(0) Obligation:

JBC Problem based on JBC Program:
Manifest-Version: 1.0 Created-By: 1.6.0_16 (Sun Microsystems Inc.) Main-Class: ListContentTail
`public class ListContentTail{  public static void main(String[] args) {    Random.args = args;    IntList l = IntList.createIntList();    int m = IntList.nth(Random.random(),l);    while (m > 0) {      l = l.next;      m = IntList.nth(Random.random(),l);    }  }}class IntList {  int value;  IntList next;  public IntList(int value, IntList next) {    this.value = value;    this.next = next;  }  public static IntList createIntList() {    int i = Random.random();    IntList l = null;    while (i > 0) {      l = new IntList(Random.random(), l);      i--;    }    return l;  }  public static int nth(int n, IntList l){    while (n > 1 && l != null) {      n--;      l = l.next;    }    if (l == null) return 0;    else           return l.value;  }}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:
ListContentTail.main([Ljava/lang/String;)V: Graph of 311 nodes with 2 SCCs.

IntList.createIntList()LIntList;: Graph of 163 nodes with 1 SCC.

(3) TerminationGraphToSCCProof (SOUND transformation)

Splitted TerminationGraph to 3 SCCss.

(5) Obligation:

SCC of termination graph based on JBC Program.
SCC contains nodes from the following methods: IntList.createIntList()LIntList;
SCC calls the following helper methods:
Performed SCC analyses: UsedFieldsAnalysis

(6) SCCToIDPv1Proof (SOUND transformation)

Transformed FIGraph SCCs to IDPs. Log:

Generated 38 rules for P and 0 rules for R.

P rules:
1827_0_createIntList_LE(EOS(STATIC_1827), i479, i479) → 1839_0_createIntList_LE(EOS(STATIC_1839), i479, i479)
1839_0_createIntList_LE(EOS(STATIC_1839), i479, i479) → 1852_0_createIntList_New(EOS(STATIC_1852), i479) | >(i479, 0)
1852_0_createIntList_New(EOS(STATIC_1852), i479) → 1862_0_createIntList_Duplicate(EOS(STATIC_1862), i479)
1862_0_createIntList_Duplicate(EOS(STATIC_1862), i479) → 1873_0_createIntList_InvokeMethod(EOS(STATIC_1873), i479)
1873_0_createIntList_InvokeMethod(EOS(STATIC_1873), i479) → 1879_0_random_FieldAccess(EOS(STATIC_1879), i479)
1879_0_random_FieldAccess(EOS(STATIC_1879), i479) → 1890_0_random_FieldAccess(EOS(STATIC_1890), i479)
1890_0_random_FieldAccess(EOS(STATIC_1890), i479) → 1895_0_random_ArrayAccess(EOS(STATIC_1895), i479)
1895_0_random_ArrayAccess(EOS(STATIC_1895), i479) → 1900_0_random_ArrayAccess(EOS(STATIC_1900), i479)
1900_0_random_ArrayAccess(EOS(STATIC_1900), i479) → 1909_0_random_Store(EOS(STATIC_1909), i479, o765)
1909_0_random_Store(EOS(STATIC_1909), i479, o765) → 1917_0_random_FieldAccess(EOS(STATIC_1917), i479, o765)
1917_0_random_FieldAccess(EOS(STATIC_1917), i479, o765) → 1927_0_random_ConstantStackPush(EOS(STATIC_1927), i479, o765)
1927_0_random_ConstantStackPush(EOS(STATIC_1927), i479, o765) → 1939_0_random_IntArithmetic(EOS(STATIC_1939), i479, o765)
1939_0_random_IntArithmetic(EOS(STATIC_1939), i479, o765) → 1950_0_random_FieldAccess(EOS(STATIC_1950), i479, o765)
1950_0_random_FieldAccess(EOS(STATIC_1950), i479, o765) → 1959_0_random_Load(EOS(STATIC_1959), i479, o765)
1959_0_random_Load(EOS(STATIC_1959), i479, o765) → 1972_0_random_InvokeMethod(EOS(STATIC_1972), i479, o765)
1972_0_random_InvokeMethod(EOS(STATIC_1972), i479, java.lang.Object(o811sub)) → 1982_0_random_InvokeMethod(EOS(STATIC_1982), i479, java.lang.Object(o811sub))
1982_0_random_InvokeMethod(EOS(STATIC_1982), i479, java.lang.Object(o811sub)) → 1994_0_length_Load(EOS(STATIC_1994), i479, java.lang.Object(o811sub), java.lang.Object(o811sub))
1994_0_length_Load(EOS(STATIC_1994), i479, java.lang.Object(o811sub), java.lang.Object(o811sub)) → 2009_0_length_FieldAccess(EOS(STATIC_2009), i479, java.lang.Object(o811sub), java.lang.Object(o811sub))
2009_0_length_FieldAccess(EOS(STATIC_2009), i479, java.lang.Object(java.lang.String(o829sub, i546)), java.lang.Object(java.lang.String(o829sub, i546))) → 2014_0_length_FieldAccess(EOS(STATIC_2014), i479, java.lang.Object(java.lang.String(o829sub, i546)), java.lang.Object(java.lang.String(o829sub, i546))) | &&(>=(i546, 0), >=(i547, 0))
2014_0_length_FieldAccess(EOS(STATIC_2014), i479, java.lang.Object(java.lang.String(o829sub, i546)), java.lang.Object(java.lang.String(o829sub, i546))) → 2023_0_length_Return(EOS(STATIC_2023), i479, java.lang.Object(java.lang.String(o829sub, i546)))
2023_0_length_Return(EOS(STATIC_2023), i479, java.lang.Object(java.lang.String(o829sub, i546))) → 2030_0_random_Return(EOS(STATIC_2030), i479)
2119_0_<init>_FieldAccess(EOS(STATIC_2119), i479) → 2131_0_<init>_Return(EOS(STATIC_2131), i479)
2131_0_<init>_Return(EOS(STATIC_2131), i479) → 2143_0_createIntList_Store(EOS(STATIC_2143), i479)
2143_0_createIntList_Store(EOS(STATIC_2143), i479) → 2159_0_createIntList_Inc(EOS(STATIC_2159), i479)
2159_0_createIntList_Inc(EOS(STATIC_2159), i479) → 2169_0_createIntList_JMP(EOS(STATIC_2169), +(i479, -1)) | >(i479, 0)
1813_0_createIntList_Load(EOS(STATIC_1813), i472) → 1827_0_createIntList_LE(EOS(STATIC_1827), i472, i472)
R rules:

