(0) Obligation:

JBC Problem based on JBC Program:
Manifest-Version: 1.0 Created-By: 1.6.0_16 (Sun Microsystems Inc.) Main-Class: GCD4
public class GCD4 {
public static int mod(int a, int b) {
while(a>=b && b > 0) {
a -= b;
}
return a;
}

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

public static void main(String[] args) {
Random.args = args;
int x = Random.random();
int y = Random.random();
gcd(x, y);
}
}


public class Random {
static String[] args;
static int index = 0;

public static int random() {
String string = args[index];
index++;
return string.length();
}
}


(1) JBCToGraph (SOUND transformation)

Constructed TerminationGraph.

(2) Obligation:

Termination Graph based on JBC Program:
GCD4.main([Ljava/lang/String;)V: Graph of 212 nodes with 1 SCC.


(3) TerminationGraphToSCCProof (SOUND transformation)

Splitted TerminationGraph to 1 SCCs.

(4) Obligation:

SCC of termination graph based on JBC Program.
SCC contains nodes from the following methods: GCD4.main([Ljava/lang/String;)V
SCC calls the following helper methods:
Performed SCC analyses: UsedFieldsAnalysis

(5) SCCToIDPv1Proof (SOUND transformation)

Transformed FIGraph SCCs to IDPs. Log:

Generated 33 rules for P and 0 rules for R.


P rules:
630_0_gcd_LE(EOS(STATIC_630), i85, i94, i94) → 634_0_gcd_LE(EOS(STATIC_634), i85, i94, i94)
634_0_gcd_LE(EOS(STATIC_634), i85, i94, i94) → 638_0_gcd_Load(EOS(STATIC_638), i85, i94) | >(i94, 0)
638_0_gcd_Load(EOS(STATIC_638), i85, i94) → 642_0_gcd_LE(EOS(STATIC_642), i85, i94, i85)
642_0_gcd_LE(EOS(STATIC_642), i97, i94, i97) → 647_0_gcd_LE(EOS(STATIC_647), i97, i94, i97)
647_0_gcd_LE(EOS(STATIC_647), i97, i94, i97) → 654_0_gcd_Load(EOS(STATIC_654), i97, i94) | >(i97, 0)
654_0_gcd_Load(EOS(STATIC_654), i97, i94) → 660_0_gcd_Store(EOS(STATIC_660), i97, i94, i94)
660_0_gcd_Store(EOS(STATIC_660), i97, i94, i94) → 665_0_gcd_Load(EOS(STATIC_665), i97, i94, i94)
665_0_gcd_Load(EOS(STATIC_665), i97, i94, i94) → 670_0_gcd_Load(EOS(STATIC_670), i94, i94, i97)
670_0_gcd_Load(EOS(STATIC_670), i94, i94, i97) → 674_0_gcd_InvokeMethod(EOS(STATIC_674), i94, i97, i94)
674_0_gcd_InvokeMethod(EOS(STATIC_674), i94, i97, i94) → 676_0_mod_Load(EOS(STATIC_676), i94, i97, i94, i97, i94)
676_0_mod_Load(EOS(STATIC_676), i94, i97, i94, i97, i94) → 722_0_mod_Load(EOS(STATIC_722), i94, i97, i94, i97, i94)
722_0_mod_Load(EOS(STATIC_722), i94, i97, i94, i106, i94) → 725_0_mod_Load(EOS(STATIC_725), i94, i97, i94, i106, i94, i106)
725_0_mod_Load(EOS(STATIC_725), i94, i97, i94, i106, i94, i106) → 727_0_mod_LT(EOS(STATIC_727), i94, i97, i94, i106, i94, i106, i94)
727_0_mod_LT(EOS(STATIC_727), i94, i97, i94, i106, i94, i106, i94) → 729_0_mod_LT(EOS(STATIC_729), i94, i97, i94, i106, i94, i106, i94)
727_0_mod_LT(EOS(STATIC_727), i94, i97, i94, i106, i94, i106, i94) → 731_0_mod_LT(EOS(STATIC_731), i94, i97, i94, i106, i94, i106, i94)
729_0_mod_LT(EOS(STATIC_729), i94, i97, i94, i106, i94, i106, i94) → 733_0_mod_Load(EOS(STATIC_733), i94, i97, i94, i106) | <(i106, i94)
733_0_mod_Load(EOS(STATIC_733), i94, i97, i94, i106) → 737_0_mod_Return(EOS(STATIC_737), i94, i97, i94, i106)
737_0_mod_Return(EOS(STATIC_737), i94, i97, i94, i106) → 740_0_gcd_Store(EOS(STATIC_740), i94, i106)
740_0_gcd_Store(EOS(STATIC_740), i94, i106) → 744_0_gcd_Load(EOS(STATIC_744), i106, i94)
744_0_gcd_Load(EOS(STATIC_744), i106, i94) → 748_0_gcd_Store(EOS(STATIC_748), i106, i94)
748_0_gcd_Store(EOS(STATIC_748), i106, i94) → 752_0_gcd_JMP(EOS(STATIC_752), i94, i106)
752_0_gcd_JMP(EOS(STATIC_752), i94, i106) → 758_0_gcd_Load(EOS(STATIC_758), i94, i106)
758_0_gcd_Load(EOS(STATIC_758), i94, i106) → 627_0_gcd_Load(EOS(STATIC_627), i94, i106)
627_0_gcd_Load(EOS(STATIC_627), i85, i87) → 630_0_gcd_LE(EOS(STATIC_630), i85, i87, i87)
731_0_mod_LT(EOS(STATIC_731), i94, i97, i94, i106, i94, i106, i94) → 735_0_mod_Load(EOS(STATIC_735), i94, i97, i94, i106, i94) | >=(i106, i94)
735_0_mod_Load(EOS(STATIC_735), i94, i97, i94, i106, i94) → 738_0_mod_LE(EOS(STATIC_738), i94, i97, i94, i106, i94, i94)
738_0_mod_LE(EOS(STATIC_738), i94, i97, i94, i106, i94, i94) → 742_0_mod_Load(EOS(STATIC_742), i94, i97, i94, i106, i94) | >(i94, 0)
742_0_mod_Load(EOS(STATIC_742), i94, i97, i94, i106, i94) → 746_0_mod_Load(EOS(STATIC_746), i94, i97, i94, i94, i106)
746_0_mod_Load(EOS(STATIC_746), i94, i97, i94, i94, i106) → 750_0_mod_IntArithmetic(EOS(STATIC_750), i94, i97, i94, i94, i106, i94)
750_0_mod_IntArithmetic(EOS(STATIC_750), i94, i97, i94, i94, i106, i94) → 754_0_mod_Store(EOS(STATIC_754), i94, i97, i94, i94, -(i106, i94)) | >(i94, 0)
754_0_mod_Store(EOS(STATIC_754), i94, i97, i94, i94, i108) → 760_0_mod_JMP(EOS(STATIC_760), i94, i97, i94, i108, i94)
760_0_mod_JMP(EOS(STATIC_760), i94, i97, i94, i108, i94) → 764_0_mod_Load(EOS(STATIC_764), i94, i97, i94, i108, i94)
764_0_mod_Load(EOS(STATIC_764), i94, i97, i94, i108, i94) → 722_0_mod_Load(EOS(STATIC_722), i94, i97, i94, i108, i94)
R rules:

