Termination of the given JBC could be proven:

0 JBC
↳1 JBCToGraph (⇒, 1070 ms)
↳2 JBCTerminationGraph
↳3 TerminationGraphToSCCProof (⇒, 0 ms)
↳4 JBCTerminationSCC
↳5 SCCToIDPv1Proof (⇒, 240 ms)
↳6 IDP
↳7 IDPNonInfProof (⇒, 1040 ms)
↳8 IDP
↳9 IDependencyGraphProof (⇔, 0 ms)
↳10 TRUE

(0) Obligation:

JBC Problem based on JBC Program:
Manifest-Version: 1.0 Created-By: 1.6.0_16 (Sun Microsystems Inc.) Main-Class: PastaC10 
/**
 * 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 PastaC10 {
    public static void main(String[] args) {
        Random.args = args;
        int i = Random.random();
        int j = Random.random();

        while (i - j >= 1) {
			i = i - Random.random();
			int r = Random.random() + 1;
			j = j + r;
        }
    } 
}


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:
PastaC10.main([Ljava/lang/String;)V: Graph of 321 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: PastaC10.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 54 rules for P and 0 rules for R.


P rules:
1757_0_main_Load(EOS(STATIC_1757), i742, i743, i742) → 1758_0_main_IntArithmetic(EOS(STATIC_1758), i742, i743, i742, i743)
1758_0_main_IntArithmetic(EOS(STATIC_1758), i742, i743, i742, i743) → 1759_0_main_ConstantStackPush(EOS(STATIC_1759), i742, i743, -(i742, i743))
1759_0_main_ConstantStackPush(EOS(STATIC_1759), i742, i743, i748) → 1761_0_main_LT(EOS(STATIC_1761), i742, i743, i748, 1)
1761_0_main_LT(EOS(STATIC_1761), i742, i743, i752, matching1) → 1764_0_main_LT(EOS(STATIC_1764), i742, i743, i752, 1) | =(matching1, 1)
1764_0_main_LT(EOS(STATIC_1764), i742, i743, i752, matching1) → 1767_0_main_Load(EOS(STATIC_1767), i742, i743) | &&(>=(i752, 1), =(matching1, 1))
1767_0_main_Load(EOS(STATIC_1767), i742, i743) → 1769_0_main_InvokeMethod(EOS(STATIC_1769), i743, i742)
1769_0_main_InvokeMethod(EOS(STATIC_1769), i743, i742) → 1770_0_random_FieldAccess(EOS(STATIC_1770), i743, i742)
1770_0_random_FieldAccess(EOS(STATIC_1770), i743, i742) → 1772_0_random_FieldAccess(EOS(STATIC_1772), i743, i742)
1772_0_random_FieldAccess(EOS(STATIC_1772), i743, i742) → 1776_0_random_ArrayAccess(EOS(STATIC_1776), i743, i742)
1776_0_random_ArrayAccess(EOS(STATIC_1776), i743, i742) → 1778_0_random_ArrayAccess(EOS(STATIC_1778), i743, i742)
1778_0_random_ArrayAccess(EOS(STATIC_1778), i743, i742) → 1781_0_random_Store(EOS(STATIC_1781), i743, i742, o792)
1781_0_random_Store(EOS(STATIC_1781), i743, i742, o792) → 1784_0_random_FieldAccess(EOS(STATIC_1784), i743, i742, o792)
1784_0_random_FieldAccess(EOS(STATIC_1784), i743, i742, o792) → 1786_0_random_ConstantStackPush(EOS(STATIC_1786), i743, i742, o792)
1786_0_random_ConstantStackPush(EOS(STATIC_1786), i743, i742, o792) → 1789_0_random_IntArithmetic(EOS(STATIC_1789), i743, i742, o792)
1789_0_random_IntArithmetic(EOS(STATIC_1789), i743, i742, o792) → 1792_0_random_FieldAccess(EOS(STATIC_1792), i743, i742, o792)
1792_0_random_FieldAccess(EOS(STATIC_1792), i743, i742, o792) → 1794_0_random_Load(EOS(STATIC_1794), i743, i742, o792)
1794_0_random_Load(EOS(STATIC_1794), i743, i742, o792) → 1802_0_random_InvokeMethod(EOS(STATIC_1802), i743, i742, o792)
1802_0_random_InvokeMethod(EOS(STATIC_1802), i743, i742, java.lang.Object(o810sub)) → 1805_0_random_InvokeMethod(EOS(STATIC_1805), i743, i742, java.lang.Object(o810sub))
1805_0_random_InvokeMethod(EOS(STATIC_1805), i743, i742, java.lang.Object(o810sub)) → 1808_0_length_Load(EOS(STATIC_1808), i743, i742, java.lang.Object(o810sub), java.lang.Object(o810sub))
1808_0_length_Load(EOS(STATIC_1808), i743, i742, java.lang.Object(o810sub), java.lang.Object(o810sub)) → 1819_0_length_FieldAccess(EOS(STATIC_1819), i743, i742, java.lang.Object(o810sub), java.lang.Object(o810sub))
1819_0_length_FieldAccess(EOS(STATIC_1819), i743, i742, java.lang.Object(java.lang.String(o818sub, i789)), java.lang.Object(java.lang.String(o818sub, i789))) → 1820_0_length_FieldAccess(EOS(STATIC_1820), i743, i742, java.lang.Object(java.lang.String(o818sub, i789)), java.lang.Object(java.lang.String(o818sub, i789))) | &&(>=(i789, 0), >=(i790, 0))
1820_0_length_FieldAccess(EOS(STATIC_1820), i743, i742, java.lang.Object(java.lang.String(o818sub, i789)), java.lang.Object(java.lang.String(o818sub, i789))) → 1826_0_length_Return(EOS(STATIC_1826), i743, i742, java.lang.Object(java.lang.String(o818sub, i789)), i789)
1826_0_length_Return(EOS(STATIC_1826), i743, i742, java.lang.Object(java.lang.String(o818sub, i789)), i789) → 1831_0_random_Return(EOS(STATIC_1831), i743, i742, i789)
