (<(0, +(x0[2], x1[2]))=TRUE∧>(x1[2], -1)=TRUE∧>(x0[2], 0)=TRUE ⇒ 304_0_MAIN_LOAD(x1[2], x0[2])≥NonInfC∧304_0_MAIN_LOAD(x1[2], x0[2])≥COND_304_0_MAIN_LOAD1(&&(&&(>(x1[2], -1), >(x0[2], 0)), <(0, +(x0[2], x1[2]))), x1[2], x0[2])∧(U^Increasing(COND_304_0_MAIN_LOAD1(&&(&&(>(x1[2], -1), >(x0[2], 0)), <(0, +(x0[2], x1[2]))), x1[2], x0[2])), ≥))

We simplified constraint (13) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(14)    (x0[2] + [-1] + x1[2] ≥ 0∧x1[2] ≥ 0∧x0[2] + [-1] ≥ 0 ⇒ (U^Increasing(COND_304_0_MAIN_LOAD1(&&(&&(>(x1[2], -1), >(x0[2], 0)), <(0, +(x0[2], x1[2]))), x1[2], x0[2])), ≥)∧[bni_16 + (-1)Bound*bni_16] + [bni_16]x0[2] + [bni_16]x1[2] ≥ 0∧[(-1)bso_17] ≥ 0)

We simplified constraint (14) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(15)    (x0[2] + [-1] + x1[2] ≥ 0∧x1[2] ≥ 0∧x0[2] + [-1] ≥ 0 ⇒ (U^Increasing(COND_304_0_MAIN_LOAD1(&&(&&(>(x1[2], -1), >(x0[2], 0)), <(0, +(x0[2], x1[2]))), x1[2], x0[2])), ≥)∧[bni_16 + (-1)Bound*bni_16] + [bni_16]x0[2] + [bni_16]x1[2] ≥ 0∧[(-1)bso_17] ≥ 0)

We simplified constraint (15) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(16)    (x0[2] + [-1] + x1[2] ≥ 0∧x1[2] ≥ 0∧x0[2] + [-1] ≥ 0 ⇒ (U^Increasing(COND_304_0_MAIN_LOAD1(&&(&&(>(x1[2], -1), >(x0[2], 0)), <(0, +(x0[2], x1[2]))), x1[2], x0[2])), ≥)∧[bni_16 + (-1)Bound*bni_16] + [bni_16]x0[2] + [bni_16]x1[2] ≥ 0∧[(-1)bso_17] ≥ 0)

We simplified constraint (16) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(17)    (x0[2] + x1[2] ≥ 0∧x1[2] ≥ 0∧x0[2] ≥ 0 ⇒ (U^Increasing(COND_304_0_MAIN_LOAD1(&&(&&(>(x1[2], -1), >(x0[2], 0)), <(0, +(x0[2], x1[2]))), x1[2], x0[2])), ≥)∧[(2)bni_16 + (-1)Bound*bni_16] + [bni_16]x0[2] + [bni_16]x1[2] ≥ 0∧[(-1)bso_17] ≥ 0)

For Pair COND_304_0_MAIN_LOAD1(TRUE, x1, x0) → 304_0_MAIN_LOAD(x1, +(x0, -1)) the following chains were created:

We consider the chain COND_304_0_MAIN_LOAD1(TRUE, x1[3], x0[3]) → 304_0_MAIN_LOAD(x1[3], +(x0[3], -1)) which results in the following constraint:
(18)    (COND_304_0_MAIN_LOAD1(TRUE, x1[3], x0[3])≥NonInfC∧COND_304_0_MAIN_LOAD1(TRUE, x1[3], x0[3])≥304_0_MAIN_LOAD(x1[3], +(x0[3], -1))∧(U^Increasing(304_0_MAIN_LOAD(x1[3], +(x0[3], -1))), ≥))

We simplified constraint (18) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(19)    ((U^Increasing(304_0_MAIN_LOAD(x1[3], +(x0[3], -1))), ≥)∧[bni_18] = 0∧[1 + (-1)bso_19] ≥ 0)

We simplified constraint (19) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(20)    ((U^Increasing(304_0_MAIN_LOAD(x1[3], +(x0[3], -1))), ≥)∧[bni_18] = 0∧[1 + (-1)bso_19] ≥ 0)

We simplified constraint (20) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(21)    ((U^Increasing(304_0_MAIN_LOAD(x1[3], +(x0[3], -1))), ≥)∧[bni_18] = 0∧[1 + (-1)bso_19] ≥ 0)

We simplified constraint (21) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint:
(22)    ((U^Increasing(304_0_MAIN_LOAD(x1[3], +(x0[3], -1))), ≥)∧[bni_18] = 0∧0 = 0∧0 = 0∧[1 + (-1)bso_19] ≥ 0)

