Termination of the given JBC could be proven:

0 JBC
↳1 JBCToGraph (⇒, 350 ms)
↳2 JBCTerminationGraph
↳3 TerminationGraphToSCCProof (⇒, 0 ms)
↳4 JBCTerminationSCC
↳5 SCCToIDPv1Proof (⇒, 70 ms)
↳6 IDP
↳7 IDPNonInfProof (⇒, 510 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: PastaC2 
/**
 * 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 PastaC2 {
    public static void main(String[] args) {
        Random.args = args;
        int x = Random.random();

        while (x >= 0) {
            x = x+1;
            int y = 1;
            while (x >= y) {
                y++;
            }
            x = x-2;
        }
    }
}


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:
PastaC2.main([Ljava/lang/String;)V: Graph of 128 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: PastaC2.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 28 rules for P and 0 rules for R.


P rules:
665_0_main_LT(EOS(STATIC_665), i114, i114) → 668_0_main_LT(EOS(STATIC_668), i114, i114)
668_0_main_LT(EOS(STATIC_668), i114, i114) → 671_0_main_Load(EOS(STATIC_671), i114) | >=(i114, 0)
671_0_main_Load(EOS(STATIC_671), i114) → 674_0_main_ConstantStackPush(EOS(STATIC_674), i114)
674_0_main_ConstantStackPush(EOS(STATIC_674), i114) → 676_0_main_IntArithmetic(EOS(STATIC_676), i114, 1)
676_0_main_IntArithmetic(EOS(STATIC_676), i114, matching1) → 678_0_main_Store(EOS(STATIC_678), +(i114, 1)) | &&(>=(i114, 0), =(matching1, 1))
678_0_main_Store(EOS(STATIC_678), i115) → 680_0_main_ConstantStackPush(EOS(STATIC_680), i115)
680_0_main_ConstantStackPush(EOS(STATIC_680), i115) → 681_0_main_Store(EOS(STATIC_681), i115, 1)
681_0_main_Store(EOS(STATIC_681), i115, matching1) → 683_0_main_Load(EOS(STATIC_683), i115, 1) | =(matching1, 1)
683_0_main_Load(EOS(STATIC_683), i115, matching1) → 696_0_main_Load(EOS(STATIC_696), i115, 1) | =(matching1, 1)
696_0_main_Load(EOS(STATIC_696), i115, i116) → 716_0_main_Load(EOS(STATIC_716), i115, i116)
716_0_main_Load(EOS(STATIC_716), i115, i119) → 734_0_main_Load(EOS(STATIC_734), i115, i119)
734_0_main_Load(EOS(STATIC_734), i115, i122) → 751_0_main_Load(EOS(STATIC_751), i115, i122)
751_0_main_Load(EOS(STATIC_751), i115, i125) → 753_0_main_Load(EOS(STATIC_753), i115, i125, i115)
753_0_main_Load(EOS(STATIC_753), i115, i125, i115) → 754_0_main_LT(EOS(STATIC_754), i115, i125, i115, i125)
754_0_main_LT(EOS(STATIC_754), i115, i125, i115, i125) → 756_0_main_LT(EOS(STATIC_756), i115, i125, i115, i125)
754_0_main_LT(EOS(STATIC_754), i115, i125, i115, i125) → 757_0_main_LT(EOS(STATIC_757), i115, i125, i115, i125)
756_0_main_LT(EOS(STATIC_756), i115, i125, i115, i125) → 758_0_main_Load(EOS(STATIC_758), i115) | <(i115, i125)
758_0_main_Load(EOS(STATIC_758), i115) → 761_0_main_ConstantStackPush(EOS(STATIC_761), i115)
761_0_main_ConstantStackPush(EOS(STATIC_761), i115) → 764_0_main_IntArithmetic(EOS(STATIC_764), i115, 2)
764_0_main_IntArithmetic(EOS(STATIC_764), i115, matching1) → 768_0_main_Store(EOS(STATIC_768), -(i115, 2)) | &&(>(i115, 0), =(matching1, 2))
768_0_main_Store(EOS(STATIC_768), i129) → 863_0_main_JMP(EOS(STATIC_863), i129)
863_0_main_JMP(EOS(STATIC_863), i129) → 865_0_main_Load(EOS(STATIC_865), i129)
865_0_main_Load(EOS(STATIC_865), i129) → 663_0_main_Load(EOS(STATIC_663), i129)
663_0_main_Load(EOS(STATIC_663), i111) → 665_0_main_LT(EOS(STATIC_665), i111, i111)
757_0_main_LT(EOS(STATIC_757), i115, i125, i115, i125) → 760_0_main_Inc(EOS(STATIC_760), i115, i125) | >=(i115, i125)
760_0_main_Inc(EOS(STATIC_760), i115, i125) → 763_0_main_JMP(EOS(STATIC_763), i115, +(i125, 1)) | >(i125, 0)
763_0_main_JMP(EOS(STATIC_763), i115, i127) → 766_0_main_Load(EOS(STATIC_766), i115, i127)
766_0_main_Load(EOS(STATIC_766), i115, i127) → 751_0_main_Load(EOS(STATIC_751), i115, i127)
R rules:

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


P rules:
754_0_main_LT(EOS(STATIC_754), x0, x1, x0, x1) → 754_0_main_LT(EOS(STATIC_754), +(-(x0, 2), 1), 1, +(-(x0, 2), 1), 1) | &&(>(x1, x0), >(+(x0, 1), 2))
754_0_main_LT(EOS(STATIC_754), x0, x1, x0, x1) → 754_0_main_LT(EOS(STATIC_754), x0, +(x1, 1), x0, +(x1, 1)) | &&(>(x1, 0), <=(x1, x0))
R rules:

Filtered ground terms:



754_0_main_LT(x1, x2, x3, x4, x5) → 754_0_main_LT(x2, x3, x4, x5)
EOS(x1) → EOS
Cond_754_0_main_LT1(x1, x2, x3, x4, x5, x6) → Cond_754_0_main_LT1(x1, x3, x4, x5, x6)
Cond_754_0_main_LT(x1, x2, x3, x4, x5, x6) → Cond_754_0_main_LT(x1, x3, x4, x5, x6)

Filtered duplicate args:



754_0_main_LT(x1, x2, x3, x4) → 754_0_main_LT(x3, x4)
Cond_754_0_main_LT(x1, x2, x3, x4, x5) → Cond_754_0_main_LT(x1, x4, x5)
Cond_754_0_main_LT1(x1, x2, x3, x4, x5) → Cond_754_0_main_LT1(x1, x4, x5)

Filtered unneeded arguments:



Cond_754_0_main_LT(x1, x2, x3) → Cond_754_0_main_LT(x1, x2)

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


P rules:
754_0_main_LT(x0, x1) → 754_0_main_LT(+(-(x0, 2), 1), 1) | &&(>(x1, x0), >(x0, 1))
754_0_main_LT(x0, x1) → 754_0_main_LT(x0, +(x1, 1)) | &&(>(x1, 0), <=(x1, x0))
R rules:

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


P rules:
754_0_MAIN_LT(x0, x1) → COND_754_0_MAIN_LT(&&(>(x1, x0), >(x0, 1)), x0, x1)
COND_754_0_MAIN_LT(TRUE, x0, x1) → 754_0_MAIN_LT(+(-(x0, 2), 1), 1)
754_0_MAIN_LT(x0, x1) → COND_754_0_MAIN_LT1(&&(>(x1, 0), <=(x1, x0)), x0, x1)
COND_754_0_MAIN_LT1(TRUE, x0, x1) → 754_0_MAIN_LT(x0, +(x1, 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): 754_0_MAIN_LT(x0[0], x1[0]) → COND_754_0_MAIN_LT(x1[0] > x0[0] && x0[0] > 1, x0[0], x1[0])
(1): COND_754_0_MAIN_LT(TRUE, x0[1], x1[1]) → 754_0_MAIN_LT(x0[1] - 2 + 1, 1)
(2): 754_0_MAIN_LT(x0[2], x1[2]) → COND_754_0_MAIN_LT1(x1[2] > 0 && x1[2] <= x0[2], x0[2], x1[2])
(3): COND_754_0_MAIN_LT1(TRUE, x0[3], x1[3]) → 754_0_MAIN_LT(x0[3], x1[3] + 1)

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


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


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


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


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


(3) -> (2), if (x0[3] →* x0[2]∧x1[3] + 1 →* x1[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@3b815cce 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 754_0_MAIN_LT(x0, x1) → COND_754_0_MAIN_LT(&&(>(x1, x0), >(x0, 1)), x0, x1) the following chains were created:
  • We consider the chain 754_0_MAIN_LT(x0[0], x1[0]) → COND_754_0_MAIN_LT(&&(>(x1[0], x0[0]), >(x0[0], 1)), x0[0], x1[0]), COND_754_0_MAIN_LT(TRUE, x0[1], x1[1]) → 754_0_MAIN_LT(+(-(x0[1], 2), 1), 1) which results in the following constraint:

    (1)    (&&(>(x1[0], x0[0]), >(x0[0], 1))=TRUE∧x0[0]=x0[1]∧x1[0]=x1[1] ⇒ 754_0_MAIN_LT(x0[0], x1[0])≥NonInfC∧754_0_MAIN_LT(x0[0], x1[0])≥COND_754_0_MAIN_LT(&&(>(x1[0], x0[0]), >(x0[0], 1)), x0[0], x1[0])∧(UIncreasing(COND_754_0_MAIN_LT(&&(>(x1[0], x0[0]), >(x0[0], 1)), x0[0], x1[0])), ≥))



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

    (2)    (>(x1[0], x0[0])=TRUE∧>(x0[0], 1)=TRUE ⇒ 754_0_MAIN_LT(x0[0], x1[0])≥NonInfC∧754_0_MAIN_LT(x0[0], x1[0])≥COND_754_0_MAIN_LT(&&(>(x1[0], x0[0]), >(x0[0], 1)), x0[0], x1[0])∧(UIncreasing(COND_754_0_MAIN_LT(&&(>(x1[0], x0[0]), >(x0[0], 1)), x0[0], x1[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] + [-2] ≥ 0 ⇒ (UIncreasing(COND_754_0_MAIN_LT(&&(>(x1[0], x0[0]), >(x0[0], 1)), x0[0], x1[0])), ≥)∧[(-1)bni_17 + (-1)Bound*bni_17] + [bni_17]x0[0] ≥ 0∧[(-1)bso_18] ≥ 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] + [-2] ≥ 0 ⇒ (UIncreasing(COND_754_0_MAIN_LT(&&(>(x1[0], x0[0]), >(x0[0], 1)), x0[0], x1[0])), ≥)∧[(-1)bni_17 + (-1)Bound*bni_17] + [bni_17]x0[0] ≥ 0∧[(-1)bso_18] ≥ 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] + [-2] ≥ 0 ⇒ (UIncreasing(COND_754_0_MAIN_LT(&&(>(x1[0], x0[0]), >(x0[0], 1)), x0[0], x1[0])), ≥)∧[(-1)bni_17 + (-1)Bound*bni_17] + [bni_17]x0[0] ≥ 0∧[(-1)bso_18] ≥ 0)



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

    (6)    (x1[0] ≥ 0∧x0[0] + [-2] ≥ 0 ⇒ (UIncreasing(COND_754_0_MAIN_LT(&&(>(x1[0], x0[0]), >(x0[0], 1)), x0[0], x1[0])), ≥)∧[(-1)bni_17 + (-1)Bound*bni_17] + [bni_17]x0[0] ≥ 0∧[(-1)bso_18] ≥ 0)