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


P rules:
727_0_mod_LT(EOS(STATIC_727), x0, x1, x0, x2, x0, x2, x0) → 727_0_mod_LT(EOS(STATIC_727), x2, x0, x2, x0, x2, x0, x2) | &&(&&(>(x2, 0), <(x2, x0)), >(x0, 0))
727_0_mod_LT(EOS(STATIC_727), x0, x1, x0, x2, x0, x2, x0) → 727_0_mod_LT(EOS(STATIC_727), x0, x1, x0, -(x2, x0), x0, -(x2, x0), x0) | &&(>=(x2, x0), >(x0, 0))
R rules:

Filtered ground terms:



727_0_mod_LT(x1, x2, x3, x4, x5, x6, x7, x8) → 727_0_mod_LT(x2, x3, x4, x5, x6, x7, x8)
EOS(x1) → EOS
Cond_727_0_mod_LT1(x1, x2, x3, x4, x5, x6, x7, x8, x9) → Cond_727_0_mod_LT1(x1, x3, x4, x5, x6, x7, x8, x9)
Cond_727_0_mod_LT(x1, x2, x3, x4, x5, x6, x7, x8, x9) → Cond_727_0_mod_LT(x1, x3, x4, x5, x6, x7, x8, x9)

Filtered duplicate args:



727_0_mod_LT(x1, x2, x3, x4, x5, x6, x7) → 727_0_mod_LT(x2, x6, x7)
Cond_727_0_mod_LT(x1, x2, x3, x4, x5, x6, x7, x8) → Cond_727_0_mod_LT(x1, x3, x7, x8)
Cond_727_0_mod_LT1(x1, x2, x3, x4, x5, x6, x7, x8) → Cond_727_0_mod_LT1(x1, x3, x7, x8)

Filtered unneeded arguments:



Cond_727_0_mod_LT(x1, x2, x3, x4) → Cond_727_0_mod_LT(x1, x3, x4)
Cond_727_0_mod_LT1(x1, x2, x3, x4) → Cond_727_0_mod_LT1(x1, x3, x4)
727_0_mod_LT(x1, x2, x3) → 727_0_mod_LT(x2, x3)

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


P rules:
727_0_mod_LT(x2, x0) → 727_0_mod_LT(x0, x2) | &&(&&(>(x2, 0), <(x2, x0)), >(x0, 0))
727_0_mod_LT(x2, x0) → 727_0_mod_LT(-(x2, x0), x0) | &&(>=(x2, x0), >(x0, 0))
R rules:

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


P rules:
727_0_MOD_LT(x2, x0) → COND_727_0_MOD_LT(&&(&&(>(x2, 0), <(x2, x0)), >(x0, 0)), x2, x0)
COND_727_0_MOD_LT(TRUE, x2, x0) → 727_0_MOD_LT(x0, x2)
727_0_MOD_LT(x2, x0) → COND_727_0_MOD_LT1(&&(>=(x2, x0), >(x0, 0)), x2, x0)
COND_727_0_MOD_LT1(TRUE, x2, x0) → 727_0_MOD_LT(-(x2, x0), x0)
R rules:

(6) Obligation:

IDP problem:
The following function symbols are pre-defined:
!=~Neq: (Integer, Integer) -> Boolean
*~Mul: (Integer, Integer) -> Integer
>=~Ge: (Integer, Integer) -> Boolean
-1~UnaryMinus: (Integer) -> Integer
|~Bwor: (Integer, Integer) -> Integer
/~Div: (Integer, Integer) -> Integer
=~Eq: (Integer, Integer) -> Boolean
~Bwxor: (Integer, Integer) -> Integer
||~Lor: (Boolean, Boolean) -> Boolean
!~Lnot: (Boolean) -> Boolean
<~Lt: (Integer, Integer) -> Boolean
-~Sub: (Integer, Integer) -> Integer
<=~Le: (Integer, Integer) -> Boolean
>~Gt: (Integer, Integer) -> Boolean
~~Bwnot: (Integer) -> Integer
%~Mod: (Integer, Integer) -> Integer
&~Bwand: (Integer, Integer) -> Integer
+~Add: (Integer, Integer) -> Integer
&&~Land: (Boolean, Boolean) -> Boolean


The following domains are used:

Boolean, Integer


R is empty.

The integer pair graph contains the following rules and edges:
(0): 727_0_MOD_LT(x2[0], x0[0]) → COND_727_0_MOD_LT(x2[0] > 0 && x2[0] < x0[0] && x0[0] > 0, x2[0], x0[0])
(1): COND_727_0_MOD_LT(TRUE, x2[1], x0[1]) → 727_0_MOD_LT(x0[1], x2[1])
(2): 727_0_MOD_LT(x2[2], x0[2]) → COND_727_0_MOD_LT1(x2[2] >= x0[2] && x0[2] > 0, x2[2], x0[2])
(3): COND_727_0_MOD_LT1(TRUE, x2[3], x0[3]) → 727_0_MOD_LT(x2[3] - x0[3], x0[3])

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


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


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


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


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


(3) -> (2), if (x2[3] - x0[3]* x2[2]x0[3]* x0[2])



The set Q is empty.

