### (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:
1739_0_main_Load(EOS(STATIC_1739), i741, i742, i741) → 1741_0_main_IntArithmetic(EOS(STATIC_1741), i741, i742, i741, i742)
1741_0_main_IntArithmetic(EOS(STATIC_1741), i741, i742, i741, i742) → 1742_0_main_ConstantStackPush(EOS(STATIC_1742), i741, i742, -(i741, i742))
1742_0_main_ConstantStackPush(EOS(STATIC_1742), i741, i742, i746) → 1744_0_main_LT(EOS(STATIC_1744), i741, i742, i746, 1)
1744_0_main_LT(EOS(STATIC_1744), i741, i742, i750, matching1) → 1746_0_main_LT(EOS(STATIC_1746), i741, i742, i750, 1) | =(matching1, 1)
1746_0_main_LT(EOS(STATIC_1746), i741, i742, i750, matching1) → 1749_0_main_Load(EOS(STATIC_1749), i741, i742) | &&(>=(i750, 1), =(matching1, 1))
1749_0_main_Load(EOS(STATIC_1749), i741, i742) → 1752_0_main_InvokeMethod(EOS(STATIC_1752), i742, i741)
1752_0_main_InvokeMethod(EOS(STATIC_1752), i742, i741) → 1753_0_random_FieldAccess(EOS(STATIC_1753), i742, i741)
1753_0_random_FieldAccess(EOS(STATIC_1753), i742, i741) → 1755_0_random_FieldAccess(EOS(STATIC_1755), i742, i741)
1755_0_random_FieldAccess(EOS(STATIC_1755), i742, i741) → 1759_0_random_ArrayAccess(EOS(STATIC_1759), i742, i741)
1759_0_random_ArrayAccess(EOS(STATIC_1759), i742, i741) → 1761_0_random_ArrayAccess(EOS(STATIC_1761), i742, i741)
1761_0_random_ArrayAccess(EOS(STATIC_1761), i742, i741) → 1763_0_random_Store(EOS(STATIC_1763), i742, i741, o792)
1763_0_random_Store(EOS(STATIC_1763), i742, i741, o792) → 1766_0_random_FieldAccess(EOS(STATIC_1766), i742, i741, o792)
1766_0_random_FieldAccess(EOS(STATIC_1766), i742, i741, o792) → 1767_0_random_ConstantStackPush(EOS(STATIC_1767), i742, i741, o792)
1767_0_random_ConstantStackPush(EOS(STATIC_1767), i742, i741, o792) → 1771_0_random_IntArithmetic(EOS(STATIC_1771), i742, i741, o792)
1771_0_random_IntArithmetic(EOS(STATIC_1771), i742, i741, o792) → 1774_0_random_FieldAccess(EOS(STATIC_1774), i742, i741, o792)
1774_0_random_FieldAccess(EOS(STATIC_1774), i742, i741, o792) → 1777_0_random_Load(EOS(STATIC_1777), i742, i741, o792)
1777_0_random_Load(EOS(STATIC_1777), i742, i741, o792) → 1784_0_random_InvokeMethod(EOS(STATIC_1784), i742, i741, o792)
1784_0_random_InvokeMethod(EOS(STATIC_1784), i742, i741, java.lang.Object(o810sub)) → 1786_0_random_InvokeMethod(EOS(STATIC_1786), i742, i741, java.lang.Object(o810sub))
1786_0_random_InvokeMethod(EOS(STATIC_1786), i742, i741, java.lang.Object(o810sub)) → 1789_0_length_Load(EOS(STATIC_1789), i742, i741, java.lang.Object(o810sub), java.lang.Object(o810sub))
1789_0_length_Load(EOS(STATIC_1789), i742, i741, java.lang.Object(o810sub), java.lang.Object(o810sub)) → 1799_0_length_FieldAccess(EOS(STATIC_1799), i742, i741, java.lang.Object(o810sub), java.lang.Object(o810sub))
1799_0_length_FieldAccess(EOS(STATIC_1799), i742, i741, java.lang.Object(java.lang.String(o818sub, i786)), java.lang.Object(java.lang.String(o818sub, i786))) → 1801_0_length_FieldAccess(EOS(STATIC_1801), i742, i741, java.lang.Object(java.lang.String(o818sub, i786)), java.lang.Object(java.lang.String(o818sub, i786))) | &&(>=(i786, 0), >=(i787, 0))
1801_0_length_FieldAccess(EOS(STATIC_1801), i742, i741, java.lang.Object(java.lang.String(o818sub, i786)), java.lang.Object(java.lang.String(o818sub, i786))) → 1806_0_length_Return(EOS(STATIC_1806), i742, i741, java.lang.Object(java.lang.String(o818sub, i786)), i786)
1806_0_length_Return(EOS(STATIC_1806), i742, i741, java.lang.Object(java.lang.String(o818sub, i786)), i786) → 1810_0_random_Return(EOS(STATIC_1810), i742, i741, i786)
1810_0_random_Return(EOS(STATIC_1810), i742, i741, i786) → 1812_0_main_IntArithmetic(EOS(STATIC_1812), i742, i741, i786)
1812_0_main_IntArithmetic(EOS(STATIC_1812), i742, i741, i786) → 1818_0_main_Store(EOS(STATIC_1818), i742, -(i741, i786)) | >=(i786, 0)