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

P rules:
1827_0_createIntList_LE(EOS(STATIC_1827), x0, x0) → 1827_0_createIntList_LE(EOS(STATIC_1827), +(x0, -1), +(x0, -1)) | >(x0, 0)
R rules:

Filtered ground terms:

1827_0_createIntList_LE(x1, x2, x3) → 1827_0_createIntList_LE(x2, x3)
EOS(x1) → EOS
Cond_1827_0_createIntList_LE(x1, x2, x3, x4) → Cond_1827_0_createIntList_LE(x1, x3, x4)

Filtered duplicate args:

1827_0_createIntList_LE(x1, x2) → 1827_0_createIntList_LE(x2)
Cond_1827_0_createIntList_LE(x1, x2, x3) → Cond_1827_0_createIntList_LE(x1, x3)

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

P rules:
1827_0_createIntList_LE(x0) → 1827_0_createIntList_LE(+(x0, -1)) | >(x0, 0)
R rules:

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

P rules:
1827_0_CREATEINTLIST_LE(x0) → COND_1827_0_CREATEINTLIST_LE(>(x0, 0), x0)
COND_1827_0_CREATEINTLIST_LE(TRUE, x0) → 1827_0_CREATEINTLIST_LE(+(x0, -1))
R rules:

(7) 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:
(0): 1827_0_CREATEINTLIST_LE(x0[0]) → COND_1827_0_CREATEINTLIST_LE(x0[0] > 0, x0[0])
(1): COND_1827_0_CREATEINTLIST_LE(TRUE, x0[1]) → 1827_0_CREATEINTLIST_LE(x0[1] + -1)

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

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

The set Q is empty.

(8) 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.IdpCand1ShapeHeuristic@21754bc Constraint Generator: NonInfConstraintGenerator: PathGenerator: MetricPathGenerator: Max Left Steps: 0 Max Right Steps: 0

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 1827_0_CREATEINTLIST_LE(x0) → COND_1827_0_CREATEINTLIST_LE(>(x0, 0), x0) the following chains were created:
• We consider the chain 1827_0_CREATEINTLIST_LE(x0[0]) → COND_1827_0_CREATEINTLIST_LE(>(x0[0], 0), x0[0]), COND_1827_0_CREATEINTLIST_LE(TRUE, x0[1]) → 1827_0_CREATEINTLIST_LE(+(x0[1], -1)) which results in the following constraint:

(1)    (>(x0[0], 0)=TRUEx0[0]=x0[1]1827_0_CREATEINTLIST_LE(x0[0])≥NonInfC∧1827_0_CREATEINTLIST_LE(x0[0])≥COND_1827_0_CREATEINTLIST_LE(>(x0[0], 0), x0[0])∧(UIncreasing(COND_1827_0_CREATEINTLIST_LE(>(x0[0], 0), x0[0])), ≥))

We simplified constraint (1) using rule (IV) which results in the following new constraint:

(2)    (>(x0[0], 0)=TRUE1827_0_CREATEINTLIST_LE(x0[0])≥NonInfC∧1827_0_CREATEINTLIST_LE(x0[0])≥COND_1827_0_CREATEINTLIST_LE(>(x0[0], 0), x0[0])∧(UIncreasing(COND_1827_0_CREATEINTLIST_LE(>(x0[0], 0), x0[0])), ≥))

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

(3)    (x0[0] + [-1] ≥ 0 ⇒ (UIncreasing(COND_1827_0_CREATEINTLIST_LE(>(x0[0], 0), x0[0])), ≥)∧[(-1)Bound*bni_8] + [(2)bni_8]x0[0] ≥ 0∧[(-1)bso_9] ≥ 0)

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

(4)    (x0[0] + [-1] ≥ 0 ⇒ (UIncreasing(COND_1827_0_CREATEINTLIST_LE(>(x0[0], 0), x0[0])), ≥)∧[(-1)Bound*bni_8] + [(2)bni_8]x0[0] ≥ 0∧[(-1)bso_9] ≥ 0)

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

(5)    (x0[0] + [-1] ≥ 0 ⇒ (UIncreasing(COND_1827_0_CREATEINTLIST_LE(>(x0[0], 0), x0[0])), ≥)∧[(-1)Bound*bni_8] + [(2)bni_8]x0[0] ≥ 0∧[(-1)bso_9] ≥ 0)

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

(6)    (x0[0] ≥ 0 ⇒ (UIncreasing(COND_1827_0_CREATEINTLIST_LE(>(x0[0], 0), x0[0])), ≥)∧[(-1)Bound*bni_8 + (2)bni_8] + [(2)bni_8]x0[0] ≥ 0∧[(-1)bso_9] ≥ 0)

For Pair COND_1827_0_CREATEINTLIST_LE(TRUE, x0) → 1827_0_CREATEINTLIST_LE(+(x0, -1)) the following chains were created:
• We consider the chain COND_1827_0_CREATEINTLIST_LE(TRUE, x0[1]) → 1827_0_CREATEINTLIST_LE(+(x0[1], -1)) which results in the following constraint:

(7)    (COND_1827_0_CREATEINTLIST_LE(TRUE, x0[1])≥NonInfC∧COND_1827_0_CREATEINTLIST_LE(TRUE, x0[1])≥1827_0_CREATEINTLIST_LE(+(x0[1], -1))∧(UIncreasing(1827_0_CREATEINTLIST_LE(+(x0[1], -1))), ≥))

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

(8)    ((UIncreasing(1827_0_CREATEINTLIST_LE(+(x0[1], -1))), ≥)∧[bni_10] = 0∧[2 + (-1)bso_11] ≥ 0)

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

(9)    ((UIncreasing(1827_0_CREATEINTLIST_LE(+(x0[1], -1))), ≥)∧[bni_10] = 0∧[2 + (-1)bso_11] ≥ 0)

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

(10)    ((UIncreasing(1827_0_CREATEINTLIST_LE(+(x0[1], -1))), ≥)∧[bni_10] = 0∧[2 + (-1)bso_11] ≥ 0)

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

(11)    ((UIncreasing(1827_0_CREATEINTLIST_LE(+(x0[1], -1))), ≥)∧[bni_10] = 0∧0 = 0∧[2 + (-1)bso_11] ≥ 0)

To summarize, we get the following constraints P for the following pairs.
• 1827_0_CREATEINTLIST_LE(x0) → COND_1827_0_CREATEINTLIST_LE(>(x0, 0), x0)
• (x0[0] ≥ 0 ⇒ (UIncreasing(COND_1827_0_CREATEINTLIST_LE(>(x0[0], 0), x0[0])), ≥)∧[(-1)Bound*bni_8 + (2)bni_8] + [(2)bni_8]x0[0] ≥ 0∧[(-1)bso_9] ≥ 0)