1831_0_random_Return(EOS(STATIC_1831), i743, i742, i789) → 1833_0_main_IntArithmetic(EOS(STATIC_1833), i743, i742, i789)
1833_0_main_IntArithmetic(EOS(STATIC_1833), i743, i742, i789) → 1840_0_main_Store(EOS(STATIC_1840), i743, -(i742, i789)) | >=(i789, 0)
1840_0_main_Store(EOS(STATIC_1840), i743, i800) → 1845_0_main_InvokeMethod(EOS(STATIC_1845), i800, i743)
1845_0_main_InvokeMethod(EOS(STATIC_1845), i800, i743) → 1849_0_random_FieldAccess(EOS(STATIC_1849), i800, i743)
1849_0_random_FieldAccess(EOS(STATIC_1849), i800, i743) → 1861_0_random_FieldAccess(EOS(STATIC_1861), i800, i743)
1861_0_random_FieldAccess(EOS(STATIC_1861), i800, i743) → 1869_0_random_ArrayAccess(EOS(STATIC_1869), i800, i743)
1869_0_random_ArrayAccess(EOS(STATIC_1869), i800, i743) → 1875_0_random_ArrayAccess(EOS(STATIC_1875), i800, i743)
1875_0_random_ArrayAccess(EOS(STATIC_1875), i800, i743) → 1884_0_random_Store(EOS(STATIC_1884), i800, i743, o852)
1884_0_random_Store(EOS(STATIC_1884), i800, i743, o852) → 1892_0_random_FieldAccess(EOS(STATIC_1892), i800, i743, o852)
1892_0_random_FieldAccess(EOS(STATIC_1892), i800, i743, o852) → 1899_0_random_ConstantStackPush(EOS(STATIC_1899), i800, i743, o852)
1899_0_random_ConstantStackPush(EOS(STATIC_1899), i800, i743, o852) → 1908_0_random_IntArithmetic(EOS(STATIC_1908), i800, i743, o852)
1908_0_random_IntArithmetic(EOS(STATIC_1908), i800, i743, o852) → 1915_0_random_FieldAccess(EOS(STATIC_1915), i800, i743, o852)
1915_0_random_FieldAccess(EOS(STATIC_1915), i800, i743, o852) → 1922_0_random_Load(EOS(STATIC_1922), i800, i743, o852)
1922_0_random_Load(EOS(STATIC_1922), i800, i743, o852) → 1933_0_random_InvokeMethod(EOS(STATIC_1933), i800, i743, o852)
1933_0_random_InvokeMethod(EOS(STATIC_1933), i800, i743, java.lang.Object(o911sub)) → 1939_0_random_InvokeMethod(EOS(STATIC_1939), i800, i743, java.lang.Object(o911sub))
1939_0_random_InvokeMethod(EOS(STATIC_1939), i800, i743, java.lang.Object(o911sub)) → 1943_0_length_Load(EOS(STATIC_1943), i800, i743, java.lang.Object(o911sub), java.lang.Object(o911sub))
1943_0_length_Load(EOS(STATIC_1943), i800, i743, java.lang.Object(o911sub), java.lang.Object(o911sub)) → 1957_0_length_FieldAccess(EOS(STATIC_1957), i800, i743, java.lang.Object(o911sub), java.lang.Object(o911sub))
1957_0_length_FieldAccess(EOS(STATIC_1957), i800, i743, java.lang.Object(java.lang.String(o936sub, i885)), java.lang.Object(java.lang.String(o936sub, i885))) → 1961_0_length_FieldAccess(EOS(STATIC_1961), i800, i743, java.lang.Object(java.lang.String(o936sub, i885)), java.lang.Object(java.lang.String(o936sub, i885))) | &&(>=(i885, 0), >=(i886, 0))
1961_0_length_FieldAccess(EOS(STATIC_1961), i800, i743, java.lang.Object(java.lang.String(o936sub, i885)), java.lang.Object(java.lang.String(o936sub, i885))) → 1969_0_length_Return(EOS(STATIC_1969), i800, i743, java.lang.Object(java.lang.String(o936sub, i885)), i885)
1969_0_length_Return(EOS(STATIC_1969), i800, i743, java.lang.Object(java.lang.String(o936sub, i885)), i885) → 1974_0_random_Return(EOS(STATIC_1974), i800, i743, i885)
1974_0_random_Return(EOS(STATIC_1974), i800, i743, i885) → 1975_0_main_ConstantStackPush(EOS(STATIC_1975), i800, i743, i885)
1975_0_main_ConstantStackPush(EOS(STATIC_1975), i800, i743, i885) → 1982_0_main_IntArithmetic(EOS(STATIC_1982), i800, i743, i885, 1)
1982_0_main_IntArithmetic(EOS(STATIC_1982), i800, i743, i885, matching1) → 1989_0_main_Store(EOS(STATIC_1989), i800, i743, +(i885, 1)) | &&(>=(i885, 0), =(matching1, 1))
1989_0_main_Store(EOS(STATIC_1989), i800, i743, i894) → 1993_0_main_Load(EOS(STATIC_1993), i800, i743, i894)
1993_0_main_Load(EOS(STATIC_1993), i800, i743, i894) → 2000_0_main_Load(EOS(STATIC_2000), i800, i894, i743)
2000_0_main_Load(EOS(STATIC_2000), i800, i894, i743) → 2007_0_main_IntArithmetic(EOS(STATIC_2007), i800, i743, i894)
2007_0_main_IntArithmetic(EOS(STATIC_2007), i800, i743, i894) → 2011_0_main_Store(EOS(STATIC_2011), i800, +(i743, i894)) | >(i894, 0)
2011_0_main_Store(EOS(STATIC_2011), i800, i905) → 2019_0_main_JMP(EOS(STATIC_2019), i800, i905)
2019_0_main_JMP(EOS(STATIC_2019), i800, i905) → 2033_0_main_Load(EOS(STATIC_2033), i800, i905)
2033_0_main_Load(EOS(STATIC_2033), i800, i905) → 1753_0_main_Load(EOS(STATIC_1753), i800, i905)
1753_0_main_Load(EOS(STATIC_1753), i742, i743) → 1757_0_main_Load(EOS(STATIC_1757), i742, i743, i742)
R rules:

