Termination of the given JBC could be proven:

0 JBC
↳1 JBCToGraph (⇒, 520 ms)
↳2 JBCTerminationGraph
↳3 TerminationGraphToSCCProof (⇒, 0 ms)
↳4 JBCTerminationSCC
↳5 SCCToIDPv1Proof (⇒, 120 ms)
↳6 IDP
↳7 IDPNonInfProof (⇒, 960 ms)
↳8 IDP
↳9 IDependencyGraphProof (⇔, 0 ms)
↳10 IDP
↳11 IDPNonInfProof (⇒, 130 ms)
↳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: PastaC3 
/**
 * 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 PastaC3 {
    public static void main(String[] args) {
        Random.args = args;
        int x = Random.random();
        int y = Random.random();
        int z = Random.random();

        while (x < y) {
            if (x < z) {
                x++;
            } else {
                z++;
            }
        }
    }
}


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:
PastaC3.main([Ljava/lang/String;)V: Graph of 249 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: PastaC3.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 16 rules for P and 0 rules for R.


P rules:
493_0_main_Load(EOS(STATIC_493), i18, i47, i87, i18) → 500_0_main_GE(EOS(STATIC_500), i18, i47, i87, i18, i47)
500_0_main_GE(EOS(STATIC_500), i18, i47, i87, i18, i47) → 517_0_main_GE(EOS(STATIC_517), i18, i47, i87, i18, i47)
517_0_main_GE(EOS(STATIC_517), i18, i47, i87, i18, i47) → 528_0_main_Load(EOS(STATIC_528), i18, i47, i87) | <(i18, i47)
528_0_main_Load(EOS(STATIC_528), i18, i47, i87) → 536_0_main_Load(EOS(STATIC_536), i18, i47, i87, i18)
536_0_main_Load(EOS(STATIC_536), i18, i47, i87, i18) → 548_0_main_GE(EOS(STATIC_548), i18, i47, i87, i18, i87)
548_0_main_GE(EOS(STATIC_548), i18, i47, i87, i18, i87) → 558_0_main_GE(EOS(STATIC_558), i18, i47, i87, i18, i87)
548_0_main_GE(EOS(STATIC_548), i18, i47, i87, i18, i87) → 559_0_main_GE(EOS(STATIC_559), i18, i47, i87, i18, i87)
558_0_main_GE(EOS(STATIC_558), i18, i47, i87, i18, i87) → 568_0_main_Inc(EOS(STATIC_568), i18, i47, i87) | >=(i18, i87)
568_0_main_Inc(EOS(STATIC_568), i18, i47, i87) → 579_0_main_JMP(EOS(STATIC_579), i18, i47, +(i87, 1)) | >=(i87, 0)
579_0_main_JMP(EOS(STATIC_579), i18, i47, i101) → 614_0_main_Load(EOS(STATIC_614), i18, i47, i101)
614_0_main_Load(EOS(STATIC_614), i18, i47, i101) → 485_0_main_Load(EOS(STATIC_485), i18, i47, i101)
485_0_main_Load(EOS(STATIC_485), i18, i47, i87) → 493_0_main_Load(EOS(STATIC_493), i18, i47, i87, i18)
559_0_main_GE(EOS(STATIC_559), i18, i47, i87, i18, i87) → 570_0_main_Inc(EOS(STATIC_570), i18, i47, i87) | <(i18, i87)
570_0_main_Inc(EOS(STATIC_570), i18, i47, i87) → 581_0_main_JMP(EOS(STATIC_581), +(i18, 1), i47, i87) | >=(i18, 0)
581_0_main_JMP(EOS(STATIC_581), i102, i47, i87) → 620_0_main_Load(EOS(STATIC_620), i102, i47, i87)
620_0_main_Load(EOS(STATIC_620), i102, i47, i87) → 485_0_main_Load(EOS(STATIC_485), i102, i47, i87)
R rules:

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


P rules:
493_0_main_Load(EOS(STATIC_493), x0, x1, x2, x0) → 493_0_main_Load(EOS(STATIC_493), x0, x1, +(x2, 1), x0) | &&(&&(>(+(x2, 1), 0), <=(x2, x0)), >(x1, x0))
493_0_main_Load(EOS(STATIC_493), x0, x1, x2, x0) → 493_0_main_Load(EOS(STATIC_493), +(x0, 1), x1, x2, +(x0, 1)) | &&(&&(>(x2, x0), >(x1, x0)), >(+(x0, 1), 0))
R rules:

Filtered ground terms:



493_0_main_Load(x1, x2, x3, x4, x5) → 493_0_main_Load(x2, x3, x4, x5)
EOS(x1) → EOS
Cond_493_0_main_Load1(x1, x2, x3, x4, x5, x6) → Cond_493_0_main_Load1(x1, x3, x4, x5, x6)
Cond_493_0_main_Load(x1, x2, x3, x4, x5, x6) → Cond_493_0_main_Load(x1, x3, x4, x5, x6)

Filtered duplicate args:



493_0_main_Load(x1, x2, x3, x4) → 493_0_main_Load(x2, x3, x4)
Cond_493_0_main_Load(x1, x2, x3, x4, x5) → Cond_493_0_main_Load(x1, x3, x4, x5)
Cond_493_0_main_Load1(x1, x2, x3, x4, x5) → Cond_493_0_main_Load1(x1, x3, x4, x5)

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


P rules:
493_0_main_Load(x1, x2, x0) → 493_0_main_Load(x1, +(x2, 1), x0) | &&(&&(>(x2, -1), <=(x2, x0)), >(x1, x0))
493_0_main_Load(x1, x2, x0) → 493_0_main_Load(x1, x2, +(x0, 1)) | &&(&&(>(x2, x0), >(x1, x0)), >(x0, -1))
R rules:

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


P rules:
493_0_MAIN_LOAD(x1, x2, x0) → COND_493_0_MAIN_LOAD(&&(&&(>(x2, -1), <=(x2, x0)), >(x1, x0)), x1, x2, x0)
COND_493_0_MAIN_LOAD(TRUE, x1, x2, x0) → 493_0_MAIN_LOAD(x1, +(x2, 1), x0)
493_0_MAIN_LOAD(x1, x2, x0) → COND_493_0_MAIN_LOAD1(&&(&&(>(x2, x0), >(x1, x0)), >(x0, -1)), x1, x2, x0)
COND_493_0_MAIN_LOAD1(TRUE, x1, x2, x0) → 493_0_MAIN_LOAD(x1, x2, +(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): 493_0_MAIN_LOAD(x1[0], x2[0], x0[0]) → COND_493_0_MAIN_LOAD(x2[0] > -1 && x2[0] <= x0[0] && x1[0] > x0[0], x1[0], x2[0], x0[0])
(1): COND_493_0_MAIN_LOAD(TRUE, x1[1], x2[1], x0[1]) → 493_0_MAIN_LOAD(x1[1], x2[1] + 1, x0[1])
(2): 493_0_MAIN_LOAD(x1[2], x2[2], x0[2]) → COND_493_0_MAIN_LOAD1(x2[2] > x0[2] && x1[2] > x0[2] && x0[2] > -1, x1[2], x2[2], x0[2])
(3): COND_493_0_MAIN_LOAD1(TRUE, x1[3], x2[3], x0[3]) → 493_0_MAIN_LOAD(x1[3], x2[3], x0[3] + 1)

(0) -> (1), if (x2[0] > -1 && x2[0] <= x0[0] && x1[0] > x0[0] ∧x1[0] →* x1[1]∧x2[0] →* x2[1]∧x0[0] →* x0[1])


(1) -> (0), if (x1[1] →* x1[0]∧x2[1] + 1 →* x2[0]∧x0[1] →* x0[0])