1818_0_main_Store(EOS(STATIC_1818), i742, i797) → 1823_0_main_InvokeMethod(EOS(STATIC_1823), i797, i742)
1823_0_main_InvokeMethod(EOS(STATIC_1823), i797, i742) → 1827_0_random_FieldAccess(EOS(STATIC_1827), i797, i742)
1827_0_random_FieldAccess(EOS(STATIC_1827), i797, i742) → 1840_0_random_FieldAccess(EOS(STATIC_1840), i797, i742)
1840_0_random_FieldAccess(EOS(STATIC_1840), i797, i742) → 1847_0_random_ArrayAccess(EOS(STATIC_1847), i797, i742)
1847_0_random_ArrayAccess(EOS(STATIC_1847), i797, i742) → 1855_0_random_ArrayAccess(EOS(STATIC_1855), i797, i742)
1855_0_random_ArrayAccess(EOS(STATIC_1855), i797, i742) → 1863_0_random_Store(EOS(STATIC_1863), i797, i742, o848)
1863_0_random_Store(EOS(STATIC_1863), i797, i742, o848) → 1872_0_random_FieldAccess(EOS(STATIC_1872), i797, i742, o848)
1872_0_random_FieldAccess(EOS(STATIC_1872), i797, i742, o848) → 1879_0_random_ConstantStackPush(EOS(STATIC_1879), i797, i742, o848)
1879_0_random_ConstantStackPush(EOS(STATIC_1879), i797, i742, o848) → 1889_0_random_IntArithmetic(EOS(STATIC_1889), i797, i742, o848)
1889_0_random_IntArithmetic(EOS(STATIC_1889), i797, i742, o848) → 1897_0_random_FieldAccess(EOS(STATIC_1897), i797, i742, o848)
1897_0_random_FieldAccess(EOS(STATIC_1897), i797, i742, o848) → 1907_0_random_Load(EOS(STATIC_1907), i797, i742, o848)
1907_0_random_Load(EOS(STATIC_1907), i797, i742, o848) → 1919_0_random_InvokeMethod(EOS(STATIC_1919), i797, i742, o848)
1919_0_random_InvokeMethod(EOS(STATIC_1919), i797, i742, java.lang.Object(o909sub)) → 1926_0_random_InvokeMethod(EOS(STATIC_1926), i797, i742, java.lang.Object(o909sub))
1926_0_random_InvokeMethod(EOS(STATIC_1926), i797, i742, java.lang.Object(o909sub)) → 1931_0_length_Load(EOS(STATIC_1931), i797, i742, java.lang.Object(o909sub), java.lang.Object(o909sub))
1931_0_length_Load(EOS(STATIC_1931), i797, i742, java.lang.Object(o909sub), java.lang.Object(o909sub)) → 1947_0_length_FieldAccess(EOS(STATIC_1947), i797, i742, java.lang.Object(o909sub), java.lang.Object(o909sub))
1947_0_length_FieldAccess(EOS(STATIC_1947), i797, i742, java.lang.Object(java.lang.String(o936sub, i885)), java.lang.Object(java.lang.String(o936sub, i885))) → 1951_0_length_FieldAccess(EOS(STATIC_1951), i797, i742, java.lang.Object(java.lang.String(o936sub, i885)), java.lang.Object(java.lang.String(o936sub, i885))) | &&(>=(i885, 0), >=(i886, 0))
1951_0_length_FieldAccess(EOS(STATIC_1951), i797, i742, java.lang.Object(java.lang.String(o936sub, i885)), java.lang.Object(java.lang.String(o936sub, i885))) → 1959_0_length_Return(EOS(STATIC_1959), i797, i742, java.lang.Object(java.lang.String(o936sub, i885)), i885)
1959_0_length_Return(EOS(STATIC_1959), i797, i742, java.lang.Object(java.lang.String(o936sub, i885)), i885) → 1963_0_random_Return(EOS(STATIC_1963), i797, i742, i885)
1963_0_random_Return(EOS(STATIC_1963), i797, i742, i885) → 1964_0_main_ConstantStackPush(EOS(STATIC_1964), i797, i742, i885)
1964_0_main_ConstantStackPush(EOS(STATIC_1964), i797, i742, i885) → 1972_0_main_IntArithmetic(EOS(STATIC_1972), i797, i742, i885, 1)
1972_0_main_IntArithmetic(EOS(STATIC_1972), i797, i742, i885, matching1) → 1977_0_main_Store(EOS(STATIC_1977), i797, i742, +(i885, 1)) | &&(>=(i885, 0), =(matching1, 1))
1977_0_main_Store(EOS(STATIC_1977), i797, i742, i894) → 1981_0_main_Load(EOS(STATIC_1981), i797, i742, i894)
1989_0_main_Load(EOS(STATIC_1989), i797, i894, i742) → 1996_0_main_IntArithmetic(EOS(STATIC_1996), i797, i742, i894)
1996_0_main_IntArithmetic(EOS(STATIC_1996), i797, i742, i894) → 2000_0_main_Store(EOS(STATIC_2000), i797, +(i742, i894)) | >(i894, 0)
2000_0_main_Store(EOS(STATIC_2000), i797, i900) → 2008_0_main_JMP(EOS(STATIC_2008), i797, i900)
2008_0_main_JMP(EOS(STATIC_2008), i797, i900) → 2015_0_main_Load(EOS(STATIC_2015), i797, i900)
R rules:

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

P rules:
1739_0_main_Load(EOS(STATIC_1739), x0, x1, x0) → 1739_0_main_Load(EOS(STATIC_1739), -(x0, x2), +(x1, +(x3, 1)), -(x0, x2)) | &&(&&(>(+(x3, 1), 0), >(+(x2, 1), 0)), <=(1, -(x0, x1)))
R rules:

Filtered ground terms:

EOS(x1) → EOS
Cond_1739_0_main_Load(x1, x2, x3, x4, x5, x6, x7) → Cond_1739_0_main_Load(x1, x3, x4, x5, x6, x7)

Filtered duplicate args:

Cond_1739_0_main_Load(x1, x2, x3, x4, x5, x6) → Cond_1739_0_main_Load(x1, x3, x4, x5, x6)

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

P rules:
1739_0_main_Load(x1, x0) → 1739_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:
1739_0_MAIN_LOAD(x1, x0) → COND_1739_0_MAIN_LOAD(&&(&&(>(x3, -1), >(x2, -1)), <=(1, -(x0, x1))), x1, x0, x3, 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): 1739_0_MAIN_LOAD(x1[0], x0[0]) → COND_1739_0_MAIN_LOAD(x3[0] > -1 && x2[0] > -1 && 1 <= x0[0] - x1[0], x1[0], x0[0], x3[0], x2[0])
(1): COND_1739_0_MAIN_LOAD(TRUE, x1[1], x0[1], x3[1], x2[1]) → 1739_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@1005ec04 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 1739_0_MAIN_LOAD(x1, x0) → COND_1739_0_MAIN_LOAD(&&(&&(>(x3, -1), >(x2, -1)), <=(1, -(x0, x1))), x1, x0, x3, x2) the following chains were created:
• We consider the chain COND_1739_0_MAIN_LOAD(TRUE, x1[1], x0[1], x3[1], x2[1]) → 1739_0_MAIN_LOAD(+(x1[1], +(x3[1], 1)), -(x0[1], x2[1])), 1739_0_MAIN_LOAD(x1[0], x0[0]) → COND_1739_0_MAIN_LOAD(&&(&&(>(x3[0], -1), >(x2[0], -1)), <=(1, -(x0[0], x1[0]))), x1[0], x0[0], x3[0], x2[0]), COND_1739_0_MAIN_LOAD(TRUE, x1[1], x0[1], x3[1], x2[1]) → 1739_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])))=TRUEx1[0]=x1[1]1x0[0]=x0[1]1x3[0]=x3[1]1x2[0]=x2[1]11739_0_MAIN_LOAD(x1[0], x0[0])≥NonInfC∧1739_0_MAIN_LOAD(x1[0], x0[0])≥COND_1739_0_MAIN_LOAD(&&(&&(>(x3[0], -1), >(x2[0], -1)), <=(1, -(x0[0], x1[0]))), x1[0], x0[0], x3[0], x2[0])∧(UIncreasing(COND_1739_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)=TRUE1739_0_MAIN_LOAD(+(x1[1], +(x3[1], 1)), -(x0[1], x2[1]))≥NonInfC∧1739_0_MAIN_LOAD(+(x1[1], +(x3[1], 1)), -(x0[1], x2[1]))≥COND_1739_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_1739_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_1739_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] + x2[0] + x3[0] ≥ 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_1739_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] + x2[0] + x3[0] ≥ 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_1739_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] + x2[0] + x3[0] ≥ 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_1739_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] + x2[0] + x3[0] ≥ 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_1739_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] + x2[0] + x3[0] ≥ 0)