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

    (7)    (x1[0] ≥ 0∧x0[0] ≥ 0 ⇒ (UIncreasing(COND_754_0_MAIN_LT(&&(>(x1[0], x0[0]), >(x0[0], 1)), x0[0], x1[0])), ≥)∧[bni_17 + (-1)Bound*bni_17] + [bni_17]x0[0] ≥ 0∧[(-1)bso_18] ≥ 0)







For Pair COND_754_0_MAIN_LT(TRUE, x0, x1) → 754_0_MAIN_LT(+(-(x0, 2), 1), 1) the following chains were created:
  • We consider the chain 754_0_MAIN_LT(x0[0], x1[0]) → COND_754_0_MAIN_LT(&&(>(x1[0], x0[0]), >(x0[0], 1)), x0[0], x1[0]), COND_754_0_MAIN_LT(TRUE, x0[1], x1[1]) → 754_0_MAIN_LT(+(-(x0[1], 2), 1), 1), 754_0_MAIN_LT(x0[0], x1[0]) → COND_754_0_MAIN_LT(&&(>(x1[0], x0[0]), >(x0[0], 1)), x0[0], x1[0]) which results in the following constraint:

    (8)    (&&(>(x1[0], x0[0]), >(x0[0], 1))=TRUE∧x0[0]=x0[1]∧x1[0]=x1[1]∧+(-(x0[1], 2), 1)=x0[0]1∧1=x1[0]1 ⇒ COND_754_0_MAIN_LT(TRUE, x0[1], x1[1])≥NonInfC∧COND_754_0_MAIN_LT(TRUE, x0[1], x1[1])≥754_0_MAIN_LT(+(-(x0[1], 2), 1), 1)∧(UIncreasing(754_0_MAIN_LT(+(-(x0[1], 2), 1), 1)), ≥))



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

    (9)    (>(x1[0], x0[0])=TRUE∧>(x0[0], 1)=TRUE ⇒ COND_754_0_MAIN_LT(TRUE, x0[0], x1[0])≥NonInfC∧COND_754_0_MAIN_LT(TRUE, x0[0], x1[0])≥754_0_MAIN_LT(+(-(x0[0], 2), 1), 1)∧(UIncreasing(754_0_MAIN_LT(+(-(x0[1], 2), 1), 1)), ≥))



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

    (10)    (x1[0] + [-1] + [-1]x0[0] ≥ 0∧x0[0] + [-2] ≥ 0 ⇒ (UIncreasing(754_0_MAIN_LT(+(-(x0[1], 2), 1), 1)), ≥)∧[(-1)bni_19 + (-1)Bound*bni_19] + [bni_19]x0[0] ≥ 0∧[1 + (-1)bso_20] ≥ 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∧x0[0] + [-2] ≥ 0 ⇒ (UIncreasing(754_0_MAIN_LT(+(-(x0[1], 2), 1), 1)), ≥)∧[(-1)bni_19 + (-1)Bound*bni_19] + [bni_19]x0[0] ≥ 0∧[1 + (-1)bso_20] ≥ 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∧x0[0] + [-2] ≥ 0 ⇒ (UIncreasing(754_0_MAIN_LT(+(-(x0[1], 2), 1), 1)), ≥)∧[(-1)bni_19 + (-1)Bound*bni_19] + [bni_19]x0[0] ≥ 0∧[1 + (-1)bso_20] ≥ 0)



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

    (13)    (x1[0] ≥ 0∧x0[0] + [-2] ≥ 0 ⇒ (UIncreasing(754_0_MAIN_LT(+(-(x0[1], 2), 1), 1)), ≥)∧[(-1)bni_19 + (-1)Bound*bni_19] + [bni_19]x0[0] ≥ 0∧[1 + (-1)bso_20] ≥ 0)



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

    (14)    (x1[0] ≥ 0∧x0[0] ≥ 0 ⇒ (UIncreasing(754_0_MAIN_LT(+(-(x0[1], 2), 1), 1)), ≥)∧[bni_19 + (-1)Bound*bni_19] + [bni_19]x0[0] ≥ 0∧[1 + (-1)bso_20] ≥ 0)



  • We consider the chain 754_0_MAIN_LT(x0[0], x1[0]) → COND_754_0_MAIN_LT(&&(>(x1[0], x0[0]), >(x0[0], 1)), x0[0], x1[0]), COND_754_0_MAIN_LT(TRUE, x0[1], x1[1]) → 754_0_MAIN_LT(+(-(x0[1], 2), 1), 1), 754_0_MAIN_LT(x0[2], x1[2]) → COND_754_0_MAIN_LT1(&&(>(x1[2], 0), <=(x1[2], x0[2])), x0[2], x1[2]) which results in the following constraint:

    (15)    (&&(>(x1[0], x0[0]), >(x0[0], 1))=TRUE∧x0[0]=x0[1]∧x1[0]=x1[1]∧+(-(x0[1], 2), 1)=x0[2]∧1=x1[2] ⇒ COND_754_0_MAIN_LT(TRUE, x0[1], x1[1])≥NonInfC∧COND_754_0_MAIN_LT(TRUE, x0[1], x1[1])≥754_0_MAIN_LT(+(-(x0[1], 2), 1), 1)∧(UIncreasing(754_0_MAIN_LT(+(-(x0[1], 2), 1), 1)), ≥))



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

    (16)    (>(x1[0], x0[0])=TRUE∧>(x0[0], 1)=TRUE ⇒ COND_754_0_MAIN_LT(TRUE, x0[0], x1[0])≥NonInfC∧COND_754_0_MAIN_LT(TRUE, x0[0], x1[0])≥754_0_MAIN_LT(+(-(x0[0], 2), 1), 1)∧(UIncreasing(754_0_MAIN_LT(+(-(x0[1], 2), 1), 1)), ≥))