(1) -> (2), if (x1[1] →* x1[2]∧x2[1] + 1 →* x2[2]∧x0[1] →* x0[2])


(2) -> (3), if (x2[2] > x0[2] && x1[2] > x0[2] && x0[2] > -1 ∧x1[2] →* x1[3]∧x2[2] →* x2[3]∧x0[2] →* x0[3])


(3) -> (0), if (x1[3] →* x1[0]∧x2[3] →* x2[0]∧x0[3] + 1 →* x0[0])


(3) -> (2), if (x1[3] →* x1[2]∧x2[3] →* x2[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.IdpDefaultShapeHeuristic@3c50bf08 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 493_0_MAIN_LOAD(x1, x2, x0) → COND_493_0_MAIN_LOAD(&&(&&(>(x2, -1), <=(x2, x0)), >(x1, x0)), x1, x2, x0) the following chains were created:
  • We consider the chain 493_0_MAIN_LOAD(x1[0], x2[0], x0[0]) → COND_493_0_MAIN_LOAD(&&(&&(>(x2[0], -1), <=(x2[0], x0[0])), >(x1[0], x0[0])), x1[0], x2[0], x0[0]), COND_493_0_MAIN_LOAD(TRUE, x1[1], x2[1], x0[1]) → 493_0_MAIN_LOAD(x1[1], +(x2[1], 1), x0[1]) which results in the following constraint:

    (1)    (&&(&&(>(x2[0], -1), <=(x2[0], x0[0])), >(x1[0], x0[0]))=TRUE∧x1[0]=x1[1]∧x2[0]=x2[1]∧x0[0]=x0[1] ⇒ 493_0_MAIN_LOAD(x1[0], x2[0], x0[0])≥NonInfC∧493_0_MAIN_LOAD(x1[0], x2[0], x0[0])≥COND_493_0_MAIN_LOAD(&&(&&(>(x2[0], -1), <=(x2[0], x0[0])), >(x1[0], x0[0])), x1[0], x2[0], x0[0])∧(UIncreasing(COND_493_0_MAIN_LOAD(&&(&&(>(x2[0], -1), <=(x2[0], x0[0])), >(x1[0], x0[0])), x1[0], x2[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∧>(x2[0], -1)=TRUE∧<=(x2[0], x0[0])=TRUE ⇒ 493_0_MAIN_LOAD(x1[0], x2[0], x0[0])≥NonInfC∧493_0_MAIN_LOAD(x1[0], x2[0], x0[0])≥COND_493_0_MAIN_LOAD(&&(&&(>(x2[0], -1), <=(x2[0], x0[0])), >(x1[0], x0[0])), x1[0], x2[0], x0[0])∧(UIncreasing(COND_493_0_MAIN_LOAD(&&(&&(>(x2[0], -1), <=(x2[0], x0[0])), >(x1[0], x0[0])), x1[0], x2[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∧x2[0] ≥ 0∧x0[0] + [-1]x2[0] ≥ 0 ⇒ (UIncreasing(COND_493_0_MAIN_LOAD(&&(&&(>(x2[0], -1), <=(x2[0], x0[0])), >(x1[0], x0[0])), x1[0], x2[0], x0[0])), ≥)∧[(-1)bni_20 + (-1)Bound*bni_20] + [(-1)bni_20]x0[0] + [bni_20]x1[0] ≥ 0∧[(-1)bso_21] ≥ 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∧x2[0] ≥ 0∧x0[0] + [-1]x2[0] ≥ 0 ⇒ (UIncreasing(COND_493_0_MAIN_LOAD(&&(&&(>(x2[0], -1), <=(x2[0], x0[0])), >(x1[0], x0[0])), x1[0], x2[0], x0[0])), ≥)∧[(-1)bni_20 + (-1)Bound*bni_20] + [(-1)bni_20]x0[0] + [bni_20]x1[0] ≥ 0∧[(-1)bso_21] ≥ 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∧x2[0] ≥ 0∧x0[0] + [-1]x2[0] ≥ 0 ⇒ (UIncreasing(COND_493_0_MAIN_LOAD(&&(&&(>(x2[0], -1), <=(x2[0], x0[0])), >(x1[0], x0[0])), x1[0], x2[0], x0[0])), ≥)∧[(-1)bni_20 + (-1)Bound*bni_20] + [(-1)bni_20]x0[0] + [bni_20]x1[0] ≥ 0∧[(-1)bso_21] ≥ 0)



    We simplified constraint (5) using rule (IDP_SMT_SPLIT) which results in the following new constraint:

    (6)    (x1[0] ≥ 0∧x2[0] ≥ 0∧x0[0] + [-1]x2[0] ≥ 0 ⇒ (UIncreasing(COND_493_0_MAIN_LOAD(&&(&&(>(x2[0], -1), <=(x2[0], x0[0])), >(x1[0], x0[0])), x1[0], x2[0], x0[0])), ≥)∧[(-1)Bound*bni_20] + [bni_20]x1[0] ≥ 0∧[(-1)bso_21] ≥ 0)



    We simplified constraint (6) using rule (IDP_SMT_SPLIT) which results in the following new constraint:

    (7)    (x1[0] ≥ 0∧x2[0] ≥ 0∧x0[0] ≥ 0 ⇒ (UIncreasing(COND_493_0_MAIN_LOAD(&&(&&(>(x2[0], -1), <=(x2[0], x0[0])), >(x1[0], x0[0])), x1[0], x2[0], x0[0])), ≥)∧[(-1)Bound*bni_20] + [bni_20]x1[0] ≥ 0∧[(-1)bso_21] ≥ 0)







For Pair COND_493_0_MAIN_LOAD(TRUE, x1, x2, x0) → 493_0_MAIN_LOAD(x1, +(x2, 1), x0) the following chains were created:
  • We consider the chain 493_0_MAIN_LOAD(x1[0], x2[0], x0[0]) → COND_493_0_MAIN_LOAD(&&(&&(>(x2[0], -1), <=(x2[0], x0[0])), >(x1[0], x0[0])), x1[0], x2[0], x0[0]), COND_493_0_MAIN_LOAD(TRUE, x1[1], x2[1], x0[1]) → 493_0_MAIN_LOAD(x1[1], +(x2[1], 1), x0[1]), 493_0_MAIN_LOAD(x1[0], x2[0], x0[0]) → COND_493_0_MAIN_LOAD(&&(&&(>(x2[0], -1), <=(x2[0], x0[0])), >(x1[0], x0[0])), x1[0], x2[0], x0[0]) which results in the following constraint:

    (8)    (&&(&&(>(x2[0], -1), <=(x2[0], x0[0])), >(x1[0], x0[0]))=TRUE∧x1[0]=x1[1]∧x2[0]=x2[1]∧x0[0]=x0[1]∧x1[1]=x1[0]1∧+(x2[1], 1)=x2[0]1∧x0[1]=x0[0]1 ⇒ COND_493_0_MAIN_LOAD(TRUE, x1[1], x2[1], x0[1])≥NonInfC∧COND_493_0_MAIN_LOAD(TRUE, x1[1], x2[1], x0[1])≥493_0_MAIN_LOAD(x1[1], +(x2[1], 1), x0[1])∧(UIncreasing(493_0_MAIN_LOAD(x1[1], +(x2[1], 1), x0[1])), ≥))



    We simplified constraint (8) using rules (III), (IV), (IDP_BOOLEAN) which results in the following new constraint:

    (9)    (>(x1[0], x0[0])=TRUE∧>(x2[0], -1)=TRUE∧<=(x2[0], x0[0])=TRUE ⇒ COND_493_0_MAIN_LOAD(TRUE, x1[0], x2[0], x0[0])≥NonInfC∧COND_493_0_MAIN_LOAD(TRUE, x1[0], x2[0], x0[0])≥493_0_MAIN_LOAD(x1[0], +(x2[0], 1), x0[0])∧(UIncreasing(493_0_MAIN_LOAD(x1[1], +(x2[1], 1), x0[1])), ≥))