• COND_1827_0_CREATEINTLIST_LE(TRUE, x0) → 1827_0_CREATEINTLIST_LE(+(x0, -1))
• ((UIncreasing(1827_0_CREATEINTLIST_LE(+(x0[1], -1))), ≥)∧[bni_10] = 0∧0 = 0∧[2 + (-1)bso_11] ≥ 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(1827_0_CREATEINTLIST_LE(x1)) = [2]x1
POL(COND_1827_0_CREATEINTLIST_LE(x1, x2)) = [2]x2
POL(>(x1, x2)) = [-1]
POL(0) = 0
POL(+(x1, x2)) = x1 + x2
POL(-1) = [-1]

The following pairs are in P>:

COND_1827_0_CREATEINTLIST_LE(TRUE, x0[1]) → 1827_0_CREATEINTLIST_LE(+(x0[1], -1))

The following pairs are in Pbound:

1827_0_CREATEINTLIST_LE(x0[0]) → COND_1827_0_CREATEINTLIST_LE(>(x0[0], 0), x0[0])

The following pairs are in P:

1827_0_CREATEINTLIST_LE(x0[0]) → COND_1827_0_CREATEINTLIST_LE(>(x0[0], 0), x0[0])

There are no usable rules.

(10) Obligation:

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

The following domains are used:

Integer

R is empty.

The integer pair graph contains the following rules and edges:
(0): 1827_0_CREATEINTLIST_LE(x0[0]) → COND_1827_0_CREATEINTLIST_LE(x0[0] > 0, x0[0])

The set Q is empty.

(11) IDependencyGraphProof (EQUIVALENT transformation)

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

(13) 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_1827_0_CREATEINTLIST_LE(TRUE, x0[1]) → 1827_0_CREATEINTLIST_LE(x0[1] + -1)

The set Q is empty.

(14) IDependencyGraphProof (EQUIVALENT transformation)

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

(16) Obligation:

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

(17) SCCToIDPv1Proof (SOUND transformation)

Transformed FIGraph SCCs to IDPs. Log:

Generated 80 rules for P and 0 rules for R.