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

304_0_MAIN_LOAD(x1, 0) → COND_304_0_MAIN_LOAD(>(x1, 0), x1, 0)
- (x1[0] ≥ 0 ⇒ (U^Increasing(COND_304_0_MAIN_LOAD(>(x1[0], 0), x1[0], 0)), ≥)∧[(2)bni_12 + (-1)Bound*bni_12] + [bni_12]x1[0] ≥ 0∧[(-1)bso_13] ≥ 0)
COND_304_0_MAIN_LOAD(TRUE, x1, 0) → 304_0_MAIN_LOAD(+(x1, -1), 0)
- ((U^Increasing(304_0_MAIN_LOAD(+(x1[1], -1), 0)), ≥)∧[bni_14] = 0∧0 = 0∧[1 + (-1)bso_15] ≥ 0)
304_0_MAIN_LOAD(x1, x0) → COND_304_0_MAIN_LOAD1(&&(&&(>(x1, -1), >(x0, 0)), <(0, +(x0, x1))), x1, x0)
- (x0[2] + x1[2] ≥ 0∧x1[2] ≥ 0∧x0[2] ≥ 0 ⇒ (U^Increasing(COND_304_0_MAIN_LOAD1(&&(&&(>(x1[2], -1), >(x0[2], 0)), <(0, +(x0[2], x1[2]))), x1[2], x0[2])), ≥)∧[(2)bni_16 + (-1)Bound*bni_16] + [bni_16]x0[2] + [bni_16]x1[2] ≥ 0∧[(-1)bso_17] ≥ 0)
COND_304_0_MAIN_LOAD1(TRUE, x1, x0) → 304_0_MAIN_LOAD(x1, +(x0, -1))
- ((U^Increasing(304_0_MAIN_LOAD(x1[3], +(x0[3], -1))), ≥)∧[bni_18] = 0∧0 = 0∧0 = 0∧[1 + (-1)bso_19] ≥ 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(304_0_MAIN_LOAD(x₁, x₂)) = [1] + x₂ + x₁
POL(0) = 0
POL(COND_304_0_MAIN_LOAD(x₁, x₂, x₃)) = [1] + x₂
POL(>(x₁, x₂)) = [-1]
POL(+(x₁, x₂)) = x₁ + x₂
POL(-1) = [-1]
POL(COND_304_0_MAIN_LOAD1(x₁, x₂, x₃)) = [1] + x₃ + x₂
POL(&&(x₁, x₂)) = [-1]
POL(<(x₁, x₂)) = [-1]

The following pairs are in P_>:

COND_304_0_MAIN_LOAD(TRUE, x1[1], 0) → 304_0_MAIN_LOAD(+(x1[1], -1), 0)
COND_304_0_MAIN_LOAD1(TRUE, x1[3], x0[3]) → 304_0_MAIN_LOAD(x1[3], +(x0[3], -1))

The following pairs are in P_bound:

304_0_MAIN_LOAD(x1[0], 0) → COND_304_0_MAIN_LOAD(>(x1[0], 0), x1[0], 0)
304_0_MAIN_LOAD(x1[2], x0[2]) → COND_304_0_MAIN_LOAD1(&&(&&(>(x1[2], -1), >(x0[2], 0)), <(0, +(x0[2], x1[2]))), x1[2], x0[2])

The following pairs are in P_≥:

304_0_MAIN_LOAD(x1[0], 0) → COND_304_0_MAIN_LOAD(>(x1[0], 0), x1[0], 0)
304_0_MAIN_LOAD(x1[2], x0[2]) → COND_304_0_MAIN_LOAD1(&&(&&(>(x1[2], -1), >(x0[2], 0)), <(0, +(x0[2], x1[2]))), x1[2], x0[2])

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:

Integer, Boolean

R is empty.

The integer pair graph contains the following rules and edges:
(0): 304_0_MAIN_LOAD(x1[0], 0) → COND_304_0_MAIN_LOAD(x1[0] > 0, x1[0], 0)
(2): 304_0_MAIN_LOAD(x1[2], x0[2]) → COND_304_0_MAIN_LOAD1(x1[2] > -1 && x0[2] > 0 && 0 < x0[2] + x1[2], x1[2], x0[2])

The set Q is empty.

(10) IDependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 2 less nodes.

(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_304_0_MAIN_LOAD(TRUE, x1[1], 0) → 304_0_MAIN_LOAD(x1[1] + -1, 0)
(3): COND_304_0_MAIN_LOAD1(TRUE, x1[3], x0[3]) → 304_0_MAIN_LOAD(x1[3], x0[3] + -1)

The set Q is empty.

(13) IDependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 2 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