(8)    (x0[1] ≥ 0∧x3[0] ≥ 0∧x2[0] ≥ 0∧x2[1] ≥ 0 ⇒ (UIncreasing(COND_1739_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] + x2[0] + x3[0] ≥ 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_1739_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] + x2[0] + x3[0] ≥ 0)

(10)    (x0[1] ≥ 0∧x3[0] ≥ 0∧x2[0] ≥ 0∧x2[1] ≥ 0∧x1[1] ≥ 0 ⇒ (UIncreasing(COND_1739_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] + x2[0] + x3[0] ≥ 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_1739_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] + x2[0] + x3[0] ≥ 0)

(12)    (x0[1] ≥ 0∧x3[0] ≥ 0∧x2[0] ≥ 0∧x2[1] ≥ 0∧x1[1] ≥ 0 ⇒ (UIncreasing(COND_1739_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] + x2[0] + x3[0] ≥ 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_1739_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] + x2[0] + x3[0] ≥ 0)

(14)    (x0[1] ≥ 0∧x3[0] ≥ 0∧x2[0] ≥ 0∧x2[1] ≥ 0∧x1[1] ≥ 0∧x3[1] ≥ 0 ⇒ (UIncreasing(COND_1739_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] + x2[0] + x3[0] ≥ 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_1739_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] + x2[0] + x3[0] ≥ 0)

(16)    (x0[1] ≥ 0∧x3[0] ≥ 0∧x2[0] ≥ 0∧x2[1] ≥ 0∧x1[1] ≥ 0∧x3[1] ≥ 0 ⇒ (UIncreasing(COND_1739_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] + x2[0] + x3[0] ≥ 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_1739_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] + x2[0] + x3[0] ≥ 0)

(18)    (x0[1] ≥ 0∧x3[0] ≥ 0∧x2[0] ≥ 0∧x2[1] ≥ 0∧x1[1] ≥ 0∧x3[1] ≥ 0 ⇒ (UIncreasing(COND_1739_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] + x2[0] + x3[0] ≥ 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_1739_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] + x2[0] + x3[0] ≥ 0)

(20)    (x0[1] ≥ 0∧x3[0] ≥ 0∧x2[0] ≥ 0∧x2[1] ≥ 0∧x1[1] ≥ 0∧x3[1] ≥ 0 ⇒ (UIncreasing(COND_1739_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] + x2[0] + x3[0] ≥ 0)