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

    (17)    (x1[0] + [-1] + [-1]x0[0] ≥ 0∧x0[0] + [-2] ≥ 0 ⇒ (UIncreasing(754_0_MAIN_LT(+(-(x0[1], 2), 1), 1)), ≥)∧[(-1)bni_19 + (-1)Bound*bni_19] + [bni_19]x0[0] ≥ 0∧[1 + (-1)bso_20] ≥ 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∧x0[0] + [-2] ≥ 0 ⇒ (UIncreasing(754_0_MAIN_LT(+(-(x0[1], 2), 1), 1)), ≥)∧[(-1)bni_19 + (-1)Bound*bni_19] + [bni_19]x0[0] ≥ 0∧[1 + (-1)bso_20] ≥ 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∧x0[0] + [-2] ≥ 0 ⇒ (UIncreasing(754_0_MAIN_LT(+(-(x0[1], 2), 1), 1)), ≥)∧[(-1)bni_19 + (-1)Bound*bni_19] + [bni_19]x0[0] ≥ 0∧[1 + (-1)bso_20] ≥ 0)



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

    (20)    (x1[0] ≥ 0∧x0[0] + [-2] ≥ 0 ⇒ (UIncreasing(754_0_MAIN_LT(+(-(x0[1], 2), 1), 1)), ≥)∧[(-1)bni_19 + (-1)Bound*bni_19] + [bni_19]x0[0] ≥ 0∧[1 + (-1)bso_20] ≥ 0)



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

    (21)    (x1[0] ≥ 0∧x0[0] ≥ 0 ⇒ (UIncreasing(754_0_MAIN_LT(+(-(x0[1], 2), 1), 1)), ≥)∧[bni_19 + (-1)Bound*bni_19] + [bni_19]x0[0] ≥ 0∧[1 + (-1)bso_20] ≥ 0)







For Pair 754_0_MAIN_LT(x0, x1) → COND_754_0_MAIN_LT1(&&(>(x1, 0), <=(x1, x0)), x0, x1) the following chains were created:
  • We consider the chain 754_0_MAIN_LT(x0[2], x1[2]) → COND_754_0_MAIN_LT1(&&(>(x1[2], 0), <=(x1[2], x0[2])), x0[2], x1[2]), COND_754_0_MAIN_LT1(TRUE, x0[3], x1[3]) → 754_0_MAIN_LT(x0[3], +(x1[3], 1)) which results in the following constraint:

    (22)    (&&(>(x1[2], 0), <=(x1[2], x0[2]))=TRUE∧x0[2]=x0[3]∧x1[2]=x1[3] ⇒ 754_0_MAIN_LT(x0[2], x1[2])≥NonInfC∧754_0_MAIN_LT(x0[2], x1[2])≥COND_754_0_MAIN_LT1(&&(>(x1[2], 0), <=(x1[2], x0[2])), x0[2], x1[2])∧(UIncreasing(COND_754_0_MAIN_LT1(&&(>(x1[2], 0), <=(x1[2], x0[2])), x0[2], x1[2])), ≥))



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

    (23)    (>(x1[2], 0)=TRUE∧<=(x1[2], x0[2])=TRUE ⇒ 754_0_MAIN_LT(x0[2], x1[2])≥NonInfC∧754_0_MAIN_LT(x0[2], x1[2])≥COND_754_0_MAIN_LT1(&&(>(x1[2], 0), <=(x1[2], x0[2])), x0[2], x1[2])∧(UIncreasing(COND_754_0_MAIN_LT1(&&(>(x1[2], 0), <=(x1[2], x0[2])), x0[2], x1[2])), ≥))



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

    (24)    (x1[2] + [-1] ≥ 0∧x0[2] + [-1]x1[2] ≥ 0 ⇒ (UIncreasing(COND_754_0_MAIN_LT1(&&(>(x1[2], 0), <=(x1[2], x0[2])), x0[2], x1[2])), ≥)∧[(-1)bni_21 + (-1)Bound*bni_21] + [bni_21]x0[2] ≥ 0∧[(-1)bso_22] ≥ 0)



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

    (25)    (x1[2] + [-1] ≥ 0∧x0[2] + [-1]x1[2] ≥ 0 ⇒ (UIncreasing(COND_754_0_MAIN_LT1(&&(>(x1[2], 0), <=(x1[2], x0[2])), x0[2], x1[2])), ≥)∧[(-1)bni_21 + (-1)Bound*bni_21] + [bni_21]x0[2] ≥ 0∧[(-1)bso_22] ≥ 0)



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

    (26)    (x1[2] + [-1] ≥ 0∧x0[2] + [-1]x1[2] ≥ 0 ⇒ (UIncreasing(COND_754_0_MAIN_LT1(&&(>(x1[2], 0), <=(x1[2], x0[2])), x0[2], x1[2])), ≥)∧[(-1)bni_21 + (-1)Bound*bni_21] + [bni_21]x0[2] ≥ 0∧[(-1)bso_22] ≥ 0)



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

    (27)    (x1[2] ≥ 0∧x0[2] + [-1] + [-1]x1[2] ≥ 0 ⇒ (UIncreasing(COND_754_0_MAIN_LT1(&&(>(x1[2], 0), <=(x1[2], x0[2])), x0[2], x1[2])), ≥)∧[(-1)bni_21 + (-1)Bound*bni_21] + [bni_21]x0[2] ≥ 0∧[(-1)bso_22] ≥ 0)



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

    (28)    (x1[2] ≥ 0∧x0[2] ≥ 0 ⇒ (UIncreasing(COND_754_0_MAIN_LT1(&&(>(x1[2], 0), <=(x1[2], x0[2])), x0[2], x1[2])), ≥)∧[(-1)Bound*bni_21] + [bni_21]x1[2] + [bni_21]x0[2] ≥ 0∧[(-1)bso_22] ≥ 0)







For Pair COND_754_0_MAIN_LT1(TRUE, x0, x1) → 754_0_MAIN_LT(x0, +(x1, 1)) the following chains were created:
  • We consider the chain 754_0_MAIN_LT(x0[2], x1[2]) → COND_754_0_MAIN_LT1(&&(>(x1[2], 0), <=(x1[2], x0[2])), x0[2], x1[2]), COND_754_0_MAIN_LT1(TRUE, x0[3], x1[3]) → 754_0_MAIN_LT(x0[3], +(x1[3], 1)), 754_0_MAIN_LT(x0[0], x1[0]) → COND_754_0_MAIN_LT(&&(>(x1[0], x0[0]), >(x0[0], 1)), x0[0], x1[0]) which results in the following constraint:

    (29)    (&&(>(x1[2], 0), <=(x1[2], x0[2]))=TRUE∧x0[2]=x0[3]∧x1[2]=x1[3]∧x0[3]=x0[0]∧+(x1[3], 1)=x1[0] ⇒ COND_754_0_MAIN_LT1(TRUE, x0[3], x1[3])≥NonInfC∧COND_754_0_MAIN_LT1(TRUE, x0[3], x1[3])≥754_0_MAIN_LT(x0[3], +(x1[3], 1))∧(UIncreasing(754_0_MAIN_LT(x0[3], +(x1[3], 1))), ≥))



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

    (30)    (>(x1[2], 0)=TRUE∧<=(x1[2], x0[2])=TRUE ⇒ COND_754_0_MAIN_LT1(TRUE, x0[2], x1[2])≥NonInfC∧COND_754_0_MAIN_LT1(TRUE, x0[2], x1[2])≥754_0_MAIN_LT(x0[2], +(x1[2], 1))∧(UIncreasing(754_0_MAIN_LT(x0[3], +(x1[3], 1))), ≥))



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

    (31)    (x1[2] + [-1] ≥ 0∧x0[2] + [-1]x1[2] ≥ 0 ⇒ (UIncreasing(754_0_MAIN_LT(x0[3], +(x1[3], 1))), ≥)∧[(-1)bni_23 + (-1)Bound*bni_23] + [bni_23]x0[2] ≥ 0∧[(-1)bso_24] ≥ 0)



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

    (32)    (x1[2] + [-1] ≥ 0∧x0[2] + [-1]x1[2] ≥ 0 ⇒ (UIncreasing(754_0_MAIN_LT(x0[3], +(x1[3], 1))), ≥)∧[(-1)bni_23 + (-1)Bound*bni_23] + [bni_23]x0[2] ≥ 0∧[(-1)bso_24] ≥ 0)



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

    (33)    (x1[2] + [-1] ≥ 0∧x0[2] + [-1]x1[2] ≥ 0 ⇒ (UIncreasing(754_0_MAIN_LT(x0[3], +(x1[3], 1))), ≥)∧[(-1)bni_23 + (-1)Bound*bni_23] + [bni_23]x0[2] ≥ 0∧[(-1)bso_24] ≥ 0)



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

    (34)    (x1[2] ≥ 0∧x0[2] + [-1] + [-1]x1[2] ≥ 0 ⇒ (UIncreasing(754_0_MAIN_LT(x0[3], +(x1[3], 1))), ≥)∧[(-1)bni_23 + (-1)Bound*bni_23] + [bni_23]x0[2] ≥ 0∧[(-1)bso_24] ≥ 0)



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

    (35)    (x1[2] ≥ 0∧x0[2] ≥ 0 ⇒ (UIncreasing(754_0_MAIN_LT(x0[3], +(x1[3], 1))), ≥)∧[(-1)Bound*bni_23] + [bni_23]x1[2] + [bni_23]x0[2] ≥ 0∧[(-1)bso_24] ≥ 0)



  • We consider the chain 754_0_MAIN_LT(x0[2], x1[2]) → COND_754_0_MAIN_LT1(&&(>(x1[2], 0), <=(x1[2], x0[2])), x0[2], x1[2]), COND_754_0_MAIN_LT1(TRUE, x0[3], x1[3]) → 754_0_MAIN_LT(x0[3], +(x1[3], 1)), 754_0_MAIN_LT(x0[2], x1[2]) → COND_754_0_MAIN_LT1(&&(>(x1[2], 0), <=(x1[2], x0[2])), x0[2], x1[2]) which results in the following constraint:

    (36)    (&&(>(x1[2], 0), <=(x1[2], x0[2]))=TRUE∧x0[2]=x0[3]∧x1[2]=x1[3]∧x0[3]=x0[2]1∧+(x1[3], 1)=x1[2]1 ⇒ COND_754_0_MAIN_LT1(TRUE, x0[3], x1[3])≥NonInfC∧COND_754_0_MAIN_LT1(TRUE, x0[3], x1[3])≥754_0_MAIN_LT(x0[3], +(x1[3], 1))∧(UIncreasing(754_0_MAIN_LT(x0[3], +(x1[3], 1))), ≥))



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

    (37)    (>(x1[2], 0)=TRUE∧<=(x1[2], x0[2])=TRUE ⇒ COND_754_0_MAIN_LT1(TRUE, x0[2], x1[2])≥NonInfC∧COND_754_0_MAIN_LT1(TRUE, x0[2], x1[2])≥754_0_MAIN_LT(x0[2], +(x1[2], 1))∧(UIncreasing(754_0_MAIN_LT(x0[3], +(x1[3], 1))), ≥))



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

    (38)    (x1[2] + [-1] ≥ 0∧x0[2] + [-1]x1[2] ≥ 0 ⇒ (UIncreasing(754_0_MAIN_LT(x0[3], +(x1[3], 1))), ≥)∧[(-1)bni_23 + (-1)Bound*bni_23] + [bni_23]x0[2] ≥ 0∧[(-1)bso_24] ≥ 0)



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

    (39)    (x1[2] + [-1] ≥ 0∧x0[2] + [-1]x1[2] ≥ 0 ⇒ (UIncreasing(754_0_MAIN_LT(x0[3], +(x1[3], 1))), ≥)∧[(-1)bni_23 + (-1)Bound*bni_23] + [bni_23]x0[2] ≥ 0∧[(-1)bso_24] ≥ 0)