Combined rules. Obtained 1 conditional rules for P and 0 conditional rules for R.


P rules:
1757_0_main_Load(EOS(STATIC_1757), x0, x1, x0) → 1757_0_main_Load(EOS(STATIC_1757), -(x0, x2), +(x1, +(x3, 1)), -(x0, x2)) | &&(&&(>(+(x3, 1), 0), >(+(x2, 1), 0)), <=(1, -(x0, x1)))
R rules:

Filtered ground terms:



1757_0_main_Load(x1, x2, x3, x4) → 1757_0_main_Load(x2, x3, x4)
EOS(x1) → EOS
Cond_1757_0_main_Load(x1, x2, x3, x4, x5, x6, x7) → Cond_1757_0_main_Load(x1, x3, x4, x5, x6, x7)

Filtered duplicate args:



1757_0_main_Load(x1, x2, x3) → 1757_0_main_Load(x2, x3)
Cond_1757_0_main_Load(x1, x2, x3, x4, x5, x6) → Cond_1757_0_main_Load(x1, x3, x4, x5, x6)

Combined rules. Obtained 1 conditional rules for P and 0 conditional rules for R.


P rules:
1757_0_main_Load(x1, x0) → 1757_0_main_Load(+(x1, +(x3, 1)), -(x0, x2)) | &&(&&(>(x3, -1), >(x2, -1)), <=(1, -(x0, x1)))
R rules:

Finished conversion. Obtained 2 rules for P and 0 rules for R. System has predefined symbols.


P rules:
1757_0_MAIN_LOAD(x1, x0) → COND_1757_0_MAIN_LOAD(&&(&&(>(x3, -1), >(x2, -1)), <=(1, -(x0, x1))), x1, x0, x3, x2)
COND_1757_0_MAIN_LOAD(TRUE, x1, x0, x3, x2) → 1757_0_MAIN_LOAD(+(x1, +(x3, 1)), -(x0, x2))
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): 1757_0_MAIN_LOAD(x1[0], x0[0]) → COND_1757_0_MAIN_LOAD(x3[0] > -1 && x2[0] > -1 && 1 <= x0[0] - x1[0], x1[0], x0[0], x3[0], x2[0])
(1): COND_1757_0_MAIN_LOAD(TRUE, x1[1], x0[1], x3[1], x2[1]) → 1757_0_MAIN_LOAD(x1[1] + x3[1] + 1, x0[1] - x2[1])

(0) -> (1), if (x3[0] > -1 && x2[0] > -1 && 1 <= x0[0] - x1[0] ∧x1[0] →* x1[1]∧x0[0] →* x0[1]∧x3[0] →* x3[1]∧x2[0] →* x2[1])


(1) -> (0), if (x1[1] + x3[1] + 1 →* x1[0]∧x0[1] - x2[1] →* x0[0])



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@156a0d22 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 1757_0_MAIN_LOAD(x1, x0) → COND_1757_0_MAIN_LOAD(&&(&&(>(x3, -1), >(x2, -1)), <=(1, -(x0, x1))), x1, x0, x3, x2) the following chains were created:
  • We consider the chain COND_1757_0_MAIN_LOAD(TRUE, x1[1], x0[1], x3[1], x2[1]) → 1757_0_MAIN_LOAD(+(x1[1], +(x3[1], 1)), -(x0[1], x2[1])), 1757_0_MAIN_LOAD(x1[0], x0[0]) → COND_1757_0_MAIN_LOAD(&&(&&(>(x3[0], -1), >(x2[0], -1)), <=(1, -(x0[0], x1[0]))), x1[0], x0[0], x3[0], x2[0]), COND_1757_0_MAIN_LOAD(TRUE, x1[1], x0[1], x3[1], x2[1]) → 1757_0_MAIN_LOAD(+(x1[1], +(x3[1], 1)), -(x0[1], x2[1])) which results in the following constraint:

    (1)    (+(x1[1], +(x3[1], 1))=x1[0]∧-(x0[1], x2[1])=x0[0]∧&&(&&(>(x3[0], -1), >(x2[0], -1)), <=(1, -(x0[0], x1[0])))=TRUE∧x1[0]=x1[1]1∧x0[0]=x0[1]1∧x3[0]=x3[1]1∧x2[0]=x2[1]1 ⇒ 1757_0_MAIN_LOAD(x1[0], x0[0])≥NonInfC∧1757_0_MAIN_LOAD(x1[0], x0[0])≥COND_1757_0_MAIN_LOAD(&&(&&(>(x3[0], -1), >(x2[0], -1)), <=(1, -(x0[0], x1[0]))), x1[0], x0[0], x3[0], x2[0])∧(UIncreasing(COND_1757_0_MAIN_LOAD(&&(&&(>(x3[0], -1), >(x2[0], -1)), <=(1, -(x0[0], x1[0]))), x1[0], x0[0], x3[0], x2[0])), ≥))



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

    (2)    (<=(1, -(-(x0[1], x2[1]), +(x1[1], +(x3[1], 1))))=TRUE∧>(x3[0], -1)=TRUE∧>(x2[0], -1)=TRUE ⇒ 1757_0_MAIN_LOAD(+(x1[1], +(x3[1], 1)), -(x0[1], x2[1]))≥NonInfC∧1757_0_MAIN_LOAD(+(x1[1], +(x3[1], 1)), -(x0[1], x2[1]))≥COND_1757_0_MAIN_LOAD(&&(&&(>(x3[0], -1), >(x2[0], -1)), <=(1, -(-(x0[1], x2[1]), +(x1[1], +(x3[1], 1))))), +(x1[1], +(x3[1], 1)), -(x0[1], x2[1]), x3[0], x2[0])∧(UIncreasing(COND_1757_0_MAIN_LOAD(&&(&&(>(x3[0], -1), >(x2[0], -1)), <=(1, -(x0[0], x1[0]))), x1[0], x0[0], x3[0], x2[0])), ≥))



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

    (3)    (x0[1] + [-2] + [-1]x2[1] + [-1]x1[1] + [-1]x3[1] ≥ 0∧x3[0] ≥ 0∧x2[0] ≥ 0 ⇒ (UIncreasing(COND_1757_0_MAIN_LOAD(&&(&&(>(x3[0], -1), >(x2[0], -1)), <=(1, -(x0[0], x1[0]))), x1[0], x0[0], x3[0], x2[0])), ≥)∧[(-2)bni_15 + (-1)Bound*bni_15] + [bni_15]x0[1] + [(-1)bni_15]x2[1] + [(-1)bni_15]x1[1] + [(-1)bni_15]x3[1] ≥ 0∧[(-1)bso_16] ≥ 0)



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

    (4)    (x0[1] + [-2] + [-1]x2[1] + [-1]x1[1] + [-1]x3[1] ≥ 0∧x3[0] ≥ 0∧x2[0] ≥ 0 ⇒ (UIncreasing(COND_1757_0_MAIN_LOAD(&&(&&(>(x3[0], -1), >(x2[0], -1)), <=(1, -(x0[0], x1[0]))), x1[0], x0[0], x3[0], x2[0])), ≥)∧[(-2)bni_15 + (-1)Bound*bni_15] + [bni_15]x0[1] + [(-1)bni_15]x2[1] + [(-1)bni_15]x1[1] + [(-1)bni_15]x3[1] ≥ 0∧[(-1)bso_16] ≥ 0)



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

    (5)    (x0[1] + [-2] + [-1]x2[1] + [-1]x1[1] + [-1]x3[1] ≥ 0∧x3[0] ≥ 0∧x2[0] ≥ 0 ⇒ (UIncreasing(COND_1757_0_MAIN_LOAD(&&(&&(>(x3[0], -1), >(x2[0], -1)), <=(1, -(x0[0], x1[0]))), x1[0], x0[0], x3[0], x2[0])), ≥)∧[(-2)bni_15 + (-1)Bound*bni_15] + [bni_15]x0[1] + [(-1)bni_15]x2[1] + [(-1)bni_15]x1[1] + [(-1)bni_15]x3[1] ≥ 0∧[(-1)bso_16] ≥ 0)



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

    (6)    (x0[1] ≥ 0∧x3[0] ≥ 0∧x2[0] ≥ 0 ⇒ (UIncreasing(COND_1757_0_MAIN_LOAD(&&(&&(>(x3[0], -1), >(x2[0], -1)), <=(1, -(x0[0], x1[0]))), x1[0], x0[0], x3[0], x2[0])), ≥)∧[(-1)Bound*bni_15] + [bni_15]x0[1] ≥ 0∧[(-1)bso_16] ≥ 0)



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

    (7)    (x0[1] ≥ 0∧x3[0] ≥ 0∧x2[0] ≥ 0∧x2[1] ≥ 0 ⇒ (UIncreasing(COND_1757_0_MAIN_LOAD(&&(&&(>(x3[0], -1), >(x2[0], -1)), <=(1, -(x0[0], x1[0]))), x1[0], x0[0], x3[0], x2[0])), ≥)∧[(-1)Bound*bni_15] + [bni_15]x0[1] ≥ 0∧[(-1)bso_16] ≥ 0)