    We simplified constraint (9) using rule (POLY_CONSTRAINTS) which results in the following new constraint:

    (10)    (x1[0] + [-1] + [-1]x0[0] ≥ 0∧x2[0] ≥ 0∧x0[0] + [-1]x2[0] ≥ 0 ⇒ (UIncreasing(493_0_MAIN_LOAD(x1[1], +(x2[1], 1), x0[1])), ≥)∧[(-1)bni_22 + (-1)Bound*bni_22] + [(-1)bni_22]x0[0] + [bni_22]x1[0] ≥ 0∧[(-1)bso_23] ≥ 0)



    We simplified constraint (10) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:

    (11)    (x1[0] + [-1] + [-1]x0[0] ≥ 0∧x2[0] ≥ 0∧x0[0] + [-1]x2[0] ≥ 0 ⇒ (UIncreasing(493_0_MAIN_LOAD(x1[1], +(x2[1], 1), x0[1])), ≥)∧[(-1)bni_22 + (-1)Bound*bni_22] + [(-1)bni_22]x0[0] + [bni_22]x1[0] ≥ 0∧[(-1)bso_23] ≥ 0)



    We simplified constraint (11) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:

    (12)    (x1[0] + [-1] + [-1]x0[0] ≥ 0∧x2[0] ≥ 0∧x0[0] + [-1]x2[0] ≥ 0 ⇒ (UIncreasing(493_0_MAIN_LOAD(x1[1], +(x2[1], 1), x0[1])), ≥)∧[(-1)bni_22 + (-1)Bound*bni_22] + [(-1)bni_22]x0[0] + [bni_22]x1[0] ≥ 0∧[(-1)bso_23] ≥ 0)



    We simplified constraint (12) using rule (IDP_SMT_SPLIT) which results in the following new constraint:

    (13)    (x1[0] ≥ 0∧x2[0] ≥ 0∧x0[0] + [-1]x2[0] ≥ 0 ⇒ (UIncreasing(493_0_MAIN_LOAD(x1[1], +(x2[1], 1), x0[1])), ≥)∧[(-1)Bound*bni_22] + [bni_22]x1[0] ≥ 0∧[(-1)bso_23] ≥ 0)



    We simplified constraint (13) using rule (IDP_SMT_SPLIT) which results in the following new constraint:

    (14)    (x1[0] ≥ 0∧x2[0] ≥ 0∧x0[0] ≥ 0 ⇒ (UIncreasing(493_0_MAIN_LOAD(x1[1], +(x2[1], 1), x0[1])), ≥)∧[(-1)Bound*bni_22] + [bni_22]x1[0] ≥ 0∧[(-1)bso_23] ≥ 0)



  • We consider the chain 493_0_MAIN_LOAD(x1[0], x2[0], x0[0]) → COND_493_0_MAIN_LOAD(&&(&&(>(x2[0], -1), <=(x2[0], x0[0])), >(x1[0], x0[0])), x1[0], x2[0], x0[0]), COND_493_0_MAIN_LOAD(TRUE, x1[1], x2[1], x0[1]) → 493_0_MAIN_LOAD(x1[1], +(x2[1], 1), x0[1]), 493_0_MAIN_LOAD(x1[2], x2[2], x0[2]) → COND_493_0_MAIN_LOAD1(&&(&&(>(x2[2], x0[2]), >(x1[2], x0[2])), >(x0[2], -1)), x1[2], x2[2], x0[2]) which results in the following constraint:

    (15)    (&&(&&(>(x2[0], -1), <=(x2[0], x0[0])), >(x1[0], x0[0]))=TRUE∧x1[0]=x1[1]∧x2[0]=x2[1]∧x0[0]=x0[1]∧x1[1]=x1[2]∧+(x2[1], 1)=x2[2]∧x0[1]=x0[2] ⇒ COND_493_0_MAIN_LOAD(TRUE, x1[1], x2[1], x0[1])≥NonInfC∧COND_493_0_MAIN_LOAD(TRUE, x1[1], x2[1], x0[1])≥493_0_MAIN_LOAD(x1[1], +(x2[1], 1), x0[1])∧(UIncreasing(493_0_MAIN_LOAD(x1[1], +(x2[1], 1), x0[1])), ≥))



    We simplified constraint (15) using rules (III), (IV), (IDP_BOOLEAN) which results in the following new constraint:

    (16)    (>(x1[0], x0[0])=TRUE∧>(x2[0], -1)=TRUE∧<=(x2[0], x0[0])=TRUE ⇒ COND_493_0_MAIN_LOAD(TRUE, x1[0], x2[0], x0[0])≥NonInfC∧COND_493_0_MAIN_LOAD(TRUE, x1[0], x2[0], x0[0])≥493_0_MAIN_LOAD(x1[0], +(x2[0], 1), x0[0])∧(UIncreasing(493_0_MAIN_LOAD(x1[1], +(x2[1], 1), x0[1])), ≥))



    We simplified constraint (16) using rule (POLY_CONSTRAINTS) which results in the following new constraint:

    (17)    (x1[0] + [-1] + [-1]x0[0] ≥ 0∧x2[0] ≥ 0∧x0[0] + [-1]x2[0] ≥ 0 ⇒ (UIncreasing(493_0_MAIN_LOAD(x1[1], +(x2[1], 1), x0[1])), ≥)∧[(-1)bni_22 + (-1)Bound*bni_22] + [(-1)bni_22]x0[0] + [bni_22]x1[0] ≥ 0∧[(-1)bso_23] ≥ 0)



    We simplified constraint (17) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:

    (18)    (x1[0] + [-1] + [-1]x0[0] ≥ 0∧x2[0] ≥ 0∧x0[0] + [-1]x2[0] ≥ 0 ⇒ (UIncreasing(493_0_MAIN_LOAD(x1[1], +(x2[1], 1), x0[1])), ≥)∧[(-1)bni_22 + (-1)Bound*bni_22] + [(-1)bni_22]x0[0] + [bni_22]x1[0] ≥ 0∧[(-1)bso_23] ≥ 0)



    We simplified constraint (18) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:

    (19)    (x1[0] + [-1] + [-1]x0[0] ≥ 0∧x2[0] ≥ 0∧x0[0] + [-1]x2[0] ≥ 0 ⇒ (UIncreasing(493_0_MAIN_LOAD(x1[1], +(x2[1], 1), x0[1])), ≥)∧[(-1)bni_22 + (-1)Bound*bni_22] + [(-1)bni_22]x0[0] + [bni_22]x1[0] ≥ 0∧[(-1)bso_23] ≥ 0)



    We simplified constraint (19) using rule (IDP_SMT_SPLIT) which results in the following new constraint:

    (20)    (x1[0] ≥ 0∧x2[0] ≥ 0∧x0[0] + [-1]x2[0] ≥ 0 ⇒ (UIncreasing(493_0_MAIN_LOAD(x1[1], +(x2[1], 1), x0[1])), ≥)∧[(-1)Bound*bni_22] + [bni_22]x1[0] ≥ 0∧[(-1)bso_23] ≥ 0)



    We simplified constraint (20) using rule (IDP_SMT_SPLIT) which results in the following new constraint:

    (21)    (x1[0] ≥ 0∧x2[0] ≥ 0∧x0[0] ≥ 0 ⇒ (UIncreasing(493_0_MAIN_LOAD(x1[1], +(x2[1], 1), x0[1])), ≥)∧[(-1)Bound*bni_22] + [bni_22]x1[0] ≥ 0∧[(-1)bso_23] ≥ 0)