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

    (40)    (x1[2] + [-1] ≥ 0∧x0[2] + [-1]x1[2] ≥ 0 ⇒ (UIncreasing(754_0_MAIN_LT(x0[3], +(x1[3], 1))), ≥)∧[(-1)bni_23 + (-1)Bound*bni_23] + [bni_23]x0[2] ≥ 0∧[(-1)bso_24] ≥ 0)



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

    (41)    (x1[2] ≥ 0∧x0[2] + [-1] + [-1]x1[2] ≥ 0 ⇒ (UIncreasing(754_0_MAIN_LT(x0[3], +(x1[3], 1))), ≥)∧[(-1)bni_23 + (-1)Bound*bni_23] + [bni_23]x0[2] ≥ 0∧[(-1)bso_24] ≥ 0)



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

    (42)    (x1[2] ≥ 0∧x0[2] ≥ 0 ⇒ (UIncreasing(754_0_MAIN_LT(x0[3], +(x1[3], 1))), ≥)∧[(-1)Bound*bni_23] + [bni_23]x1[2] + [bni_23]x0[2] ≥ 0∧[(-1)bso_24] ≥ 0)







To summarize, we get the following constraints P≥ for the following pairs.
  • 754_0_MAIN_LT(x0, x1) → COND_754_0_MAIN_LT(&&(>(x1, x0), >(x0, 1)), x0, x1)
    • (x1[0] ≥ 0∧x0[0] ≥ 0 ⇒ (UIncreasing(COND_754_0_MAIN_LT(&&(>(x1[0], x0[0]), >(x0[0], 1)), x0[0], x1[0])), ≥)∧[bni_17 + (-1)Bound*bni_17] + [bni_17]x0[0] ≥ 0∧[(-1)bso_18] ≥ 0)

  • COND_754_0_MAIN_LT(TRUE, x0, x1) → 754_0_MAIN_LT(+(-(x0, 2), 1), 1)
    • (x1[0] ≥ 0∧x0[0] ≥ 0 ⇒ (UIncreasing(754_0_MAIN_LT(+(-(x0[1], 2), 1), 1)), ≥)∧[bni_19 + (-1)Bound*bni_19] + [bni_19]x0[0] ≥ 0∧[1 + (-1)bso_20] ≥ 0)
    • (x1[0] ≥ 0∧x0[0] ≥ 0 ⇒ (UIncreasing(754_0_MAIN_LT(+(-(x0[1], 2), 1), 1)), ≥)∧[bni_19 + (-1)Bound*bni_19] + [bni_19]x0[0] ≥ 0∧[1 + (-1)bso_20] ≥ 0)

  • 754_0_MAIN_LT(x0, x1) → COND_754_0_MAIN_LT1(&&(>(x1, 0), <=(x1, x0)), x0, x1)
    • (x1[2] ≥ 0∧x0[2] ≥ 0 ⇒ (UIncreasing(COND_754_0_MAIN_LT1(&&(>(x1[2], 0), <=(x1[2], x0[2])), x0[2], x1[2])), ≥)∧[(-1)Bound*bni_21] + [bni_21]x1[2] + [bni_21]x0[2] ≥ 0∧[(-1)bso_22] ≥ 0)

  • COND_754_0_MAIN_LT1(TRUE, x0, x1) → 754_0_MAIN_LT(x0, +(x1, 1))
    • (x1[2] ≥ 0∧x0[2] ≥ 0 ⇒ (UIncreasing(754_0_MAIN_LT(x0[3], +(x1[3], 1))), ≥)∧[(-1)Bound*bni_23] + [bni_23]x1[2] + [bni_23]x0[2] ≥ 0∧[(-1)bso_24] ≥ 0)
    • (x1[2] ≥ 0∧x0[2] ≥ 0 ⇒ (UIncreasing(754_0_MAIN_LT(x0[3], +(x1[3], 1))), ≥)∧[(-1)Bound*bni_23] + [bni_23]x1[2] + [bni_23]x0[2] ≥ 0∧[(-1)bso_24] ≥ 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(754_0_MAIN_LT(x1, x2)) = [-1] + x1   
POL(COND_754_0_MAIN_LT(x1, x2, x3)) = [-1] + x2   
POL(&&(x1, x2)) = [-1]   
POL(>(x1, x2)) = [-1]   
POL(1) = [1]   
POL(+(x1, x2)) = x1 + x2   
POL(-(x1, x2)) = x1 + [-1]x2   
POL(2) = [2]   
POL(COND_754_0_MAIN_LT1(x1, x2, x3)) = [-1] + x2   
POL(0) = 0   
POL(<=(x1, x2)) = [-1]   

The following pairs are in P>:

COND_754_0_MAIN_LT(TRUE, x0[1], x1[1]) → 754_0_MAIN_LT(+(-(x0[1], 2), 1), 1)

The following pairs are in Pbound:

754_0_MAIN_LT(x0[0], x1[0]) → COND_754_0_MAIN_LT(&&(>(x1[0], x0[0]), >(x0[0], 1)), x0[0], x1[0])
COND_754_0_MAIN_LT(TRUE, x0[1], x1[1]) → 754_0_MAIN_LT(+(-(x0[1], 2), 1), 1)
754_0_MAIN_LT(x0[2], x1[2]) → COND_754_0_MAIN_LT1(&&(>(x1[2], 0), <=(x1[2], x0[2])), x0[2], x1[2])
COND_754_0_MAIN_LT1(TRUE, x0[3], x1[3]) → 754_0_MAIN_LT(x0[3], +(x1[3], 1))

The following pairs are in P≥:

754_0_MAIN_LT(x0[0], x1[0]) → COND_754_0_MAIN_LT(&&(>(x1[0], x0[0]), >(x0[0], 1)), x0[0], x1[0])
754_0_MAIN_LT(x0[2], x1[2]) → COND_754_0_MAIN_LT1(&&(>(x1[2], 0), <=(x1[2], x0[2])), x0[2], x1[2])
COND_754_0_MAIN_LT1(TRUE, x0[3], x1[3]) → 754_0_MAIN_LT(x0[3], +(x1[3], 1))