    (8)    (x0[1] ≥ 0∧x3[0] ≥ 0∧x2[0] ≥ 0∧x2[1] ≥ 0 ⇒ (UIncreasing(COND_1757_0_MAIN_LOAD(&&(&&(>(x3[0], -1), >(x2[0], -1)), <=(1, -(x0[0], x1[0]))), x1[0], x0[0], x3[0], x2[0])), ≥)∧[(-1)Bound*bni_15] + [bni_15]x0[1] ≥ 0∧[(-1)bso_16] ≥ 0)



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

    (9)    (x0[1] ≥ 0∧x3[0] ≥ 0∧x2[0] ≥ 0∧x2[1] ≥ 0∧x1[1] ≥ 0 ⇒ (UIncreasing(COND_1757_0_MAIN_LOAD(&&(&&(>(x3[0], -1), >(x2[0], -1)), <=(1, -(x0[0], x1[0]))), x1[0], x0[0], x3[0], x2[0])), ≥)∧[(-1)Bound*bni_15] + [bni_15]x0[1] ≥ 0∧[(-1)bso_16] ≥ 0)


    (10)    (x0[1] ≥ 0∧x3[0] ≥ 0∧x2[0] ≥ 0∧x2[1] ≥ 0∧x1[1] ≥ 0 ⇒ (UIncreasing(COND_1757_0_MAIN_LOAD(&&(&&(>(x3[0], -1), >(x2[0], -1)), <=(1, -(x0[0], x1[0]))), x1[0], x0[0], x3[0], x2[0])), ≥)∧[(-1)Bound*bni_15] + [bni_15]x0[1] ≥ 0∧[(-1)bso_16] ≥ 0)



    We simplified constraint (8) using rule (IDP_SMT_SPLIT) which results in the following new constraints:

    (11)    (x0[1] ≥ 0∧x3[0] ≥ 0∧x2[0] ≥ 0∧x2[1] ≥ 0∧x1[1] ≥ 0 ⇒ (UIncreasing(COND_1757_0_MAIN_LOAD(&&(&&(>(x3[0], -1), >(x2[0], -1)), <=(1, -(x0[0], x1[0]))), x1[0], x0[0], x3[0], x2[0])), ≥)∧[(-1)Bound*bni_15] + [bni_15]x0[1] ≥ 0∧[(-1)bso_16] ≥ 0)


    (12)    (x0[1] ≥ 0∧x3[0] ≥ 0∧x2[0] ≥ 0∧x2[1] ≥ 0∧x1[1] ≥ 0 ⇒ (UIncreasing(COND_1757_0_MAIN_LOAD(&&(&&(>(x3[0], -1), >(x2[0], -1)), <=(1, -(x0[0], x1[0]))), x1[0], x0[0], x3[0], x2[0])), ≥)∧[(-1)Bound*bni_15] + [bni_15]x0[1] ≥ 0∧[(-1)bso_16] ≥ 0)



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

    (13)    (x0[1] ≥ 0∧x3[0] ≥ 0∧x2[0] ≥ 0∧x2[1] ≥ 0∧x1[1] ≥ 0∧x3[1] ≥ 0 ⇒ (UIncreasing(COND_1757_0_MAIN_LOAD(&&(&&(>(x3[0], -1), >(x2[0], -1)), <=(1, -(x0[0], x1[0]))), x1[0], x0[0], x3[0], x2[0])), ≥)∧[(-1)Bound*bni_15] + [bni_15]x0[1] ≥ 0∧[(-1)bso_16] ≥ 0)


    (14)    (x0[1] ≥ 0∧x3[0] ≥ 0∧x2[0] ≥ 0∧x2[1] ≥ 0∧x1[1] ≥ 0∧x3[1] ≥ 0 ⇒ (UIncreasing(COND_1757_0_MAIN_LOAD(&&(&&(>(x3[0], -1), >(x2[0], -1)), <=(1, -(x0[0], x1[0]))), x1[0], x0[0], x3[0], x2[0])), ≥)∧[(-1)Bound*bni_15] + [bni_15]x0[1] ≥ 0∧[(-1)bso_16] ≥ 0)



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

    (15)    (x0[1] ≥ 0∧x3[0] ≥ 0∧x2[0] ≥ 0∧x2[1] ≥ 0∧x1[1] ≥ 0∧x3[1] ≥ 0 ⇒ (UIncreasing(COND_1757_0_MAIN_LOAD(&&(&&(>(x3[0], -1), >(x2[0], -1)), <=(1, -(x0[0], x1[0]))), x1[0], x0[0], x3[0], x2[0])), ≥)∧[(-1)Bound*bni_15] + [bni_15]x0[1] ≥ 0∧[(-1)bso_16] ≥ 0)


    (16)    (x0[1] ≥ 0∧x3[0] ≥ 0∧x2[0] ≥ 0∧x2[1] ≥ 0∧x1[1] ≥ 0∧x3[1] ≥ 0 ⇒ (UIncreasing(COND_1757_0_MAIN_LOAD(&&(&&(>(x3[0], -1), >(x2[0], -1)), <=(1, -(x0[0], x1[0]))), x1[0], x0[0], x3[0], x2[0])), ≥)∧[(-1)Bound*bni_15] + [bni_15]x0[1] ≥ 0∧[(-1)bso_16] ≥ 0)