P rules:
1985_0_main_LE(EOS(STATIC_1985), o740, i535) → 2000_0_main_LE(EOS(STATIC_2000), o740, i535)
2000_0_main_LE(EOS(STATIC_2000), o740, i535) → 2006_0_main_Load(EOS(STATIC_2006), o740) | >(i535, 0)
2012_0_main_FieldAccess(EOS(STATIC_2012), java.lang.Object(o833sub)) → 2020_0_main_FieldAccess(EOS(STATIC_2020), java.lang.Object(o833sub))
2020_0_main_FieldAccess(EOS(STATIC_2020), java.lang.Object(IntList(EOC, o836, i552))) → 2027_0_main_FieldAccess(EOS(STATIC_2027), java.lang.Object(IntList(EOC, o836, i552)))
2027_0_main_FieldAccess(EOS(STATIC_2027), java.lang.Object(IntList(EOC, o836, i552))) → 2032_0_main_Store(EOS(STATIC_2032), o836)
2032_0_main_Store(EOS(STATIC_2032), o836) → 2041_0_main_InvokeMethod(EOS(STATIC_2041), o836)
2041_0_main_InvokeMethod(EOS(STATIC_2041), o836) → 2046_0_random_FieldAccess(EOS(STATIC_2046), o836)
2046_0_random_FieldAccess(EOS(STATIC_2046), o836) → 2060_0_random_FieldAccess(EOS(STATIC_2060), o836)
2060_0_random_FieldAccess(EOS(STATIC_2060), o836) → 2069_0_random_ArrayAccess(EOS(STATIC_2069), o836)
2069_0_random_ArrayAccess(EOS(STATIC_2069), o836) → 2076_0_random_ArrayAccess(EOS(STATIC_2076), o836)
2076_0_random_ArrayAccess(EOS(STATIC_2076), o836) → 2086_0_random_Store(EOS(STATIC_2086), o836, o870)
2086_0_random_Store(EOS(STATIC_2086), o836, o870) → 2095_0_random_FieldAccess(EOS(STATIC_2095), o836, o870)
2095_0_random_FieldAccess(EOS(STATIC_2095), o836, o870) → 2104_0_random_ConstantStackPush(EOS(STATIC_2104), o836, o870)
2104_0_random_ConstantStackPush(EOS(STATIC_2104), o836, o870) → 2116_0_random_IntArithmetic(EOS(STATIC_2116), o836, o870)
2116_0_random_IntArithmetic(EOS(STATIC_2116), o836, o870) → 2127_0_random_FieldAccess(EOS(STATIC_2127), o836, o870)
2127_0_random_FieldAccess(EOS(STATIC_2127), o836, o870) → 2137_0_random_Load(EOS(STATIC_2137), o836, o870)
2137_0_random_Load(EOS(STATIC_2137), o836, o870) → 2154_0_random_InvokeMethod(EOS(STATIC_2154), o836, o870)
2154_0_random_InvokeMethod(EOS(STATIC_2154), o836, java.lang.Object(o933sub)) → 2164_0_random_InvokeMethod(EOS(STATIC_2164), o836, java.lang.Object(o933sub))
2164_0_random_InvokeMethod(EOS(STATIC_2164), o836, java.lang.Object(o933sub)) → 2175_0_length_Load(EOS(STATIC_2175), o836, java.lang.Object(o933sub), java.lang.Object(o933sub))
2175_0_length_Load(EOS(STATIC_2175), o836, java.lang.Object(o933sub), java.lang.Object(o933sub)) → 2204_0_length_FieldAccess(EOS(STATIC_2204), o836, java.lang.Object(o933sub), java.lang.Object(o933sub))
2204_0_length_FieldAccess(EOS(STATIC_2204), o836, java.lang.Object(java.lang.String(o970sub, i618)), java.lang.Object(java.lang.String(o970sub, i618))) → 2214_0_length_FieldAccess(EOS(STATIC_2214), o836, java.lang.Object(java.lang.String(o970sub, i618)), java.lang.Object(java.lang.String(o970sub, i618))) | &&(>=(i618, 0), >=(i619, 0))
2214_0_length_FieldAccess(EOS(STATIC_2214), o836, java.lang.Object(java.lang.String(o970sub, i618)), java.lang.Object(java.lang.String(o970sub, i618))) → 2226_0_length_Return(EOS(STATIC_2226), o836, java.lang.Object(java.lang.String(o970sub, i618)), i618)
2226_0_length_Return(EOS(STATIC_2226), o836, java.lang.Object(java.lang.String(o970sub, i618)), i618) → 2234_0_random_Return(EOS(STATIC_2234), o836, i618)
2234_0_random_Return(EOS(STATIC_2234), o836, i618) → 2239_0_main_Load(EOS(STATIC_2239), o836, i618)
2239_0_main_Load(EOS(STATIC_2239), o836, i618) → 2252_0_main_InvokeMethod(EOS(STATIC_2252), o836, i618, o836)