For Pair 493_0_MAIN_LOAD(x1, x2, x0) → COND_493_0_MAIN_LOAD1(&&(&&(>(x2, x0), >(x1, x0)), >(x0, -1)), x1, x2, x0) the following chains were created:
  • We consider the chain 493_0_MAIN_LOAD(x1[2], x2[2], x0[2]) → COND_493_0_MAIN_LOAD1(&&(&&(>(x2[2], x0[2]), >(x1[2], x0[2])), >(x0[2], -1)), x1[2], x2[2], x0[2]), COND_493_0_MAIN_LOAD1(TRUE, x1[3], x2[3], x0[3]) → 493_0_MAIN_LOAD(x1[3], x2[3], +(x0[3], 1)) which results in the following constraint:

    (22)    (&&(&&(>(x2[2], x0[2]), >(x1[2], x0[2])), >(x0[2], -1))=TRUE∧x1[2]=x1[3]∧x2[2]=x2[3]∧x0[2]=x0[3] ⇒ 493_0_MAIN_LOAD(x1[2], x2[2], x0[2])≥NonInfC∧493_0_MAIN_LOAD(x1[2], x2[2], x0[2])≥COND_493_0_MAIN_LOAD1(&&(&&(>(x2[2], x0[2]), >(x1[2], x0[2])), >(x0[2], -1)), x1[2], x2[2], x0[2])∧(UIncreasing(COND_493_0_MAIN_LOAD1(&&(&&(>(x2[2], x0[2]), >(x1[2], x0[2])), >(x0[2], -1)), x1[2], x2[2], x0[2])), ≥))



    We simplified constraint (22) using rules (IV), (IDP_BOOLEAN) which results in the following new constraint:

    (23)    (>(x0[2], -1)=TRUE∧>(x2[2], x0[2])=TRUE∧>(x1[2], x0[2])=TRUE ⇒ 493_0_MAIN_LOAD(x1[2], x2[2], x0[2])≥NonInfC∧493_0_MAIN_LOAD(x1[2], x2[2], x0[2])≥COND_493_0_MAIN_LOAD1(&&(&&(>(x2[2], x0[2]), >(x1[2], x0[2])), >(x0[2], -1)), x1[2], x2[2], x0[2])∧(UIncreasing(COND_493_0_MAIN_LOAD1(&&(&&(>(x2[2], x0[2]), >(x1[2], x0[2])), >(x0[2], -1)), x1[2], x2[2], x0[2])), ≥))



    We simplified constraint (23) using rule (POLY_CONSTRAINTS) which results in the following new constraint:

    (24)    (x0[2] ≥ 0∧x2[2] + [-1] + [-1]x0[2] ≥ 0∧x1[2] + [-1] + [-1]x0[2] ≥ 0 ⇒ (UIncreasing(COND_493_0_MAIN_LOAD1(&&(&&(>(x2[2], x0[2]), >(x1[2], x0[2])), >(x0[2], -1)), x1[2], x2[2], x0[2])), ≥)∧[(-1)bni_24 + (-1)Bound*bni_24] + [(-1)bni_24]x0[2] + [bni_24]x1[2] ≥ 0∧[(-1)bso_25] ≥ 0)



    We simplified constraint (24) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:

    (25)    (x0[2] ≥ 0∧x2[2] + [-1] + [-1]x0[2] ≥ 0∧x1[2] + [-1] + [-1]x0[2] ≥ 0 ⇒ (UIncreasing(COND_493_0_MAIN_LOAD1(&&(&&(>(x2[2], x0[2]), >(x1[2], x0[2])), >(x0[2], -1)), x1[2], x2[2], x0[2])), ≥)∧[(-1)bni_24 + (-1)Bound*bni_24] + [(-1)bni_24]x0[2] + [bni_24]x1[2] ≥ 0∧[(-1)bso_25] ≥ 0)



    We simplified constraint (25) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:

    (26)    (x0[2] ≥ 0∧x2[2] + [-1] + [-1]x0[2] ≥ 0∧x1[2] + [-1] + [-1]x0[2] ≥ 0 ⇒ (UIncreasing(COND_493_0_MAIN_LOAD1(&&(&&(>(x2[2], x0[2]), >(x1[2], x0[2])), >(x0[2], -1)), x1[2], x2[2], x0[2])), ≥)∧[(-1)bni_24 + (-1)Bound*bni_24] + [(-1)bni_24]x0[2] + [bni_24]x1[2] ≥ 0∧[(-1)bso_25] ≥ 0)



    We simplified constraint (26) using rule (IDP_SMT_SPLIT) which results in the following new constraint:

    (27)    (x0[2] ≥ 0∧x2[2] ≥ 0∧x1[2] + [-1] + [-1]x0[2] ≥ 0 ⇒ (UIncreasing(COND_493_0_MAIN_LOAD1(&&(&&(>(x2[2], x0[2]), >(x1[2], x0[2])), >(x0[2], -1)), x1[2], x2[2], x0[2])), ≥)∧[(-1)bni_24 + (-1)Bound*bni_24] + [(-1)bni_24]x0[2] + [bni_24]x1[2] ≥ 0∧[(-1)bso_25] ≥ 0)



    We simplified constraint (27) using rule (IDP_SMT_SPLIT) which results in the following new constraint:

    (28)    (x0[2] ≥ 0∧x2[2] ≥ 0∧x1[2] ≥ 0 ⇒ (UIncreasing(COND_493_0_MAIN_LOAD1(&&(&&(>(x2[2], x0[2]), >(x1[2], x0[2])), >(x0[2], -1)), x1[2], x2[2], x0[2])), ≥)∧[(-1)Bound*bni_24] + [bni_24]x1[2] ≥ 0∧[(-1)bso_25] ≥ 0)







For Pair COND_493_0_MAIN_LOAD1(TRUE, x1, x2, x0) → 493_0_MAIN_LOAD(x1, x2, +(x0, 1)) the following chains were created:
  • We consider the chain 493_0_MAIN_LOAD(x1[2], x2[2], x0[2]) → COND_493_0_MAIN_LOAD1(&&(&&(>(x2[2], x0[2]), >(x1[2], x0[2])), >(x0[2], -1)), x1[2], x2[2], x0[2]), COND_493_0_MAIN_LOAD1(TRUE, x1[3], x2[3], x0[3]) → 493_0_MAIN_LOAD(x1[3], x2[3], +(x0[3], 1)), 493_0_MAIN_LOAD(x1[0], x2[0], x0[0]) → COND_493_0_MAIN_LOAD(&&(&&(>(x2[0], -1), <=(x2[0], x0[0])), >(x1[0], x0[0])), x1[0], x2[0], x0[0]) which results in the following constraint:

    (29)    (&&(&&(>(x2[2], x0[2]), >(x1[2], x0[2])), >(x0[2], -1))=TRUE∧x1[2]=x1[3]∧x2[2]=x2[3]∧x0[2]=x0[3]∧x1[3]=x1[0]∧x2[3]=x2[0]∧+(x0[3], 1)=x0[0] ⇒ COND_493_0_MAIN_LOAD1(TRUE, x1[3], x2[3], x0[3])≥NonInfC∧COND_493_0_MAIN_LOAD1(TRUE, x1[3], x2[3], x0[3])≥493_0_MAIN_LOAD(x1[3], x2[3], +(x0[3], 1))∧(UIncreasing(493_0_MAIN_LOAD(x1[3], x2[3], +(x0[3], 1))), ≥))



    We simplified constraint (29) using rules (III), (IV), (IDP_BOOLEAN) which results in the following new constraint:

    (30)    (>(x0[2], -1)=TRUE∧>(x2[2], x0[2])=TRUE∧>(x1[2], x0[2])=TRUE ⇒ COND_493_0_MAIN_LOAD1(TRUE, x1[2], x2[2], x0[2])≥NonInfC∧COND_493_0_MAIN_LOAD1(TRUE, x1[2], x2[2], x0[2])≥493_0_MAIN_LOAD(x1[2], x2[2], +(x0[2], 1))∧(UIncreasing(493_0_MAIN_LOAD(x1[3], x2[3], +(x0[3], 1))), ≥))



    We simplified constraint (30) using rule (POLY_CONSTRAINTS) which results in the following new constraint:

    (31)    (x0[2] ≥ 0∧x2[2] + [-1] + [-1]x0[2] ≥ 0∧x1[2] + [-1] + [-1]x0[2] ≥ 0 ⇒ (UIncreasing(493_0_MAIN_LOAD(x1[3], x2[3], +(x0[3], 1))), ≥)∧[(-1)bni_26 + (-1)Bound*bni_26] + [(-1)bni_26]x0[2] + [bni_26]x1[2] ≥ 0∧[1 + (-1)bso_27] ≥ 0)



    We simplified constraint (31) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:

    (32)    (x0[2] ≥ 0∧x2[2] + [-1] + [-1]x0[2] ≥ 0∧x1[2] + [-1] + [-1]x0[2] ≥ 0 ⇒ (UIncreasing(493_0_MAIN_LOAD(x1[3], x2[3], +(x0[3], 1))), ≥)∧[(-1)bni_26 + (-1)Bound*bni_26] + [(-1)bni_26]x0[2] + [bni_26]x1[2] ≥ 0∧[1 + (-1)bso_27] ≥ 0)



    We simplified constraint (32) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:

    (33)    (x0[2] ≥ 0∧x2[2] + [-1] + [-1]x0[2] ≥ 0∧x1[2] + [-1] + [-1]x0[2] ≥ 0 ⇒ (UIncreasing(493_0_MAIN_LOAD(x1[3], x2[3], +(x0[3], 1))), ≥)∧[(-1)bni_26 + (-1)Bound*bni_26] + [(-1)bni_26]x0[2] + [bni_26]x1[2] ≥ 0∧[1 + (-1)bso_27] ≥ 0)



    We simplified constraint (33) using rule (IDP_SMT_SPLIT) which results in the following new constraint:

    (34)    (x0[2] ≥ 0∧x2[2] ≥ 0∧x1[2] + [-1] + [-1]x0[2] ≥ 0 ⇒ (UIncreasing(493_0_MAIN_LOAD(x1[3], x2[3], +(x0[3], 1))), ≥)∧[(-1)bni_26 + (-1)Bound*bni_26] + [(-1)bni_26]x0[2] + [bni_26]x1[2] ≥ 0∧[1 + (-1)bso_27] ≥ 0)



    We simplified constraint (34) using rule (IDP_SMT_SPLIT) which results in the following new constraint:

    (35)    (x0[2] ≥ 0∧x2[2] ≥ 0∧x1[2] ≥ 0 ⇒ (UIncreasing(493_0_MAIN_LOAD(x1[3], x2[3], +(x0[3], 1))), ≥)∧[(-1)Bound*bni_26] + [bni_26]x1[2] ≥ 0∧[1 + (-1)bso_27] ≥ 0)



  • We consider the chain 493_0_MAIN_LOAD(x1[2], x2[2], x0[2]) → COND_493_0_MAIN_LOAD1(&&(&&(>(x2[2], x0[2]), >(x1[2], x0[2])), >(x0[2], -1)), x1[2], x2[2], x0[2]), COND_493_0_MAIN_LOAD1(TRUE, x1[3], x2[3], x0[3]) → 493_0_MAIN_LOAD(x1[3], x2[3], +(x0[3], 1)), 493_0_MAIN_LOAD(x1[2], x2[2], x0[2]) → COND_493_0_MAIN_LOAD1(&&(&&(>(x2[2], x0[2]), >(x1[2], x0[2])), >(x0[2], -1)), x1[2], x2[2], x0[2]) which results in the following constraint:

    (36)    (&&(&&(>(x2[2], x0[2]), >(x1[2], x0[2])), >(x0[2], -1))=TRUE∧x1[2]=x1[3]∧x2[2]=x2[3]∧x0[2]=x0[3]∧x1[3]=x1[2]1∧x2[3]=x2[2]1∧+(x0[3], 1)=x0[2]1 ⇒ COND_493_0_MAIN_LOAD1(TRUE, x1[3], x2[3], x0[3])≥NonInfC∧COND_493_0_MAIN_LOAD1(TRUE, x1[3], x2[3], x0[3])≥493_0_MAIN_LOAD(x1[3], x2[3], +(x0[3], 1))∧(UIncreasing(493_0_MAIN_LOAD(x1[3], x2[3], +(x0[3], 1))), ≥))



    We simplified constraint (36) using rules (III), (IV), (IDP_BOOLEAN) which results in the following new constraint:

    (37)    (>(x0[2], -1)=TRUE∧>(x2[2], x0[2])=TRUE∧>(x1[2], x0[2])=TRUE ⇒ COND_493_0_MAIN_LOAD1(TRUE, x1[2], x2[2], x0[2])≥NonInfC∧COND_493_0_MAIN_LOAD1(TRUE, x1[2], x2[2], x0[2])≥493_0_MAIN_LOAD(x1[2], x2[2], +(x0[2], 1))∧(UIncreasing(493_0_MAIN_LOAD(x1[3], x2[3], +(x0[3], 1))), ≥))



    We simplified constraint (37) using rule (POLY_CONSTRAINTS) which results in the following new constraint:

    (38)    (x0[2] ≥ 0∧x2[2] + [-1] + [-1]x0[2] ≥ 0∧x1[2] + [-1] + [-1]x0[2] ≥ 0 ⇒ (UIncreasing(493_0_MAIN_LOAD(x1[3], x2[3], +(x0[3], 1))), ≥)∧[(-1)bni_26 + (-1)Bound*bni_26] + [(-1)bni_26]x0[2] + [bni_26]x1[2] ≥ 0∧[1 + (-1)bso_27] ≥ 0)



    We simplified constraint (38) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:

    (39)    (x0[2] ≥ 0∧x2[2] + [-1] + [-1]x0[2] ≥ 0∧x1[2] + [-1] + [-1]x0[2] ≥ 0 ⇒ (UIncreasing(493_0_MAIN_LOAD(x1[3], x2[3], +(x0[3], 1))), ≥)∧[(-1)bni_26 + (-1)Bound*bni_26] + [(-1)bni_26]x0[2] + [bni_26]x1[2] ≥ 0∧[1 + (-1)bso_27] ≥ 0)



    We simplified constraint (39) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:

    (40)    (x0[2] ≥ 0∧x2[2] + [-1] + [-1]x0[2] ≥ 0∧x1[2] + [-1] + [-1]x0[2] ≥ 0 ⇒ (UIncreasing(493_0_MAIN_LOAD(x1[3], x2[3], +(x0[3], 1))), ≥)∧[(-1)bni_26 + (-1)Bound*bni_26] + [(-1)bni_26]x0[2] + [bni_26]x1[2] ≥ 0∧[1 + (-1)bso_27] ≥ 0)



    We simplified constraint (40) using rule (IDP_SMT_SPLIT) which results in the following new constraint:

    (41)    (x0[2] ≥ 0∧x2[2] ≥ 0∧x1[2] + [-1] + [-1]x0[2] ≥ 0 ⇒ (UIncreasing(493_0_MAIN_LOAD(x1[3], x2[3], +(x0[3], 1))), ≥)∧[(-1)bni_26 + (-1)Bound*bni_26] + [(-1)bni_26]x0[2] + [bni_26]x1[2] ≥ 0∧[1 + (-1)bso_27] ≥ 0)