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

    (17)    (x0[1] ≥ 0∧x3[0] ≥ 0∧x2[0] ≥ 0∧x2[1] ≥ 0∧x1[1] ≥ 0∧x3[1] ≥ 0 ⇒ (UIncreasing(COND_1757_0_MAIN_LOAD(&&(&&(>(x3[0], -1), >(x2[0], -1)), <=(1, -(x0[0], x1[0]))), x1[0], x0[0], x3[0], x2[0])), ≥)∧[(-1)Bound*bni_15] + [bni_15]x0[1] ≥ 0∧[(-1)bso_16] ≥ 0)


    (18)    (x0[1] ≥ 0∧x3[0] ≥ 0∧x2[0] ≥ 0∧x2[1] ≥ 0∧x1[1] ≥ 0∧x3[1] ≥ 0 ⇒ (UIncreasing(COND_1757_0_MAIN_LOAD(&&(&&(>(x3[0], -1), >(x2[0], -1)), <=(1, -(x0[0], x1[0]))), x1[0], x0[0], x3[0], x2[0])), ≥)∧[(-1)Bound*bni_15] + [bni_15]x0[1] ≥ 0∧[(-1)bso_16] ≥ 0)



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

    (19)    (x0[1] ≥ 0∧x3[0] ≥ 0∧x2[0] ≥ 0∧x2[1] ≥ 0∧x1[1] ≥ 0∧x3[1] ≥ 0 ⇒ (UIncreasing(COND_1757_0_MAIN_LOAD(&&(&&(>(x3[0], -1), >(x2[0], -1)), <=(1, -(x0[0], x1[0]))), x1[0], x0[0], x3[0], x2[0])), ≥)∧[(-1)Bound*bni_15] + [bni_15]x0[1] ≥ 0∧[(-1)bso_16] ≥ 0)


    (20)    (x0[1] ≥ 0∧x3[0] ≥ 0∧x2[0] ≥ 0∧x2[1] ≥ 0∧x1[1] ≥ 0∧x3[1] ≥ 0 ⇒ (UIncreasing(COND_1757_0_MAIN_LOAD(&&(&&(>(x3[0], -1), >(x2[0], -1)), <=(1, -(x0[0], x1[0]))), x1[0], x0[0], x3[0], x2[0])), ≥)∧[(-1)Bound*bni_15] + [bni_15]x0[1] ≥ 0∧[(-1)bso_16] ≥ 0)







For Pair COND_1757_0_MAIN_LOAD(TRUE, x1, x0, x3, x2) → 1757_0_MAIN_LOAD(+(x1, +(x3, 1)), -(x0, x2)) the following chains were created:
  • We consider the chain 1757_0_MAIN_LOAD(x1[0], x0[0]) → COND_1757_0_MAIN_LOAD(&&(&&(>(x3[0], -1), >(x2[0], -1)), <=(1, -(x0[0], x1[0]))), x1[0], x0[0], x3[0], x2[0]), COND_1757_0_MAIN_LOAD(TRUE, x1[1], x0[1], x3[1], x2[1]) → 1757_0_MAIN_LOAD(+(x1[1], +(x3[1], 1)), -(x0[1], x2[1])), 1757_0_MAIN_LOAD(x1[0], x0[0]) → COND_1757_0_MAIN_LOAD(&&(&&(>(x3[0], -1), >(x2[0], -1)), <=(1, -(x0[0], x1[0]))), x1[0], x0[0], x3[0], x2[0]) which results in the following constraint:

    (21)    (&&(&&(>(x3[0], -1), >(x2[0], -1)), <=(1, -(x0[0], x1[0])))=TRUE∧x1[0]=x1[1]∧x0[0]=x0[1]∧x3[0]=x3[1]∧x2[0]=x2[1]∧+(x1[1], +(x3[1], 1))=x1[0]1∧-(x0[1], x2[1])=x0[0]1 ⇒ COND_1757_0_MAIN_LOAD(TRUE, x1[1], x0[1], x3[1], x2[1])≥NonInfC∧COND_1757_0_MAIN_LOAD(TRUE, x1[1], x0[1], x3[1], x2[1])≥1757_0_MAIN_LOAD(+(x1[1], +(x3[1], 1)), -(x0[1], x2[1]))∧(UIncreasing(1757_0_MAIN_LOAD(+(x1[1], +(x3[1], 1)), -(x0[1], x2[1]))), ≥))



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

    (22)    (<=(1, -(x0[0], x1[0]))=TRUE∧>(x3[0], -1)=TRUE∧>(x2[0], -1)=TRUE ⇒ COND_1757_0_MAIN_LOAD(TRUE, x1[0], x0[0], x3[0], x2[0])≥NonInfC∧COND_1757_0_MAIN_LOAD(TRUE, x1[0], x0[0], x3[0], x2[0])≥1757_0_MAIN_LOAD(+(x1[0], +(x3[0], 1)), -(x0[0], x2[0]))∧(UIncreasing(1757_0_MAIN_LOAD(+(x1[1], +(x3[1], 1)), -(x0[1], x2[1]))), ≥))



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

    (23)    (x0[0] + [-1] + [-1]x1[0] ≥ 0∧x3[0] ≥ 0∧x2[0] ≥ 0 ⇒ (UIncreasing(1757_0_MAIN_LOAD(+(x1[1], +(x3[1], 1)), -(x0[1], x2[1]))), ≥)∧[(-1)bni_17 + (-1)Bound*bni_17] + [bni_17]x0[0] + [(-1)bni_17]x1[0] ≥ 0∧[1 + (-1)bso_18] + x2[0] + x3[0] ≥ 0)