For Pair COND_1739_0_MAIN_LOAD(TRUE, x1, x0, x3, x2) → 1739_0_MAIN_LOAD(+(x1, +(x3, 1)), -(x0, x2)) the following chains were created:
• We consider the chain 1739_0_MAIN_LOAD(x1[0], x0[0]) → COND_1739_0_MAIN_LOAD(&&(&&(>(x3[0], -1), >(x2[0], -1)), <=(1, -(x0[0], x1[0]))), x1[0], x0[0], x3[0], x2[0]), COND_1739_0_MAIN_LOAD(TRUE, x1[1], x0[1], x3[1], x2[1]) → 1739_0_MAIN_LOAD(+(x1[1], +(x3[1], 1)), -(x0[1], x2[1])), 1739_0_MAIN_LOAD(x1[0], x0[0]) → COND_1739_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])))=TRUEx1[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]1COND_1739_0_MAIN_LOAD(TRUE, x1[1], x0[1], x3[1], x2[1])≥NonInfC∧COND_1739_0_MAIN_LOAD(TRUE, x1[1], x0[1], x3[1], x2[1])≥1739_0_MAIN_LOAD(+(x1[1], +(x3[1], 1)), -(x0[1], x2[1]))∧(UIncreasing(1739_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)=TRUECOND_1739_0_MAIN_LOAD(TRUE, x1[0], x0[0], x3[0], x2[0])≥NonInfC∧COND_1739_0_MAIN_LOAD(TRUE, x1[0], x0[0], x3[0], x2[0])≥1739_0_MAIN_LOAD(+(x1[0], +(x3[0], 1)), -(x0[0], x2[0]))∧(UIncreasing(1739_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(1739_0_MAIN_LOAD(+(x1[1], +(x3[1], 1)), -(x0[1], x2[1]))), ≥)∧[(-1)bni_17 + (-1)Bound*bni_17] + [(-1)bni_17]x2[0] + [(-1)bni_17]x3[0] + [bni_17]x0[0] + [(-1)bni_17]x1[0] ≥ 0∧[1 + (-1)bso_18] ≥ 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(1739_0_MAIN_LOAD(+(x1[1], +(x3[1], 1)), -(x0[1], x2[1]))), ≥)∧[(-1)bni_17 + (-1)Bound*bni_17] + [(-1)bni_17]x2[0] + [(-1)bni_17]x3[0] + [bni_17]x0[0] + [(-1)bni_17]x1[0] ≥ 0∧[1 + (-1)bso_18] ≥ 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(1739_0_MAIN_LOAD(+(x1[1], +(x3[1], 1)), -(x0[1], x2[1]))), ≥)∧[(-1)bni_17 + (-1)Bound*bni_17] + [(-1)bni_17]x2[0] + [(-1)bni_17]x3[0] + [bni_17]x0[0] + [(-1)bni_17]x1[0] ≥ 0∧[1 + (-1)bso_18] ≥ 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(1739_0_MAIN_LOAD(+(x1[1], +(x3[1], 1)), -(x0[1], x2[1]))), ≥)∧[(-1)Bound*bni_17] + [(-1)bni_17]x2[0] + [(-1)bni_17]x3[0] + [bni_17]x0[0] ≥ 0∧[1 + (-1)bso_18] ≥ 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(1739_0_MAIN_LOAD(+(x1[1], +(x3[1], 1)), -(x0[1], x2[1]))), ≥)∧[(-1)Bound*bni_17] + [(-1)bni_17]x2[0] + [(-1)bni_17]x3[0] + [bni_17]x0[0] ≥ 0∧[1 + (-1)bso_18] ≥ 0)

(28)    (x0[0] ≥ 0∧x3[0] ≥ 0∧x2[0] ≥ 0∧x1[0] ≥ 0 ⇒ (UIncreasing(1739_0_MAIN_LOAD(+(x1[1], +(x3[1], 1)), -(x0[1], x2[1]))), ≥)∧[(-1)Bound*bni_17] + [(-1)bni_17]x2[0] + [(-1)bni_17]x3[0] + [bni_17]x0[0] ≥ 0∧[1 + (-1)bso_18] ≥ 0)