(7) IDPNonInfProof (SOUND transformation)

Used the following options for this NonInfProof:
IDPGPoloSolver: Range: [(-1,2)] IsNat: false Interpretation Shape Heuristic: aprove.DPFramework.IDPProblem.Processors.nonInf.poly.IdpDefaultShapeHeuristic@72865d61 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 727_0_MOD_LT(x2, x0) → COND_727_0_MOD_LT(&&(&&(>(x2, 0), <(x2, x0)), >(x0, 0)), x2, x0) the following chains were created:
  • We consider the chain 727_0_MOD_LT(x2[0], x0[0]) → COND_727_0_MOD_LT(&&(&&(>(x2[0], 0), <(x2[0], x0[0])), >(x0[0], 0)), x2[0], x0[0]), COND_727_0_MOD_LT(TRUE, x2[1], x0[1]) → 727_0_MOD_LT(x0[1], x2[1]) which results in the following constraint:

    (1)    (&&(&&(>(x2[0], 0), <(x2[0], x0[0])), >(x0[0], 0))=TRUEx2[0]=x2[1]x0[0]=x0[1]727_0_MOD_LT(x2[0], x0[0])≥NonInfC∧727_0_MOD_LT(x2[0], x0[0])≥COND_727_0_MOD_LT(&&(&&(>(x2[0], 0), <(x2[0], x0[0])), >(x0[0], 0)), x2[0], x0[0])∧(UIncreasing(COND_727_0_MOD_LT(&&(&&(>(x2[0], 0), <(x2[0], x0[0])), >(x0[0], 0)), x2[0], x0[0])), ≥))



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

    (2)    (>(x0[0], 0)=TRUE>(x2[0], 0)=TRUE<(x2[0], x0[0])=TRUE727_0_MOD_LT(x2[0], x0[0])≥NonInfC∧727_0_MOD_LT(x2[0], x0[0])≥COND_727_0_MOD_LT(&&(&&(>(x2[0], 0), <(x2[0], x0[0])), >(x0[0], 0)), x2[0], x0[0])∧(UIncreasing(COND_727_0_MOD_LT(&&(&&(>(x2[0], 0), <(x2[0], x0[0])), >(x0[0], 0)), x2[0], x0[0])), ≥))



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

    (3)    (x0[0] + [-1] ≥ 0∧x2[0] + [-1] ≥ 0∧x0[0] + [-1] + [-1]x2[0] ≥ 0 ⇒ (UIncreasing(COND_727_0_MOD_LT(&&(&&(>(x2[0], 0), <(x2[0], x0[0])), >(x0[0], 0)), x2[0], x0[0])), ≥)∧[(-1)bni_18 + (-1)Bound*bni_18] + [bni_18]x0[0] + [bni_18]x2[0] ≥ 0∧[(-1)bso_19] ≥ 0)



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

    (4)    (x0[0] + [-1] ≥ 0∧x2[0] + [-1] ≥ 0∧x0[0] + [-1] + [-1]x2[0] ≥ 0 ⇒ (UIncreasing(COND_727_0_MOD_LT(&&(&&(>(x2[0], 0), <(x2[0], x0[0])), >(x0[0], 0)), x2[0], x0[0])), ≥)∧[(-1)bni_18 + (-1)Bound*bni_18] + [bni_18]x0[0] + [bni_18]x2[0] ≥ 0∧[(-1)bso_19] ≥ 0)



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

    (5)    (x0[0] + [-1] ≥ 0∧x2[0] + [-1] ≥ 0∧x0[0] + [-1] + [-1]x2[0] ≥ 0 ⇒ (UIncreasing(COND_727_0_MOD_LT(&&(&&(>(x2[0], 0), <(x2[0], x0[0])), >(x0[0], 0)), x2[0], x0[0])), ≥)∧[(-1)bni_18 + (-1)Bound*bni_18] + [bni_18]x0[0] + [bni_18]x2[0] ≥ 0∧[(-1)bso_19] ≥ 0)



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

    (6)    (x0[0] ≥ 0∧x2[0] + [-1] ≥ 0∧x0[0] + [-1]x2[0] ≥ 0 ⇒ (UIncreasing(COND_727_0_MOD_LT(&&(&&(>(x2[0], 0), <(x2[0], x0[0])), >(x0[0], 0)), x2[0], x0[0])), ≥)∧[(-1)Bound*bni_18] + [bni_18]x0[0] + [bni_18]x2[0] ≥ 0∧[(-1)bso_19] ≥ 0)



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

    (7)    (x2[0] + x0[0] ≥ 0∧x2[0] + [-1] ≥ 0∧x0[0] ≥ 0 ⇒ (UIncreasing(COND_727_0_MOD_LT(&&(&&(>(x2[0], 0), <(x2[0], x0[0])), >(x0[0], 0)), x2[0], x0[0])), ≥)∧[(-1)Bound*bni_18] + [(2)bni_18]x2[0] + [bni_18]x0[0] ≥ 0∧[(-1)bso_19] ≥ 0)



    We simplified constraint (7) using rule (IDP_SMT_SPLIT) which results in the following new constraint:

    (8)    ([1] + x2[0] + x0[0] ≥ 0∧x2[0] ≥ 0∧x0[0] ≥ 0 ⇒ (UIncreasing(COND_727_0_MOD_LT(&&(&&(>(x2[0], 0), <(x2[0], x0[0])), >(x0[0], 0)), x2[0], x0[0])), ≥)∧[(2)bni_18 + (-1)Bound*bni_18] + [(2)bni_18]x2[0] + [bni_18]x0[0] ≥ 0∧[(-1)bso_19] ≥ 0)







For Pair COND_727_0_MOD_LT(TRUE, x2, x0) → 727_0_MOD_LT(x0, x2) the following chains were created:
  • We consider the chain COND_727_0_MOD_LT(TRUE, x2[1], x0[1]) → 727_0_MOD_LT(x0[1], x2[1]), 727_0_MOD_LT(x2[0], x0[0]) → COND_727_0_MOD_LT(&&(&&(>(x2[0], 0), <(x2[0], x0[0])), >(x0[0], 0)), x2[0], x0[0]) which results in the following constraint:

    (9)    (x0[1]=x2[0]x2[1]=x0[0]COND_727_0_MOD_LT(TRUE, x2[1], x0[1])≥NonInfC∧COND_727_0_MOD_LT(TRUE, x2[1], x0[1])≥727_0_MOD_LT(x0[1], x2[1])∧(UIncreasing(727_0_MOD_LT(x0[1], x2[1])), ≥))



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

    (10)    (COND_727_0_MOD_LT(TRUE, x2[1], x0[1])≥NonInfC∧COND_727_0_MOD_LT(TRUE, x2[1], x0[1])≥727_0_MOD_LT(x0[1], x2[1])∧(UIncreasing(727_0_MOD_LT(x0[1], x2[1])), ≥))



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

    (11)    ((UIncreasing(727_0_MOD_LT(x0[1], x2[1])), ≥)∧[bni_20] = 0∧[(-1)bso_21] ≥ 0)



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

    (12)    ((UIncreasing(727_0_MOD_LT(x0[1], x2[1])), ≥)∧[bni_20] = 0∧[(-1)bso_21] ≥ 0)



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

    (13)    ((UIncreasing(727_0_MOD_LT(x0[1], x2[1])), ≥)∧[bni_20] = 0∧[(-1)bso_21] ≥ 0)