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

    (24)    (x0[0] + [-1] + [-1]x1[0] ≥ 0∧x3[0] ≥ 0∧x2[0] ≥ 0 ⇒ (UIncreasing(1757_0_MAIN_LOAD(+(x1[1], +(x3[1], 1)), -(x0[1], x2[1]))), ≥)∧[(-1)bni_17 + (-1)Bound*bni_17] + [bni_17]x0[0] + [(-1)bni_17]x1[0] ≥ 0∧[1 + (-1)bso_18] + x2[0] + x3[0] ≥ 0)



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

    (25)    (x0[0] + [-1] + [-1]x1[0] ≥ 0∧x3[0] ≥ 0∧x2[0] ≥ 0 ⇒ (UIncreasing(1757_0_MAIN_LOAD(+(x1[1], +(x3[1], 1)), -(x0[1], x2[1]))), ≥)∧[(-1)bni_17 + (-1)Bound*bni_17] + [bni_17]x0[0] + [(-1)bni_17]x1[0] ≥ 0∧[1 + (-1)bso_18] + x2[0] + x3[0] ≥ 0)



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

    (26)    (x0[0] ≥ 0∧x3[0] ≥ 0∧x2[0] ≥ 0 ⇒ (UIncreasing(1757_0_MAIN_LOAD(+(x1[1], +(x3[1], 1)), -(x0[1], x2[1]))), ≥)∧[(-1)Bound*bni_17] + [bni_17]x0[0] ≥ 0∧[1 + (-1)bso_18] + x2[0] + x3[0] ≥ 0)



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

    (27)    (x0[0] ≥ 0∧x3[0] ≥ 0∧x2[0] ≥ 0∧x1[0] ≥ 0 ⇒ (UIncreasing(1757_0_MAIN_LOAD(+(x1[1], +(x3[1], 1)), -(x0[1], x2[1]))), ≥)∧[(-1)Bound*bni_17] + [bni_17]x0[0] ≥ 0∧[1 + (-1)bso_18] + x2[0] + x3[0] ≥ 0)


    (28)    (x0[0] ≥ 0∧x3[0] ≥ 0∧x2[0] ≥ 0∧x1[0] ≥ 0 ⇒ (UIncreasing(1757_0_MAIN_LOAD(+(x1[1], +(x3[1], 1)), -(x0[1], x2[1]))), ≥)∧[(-1)Bound*bni_17] + [bni_17]x0[0] ≥ 0∧[1 + (-1)bso_18] + x2[0] + x3[0] ≥ 0)







To summarize, we get the following constraints P≥ for the following pairs.
  • 1757_0_MAIN_LOAD(x1, x0) → COND_1757_0_MAIN_LOAD(&&(&&(>(x3, -1), >(x2, -1)), <=(1, -(x0, x1))), x1, x0, x3, x2)
    • (x0[1] ≥ 0∧x3[0] ≥ 0∧x2[0] ≥ 0∧x2[1] ≥ 0∧x1[1] ≥ 0∧x3[1] ≥ 0 ⇒ (UIncreasing(COND_1757_0_MAIN_LOAD(&&(&&(>(x3[0], -1), >(x2[0], -1)), <=(1, -(x0[0], x1[0]))), x1[0], x0[0], x3[0], x2[0])), ≥)∧[(-1)Bound*bni_15] + [bni_15]x0[1] ≥ 0∧[(-1)bso_16] ≥ 0)
    • (x0[1] ≥ 0∧x3[0] ≥ 0∧x2[0] ≥ 0∧x2[1] ≥ 0∧x1[1] ≥ 0∧x3[1] ≥ 0 ⇒ (UIncreasing(COND_1757_0_MAIN_LOAD(&&(&&(>(x3[0], -1), >(x2[0], -1)), <=(1, -(x0[0], x1[0]))), x1[0], x0[0], x3[0], x2[0])), ≥)∧[(-1)Bound*bni_15] + [bni_15]x0[1] ≥ 0∧[(-1)bso_16] ≥ 0)
    • (x0[1] ≥ 0∧x3[0] ≥ 0∧x2[0] ≥ 0∧x2[1] ≥ 0∧x1[1] ≥ 0∧x3[1] ≥ 0 ⇒ (UIncreasing(COND_1757_0_MAIN_LOAD(&&(&&(>(x3[0], -1), >(x2[0], -1)), <=(1, -(x0[0], x1[0]))), x1[0], x0[0], x3[0], x2[0])), ≥)∧[(-1)Bound*bni_15] + [bni_15]x0[1] ≥ 0∧[(-1)bso_16] ≥ 0)
    • (x0[1] ≥ 0∧x3[0] ≥ 0∧x2[0] ≥ 0∧x2[1] ≥ 0∧x1[1] ≥ 0∧x3[1] ≥ 0 ⇒ (UIncreasing(COND_1757_0_MAIN_LOAD(&&(&&(>(x3[0], -1), >(x2[0], -1)), <=(1, -(x0[0], x1[0]))), x1[0], x0[0], x3[0], x2[0])), ≥)∧[(-1)Bound*bni_15] + [bni_15]x0[1] ≥ 0∧[(-1)bso_16] ≥ 0)
    • (x0[1] ≥ 0∧x3[0] ≥ 0∧x2[0] ≥ 0∧x2[1] ≥ 0∧x1[1] ≥ 0∧x3[1] ≥ 0 ⇒ (UIncreasing(COND_1757_0_MAIN_LOAD(&&(&&(>(x3[0], -1), >(x2[0], -1)), <=(1, -(x0[0], x1[0]))), x1[0], x0[0], x3[0], x2[0])), ≥)∧[(-1)Bound*bni_15] + [bni_15]x0[1] ≥ 0∧[(-1)bso_16] ≥ 0)
    • (x0[1] ≥ 0∧x3[0] ≥ 0∧x2[0] ≥ 0∧x2[1] ≥ 0∧x1[1] ≥ 0∧x3[1] ≥ 0 ⇒ (UIncreasing(COND_1757_0_MAIN_LOAD(&&(&&(>(x3[0], -1), >(x2[0], -1)), <=(1, -(x0[0], x1[0]))), x1[0], x0[0], x3[0], x2[0])), ≥)∧[(-1)Bound*bni_15] + [bni_15]x0[1] ≥ 0∧[(-1)bso_16] ≥ 0)
    • (x0[1] ≥ 0∧x3[0] ≥ 0∧x2[0] ≥ 0∧x2[1] ≥ 0∧x1[1] ≥ 0∧x3[1] ≥ 0 ⇒ (UIncreasing(COND_1757_0_MAIN_LOAD(&&(&&(>(x3[0], -1), >(x2[0], -1)), <=(1, -(x0[0], x1[0]))), x1[0], x0[0], x3[0], x2[0])), ≥)∧[(-1)Bound*bni_15] + [bni_15]x0[1] ≥ 0∧[(-1)bso_16] ≥ 0)
    • (x0[1] ≥ 0∧x3[0] ≥ 0∧x2[0] ≥ 0∧x2[1] ≥ 0∧x1[1] ≥ 0∧x3[1] ≥ 0 ⇒ (UIncreasing(COND_1757_0_MAIN_LOAD(&&(&&(>(x3[0], -1), >(x2[0], -1)), <=(1, -(x0[0], x1[0]))), x1[0], x0[0], x3[0], x2[0])), ≥)∧[(-1)Bound*bni_15] + [bni_15]x0[1] ≥ 0∧[(-1)bso_16] ≥ 0)