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): 754_0_MAIN_LT(x0[0], x1[0]) → COND_754_0_MAIN_LT(x1[0] > x0[0] && x0[0] > 1, x0[0], x1[0])
(2): 754_0_MAIN_LT(x0[2], x1[2]) → COND_754_0_MAIN_LT1(x1[2] > 0 && x1[2] <= x0[2], x0[2], x1[2])
(3): COND_754_0_MAIN_LT1(TRUE, x0[3], x1[3]) → 754_0_MAIN_LT(x0[3], x1[3] + 1)

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


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


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



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:
(3): COND_754_0_MAIN_LT1(TRUE, x0[3], x1[3]) → 754_0_MAIN_LT(x0[3], x1[3] + 1)
(2): 754_0_MAIN_LT(x0[2], x1[2]) → COND_754_0_MAIN_LT1(x1[2] > 0 && x1[2] <= x0[2], x0[2], x1[2])

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


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



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@3b815cce 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_754_0_MAIN_LT1(TRUE, x0[3], x1[3]) → 754_0_MAIN_LT(x0[3], +(x1[3], 1)) the following chains were created:
  • We consider the chain 754_0_MAIN_LT(x0[2], x1[2]) → COND_754_0_MAIN_LT1(&&(>(x1[2], 0), <=(x1[2], x0[2])), x0[2], x1[2]), COND_754_0_MAIN_LT1(TRUE, x0[3], x1[3]) → 754_0_MAIN_LT(x0[3], +(x1[3], 1)), 754_0_MAIN_LT(x0[2], x1[2]) → COND_754_0_MAIN_LT1(&&(>(x1[2], 0), <=(x1[2], x0[2])), x0[2], x1[2]) which results in the following constraint:

    (1)    (&&(>(x1[2], 0), <=(x1[2], x0[2]))=TRUE∧x0[2]=x0[3]∧x1[2]=x1[3]∧x0[3]=x0[2]1∧+(x1[3], 1)=x1[2]1 ⇒ COND_754_0_MAIN_LT1(TRUE, x0[3], x1[3])≥NonInfC∧COND_754_0_MAIN_LT1(TRUE, x0[3], x1[3])≥754_0_MAIN_LT(x0[3], +(x1[3], 1))∧(UIncreasing(754_0_MAIN_LT(x0[3], +(x1[3], 1))), ≥))



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

    (2)    (>(x1[2], 0)=TRUE∧<=(x1[2], x0[2])=TRUE ⇒ COND_754_0_MAIN_LT1(TRUE, x0[2], x1[2])≥NonInfC∧COND_754_0_MAIN_LT1(TRUE, x0[2], x1[2])≥754_0_MAIN_LT(x0[2], +(x1[2], 1))∧(UIncreasing(754_0_MAIN_LT(x0[3], +(x1[3], 1))), ≥))



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

    (3)    (x1[2] + [-1] ≥ 0∧x0[2] + [-1]x1[2] ≥ 0 ⇒ (UIncreasing(754_0_MAIN_LT(x0[3], +(x1[3], 1))), ≥)∧[(2)bni_13 + (-1)Bound*bni_13] + [(-1)bni_13]x1[2] + [bni_13]x0[2] ≥ 0∧[1 + (-1)bso_14] ≥ 0)



    We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:

    (4)    (x1[2] + [-1] ≥ 0∧x0[2] + [-1]x1[2] ≥ 0 ⇒ (UIncreasing(754_0_MAIN_LT(x0[3], +(x1[3], 1))), ≥)∧[(2)bni_13 + (-1)Bound*bni_13] + [(-1)bni_13]x1[2] + [bni_13]x0[2] ≥ 0∧[1 + (-1)bso_14] ≥ 0)



    We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:

    (5)    (x1[2] + [-1] ≥ 0∧x0[2] + [-1]x1[2] ≥ 0 ⇒ (UIncreasing(754_0_MAIN_LT(x0[3], +(x1[3], 1))), ≥)∧[(2)bni_13 + (-1)Bound*bni_13] + [(-1)bni_13]x1[2] + [bni_13]x0[2] ≥ 0∧[1 + (-1)bso_14] ≥ 0)



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

    (6)    (x1[2] ≥ 0∧x0[2] + [-1] + [-1]x1[2] ≥ 0 ⇒ (UIncreasing(754_0_MAIN_LT(x0[3], +(x1[3], 1))), ≥)∧[bni_13 + (-1)Bound*bni_13] + [(-1)bni_13]x1[2] + [bni_13]x0[2] ≥ 0∧[1 + (-1)bso_14] ≥ 0)



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

    (7)    (x1[2] ≥ 0∧x0[2] ≥ 0 ⇒ (UIncreasing(754_0_MAIN_LT(x0[3], +(x1[3], 1))), ≥)∧[(2)bni_13 + (-1)Bound*bni_13] + [bni_13]x0[2] ≥ 0∧[1 + (-1)bso_14] ≥ 0)







For Pair 754_0_MAIN_LT(x0[2], x1[2]) → COND_754_0_MAIN_LT1(&&(>(x1[2], 0), <=(x1[2], x0[2])), x0[2], x1[2]) the following chains were created:
  • We consider the chain 754_0_MAIN_LT(x0[2], x1[2]) → COND_754_0_MAIN_LT1(&&(>(x1[2], 0), <=(x1[2], x0[2])), x0[2], x1[2]), COND_754_0_MAIN_LT1(TRUE, x0[3], x1[3]) → 754_0_MAIN_LT(x0[3], +(x1[3], 1)) which results in the following constraint:

    (8)    (&&(>(x1[2], 0), <=(x1[2], x0[2]))=TRUE∧x0[2]=x0[3]∧x1[2]=x1[3] ⇒ 754_0_MAIN_LT(x0[2], x1[2])≥NonInfC∧754_0_MAIN_LT(x0[2], x1[2])≥COND_754_0_MAIN_LT1(&&(>(x1[2], 0), <=(x1[2], x0[2])), x0[2], x1[2])∧(UIncreasing(COND_754_0_MAIN_LT1(&&(>(x1[2], 0), <=(x1[2], x0[2])), x0[2], x1[2])), ≥))