    We simplified constraint (41) using rule (IDP_SMT_SPLIT) which results in the following new constraint:

    (42)    (x0[2] ≥ 0∧x2[2] ≥ 0∧x1[2] ≥ 0 ⇒ (UIncreasing(493_0_MAIN_LOAD(x1[3], x2[3], +(x0[3], 1))), ≥)∧[(-1)Bound*bni_26] + [bni_26]x1[2] ≥ 0∧[1 + (-1)bso_27] ≥ 0)







To summarize, we get the following constraints P≥ for the following pairs.
  • 493_0_MAIN_LOAD(x1, x2, x0) → COND_493_0_MAIN_LOAD(&&(&&(>(x2, -1), <=(x2, x0)), >(x1, x0)), x1, x2, x0)
    • (x1[0] ≥ 0∧x2[0] ≥ 0∧x0[0] ≥ 0 ⇒ (UIncreasing(COND_493_0_MAIN_LOAD(&&(&&(>(x2[0], -1), <=(x2[0], x0[0])), >(x1[0], x0[0])), x1[0], x2[0], x0[0])), ≥)∧[(-1)Bound*bni_20] + [bni_20]x1[0] ≥ 0∧[(-1)bso_21] ≥ 0)

  • COND_493_0_MAIN_LOAD(TRUE, x1, x2, x0) → 493_0_MAIN_LOAD(x1, +(x2, 1), x0)
    • (x1[0] ≥ 0∧x2[0] ≥ 0∧x0[0] ≥ 0 ⇒ (UIncreasing(493_0_MAIN_LOAD(x1[1], +(x2[1], 1), x0[1])), ≥)∧[(-1)Bound*bni_22] + [bni_22]x1[0] ≥ 0∧[(-1)bso_23] ≥ 0)
    • (x1[0] ≥ 0∧x2[0] ≥ 0∧x0[0] ≥ 0 ⇒ (UIncreasing(493_0_MAIN_LOAD(x1[1], +(x2[1], 1), x0[1])), ≥)∧[(-1)Bound*bni_22] + [bni_22]x1[0] ≥ 0∧[(-1)bso_23] ≥ 0)

  • 493_0_MAIN_LOAD(x1, x2, x0) → COND_493_0_MAIN_LOAD1(&&(&&(>(x2, x0), >(x1, x0)), >(x0, -1)), x1, x2, x0)
    • (x0[2] ≥ 0∧x2[2] ≥ 0∧x1[2] ≥ 0 ⇒ (UIncreasing(COND_493_0_MAIN_LOAD1(&&(&&(>(x2[2], x0[2]), >(x1[2], x0[2])), >(x0[2], -1)), x1[2], x2[2], x0[2])), ≥)∧[(-1)Bound*bni_24] + [bni_24]x1[2] ≥ 0∧[(-1)bso_25] ≥ 0)

  • COND_493_0_MAIN_LOAD1(TRUE, x1, x2, x0) → 493_0_MAIN_LOAD(x1, x2, +(x0, 1))
    • (x0[2] ≥ 0∧x2[2] ≥ 0∧x1[2] ≥ 0 ⇒ (UIncreasing(493_0_MAIN_LOAD(x1[3], x2[3], +(x0[3], 1))), ≥)∧[(-1)Bound*bni_26] + [bni_26]x1[2] ≥ 0∧[1 + (-1)bso_27] ≥ 0)
    • (x0[2] ≥ 0∧x2[2] ≥ 0∧x1[2] ≥ 0 ⇒ (UIncreasing(493_0_MAIN_LOAD(x1[3], x2[3], +(x0[3], 1))), ≥)∧[(-1)Bound*bni_26] + [bni_26]x1[2] ≥ 0∧[1 + (-1)bso_27] ≥ 0)




The constraints for P> respective Pbound 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 Pbound.
Using the following integer polynomial ordering the resulting constraints can be solved
Polynomial interpretation over integers[POLO]:

POL(TRUE) = 0   
POL(FALSE) = [3]   
POL(493_0_MAIN_LOAD(x1, x2, x3)) = [-1] + [-1]x3 + x1   
POL(COND_493_0_MAIN_LOAD(x1, x2, x3, x4)) = [-1] + [-1]x4 + x2 + [-1]x1   
POL(&&(x1, x2)) = 0   
POL(>(x1, x2)) = [-1]   
POL(-1) = [-1]   
POL(<=(x1, x2)) = [-1]   
POL(+(x1, x2)) = x1 + x2   
POL(1) = [1]   
POL(COND_493_0_MAIN_LOAD1(x1, x2, x3, x4)) = [-1] + [-1]x4 + x2 + [-1]x1   

The following pairs are in P>:

COND_493_0_MAIN_LOAD1(TRUE, x1[3], x2[3], x0[3]) → 493_0_MAIN_LOAD(x1[3], x2[3], +(x0[3], 1))

The following pairs are in Pbound:

493_0_MAIN_LOAD(x1[0], x2[0], x0[0]) → COND_493_0_MAIN_LOAD(&&(&&(>(x2[0], -1), <=(x2[0], x0[0])), >(x1[0], x0[0])), x1[0], x2[0], x0[0])
COND_493_0_MAIN_LOAD(TRUE, x1[1], x2[1], x0[1]) → 493_0_MAIN_LOAD(x1[1], +(x2[1], 1), x0[1])
493_0_MAIN_LOAD(x1[2], x2[2], x0[2]) → COND_493_0_MAIN_LOAD1(&&(&&(>(x2[2], x0[2]), >(x1[2], x0[2])), >(x0[2], -1)), x1[2], x2[2], x0[2])
COND_493_0_MAIN_LOAD1(TRUE, x1[3], x2[3], x0[3]) → 493_0_MAIN_LOAD(x1[3], x2[3], +(x0[3], 1))

The following pairs are in P≥:

493_0_MAIN_LOAD(x1[0], x2[0], x0[0]) → COND_493_0_MAIN_LOAD(&&(&&(>(x2[0], -1), <=(x2[0], x0[0])), >(x1[0], x0[0])), x1[0], x2[0], x0[0])
COND_493_0_MAIN_LOAD(TRUE, x1[1], x2[1], x0[1]) → 493_0_MAIN_LOAD(x1[1], +(x2[1], 1), x0[1])
493_0_MAIN_LOAD(x1[2], x2[2], x0[2]) → COND_493_0_MAIN_LOAD1(&&(&&(>(x2[2], x0[2]), >(x1[2], x0[2])), >(x0[2], -1)), x1[2], x2[2], x0[2])

At least the following rules have been oriented under context sensitive arithmetic replacement:

TRUE1 → &&(TRUE, TRUE)1
FALSE1 → &&(TRUE, FALSE)1
FALSE1 → &&(FALSE, TRUE)1
FALSE1 → &&(FALSE, FALSE)1

(8) Obligation:

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): 493_0_MAIN_LOAD(x1[0], x2[0], x0[0]) → COND_493_0_MAIN_LOAD(x2[0] > -1 && x2[0] <= x0[0] && x1[0] > x0[0], x1[0], x2[0], x0[0])
(1): COND_493_0_MAIN_LOAD(TRUE, x1[1], x2[1], x0[1]) → 493_0_MAIN_LOAD(x1[1], x2[1] + 1, x0[1])
(2): 493_0_MAIN_LOAD(x1[2], x2[2], x0[2]) → COND_493_0_MAIN_LOAD1(x2[2] > x0[2] && x1[2] > x0[2] && x0[2] > -1, x1[2], x2[2], x0[2])

(1) -> (0), if (x1[1] →* x1[0]∧x2[1] + 1 →* x2[0]∧x0[1] →* x0[0])


(0) -> (1), if (x2[0] > -1 && x2[0] <= x0[0] && x1[0] > x0[0] ∧x1[0] →* x1[1]∧x2[0] →* x2[1]∧x0[0] →* x0[1])


(1) -> (2), if (x1[1] →* x1[2]∧x2[1] + 1 →* x2[2]∧x0[1] →* x0[2])



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:

Integer, Boolean


R is empty.