2252_0_main_InvokeMethod(EOS(STATIC_2252), o836, i618, o836) → 2260_0_nth_Load(EOS(STATIC_2260), o836, i618, o836, i618, o836)
2260_0_nth_Load(EOS(STATIC_2260), o836, i618, o836, i618, o836) → 2408_0_nth_Load(EOS(STATIC_2408), o836, i618, o836, i618, o836)
2408_0_nth_Load(EOS(STATIC_2408), o1164, i664, o1164, i663, o1163) → 2414_0_nth_ConstantStackPush(EOS(STATIC_2414), o1164, i664, o1164, i663, o1163, i663)
2414_0_nth_ConstantStackPush(EOS(STATIC_2414), o1164, i664, o1164, i663, o1163, i663) → 2422_0_nth_LE(EOS(STATIC_2422), o1164, i664, o1164, i663, o1163, i663, 1)
2422_0_nth_LE(EOS(STATIC_2422), o1164, i664, o1164, i675, o1163, i675, matching1) → 2427_0_nth_LE(EOS(STATIC_2427), o1164, i664, o1164, i675, o1163, i675, 1) | =(matching1, 1)
2422_0_nth_LE(EOS(STATIC_2422), o1164, i664, o1164, i676, o1163, i676, matching1) → 2428_0_nth_LE(EOS(STATIC_2428), o1164, i664, o1164, i676, o1163, i676, 1) | =(matching1, 1)
2427_0_nth_LE(EOS(STATIC_2427), o1164, i664, o1164, i675, o1163, i675, matching1) → 2432_0_nth_Load(EOS(STATIC_2432), o1164, i664, o1164, o1163) | &&(<=(i675, 1), =(matching1, 1))
2432_0_nth_Load(EOS(STATIC_2432), o1164, i664, o1164, o1163) → 2438_0_nth_NONNULL(EOS(STATIC_2438), o1164, i664, o1164, o1163, o1163)
2438_0_nth_NONNULL(EOS(STATIC_2438), o1164, i664, o1164, java.lang.Object(o1201sub), java.lang.Object(o1201sub)) → 2443_0_nth_NONNULL(EOS(STATIC_2443), o1164, i664, o1164, java.lang.Object(o1201sub), java.lang.Object(o1201sub))
2438_0_nth_NONNULL(EOS(STATIC_2438), o1164, i664, o1164, NULL, NULL) → 2444_0_nth_NONNULL(EOS(STATIC_2444), o1164, i664, o1164, NULL, NULL)
2443_0_nth_NONNULL(EOS(STATIC_2443), o1164, i664, o1164, java.lang.Object(o1201sub), java.lang.Object(o1201sub)) → 2449_0_nth_Load(EOS(STATIC_2449), o1164, i664, o1164, java.lang.Object(o1201sub))
2449_0_nth_Load(EOS(STATIC_2449), o1164, i664, o1164, java.lang.Object(o1201sub)) → 2458_0_nth_FieldAccess(EOS(STATIC_2458), o1164, i664, o1164, java.lang.Object(o1201sub))
2458_0_nth_FieldAccess(EOS(STATIC_2458), o1164, i664, o1164, java.lang.Object(o1201sub)) → 2466_0_nth_FieldAccess(EOS(STATIC_2466), o1164, i664, o1164, java.lang.Object(o1201sub))
2458_0_nth_FieldAccess(EOS(STATIC_2458), java.lang.Object(o1201sub), i664, java.lang.Object(o1201sub), java.lang.Object(o1201sub)) → 2467_0_nth_FieldAccess(EOS(STATIC_2467), java.lang.Object(o1201sub), i664, java.lang.Object(o1201sub), java.lang.Object(o1201sub))
2466_0_nth_FieldAccess(EOS(STATIC_2466), o1164, i664, o1164, java.lang.Object(IntList(EOC, o1230, i687))) → 2476_0_nth_FieldAccess(EOS(STATIC_2476), o1164, i664, o1164, java.lang.Object(IntList(EOC, o1230, i687)))
2476_0_nth_FieldAccess(EOS(STATIC_2476), o1164, i664, o1164, java.lang.Object(IntList(EOC, o1230, i687))) → 2485_0_nth_Return(EOS(STATIC_2485), o1164, i664, o1164, i687)
2485_0_nth_Return(EOS(STATIC_2485), o1164, i664, o1164, i687) → 2494_0_main_Store(EOS(STATIC_2494), o1164, i687)
2494_0_main_Store(EOS(STATIC_2494), o1164, i687) → 2503_0_main_JMP(EOS(STATIC_2503), o1164, i687)
2503_0_main_JMP(EOS(STATIC_2503), o1164, i687) → 2511_0_main_Load(EOS(STATIC_2511), o1164, i687)
1975_0_main_Load(EOS(STATIC_1975), o740, i503) → 1985_0_main_LE(EOS(STATIC_1985), o740, i503)