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

    (14)    ((UIncreasing(727_0_MOD_LT(x0[1], x2[1])), ≥)∧[bni_20] = 0∧0 = 0∧0 = 0∧[(-1)bso_21] ≥ 0)



  • We consider the chain COND_727_0_MOD_LT(TRUE, x2[1], x0[1]) → 727_0_MOD_LT(x0[1], x2[1]), 727_0_MOD_LT(x2[2], x0[2]) → COND_727_0_MOD_LT1(&&(>=(x2[2], x0[2]), >(x0[2], 0)), x2[2], x0[2]) which results in the following constraint:

    (15)    (x0[1]=x2[2]x2[1]=x0[2]COND_727_0_MOD_LT(TRUE, x2[1], x0[1])≥NonInfC∧COND_727_0_MOD_LT(TRUE, x2[1], x0[1])≥727_0_MOD_LT(x0[1], x2[1])∧(UIncreasing(727_0_MOD_LT(x0[1], x2[1])), ≥))



    We simplified constraint (15) using rule (IV) which results in the following new constraint:

    (16)    (COND_727_0_MOD_LT(TRUE, x2[1], x0[1])≥NonInfC∧COND_727_0_MOD_LT(TRUE, x2[1], x0[1])≥727_0_MOD_LT(x0[1], x2[1])∧(UIncreasing(727_0_MOD_LT(x0[1], x2[1])), ≥))



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

    (17)    ((UIncreasing(727_0_MOD_LT(x0[1], x2[1])), ≥)∧[bni_20] = 0∧[(-1)bso_21] ≥ 0)



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

    (18)    ((UIncreasing(727_0_MOD_LT(x0[1], x2[1])), ≥)∧[bni_20] = 0∧[(-1)bso_21] ≥ 0)



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

    (19)    ((UIncreasing(727_0_MOD_LT(x0[1], x2[1])), ≥)∧[bni_20] = 0∧[(-1)bso_21] ≥ 0)



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

    (20)    ((UIncreasing(727_0_MOD_LT(x0[1], x2[1])), ≥)∧[bni_20] = 0∧0 = 0∧0 = 0∧[(-1)bso_21] ≥ 0)







For Pair 727_0_MOD_LT(x2, x0) → COND_727_0_MOD_LT1(&&(>=(x2, x0), >(x0, 0)), x2, x0) the following chains were created:
  • We consider the chain 727_0_MOD_LT(x2[2], x0[2]) → COND_727_0_MOD_LT1(&&(>=(x2[2], x0[2]), >(x0[2], 0)), x2[2], x0[2]), COND_727_0_MOD_LT1(TRUE, x2[3], x0[3]) → 727_0_MOD_LT(-(x2[3], x0[3]), x0[3]) which results in the following constraint:

    (21)    (&&(>=(x2[2], x0[2]), >(x0[2], 0))=TRUEx2[2]=x2[3]x0[2]=x0[3]727_0_MOD_LT(x2[2], x0[2])≥NonInfC∧727_0_MOD_LT(x2[2], x0[2])≥COND_727_0_MOD_LT1(&&(>=(x2[2], x0[2]), >(x0[2], 0)), x2[2], x0[2])∧(UIncreasing(COND_727_0_MOD_LT1(&&(>=(x2[2], x0[2]), >(x0[2], 0)), x2[2], x0[2])), ≥))



    We simplified constraint (21) using rules (IV), (IDP_BOOLEAN) which results in the following new constraint:

    (22)    (>=(x2[2], x0[2])=TRUE>(x0[2], 0)=TRUE727_0_MOD_LT(x2[2], x0[2])≥NonInfC∧727_0_MOD_LT(x2[2], x0[2])≥COND_727_0_MOD_LT1(&&(>=(x2[2], x0[2]), >(x0[2], 0)), x2[2], x0[2])∧(UIncreasing(COND_727_0_MOD_LT1(&&(>=(x2[2], x0[2]), >(x0[2], 0)), x2[2], x0[2])), ≥))



    We simplified constraint (22) using rule (POLY_CONSTRAINTS) which results in the following new constraint:

    (23)    (x2[2] + [-1]x0[2] ≥ 0∧x0[2] + [-1] ≥ 0 ⇒ (UIncreasing(COND_727_0_MOD_LT1(&&(>=(x2[2], x0[2]), >(x0[2], 0)), x2[2], x0[2])), ≥)∧[(-1)bni_22 + (-1)Bound*bni_22] + [bni_22]x0[2] + [bni_22]x2[2] ≥ 0∧[(-1)bso_23] ≥ 0)



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

    (24)    (x2[2] + [-1]x0[2] ≥ 0∧x0[2] + [-1] ≥ 0 ⇒ (UIncreasing(COND_727_0_MOD_LT1(&&(>=(x2[2], x0[2]), >(x0[2], 0)), x2[2], x0[2])), ≥)∧[(-1)bni_22 + (-1)Bound*bni_22] + [bni_22]x0[2] + [bni_22]x2[2] ≥ 0∧[(-1)bso_23] ≥ 0)



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

    (25)    (x2[2] + [-1]x0[2] ≥ 0∧x0[2] + [-1] ≥ 0 ⇒ (UIncreasing(COND_727_0_MOD_LT1(&&(>=(x2[2], x0[2]), >(x0[2], 0)), x2[2], x0[2])), ≥)∧[(-1)bni_22 + (-1)Bound*bni_22] + [bni_22]x0[2] + [bni_22]x2[2] ≥ 0∧[(-1)bso_23] ≥ 0)



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

    (26)    (x2[2] ≥ 0∧x0[2] + [-1] ≥ 0 ⇒ (UIncreasing(COND_727_0_MOD_LT1(&&(>=(x2[2], x0[2]), >(x0[2], 0)), x2[2], x0[2])), ≥)∧[(-1)bni_22 + (-1)Bound*bni_22] + [(2)bni_22]x0[2] + [bni_22]x2[2] ≥ 0∧[(-1)bso_23] ≥ 0)



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

    (27)    (x2[2] ≥ 0∧x0[2] ≥ 0 ⇒ (UIncreasing(COND_727_0_MOD_LT1(&&(>=(x2[2], x0[2]), >(x0[2], 0)), x2[2], x0[2])), ≥)∧[(-1)Bound*bni_22 + bni_22] + [(2)bni_22]x0[2] + [bni_22]x2[2] ≥ 0∧[(-1)bso_23] ≥ 0)