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

    (9)    (>(x1[2], 0)=TRUE∧<=(x1[2], x0[2])=TRUE ⇒ 754_0_MAIN_LT(x0[2], x1[2])≥NonInfC∧754_0_MAIN_LT(x0[2], x1[2])≥COND_754_0_MAIN_LT1(&&(>(x1[2], 0), <=(x1[2], x0[2])), x0[2], x1[2])∧(UIncreasing(COND_754_0_MAIN_LT1(&&(>(x1[2], 0), <=(x1[2], x0[2])), x0[2], x1[2])), ≥))



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

    (10)    (x1[2] + [-1] ≥ 0∧x0[2] + [-1]x1[2] ≥ 0 ⇒ (UIncreasing(COND_754_0_MAIN_LT1(&&(>(x1[2], 0), <=(x1[2], x0[2])), x0[2], x1[2])), ≥)∧[(2)bni_15 + (-1)Bound*bni_15] + [(-1)bni_15]x1[2] + [bni_15]x0[2] ≥ 0∧[(-1)bso_16] ≥ 0)



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

    (11)    (x1[2] + [-1] ≥ 0∧x0[2] + [-1]x1[2] ≥ 0 ⇒ (UIncreasing(COND_754_0_MAIN_LT1(&&(>(x1[2], 0), <=(x1[2], x0[2])), x0[2], x1[2])), ≥)∧[(2)bni_15 + (-1)Bound*bni_15] + [(-1)bni_15]x1[2] + [bni_15]x0[2] ≥ 0∧[(-1)bso_16] ≥ 0)



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

    (12)    (x1[2] + [-1] ≥ 0∧x0[2] + [-1]x1[2] ≥ 0 ⇒ (UIncreasing(COND_754_0_MAIN_LT1(&&(>(x1[2], 0), <=(x1[2], x0[2])), x0[2], x1[2])), ≥)∧[(2)bni_15 + (-1)Bound*bni_15] + [(-1)bni_15]x1[2] + [bni_15]x0[2] ≥ 0∧[(-1)bso_16] ≥ 0)



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

    (13)    (x1[2] ≥ 0∧x0[2] + [-1] + [-1]x1[2] ≥ 0 ⇒ (UIncreasing(COND_754_0_MAIN_LT1(&&(>(x1[2], 0), <=(x1[2], x0[2])), x0[2], x1[2])), ≥)∧[bni_15 + (-1)Bound*bni_15] + [(-1)bni_15]x1[2] + [bni_15]x0[2] ≥ 0∧[(-1)bso_16] ≥ 0)



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

    (14)    (x1[2] ≥ 0∧x0[2] ≥ 0 ⇒ (UIncreasing(COND_754_0_MAIN_LT1(&&(>(x1[2], 0), <=(x1[2], x0[2])), x0[2], x1[2])), ≥)∧[(2)bni_15 + (-1)Bound*bni_15] + [bni_15]x0[2] ≥ 0∧[(-1)bso_16] ≥ 0)







To summarize, we get the following constraints P≥ for the following pairs.
  • COND_754_0_MAIN_LT1(TRUE, x0[3], x1[3]) → 754_0_MAIN_LT(x0[3], +(x1[3], 1))
    • (x1[2] ≥ 0∧x0[2] ≥ 0 ⇒ (UIncreasing(754_0_MAIN_LT(x0[3], +(x1[3], 1))), ≥)∧[(2)bni_13 + (-1)Bound*bni_13] + [bni_13]x0[2] ≥ 0∧[1 + (-1)bso_14] ≥ 0)

  • 754_0_MAIN_LT(x0[2], x1[2]) → COND_754_0_MAIN_LT1(&&(>(x1[2], 0), <=(x1[2], x0[2])), x0[2], x1[2])
    • (x1[2] ≥ 0∧x0[2] ≥ 0 ⇒ (UIncreasing(COND_754_0_MAIN_LT1(&&(>(x1[2], 0), <=(x1[2], x0[2])), x0[2], x1[2])), ≥)∧[(2)bni_15 + (-1)Bound*bni_15] + [bni_15]x0[2] ≥ 0∧[(-1)bso_16] ≥ 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_754_0_MAIN_LT1(x1, x2, x3)) = [2] + [-1]x3 + x2   
POL(754_0_MAIN_LT(x1, x2)) = [2] + [-1]x2 + x1   
POL(+(x1, x2)) = x1 + x2   
POL(1) = [1]   
POL(&&(x1, x2)) = [-1]   
POL(>(x1, x2)) = [-1]   
POL(0) = 0   
POL(<=(x1, x2)) = [-1]   

The following pairs are in P>:

COND_754_0_MAIN_LT1(TRUE, x0[3], x1[3]) → 754_0_MAIN_LT(x0[3], +(x1[3], 1))

The following pairs are in Pbound:

COND_754_0_MAIN_LT1(TRUE, x0[3], x1[3]) → 754_0_MAIN_LT(x0[3], +(x1[3], 1))
754_0_MAIN_LT(x0[2], x1[2]) → COND_754_0_MAIN_LT1(&&(>(x1[2], 0), <=(x1[2], x0[2])), x0[2], x1[2])

The following pairs are in P≥:

754_0_MAIN_LT(x0[2], x1[2]) → COND_754_0_MAIN_LT1(&&(>(x1[2], 0), <=(x1[2], x0[2])), x0[2], x1[2])

There are no usable rules.

(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:

Boolean, Integer


R is empty.

The integer pair graph contains the following rules and edges:
(2): 754_0_MAIN_LT(x0[2], x1[2]) → COND_754_0_MAIN_LT1(x1[2] > 0 && x1[2] <= x0[2], x0[2], x1[2])


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