2467_0_nth_FieldAccess(EOS(STATIC_2467), java.lang.Object(IntList(EOC, o1232, i688)), i664, java.lang.Object(IntList(EOC, o1232, i688)), java.lang.Object(IntList(EOC, o1232, i688))) → 2478_0_nth_FieldAccess(EOS(STATIC_2478), java.lang.Object(IntList(EOC, o1232, i688)), i664, java.lang.Object(IntList(EOC, o1232, i688)), java.lang.Object(IntList(EOC, o1232, i688)))
2478_0_nth_FieldAccess(EOS(STATIC_2478), java.lang.Object(IntList(EOC, o1232, i688)), i664, java.lang.Object(IntList(EOC, o1232, i688)), java.lang.Object(IntList(EOC, o1232, i688))) → 2488_0_nth_Return(EOS(STATIC_2488), java.lang.Object(IntList(EOC, o1232, i688)), i664, java.lang.Object(IntList(EOC, o1232, i688)), i688)
2488_0_nth_Return(EOS(STATIC_2488), java.lang.Object(IntList(EOC, o1232, i688)), i664, java.lang.Object(IntList(EOC, o1232, i688)), i688) → 2496_0_main_Store(EOS(STATIC_2496), java.lang.Object(IntList(EOC, o1232, i688)), i688)
2496_0_main_Store(EOS(STATIC_2496), java.lang.Object(IntList(EOC, o1232, i688)), i688) → 2504_0_main_JMP(EOS(STATIC_2504), java.lang.Object(IntList(EOC, o1232, i688)), i688)
2504_0_main_JMP(EOS(STATIC_2504), java.lang.Object(IntList(EOC, o1232, i688)), i688) → 2514_0_main_Load(EOS(STATIC_2514), java.lang.Object(IntList(EOC, o1232, i688)), i688)
2444_0_nth_NONNULL(EOS(STATIC_2444), o1164, i664, o1164, NULL, NULL) → 2450_0_nth_ConstantStackPush(EOS(STATIC_2450), o1164, i664, o1164, NULL)
2450_0_nth_ConstantStackPush(EOS(STATIC_2450), o1164, i664, o1164, NULL) → 2475_0_nth_ConstantStackPush(EOS(STATIC_2475), o1164, i664, o1164, NULL)
2475_0_nth_ConstantStackPush(EOS(STATIC_2475), o1164, i664, o1164, NULL) → 2483_0_nth_Return(EOS(STATIC_2483), o1164, i664, o1164, NULL, 0)
2483_0_nth_Return(EOS(STATIC_2483), o1164, i664, o1164, NULL, matching1) → 2492_0_main_Store(EOS(STATIC_2492), o1164, 0) | =(matching1, 0)
2492_0_main_Store(EOS(STATIC_2492), o1164, matching1) → 2494_0_main_Store(EOS(STATIC_2494), o1164, 0) | =(matching1, 0)
2428_0_nth_LE(EOS(STATIC_2428), o1164, i664, o1164, i676, o1163, i676, matching1) → 2435_0_nth_Load(EOS(STATIC_2435), o1164, i664, o1164, i676, o1163) | &&(>(i676, 1), =(matching1, 1))
2435_0_nth_Load(EOS(STATIC_2435), o1164, i664, o1164, i676, o1163) → 2441_0_nth_NULL(EOS(STATIC_2441), o1164, i664, o1164, i676, o1163, o1163)
2441_0_nth_NULL(EOS(STATIC_2441), o1164, i664, o1164, i676, java.lang.Object(o1202sub), java.lang.Object(o1202sub)) → 2446_0_nth_NULL(EOS(STATIC_2446), o1164, i664, o1164, i676, java.lang.Object(o1202sub), java.lang.Object(o1202sub))
2441_0_nth_NULL(EOS(STATIC_2441), o1164, i664, o1164, i676, NULL, NULL) → 2447_0_nth_NULL(EOS(STATIC_2447), o1164, i664, o1164, i676, NULL, NULL)
2446_0_nth_NULL(EOS(STATIC_2446), o1164, i664, o1164, i676, java.lang.Object(o1202sub), java.lang.Object(o1202sub)) → 2452_0_nth_Inc(EOS(STATIC_2452), o1164, i664, o1164, i676, java.lang.Object(o1202sub))
2452_0_nth_Inc(EOS(STATIC_2452), o1164, i664, o1164, i676, java.lang.Object(o1202sub)) → 2462_0_nth_Load(EOS(STATIC_2462), o1164, i664, o1164, +(i676, -1), java.lang.Object(o1202sub)) | >(i676, 0)
2462_0_nth_Load(EOS(STATIC_2462), o1164, i664, o1164, i684, java.lang.Object(o1202sub)) → 2472_0_nth_FieldAccess(EOS(STATIC_2472), o1164, i664, o1164, i684, java.lang.Object(o1202sub))
2472_0_nth_FieldAccess(EOS(STATIC_2472), o1164, i664, o1164, i684, java.lang.Object(o1202sub)) → 2480_0_nth_FieldAccess(EOS(STATIC_2480), o1164, i664, o1164, i684, java.lang.Object(o1202sub))