For Pair COND_727_0_MOD_LT1(TRUE, x2, x0) → 727_0_MOD_LT(-(x2, x0), x0) the following chains were created:
  • We consider the chain 727_0_MOD_LT(x2[2], x0[2]) → COND_727_0_MOD_LT1(&&(>=(x2[2], x0[2]), >(x0[2], 0)), x2[2], x0[2]), COND_727_0_MOD_LT1(TRUE, x2[3], x0[3]) → 727_0_MOD_LT(-(x2[3], x0[3]), x0[3]), 727_0_MOD_LT(x2[0], x0[0]) → COND_727_0_MOD_LT(&&(&&(>(x2[0], 0), <(x2[0], x0[0])), >(x0[0], 0)), x2[0], x0[0]) which results in the following constraint:

    (28)    (&&(>=(x2[2], x0[2]), >(x0[2], 0))=TRUEx2[2]=x2[3]x0[2]=x0[3]-(x2[3], x0[3])=x2[0]x0[3]=x0[0]COND_727_0_MOD_LT1(TRUE, x2[3], x0[3])≥NonInfC∧COND_727_0_MOD_LT1(TRUE, x2[3], x0[3])≥727_0_MOD_LT(-(x2[3], x0[3]), x0[3])∧(UIncreasing(727_0_MOD_LT(-(x2[3], x0[3]), x0[3])), ≥))



    We simplified constraint (28) using rules (III), (IV), (IDP_BOOLEAN) which results in the following new constraint:

    (29)    (>=(x2[2], x0[2])=TRUE>(x0[2], 0)=TRUECOND_727_0_MOD_LT1(TRUE, x2[2], x0[2])≥NonInfC∧COND_727_0_MOD_LT1(TRUE, x2[2], x0[2])≥727_0_MOD_LT(-(x2[2], x0[2]), x0[2])∧(UIncreasing(727_0_MOD_LT(-(x2[3], x0[3]), x0[3])), ≥))



    We simplified constraint (29) using rule (POLY_CONSTRAINTS) which results in the following new constraint:

    (30)    (x2[2] + [-1]x0[2] ≥ 0∧x0[2] + [-1] ≥ 0 ⇒ (UIncreasing(727_0_MOD_LT(-(x2[3], x0[3]), x0[3])), ≥)∧[(-1)bni_24 + (-1)Bound*bni_24] + [bni_24]x0[2] + [bni_24]x2[2] ≥ 0∧[(-1)bso_25] + x0[2] ≥ 0)



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

    (31)    (x2[2] + [-1]x0[2] ≥ 0∧x0[2] + [-1] ≥ 0 ⇒ (UIncreasing(727_0_MOD_LT(-(x2[3], x0[3]), x0[3])), ≥)∧[(-1)bni_24 + (-1)Bound*bni_24] + [bni_24]x0[2] + [bni_24]x2[2] ≥ 0∧[(-1)bso_25] + x0[2] ≥ 0)



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

    (32)    (x2[2] + [-1]x0[2] ≥ 0∧x0[2] + [-1] ≥ 0 ⇒ (UIncreasing(727_0_MOD_LT(-(x2[3], x0[3]), x0[3])), ≥)∧[(-1)bni_24 + (-1)Bound*bni_24] + [bni_24]x0[2] + [bni_24]x2[2] ≥ 0∧[(-1)bso_25] + x0[2] ≥ 0)



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

    (33)    (x2[2] ≥ 0∧x0[2] + [-1] ≥ 0 ⇒ (UIncreasing(727_0_MOD_LT(-(x2[3], x0[3]), x0[3])), ≥)∧[(-1)bni_24 + (-1)Bound*bni_24] + [(2)bni_24]x0[2] + [bni_24]x2[2] ≥ 0∧[(-1)bso_25] + x0[2] ≥ 0)



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

    (34)    (x2[2] ≥ 0∧x0[2] ≥ 0 ⇒ (UIncreasing(727_0_MOD_LT(-(x2[3], x0[3]), x0[3])), ≥)∧[(-1)Bound*bni_24 + bni_24] + [(2)bni_24]x0[2] + [bni_24]x2[2] ≥ 0∧[1 + (-1)bso_25] + x0[2] ≥ 0)



  • We consider the chain 727_0_MOD_LT(x2[2], x0[2]) → COND_727_0_MOD_LT1(&&(>=(x2[2], x0[2]), >(x0[2], 0)), x2[2], x0[2]), COND_727_0_MOD_LT1(TRUE, x2[3], x0[3]) → 727_0_MOD_LT(-(x2[3], x0[3]), x0[3]), 727_0_MOD_LT(x2[2], x0[2]) → COND_727_0_MOD_LT1(&&(>=(x2[2], x0[2]), >(x0[2], 0)), x2[2], x0[2]) which results in the following constraint:

    (35)    (&&(>=(x2[2], x0[2]), >(x0[2], 0))=TRUEx2[2]=x2[3]x0[2]=x0[3]-(x2[3], x0[3])=x2[2]1x0[3]=x0[2]1COND_727_0_MOD_LT1(TRUE, x2[3], x0[3])≥NonInfC∧COND_727_0_MOD_LT1(TRUE, x2[3], x0[3])≥727_0_MOD_LT(-(x2[3], x0[3]), x0[3])∧(UIncreasing(727_0_MOD_LT(-(x2[3], x0[3]), x0[3])), ≥))



    We simplified constraint (35) using rules (III), (IV), (IDP_BOOLEAN) which results in the following new constraint:

    (36)    (>=(x2[2], x0[2])=TRUE>(x0[2], 0)=TRUECOND_727_0_MOD_LT1(TRUE, x2[2], x0[2])≥NonInfC∧COND_727_0_MOD_LT1(TRUE, x2[2], x0[2])≥727_0_MOD_LT(-(x2[2], x0[2]), x0[2])∧(UIncreasing(727_0_MOD_LT(-(x2[3], x0[3]), x0[3])), ≥))



    We simplified constraint (36) using rule (POLY_CONSTRAINTS) which results in the following new constraint:

    (37)    (x2[2] + [-1]x0[2] ≥ 0∧x0[2] + [-1] ≥ 0 ⇒ (UIncreasing(727_0_MOD_LT(-(x2[3], x0[3]), x0[3])), ≥)∧[(-1)bni_24 + (-1)Bound*bni_24] + [bni_24]x0[2] + [bni_24]x2[2] ≥ 0∧[(-1)bso_25] + x0[2] ≥ 0)



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

    (38)    (x2[2] + [-1]x0[2] ≥ 0∧x0[2] + [-1] ≥ 0 ⇒ (UIncreasing(727_0_MOD_LT(-(x2[3], x0[3]), x0[3])), ≥)∧[(-1)bni_24 + (-1)Bound*bni_24] + [bni_24]x0[2] + [bni_24]x2[2] ≥ 0∧[(-1)bso_25] + x0[2] ≥ 0)