  • COND_1757_0_MAIN_LOAD(TRUE, x1, x0, x3, x2) → 1757_0_MAIN_LOAD(+(x1, +(x3, 1)), -(x0, x2))
    • (x0[0] ≥ 0∧x3[0] ≥ 0∧x2[0] ≥ 0∧x1[0] ≥ 0 ⇒ (UIncreasing(1757_0_MAIN_LOAD(+(x1[1], +(x3[1], 1)), -(x0[1], x2[1]))), ≥)∧[(-1)Bound*bni_17] + [bni_17]x0[0] ≥ 0∧[1 + (-1)bso_18] + x2[0] + x3[0] ≥ 0)
    • (x0[0] ≥ 0∧x3[0] ≥ 0∧x2[0] ≥ 0∧x1[0] ≥ 0 ⇒ (UIncreasing(1757_0_MAIN_LOAD(+(x1[1], +(x3[1], 1)), -(x0[1], x2[1]))), ≥)∧[(-1)Bound*bni_17] + [bni_17]x0[0] ≥ 0∧[1 + (-1)bso_18] + x2[0] + x3[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) = [1]   
POL(FALSE) = [2]   
POL(1757_0_MAIN_LOAD(x1, x2)) = [-1] + x2 + [-1]x1   
POL(COND_1757_0_MAIN_LOAD(x1, x2, x3, x4, x5)) = [-1] + x3 + [-1]x2   
POL(&&(x1, x2)) = [-1]   
POL(>(x1, x2)) = [-1]   
POL(-1) = [-1]   
POL(<=(x1, x2)) = [-1]   
POL(1) = [1]   
POL(-(x1, x2)) = x1 + [-1]x2   
POL(+(x1, x2)) = x1 + x2   

The following pairs are in P>:

COND_1757_0_MAIN_LOAD(TRUE, x1[1], x0[1], x3[1], x2[1]) → 1757_0_MAIN_LOAD(+(x1[1], +(x3[1], 1)), -(x0[1], x2[1]))

The following pairs are in Pbound:

1757_0_MAIN_LOAD(x1[0], x0[0]) → COND_1757_0_MAIN_LOAD(&&(&&(>(x3[0], -1), >(x2[0], -1)), <=(1, -(x0[0], x1[0]))), x1[0], x0[0], x3[0], x2[0])
COND_1757_0_MAIN_LOAD(TRUE, x1[1], x0[1], x3[1], x2[1]) → 1757_0_MAIN_LOAD(+(x1[1], +(x3[1], 1)), -(x0[1], x2[1]))

The following pairs are in P≥:

1757_0_MAIN_LOAD(x1[0], x0[0]) → COND_1757_0_MAIN_LOAD(&&(&&(>(x3[0], -1), >(x2[0], -1)), <=(1, -(x0[0], x1[0]))), x1[0], x0[0], x3[0], x2[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

(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): 1757_0_MAIN_LOAD(x1[0], x0[0]) → COND_1757_0_MAIN_LOAD(x3[0] > -1 && x2[0] > -1 && 1 <= x0[0] - x1[0], x1[0], x0[0], x3[0], x2[0])


The set Q is empty.

(9) IDependencyGraphProof (EQUIVALENT transformation)

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

(10) TRUE