The integer pair graph contains the following rules and edges:
(1): COND_493_0_MAIN_LOAD(TRUE, x1[1], x2[1], x0[1]) → 493_0_MAIN_LOAD(x1[1], x2[1] + 1, x0[1])
(0): 493_0_MAIN_LOAD(x1[0], x2[0], x0[0]) → COND_493_0_MAIN_LOAD(x2[0] > -1 && x2[0] <= x0[0] && x1[0] > x0[0], x1[0], x2[0], x0[0])

(1) -> (0), if (x1[1] →* x1[0]∧x2[1] + 1 →* x2[0]∧x0[1] →* x0[0])


(0) -> (1), if (x2[0] > -1 && x2[0] <= x0[0] && x1[0] > x0[0] ∧x1[0] →* x1[1]∧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@3c50bf08 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_493_0_MAIN_LOAD(TRUE, x1[1], x2[1], x0[1]) → 493_0_MAIN_LOAD(x1[1], +(x2[1], 1), x0[1]) the following chains were created:
  • We consider the chain 493_0_MAIN_LOAD(x1[0], x2[0], x0[0]) → COND_493_0_MAIN_LOAD(&&(&&(>(x2[0], -1), <=(x2[0], x0[0])), >(x1[0], x0[0])), x1[0], x2[0], x0[0]), COND_493_0_MAIN_LOAD(TRUE, x1[1], x2[1], x0[1]) → 493_0_MAIN_LOAD(x1[1], +(x2[1], 1), x0[1]), 493_0_MAIN_LOAD(x1[0], x2[0], x0[0]) → COND_493_0_MAIN_LOAD(&&(&&(>(x2[0], -1), <=(x2[0], x0[0])), >(x1[0], x0[0])), x1[0], x2[0], x0[0]) which results in the following constraint:

    (1)    (&&(&&(>(x2[0], -1), <=(x2[0], x0[0])), >(x1[0], x0[0]))=TRUE∧x1[0]=x1[1]∧x2[0]=x2[1]∧x0[0]=x0[1]∧x1[1]=x1[0]1∧+(x2[1], 1)=x2[0]1∧x0[1]=x0[0]1 ⇒ COND_493_0_MAIN_LOAD(TRUE, x1[1], x2[1], x0[1])≥NonInfC∧COND_493_0_MAIN_LOAD(TRUE, x1[1], x2[1], x0[1])≥493_0_MAIN_LOAD(x1[1], +(x2[1], 1), x0[1])∧(UIncreasing(493_0_MAIN_LOAD(x1[1], +(x2[1], 1), x0[1])), ≥))



    We simplified constraint (1) using rules (III), (IV), (IDP_BOOLEAN) which results in the following new constraint:

    (2)    (>(x1[0], x0[0])=TRUE∧>(x2[0], -1)=TRUE∧<=(x2[0], x0[0])=TRUE ⇒ COND_493_0_MAIN_LOAD(TRUE, x1[0], x2[0], x0[0])≥NonInfC∧COND_493_0_MAIN_LOAD(TRUE, x1[0], x2[0], x0[0])≥493_0_MAIN_LOAD(x1[0], +(x2[0], 1), x0[0])∧(UIncreasing(493_0_MAIN_LOAD(x1[1], +(x2[1], 1), x0[1])), ≥))



    We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint:

    (3)    (x1[0] + [-1] + [-1]x0[0] ≥ 0∧x2[0] ≥ 0∧x0[0] + [-1]x2[0] ≥ 0 ⇒ (UIncreasing(493_0_MAIN_LOAD(x1[1], +(x2[1], 1), x0[1])), ≥)∧[(-2)bni_15 + (-1)Bound*bni_15] + [bni_15]x0[0] + [(-1)bni_15]x2[0] ≥ 0∧[(-1)bso_16] ≥ 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∧x2[0] ≥ 0∧x0[0] + [-1]x2[0] ≥ 0 ⇒ (UIncreasing(493_0_MAIN_LOAD(x1[1], +(x2[1], 1), x0[1])), ≥)∧[(-2)bni_15 + (-1)Bound*bni_15] + [bni_15]x0[0] + [(-1)bni_15]x2[0] ≥ 0∧[(-1)bso_16] ≥ 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∧x2[0] ≥ 0∧x0[0] + [-1]x2[0] ≥ 0 ⇒ (UIncreasing(493_0_MAIN_LOAD(x1[1], +(x2[1], 1), x0[1])), ≥)∧[(-2)bni_15 + (-1)Bound*bni_15] + [bni_15]x0[0] + [(-1)bni_15]x2[0] ≥ 0∧[(-1)bso_16] ≥ 0)



    We simplified constraint (5) using rule (IDP_SMT_SPLIT) which results in the following new constraint:

    (6)    (x1[0] ≥ 0∧x2[0] ≥ 0∧x0[0] + [-1]x2[0] ≥ 0 ⇒ (UIncreasing(493_0_MAIN_LOAD(x1[1], +(x2[1], 1), x0[1])), ≥)∧[(-2)bni_15 + (-1)Bound*bni_15] + [bni_15]x0[0] + [(-1)bni_15]x2[0] ≥ 0∧[(-1)bso_16] ≥ 0)



    We simplified constraint (6) using rule (IDP_SMT_SPLIT) which results in the following new constraint:

    (7)    (x1[0] ≥ 0∧x2[0] ≥ 0∧x0[0] ≥ 0 ⇒ (UIncreasing(493_0_MAIN_LOAD(x1[1], +(x2[1], 1), x0[1])), ≥)∧[(-2)bni_15 + (-1)Bound*bni_15] + [bni_15]x0[0] ≥ 0∧[(-1)bso_16] ≥ 0)







For Pair 493_0_MAIN_LOAD(x1[0], x2[0], x0[0]) → COND_493_0_MAIN_LOAD(&&(&&(>(x2[0], -1), <=(x2[0], x0[0])), >(x1[0], x0[0])), x1[0], x2[0], x0[0]) the following chains were created:
  • We consider the chain 493_0_MAIN_LOAD(x1[0], x2[0], x0[0]) → COND_493_0_MAIN_LOAD(&&(&&(>(x2[0], -1), <=(x2[0], x0[0])), >(x1[0], x0[0])), x1[0], x2[0], x0[0]), COND_493_0_MAIN_LOAD(TRUE, x1[1], x2[1], x0[1]) → 493_0_MAIN_LOAD(x1[1], +(x2[1], 1), x0[1]) which results in the following constraint:

    (8)    (&&(&&(>(x2[0], -1), <=(x2[0], x0[0])), >(x1[0], x0[0]))=TRUE∧x1[0]=x1[1]∧x2[0]=x2[1]∧x0[0]=x0[1] ⇒ 493_0_MAIN_LOAD(x1[0], x2[0], x0[0])≥NonInfC∧493_0_MAIN_LOAD(x1[0], x2[0], x0[0])≥COND_493_0_MAIN_LOAD(&&(&&(>(x2[0], -1), <=(x2[0], x0[0])), >(x1[0], x0[0])), x1[0], x2[0], x0[0])∧(UIncreasing(COND_493_0_MAIN_LOAD(&&(&&(>(x2[0], -1), <=(x2[0], x0[0])), >(x1[0], x0[0])), x1[0], x2[0], x0[0])), ≥))