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

    (39)    (x2[2] + [-1]x0[2] ≥ 0∧x0[2] + [-1] ≥ 0 ⇒ (UIncreasing(727_0_MOD_LT(-(x2[3], x0[3]), x0[3])), ≥)∧[(-1)bni_24 + (-1)Bound*bni_24] + [bni_24]x0[2] + [bni_24]x2[2] ≥ 0∧[(-1)bso_25] + x0[2] ≥ 0)



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

    (40)    (x2[2] ≥ 0∧x0[2] + [-1] ≥ 0 ⇒ (UIncreasing(727_0_MOD_LT(-(x2[3], x0[3]), x0[3])), ≥)∧[(-1)bni_24 + (-1)Bound*bni_24] + [(2)bni_24]x0[2] + [bni_24]x2[2] ≥ 0∧[(-1)bso_25] + x0[2] ≥ 0)



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

    (41)    (x2[2] ≥ 0∧x0[2] ≥ 0 ⇒ (UIncreasing(727_0_MOD_LT(-(x2[3], x0[3]), x0[3])), ≥)∧[(-1)Bound*bni_24 + bni_24] + [(2)bni_24]x0[2] + [bni_24]x2[2] ≥ 0∧[1 + (-1)bso_25] + x0[2] ≥ 0)







To summarize, we get the following constraints P for the following pairs.
  • 727_0_MOD_LT(x2, x0) → COND_727_0_MOD_LT(&&(&&(>(x2, 0), <(x2, x0)), >(x0, 0)), x2, x0)
    • ([1] + x2[0] + x0[0] ≥ 0∧x2[0] ≥ 0∧x0[0] ≥ 0 ⇒ (UIncreasing(COND_727_0_MOD_LT(&&(&&(>(x2[0], 0), <(x2[0], x0[0])), >(x0[0], 0)), x2[0], x0[0])), ≥)∧[(2)bni_18 + (-1)Bound*bni_18] + [(2)bni_18]x2[0] + [bni_18]x0[0] ≥ 0∧[(-1)bso_19] ≥ 0)

  • COND_727_0_MOD_LT(TRUE, x2, x0) → 727_0_MOD_LT(x0, x2)
    • ((UIncreasing(727_0_MOD_LT(x0[1], x2[1])), ≥)∧[bni_20] = 0∧0 = 0∧0 = 0∧[(-1)bso_21] ≥ 0)
    • ((UIncreasing(727_0_MOD_LT(x0[1], x2[1])), ≥)∧[bni_20] = 0∧0 = 0∧0 = 0∧[(-1)bso_21] ≥ 0)

  • 727_0_MOD_LT(x2, x0) → COND_727_0_MOD_LT1(&&(>=(x2, x0), >(x0, 0)), x2, x0)
    • (x2[2] ≥ 0∧x0[2] ≥ 0 ⇒ (UIncreasing(COND_727_0_MOD_LT1(&&(>=(x2[2], x0[2]), >(x0[2], 0)), x2[2], x0[2])), ≥)∧[(-1)Bound*bni_22 + bni_22] + [(2)bni_22]x0[2] + [bni_22]x2[2] ≥ 0∧[(-1)bso_23] ≥ 0)

  • COND_727_0_MOD_LT1(TRUE, x2, x0) → 727_0_MOD_LT(-(x2, x0), x0)
    • (x2[2] ≥ 0∧x0[2] ≥ 0 ⇒ (UIncreasing(727_0_MOD_LT(-(x2[3], x0[3]), x0[3])), ≥)∧[(-1)Bound*bni_24 + bni_24] + [(2)bni_24]x0[2] + [bni_24]x2[2] ≥ 0∧[1 + (-1)bso_25] + x0[2] ≥ 0)
    • (x2[2] ≥ 0∧x0[2] ≥ 0 ⇒ (UIncreasing(727_0_MOD_LT(-(x2[3], x0[3]), x0[3])), ≥)∧[(-1)Bound*bni_24 + bni_24] + [(2)bni_24]x0[2] + [bni_24]x2[2] ≥ 0∧[1 + (-1)bso_25] + x0[2] ≥ 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) = [1]   
POL(727_0_MOD_LT(x1, x2)) = [-1] + x2 + x1   
POL(COND_727_0_MOD_LT(x1, x2, x3)) = [-1] + x3 + x2   
POL(&&(x1, x2)) = [-1]   
POL(>(x1, x2)) = [-1]   
POL(0) = 0   
POL(<(x1, x2)) = [-1]   
POL(COND_727_0_MOD_LT1(x1, x2, x3)) = [-1] + x3 + x2   
POL(>=(x1, x2)) = [-1]   
POL(-(x1, x2)) = x1 + [-1]x2   

The following pairs are in P>:

COND_727_0_MOD_LT1(TRUE, x2[3], x0[3]) → 727_0_MOD_LT(-(x2[3], x0[3]), x0[3])

The following pairs are in Pbound:

727_0_MOD_LT(x2[0], x0[0]) → COND_727_0_MOD_LT(&&(&&(>(x2[0], 0), <(x2[0], x0[0])), >(x0[0], 0)), x2[0], x0[0])
727_0_MOD_LT(x2[2], x0[2]) → COND_727_0_MOD_LT1(&&(>=(x2[2], x0[2]), >(x0[2], 0)), x2[2], x0[2])
COND_727_0_MOD_LT1(TRUE, x2[3], x0[3]) → 727_0_MOD_LT(-(x2[3], x0[3]), x0[3])

The following pairs are in P:

727_0_MOD_LT(x2[0], x0[0]) → COND_727_0_MOD_LT(&&(&&(>(x2[0], 0), <(x2[0], x0[0])), >(x0[0], 0)), x2[0], x0[0])
COND_727_0_MOD_LT(TRUE, x2[1], x0[1]) → 727_0_MOD_LT(x0[1], x2[1])
727_0_MOD_LT(x2[2], x0[2]) → COND_727_0_MOD_LT1(&&(>=(x2[2], x0[2]), >(x0[2], 0)), x2[2], x0[2])

There are no usable rules.

(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): 727_0_MOD_LT(x2[0], x0[0]) → COND_727_0_MOD_LT(x2[0] > 0 && x2[0] < x0[0] && x0[0] > 0, x2[0], x0[0])
(1): COND_727_0_MOD_LT(TRUE, x2[1], x0[1]) → 727_0_MOD_LT(x0[1], x2[1])
(2): 727_0_MOD_LT(x2[2], x0[2]) → COND_727_0_MOD_LT1(x2[2] >= x0[2] && x0[2] > 0, x2[2], x0[2])

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


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


(1) -> (2), if (x0[1]* x2[2]x2[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:

Boolean, Integer


R is empty.