To summarize, we get the following constraints P for the following pairs.
• 1739_0_MAIN_LOAD(x1, x0) → COND_1739_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_1739_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] + x2[0] + x3[0] ≥ 0)
• (x0[1] ≥ 0∧x3[0] ≥ 0∧x2[0] ≥ 0∧x2[1] ≥ 0∧x1[1] ≥ 0∧x3[1] ≥ 0 ⇒ (UIncreasing(COND_1739_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] + x2[0] + x3[0] ≥ 0)
• (x0[1] ≥ 0∧x3[0] ≥ 0∧x2[0] ≥ 0∧x2[1] ≥ 0∧x1[1] ≥ 0∧x3[1] ≥ 0 ⇒ (UIncreasing(COND_1739_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] + x2[0] + x3[0] ≥ 0)
• (x0[1] ≥ 0∧x3[0] ≥ 0∧x2[0] ≥ 0∧x2[1] ≥ 0∧x1[1] ≥ 0∧x3[1] ≥ 0 ⇒ (UIncreasing(COND_1739_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] + x2[0] + x3[0] ≥ 0)
• (x0[1] ≥ 0∧x3[0] ≥ 0∧x2[0] ≥ 0∧x2[1] ≥ 0∧x1[1] ≥ 0∧x3[1] ≥ 0 ⇒ (UIncreasing(COND_1739_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] + x2[0] + x3[0] ≥ 0)
• (x0[1] ≥ 0∧x3[0] ≥ 0∧x2[0] ≥ 0∧x2[1] ≥ 0∧x1[1] ≥ 0∧x3[1] ≥ 0 ⇒ (UIncreasing(COND_1739_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] + x2[0] + x3[0] ≥ 0)
• (x0[1] ≥ 0∧x3[0] ≥ 0∧x2[0] ≥ 0∧x2[1] ≥ 0∧x1[1] ≥ 0∧x3[1] ≥ 0 ⇒ (UIncreasing(COND_1739_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] + x2[0] + x3[0] ≥ 0)
• (x0[1] ≥ 0∧x3[0] ≥ 0∧x2[0] ≥ 0∧x2[1] ≥ 0∧x1[1] ≥ 0∧x3[1] ≥ 0 ⇒ (UIncreasing(COND_1739_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] + x2[0] + x3[0] ≥ 0)

• COND_1739_0_MAIN_LOAD(TRUE, x1, x0, x3, x2) → 1739_0_MAIN_LOAD(+(x1, +(x3, 1)), -(x0, x2))
• (x0[0] ≥ 0∧x3[0] ≥ 0∧x2[0] ≥ 0∧x1[0] ≥ 0 ⇒ (UIncreasing(1739_0_MAIN_LOAD(+(x1[1], +(x3[1], 1)), -(x0[1], x2[1]))), ≥)∧[(-1)Bound*bni_17] + [(-1)bni_17]x2[0] + [(-1)bni_17]x3[0] + [bni_17]x0[0] ≥ 0∧[1 + (-1)bso_18] ≥ 0)
• (x0[0] ≥ 0∧x3[0] ≥ 0∧x2[0] ≥ 0∧x1[0] ≥ 0 ⇒ (UIncreasing(1739_0_MAIN_LOAD(+(x1[1], +(x3[1], 1)), -(x0[1], x2[1]))), ≥)∧[(-1)Bound*bni_17] + [(-1)bni_17]x2[0] + [(-1)bni_17]x3[0] + [bni_17]x0[0] ≥ 0∧[1 + (-1)bso_18] ≥ 0)

The constraints for P> respective Pbound are constructed from P where we just replace every occurence of "t ≥ s" in P by "t > s" respective "t ≥ c". Here c stands for the fresh constant used for Pbound.
Using the following integer polynomial ordering the resulting constraints can be solved
Polynomial interpretation over integers[POLO]:

POL(TRUE) = 0
POL(FALSE) = 0
POL(1739_0_MAIN_LOAD(x1, x2)) = [-1] + x2 + [-1]x1
POL(COND_1739_0_MAIN_LOAD(x1, x2, x3, x4, x5)) = [-1] + [-1]x5 + [-1]x4 + x3 + [-1]x2 + [-1]x1
POL(&&(x1, x2)) = 0
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>:

The following pairs are in Pbound:

1739_0_MAIN_LOAD(x1[0], x0[0]) → COND_1739_0_MAIN_LOAD(&&(&&(>(x3[0], -1), >(x2[0], -1)), <=(1, -(x0[0], x1[0]))), x1[0], x0[0], x3[0], x2[0])

The following pairs are in P:

1739_0_MAIN_LOAD(x1[0], x0[0]) → COND_1739_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:

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

### (9) Obligation:

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

The following domains are used:

Boolean, Integer

R is empty.

The integer pair graph contains the following rules and edges:
(0): 1739_0_MAIN_LOAD(x1[0], x0[0]) → COND_1739_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.

### (10) IDependencyGraphProof (EQUIVALENT transformation)

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

### (12) Obligation:

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

The following domains are used:

Integer

R is empty.

The integer pair graph contains the following rules and edges:
(1): COND_1739_0_MAIN_LOAD(TRUE, x1[1], x0[1], x3[1], x2[1]) → 1739_0_MAIN_LOAD(x1[1] + x3[1] + 1, x0[1] - x2[1])

The set Q is empty.

### (13) IDependencyGraphProof (EQUIVALENT transformation)

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