    We simplified constraint (8) using rules (IV), (IDP_BOOLEAN) which results in the following new constraint:

    (9)    (>(x1[0], x0[0])=TRUE∧>(x2[0], -1)=TRUE∧<=(x2[0], x0[0])=TRUE ⇒ 493_0_MAIN_LOAD(x1[0], x2[0], x0[0])≥NonInfC∧493_0_MAIN_LOAD(x1[0], x2[0], x0[0])≥COND_493_0_MAIN_LOAD(&&(&&(>(x2[0], -1), <=(x2[0], x0[0])), >(x1[0], x0[0])), x1[0], x2[0], x0[0])∧(UIncreasing(COND_493_0_MAIN_LOAD(&&(&&(>(x2[0], -1), <=(x2[0], x0[0])), >(x1[0], x0[0])), x1[0], x2[0], x0[0])), ≥))



    We simplified constraint (9) using rule (POLY_CONSTRAINTS) which results in the following new constraint:

    (10)    (x1[0] + [-1] + [-1]x0[0] ≥ 0∧x2[0] ≥ 0∧x0[0] + [-1]x2[0] ≥ 0 ⇒ (UIncreasing(COND_493_0_MAIN_LOAD(&&(&&(>(x2[0], -1), <=(x2[0], x0[0])), >(x1[0], x0[0])), x1[0], x2[0], x0[0])), ≥)∧[(-1)bni_17 + (-1)Bound*bni_17] + [bni_17]x0[0] + [(-1)bni_17]x2[0] ≥ 0∧[1 + (-1)bso_18] ≥ 0)



    We simplified constraint (10) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:

    (11)    (x1[0] + [-1] + [-1]x0[0] ≥ 0∧x2[0] ≥ 0∧x0[0] + [-1]x2[0] ≥ 0 ⇒ (UIncreasing(COND_493_0_MAIN_LOAD(&&(&&(>(x2[0], -1), <=(x2[0], x0[0])), >(x1[0], x0[0])), x1[0], x2[0], x0[0])), ≥)∧[(-1)bni_17 + (-1)Bound*bni_17] + [bni_17]x0[0] + [(-1)bni_17]x2[0] ≥ 0∧[1 + (-1)bso_18] ≥ 0)



    We simplified constraint (11) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:

    (12)    (x1[0] + [-1] + [-1]x0[0] ≥ 0∧x2[0] ≥ 0∧x0[0] + [-1]x2[0] ≥ 0 ⇒ (UIncreasing(COND_493_0_MAIN_LOAD(&&(&&(>(x2[0], -1), <=(x2[0], x0[0])), >(x1[0], x0[0])), x1[0], x2[0], x0[0])), ≥)∧[(-1)bni_17 + (-1)Bound*bni_17] + [bni_17]x0[0] + [(-1)bni_17]x2[0] ≥ 0∧[1 + (-1)bso_18] ≥ 0)



    We simplified constraint (12) using rule (IDP_SMT_SPLIT) which results in the following new constraint:

    (13)    (x1[0] ≥ 0∧x2[0] ≥ 0∧x0[0] + [-1]x2[0] ≥ 0 ⇒ (UIncreasing(COND_493_0_MAIN_LOAD(&&(&&(>(x2[0], -1), <=(x2[0], x0[0])), >(x1[0], x0[0])), x1[0], x2[0], x0[0])), ≥)∧[(-1)bni_17 + (-1)Bound*bni_17] + [bni_17]x0[0] + [(-1)bni_17]x2[0] ≥ 0∧[1 + (-1)bso_18] ≥ 0)



    We simplified constraint (13) using rule (IDP_SMT_SPLIT) which results in the following new constraint:

    (14)    (x1[0] ≥ 0∧x2[0] ≥ 0∧x0[0] ≥ 0 ⇒ (UIncreasing(COND_493_0_MAIN_LOAD(&&(&&(>(x2[0], -1), <=(x2[0], x0[0])), >(x1[0], x0[0])), x1[0], x2[0], x0[0])), ≥)∧[(-1)bni_17 + (-1)Bound*bni_17] + [bni_17]x0[0] ≥ 0∧[1 + (-1)bso_18] ≥ 0)







To summarize, we get the following constraints P≥ for the following pairs.
  • COND_493_0_MAIN_LOAD(TRUE, x1[1], x2[1], x0[1]) → 493_0_MAIN_LOAD(x1[1], +(x2[1], 1), x0[1])
    • (x1[0] ≥ 0∧x2[0] ≥ 0∧x0[0] ≥ 0 ⇒ (UIncreasing(493_0_MAIN_LOAD(x1[1], +(x2[1], 1), x0[1])), ≥)∧[(-2)bni_15 + (-1)Bound*bni_15] + [bni_15]x0[0] ≥ 0∧[(-1)bso_16] ≥ 0)

  • 493_0_MAIN_LOAD(x1[0], x2[0], x0[0]) → COND_493_0_MAIN_LOAD(&&(&&(>(x2[0], -1), <=(x2[0], x0[0])), >(x1[0], x0[0])), x1[0], x2[0], x0[0])
    • (x1[0] ≥ 0∧x2[0] ≥ 0∧x0[0] ≥ 0 ⇒ (UIncreasing(COND_493_0_MAIN_LOAD(&&(&&(>(x2[0], -1), <=(x2[0], x0[0])), >(x1[0], x0[0])), x1[0], x2[0], x0[0])), ≥)∧[(-1)bni_17 + (-1)Bound*bni_17] + [bni_17]x0[0] ≥ 0∧[1 + (-1)bso_18] ≥ 0)




The constraints for P> respective Pbound 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 Pbound.
Using the following integer polynomial ordering the resulting constraints can be solved
Polynomial interpretation over integers[POLO]:

POL(TRUE) = [1]   
POL(FALSE) = [1]   
POL(COND_493_0_MAIN_LOAD(x1, x2, x3, x4)) = [-1] + x4 + [-1]x3 + [-1]x1   
POL(493_0_MAIN_LOAD(x1, x2, x3)) = [-1] + x3 + [-1]x2   
POL(+(x1, x2)) = x1 + x2   
POL(1) = [1]   
POL(&&(x1, x2)) = [1]   
POL(>(x1, x2)) = [-1]   
POL(-1) = [-1]   
POL(<=(x1, x2)) = [-1]   

The following pairs are in P>:

493_0_MAIN_LOAD(x1[0], x2[0], x0[0]) → COND_493_0_MAIN_LOAD(&&(&&(>(x2[0], -1), <=(x2[0], x0[0])), >(x1[0], x0[0])), x1[0], x2[0], x0[0])

The following pairs are in Pbound:

COND_493_0_MAIN_LOAD(TRUE, x1[1], x2[1], x0[1]) → 493_0_MAIN_LOAD(x1[1], +(x2[1], 1), x0[1])
493_0_MAIN_LOAD(x1[0], x2[0], x0[0]) → COND_493_0_MAIN_LOAD(&&(&&(>(x2[0], -1), <=(x2[0], x0[0])), >(x1[0], x0[0])), x1[0], x2[0], x0[0])

The following pairs are in P≥:

COND_493_0_MAIN_LOAD(TRUE, x1[1], x2[1], x0[1]) → 493_0_MAIN_LOAD(x1[1], +(x2[1], 1), x0[1])

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) 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_493_0_MAIN_LOAD(TRUE, x1[1], x2[1], x0[1]) → 493_0_MAIN_LOAD(x1[1], x2[1] + 1, x0[1])


The set Q is empty.

(13) IDependencyGraphProof (EQUIVALENT transformation)

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

(14) TRUE