The integer pair graph contains the following rules and edges:
(1): COND_727_0_MOD_LT(TRUE, x2[1], x0[1]) → 727_0_MOD_LT(x0[1], x2[1])
(0): 727_0_MOD_LT(x2[0], x0[0]) → COND_727_0_MOD_LT(x2[0] > 0 && x2[0] < x0[0] && x0[0] > 0, x2[0], x0[0])

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


(0) -> (1), if (x2[0] > 0 && x2[0] < x0[0] && x0[0] > 0x2[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@72865d61 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_727_0_MOD_LT(TRUE, x2[1], x0[1]) → 727_0_MOD_LT(x0[1], x2[1]) the following chains were created:
  • We consider the chain 727_0_MOD_LT(x2[0], x0[0]) → COND_727_0_MOD_LT(&&(&&(>(x2[0], 0), <(x2[0], x0[0])), >(x0[0], 0)), x2[0], x0[0]), COND_727_0_MOD_LT(TRUE, x2[1], x0[1]) → 727_0_MOD_LT(x0[1], x2[1]), 727_0_MOD_LT(x2[0], x0[0]) → COND_727_0_MOD_LT(&&(&&(>(x2[0], 0), <(x2[0], x0[0])), >(x0[0], 0)), x2[0], x0[0]) which results in the following constraint:

    (1)    (&&(&&(>(x2[0], 0), <(x2[0], x0[0])), >(x0[0], 0))=TRUEx2[0]=x2[1]x0[0]=x0[1]x0[1]=x2[0]1x2[1]=x0[0]1COND_727_0_MOD_LT(TRUE, x2[1], x0[1])≥NonInfC∧COND_727_0_MOD_LT(TRUE, x2[1], x0[1])≥727_0_MOD_LT(x0[1], x2[1])∧(UIncreasing(727_0_MOD_LT(x0[1], x2[1])), ≥))



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

    (2)    (>(x0[0], 0)=TRUE>(x2[0], 0)=TRUE<(x2[0], x0[0])=TRUECOND_727_0_MOD_LT(TRUE, x2[0], x0[0])≥NonInfC∧COND_727_0_MOD_LT(TRUE, x2[0], x0[0])≥727_0_MOD_LT(x0[0], x2[0])∧(UIncreasing(727_0_MOD_LT(x0[1], x2[1])), ≥))



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

    (3)    (x0[0] + [-1] ≥ 0∧x2[0] + [-1] ≥ 0∧x0[0] + [-1] + [-1]x2[0] ≥ 0 ⇒ (UIncreasing(727_0_MOD_LT(x0[1], x2[1])), ≥)∧[(-1)Bound*bni_13] ≥ 0∧[(-1)bso_14] + x0[0] + [-1]x2[0] ≥ 0)



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

    (4)    (x0[0] + [-1] ≥ 0∧x2[0] + [-1] ≥ 0∧x0[0] + [-1] + [-1]x2[0] ≥ 0 ⇒ (UIncreasing(727_0_MOD_LT(x0[1], x2[1])), ≥)∧[(-1)Bound*bni_13] ≥ 0∧[(-1)bso_14] + x0[0] + [-1]x2[0] ≥ 0)



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

    (5)    (x0[0] + [-1] ≥ 0∧x2[0] + [-1] ≥ 0∧x0[0] + [-1] + [-1]x2[0] ≥ 0 ⇒ (UIncreasing(727_0_MOD_LT(x0[1], x2[1])), ≥)∧[(-1)Bound*bni_13] ≥ 0∧[(-1)bso_14] + x0[0] + [-1]x2[0] ≥ 0)



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

    (6)    (x0[0] ≥ 0∧x2[0] + [-1] ≥ 0∧x0[0] + [-1]x2[0] ≥ 0 ⇒ (UIncreasing(727_0_MOD_LT(x0[1], x2[1])), ≥)∧[(-1)Bound*bni_13] ≥ 0∧[1 + (-1)bso_14] + x0[0] + [-1]x2[0] ≥ 0)



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

    (7)    (x2[0] + x0[0] ≥ 0∧x2[0] + [-1] ≥ 0∧x0[0] ≥ 0 ⇒ (UIncreasing(727_0_MOD_LT(x0[1], x2[1])), ≥)∧[(-1)Bound*bni_13] ≥ 0∧[1 + (-1)bso_14] + x0[0] ≥ 0)



    We simplified constraint (7) using rule (IDP_SMT_SPLIT) which results in the following new constraint:

    (8)    ([1] + x2[0] + x0[0] ≥ 0∧x2[0] ≥ 0∧x0[0] ≥ 0 ⇒ (UIncreasing(727_0_MOD_LT(x0[1], x2[1])), ≥)∧[(-1)Bound*bni_13] ≥ 0∧[1 + (-1)bso_14] + x0[0] ≥ 0)







For Pair 727_0_MOD_LT(x2[0], x0[0]) → COND_727_0_MOD_LT(&&(&&(>(x2[0], 0), <(x2[0], x0[0])), >(x0[0], 0)), x2[0], x0[0]) the following chains were created:
  • We consider the chain COND_727_0_MOD_LT(TRUE, x2[1], x0[1]) → 727_0_MOD_LT(x0[1], x2[1]), 727_0_MOD_LT(x2[0], x0[0]) → COND_727_0_MOD_LT(&&(&&(>(x2[0], 0), <(x2[0], x0[0])), >(x0[0], 0)), x2[0], x0[0]), COND_727_0_MOD_LT(TRUE, x2[1], x0[1]) → 727_0_MOD_LT(x0[1], x2[1]) which results in the following constraint:

    (9)    (x0[1]=x2[0]x2[1]=x0[0]&&(&&(>(x2[0], 0), <(x2[0], x0[0])), >(x0[0], 0))=TRUEx2[0]=x2[1]1x0[0]=x0[1]1727_0_MOD_LT(x2[0], x0[0])≥NonInfC∧727_0_MOD_LT(x2[0], x0[0])≥COND_727_0_MOD_LT(&&(&&(>(x2[0], 0), <(x2[0], x0[0])), >(x0[0], 0)), x2[0], x0[0])∧(UIncreasing(COND_727_0_MOD_LT(&&(&&(>(x2[0], 0), <(x2[0], x0[0])), >(x0[0], 0)), x2[0], x0[0])), ≥))



    We simplified constraint (9) using rules (III), (IV), (IDP_BOOLEAN) which results in the following new constraint:

    (10)    (>(x0[0], 0)=TRUE>(x2[0], 0)=TRUE<(x2[0], x0[0])=TRUE727_0_MOD_LT(x2[0], x0[0])≥NonInfC∧727_0_MOD_LT(x2[0], x0[0])≥COND_727_0_MOD_LT(&&(&&(>(x2[0], 0), <(x2[0], x0[0])), >(x0[0], 0)), x2[0], x0[0])∧(UIncreasing(COND_727_0_MOD_LT(&&(&&(>(x2[0], 0), <(x2[0], x0[0])), >(x0[0], 0)), x2[0], x0[0])), ≥))



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

    (11)    (x0[0] + [-1] ≥ 0∧x2[0] + [-1] ≥ 0∧x0[0] + [-1] + [-1]x2[0] ≥ 0 ⇒ (UIncreasing(COND_727_0_MOD_LT(&&(&&(>(x2[0], 0), <(x2[0], x0[0])), >(x0[0], 0)), x2[0], x0[0])), ≥)∧[(-1)Bound*bni_15] + [bni_15]x0[0] + [(-1)bni_15]x2[0] ≥ 0∧[-1 + (-1)bso_16] + x0[0] + [-1]x2[0] ≥ 0)



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

    (12)    (x0[0] + [-1] ≥ 0∧x2[0] + [-1] ≥ 0∧x0[0] + [-1] + [-1]x2[0] ≥ 0 ⇒ (UIncreasing(COND_727_0_MOD_LT(&&(&&(>(x2[0], 0), <(x2[0], x0[0])), >(x0[0], 0)), x2[0], x0[0])), ≥)∧[(-1)Bound*bni_15] + [bni_15]x0[0] + [(-1)bni_15]x2[0] ≥ 0∧[-1 + (-1)bso_16] + x0[0] + [-1]x2[0] ≥ 0)



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

    (13)    (x0[0] + [-1] ≥ 0∧x2[0] + [-1] ≥ 0∧x0[0] + [-1] + [-1]x2[0] ≥ 0 ⇒ (UIncreasing(COND_727_0_MOD_LT(&&(&&(>(x2[0], 0), <(x2[0], x0[0])), >(x0[0], 0)), x2[0], x0[0])), ≥)∧[(-1)Bound*bni_15] + [bni_15]x0[0] + [(-1)bni_15]x2[0] ≥ 0∧[-1 + (-1)bso_16] + x0[0] + [-1]x2[0] ≥ 0)



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

    (14)    (x0[0] ≥ 0∧x2[0] + [-1] ≥ 0∧x0[0] + [-1]x2[0] ≥ 0 ⇒ (UIncreasing(COND_727_0_MOD_LT(&&(&&(>(x2[0], 0), <(x2[0], x0[0])), >(x0[0], 0)), x2[0], x0[0])), ≥)∧[(-1)Bound*bni_15 + bni_15] + [bni_15]x0[0] + [(-1)bni_15]x2[0] ≥ 0∧[(-1)bso_16] + x0[0] + [-1]x2[0] ≥ 0)



    We simplified constraint (14) using rule (IDP_SMT_SPLIT) which results in the following new constraint:

    (15)    (x2[0] + x0[0] ≥ 0∧x2[0] + [-1] ≥ 0∧x0[0] ≥ 0 ⇒ (UIncreasing(COND_727_0_MOD_LT(&&(&&(>(x2[0], 0), <(x2[0], x0[0])), >(x0[0], 0)), x2[0], x0[0])), ≥)∧[(-1)Bound*bni_15 + bni_15] + [bni_15]x0[0] ≥ 0∧[(-1)bso_16] + x0[0] ≥ 0)



    We simplified constraint (15) using rule (IDP_SMT_SPLIT) which results in the following new constraint:

    (16)    ([1] + x2[0] + x0[0] ≥ 0∧x2[0] ≥ 0∧x0[0] ≥ 0 ⇒ (UIncreasing(COND_727_0_MOD_LT(&&(&&(>(x2[0], 0), <(x2[0], x0[0])), >(x0[0], 0)), x2[0], x0[0])), ≥)∧[(-1)Bound*bni_15 + bni_15] + [bni_15]x0[0] ≥ 0∧[(-1)bso_16] + x0[0] ≥ 0)







To summarize, we get the following constraints P for the following pairs.
  • COND_727_0_MOD_LT(TRUE, x2[1], x0[1]) → 727_0_MOD_LT(x0[1], x2[1])
    • ([1] + x2[0] + x0[0] ≥ 0∧x2[0] ≥ 0∧x0[0] ≥ 0 ⇒ (UIncreasing(727_0_MOD_LT(x0[1], x2[1])), ≥)∧[(-1)Bound*bni_13] ≥ 0∧[1 + (-1)bso_14] + x0[0] ≥ 0)

  • 727_0_MOD_LT(x2[0], x0[0]) → COND_727_0_MOD_LT(&&(&&(>(x2[0], 0), <(x2[0], x0[0])), >(x0[0], 0)), x2[0], x0[0])
    • ([1] + x2[0] + x0[0] ≥ 0∧x2[0] ≥ 0∧x0[0] ≥ 0 ⇒ (UIncreasing(COND_727_0_MOD_LT(&&(&&(>(x2[0], 0), <(x2[0], x0[0])), >(x0[0], 0)), x2[0], x0[0])), ≥)∧[(-1)Bound*bni_15 + bni_15] + [bni_15]x0[0] ≥ 0∧[(-1)bso_16] + x0[0] ≥ 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) = 0   
POL(COND_727_0_MOD_LT(x1, x2, x3)) = [-1]x1   
POL(727_0_MOD_LT(x1, x2)) = x2 + [-1]x1   
POL(&&(x1, x2)) = [-1]   
POL(>(x1, x2)) = [-1]   
POL(0) = 0   
POL(<(x1, x2)) = [-1]   

The following pairs are in P>:

COND_727_0_MOD_LT(TRUE, x2[1], x0[1]) → 727_0_MOD_LT(x0[1], x2[1])

The following pairs are in Pbound:

COND_727_0_MOD_LT(TRUE, x2[1], x0[1]) → 727_0_MOD_LT(x0[1], x2[1])
727_0_MOD_LT(x2[0], x0[0]) → COND_727_0_MOD_LT(&&(&&(>(x2[0], 0), <(x2[0], x0[0])), >(x0[0], 0)), x2[0], x0[0])

The following pairs are in P:

727_0_MOD_LT(x2[0], x0[0]) → COND_727_0_MOD_LT(&&(&&(>(x2[0], 0), <(x2[0], x0[0])), >(x0[0], 0)), x2[0], x0[0])

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

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

(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:
(0): 727_0_MOD_LT(x2[0], x0[0]) → COND_727_0_MOD_LT(x2[0] > 0 && x2[0] < x0[0] && x0[0] > 0, x2[0], x0[0])


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