2472_0_nth_FieldAccess(EOS(STATIC_2472), java.lang.Object(o1202sub), i664, java.lang.Object(o1202sub), i684, java.lang.Object(o1202sub)) → 2481_0_nth_FieldAccess(EOS(STATIC_2481), java.lang.Object(o1202sub), i664, java.lang.Object(o1202sub), i684, java.lang.Object(o1202sub))
2480_0_nth_FieldAccess(EOS(STATIC_2480), o1164, i664, o1164, i684, java.lang.Object(IntList(EOC, o1243, i691))) → 2489_0_nth_FieldAccess(EOS(STATIC_2489), o1164, i664, o1164, i684, java.lang.Object(IntList(EOC, o1243, i691)))
2489_0_nth_FieldAccess(EOS(STATIC_2489), o1164, i664, o1164, i684, java.lang.Object(IntList(EOC, o1243, i691))) → 2498_0_nth_Store(EOS(STATIC_2498), o1164, i664, o1164, i684, o1243)
2498_0_nth_Store(EOS(STATIC_2498), o1164, i664, o1164, i684, o1243) → 2506_0_nth_JMP(EOS(STATIC_2506), o1164, i664, o1164, i684, o1243)
2506_0_nth_JMP(EOS(STATIC_2506), o1164, i664, o1164, i684, o1243) → 2517_0_nth_Load(EOS(STATIC_2517), o1164, i664, o1164, i684, o1243)
2517_0_nth_Load(EOS(STATIC_2517), o1164, i664, o1164, i684, o1243) → 2408_0_nth_Load(EOS(STATIC_2408), o1164, i664, o1164, i684, o1243)
2481_0_nth_FieldAccess(EOS(STATIC_2481), java.lang.Object(IntList(EOC, o1245, i692)), i664, java.lang.Object(IntList(EOC, o1245, i692)), i684, java.lang.Object(IntList(EOC, o1245, i692))) → 2490_0_nth_FieldAccess(EOS(STATIC_2490), java.lang.Object(IntList(EOC, o1245, i692)), i664, java.lang.Object(IntList(EOC, o1245, i692)), i684, java.lang.Object(IntList(EOC, o1245, i692)))
2490_0_nth_FieldAccess(EOS(STATIC_2490), java.lang.Object(IntList(EOC, o1245, i692)), i664, java.lang.Object(IntList(EOC, o1245, i692)), i684, java.lang.Object(IntList(EOC, o1245, i692))) → 2501_0_nth_Store(EOS(STATIC_2501), java.lang.Object(IntList(EOC, o1245, i692)), i664, java.lang.Object(IntList(EOC, o1245, i692)), i684, o1245)
2501_0_nth_Store(EOS(STATIC_2501), java.lang.Object(IntList(EOC, o1245, i692)), i664, java.lang.Object(IntList(EOC, o1245, i692)), i684, o1245) → 2509_0_nth_JMP(EOS(STATIC_2509), java.lang.Object(IntList(EOC, o1245, i692)), i664, java.lang.Object(IntList(EOC, o1245, i692)), i684, o1245)
2509_0_nth_JMP(EOS(STATIC_2509), java.lang.Object(IntList(EOC, o1245, i692)), i664, java.lang.Object(IntList(EOC, o1245, i692)), i684, o1245) → 2521_0_nth_Load(EOS(STATIC_2521), java.lang.Object(IntList(EOC, o1245, i692)), i664, java.lang.Object(IntList(EOC, o1245, i692)), i684, o1245)
2521_0_nth_Load(EOS(STATIC_2521), java.lang.Object(IntList(EOC, o1245, i692)), i664, java.lang.Object(IntList(EOC, o1245, i692)), i684, o1245) → 2408_0_nth_Load(EOS(STATIC_2408), java.lang.Object(IntList(EOC, o1245, i692)), i664, java.lang.Object(IntList(EOC, o1245, i692)), i684, o1245)
2447_0_nth_NULL(EOS(STATIC_2447), o1164, i664, o1164, i676, NULL, NULL) → 2454_0_nth_Load(EOS(STATIC_2454), o1164, i664, o1164, NULL)
2454_0_nth_Load(EOS(STATIC_2454), o1164, i664, o1164, NULL) → 2464_0_nth_NONNULL(EOS(STATIC_2464), o1164, i664, o1164, NULL, NULL)
2464_0_nth_NONNULL(EOS(STATIC_2464), o1164, i664, o1164, NULL, NULL) → 2475_0_nth_ConstantStackPush(EOS(STATIC_2475), o1164, i664, o1164, NULL)
R rules:

Combined rules. Obtained 5 conditional rules for P and 0 conditional rules for R.

P rules:
2422_0_nth_LE(EOS(STATIC_2422), java.lang.Object(IntList(EOC, x0, x1)), x2, java.lang.Object(IntList(EOC, x0, x1)), x3, java.lang.Object(IntList(EOC, x4, x5)), x3, 1) → 2422_0_nth_LE(EOS(STATIC_2422), x0, x7, x0, x7, x0, x7, 1) | &&(&&(>(+(x7, 1), 0), >(x5, 0)), <=(x3, 1))
2422_0_nth_LE(EOS(STATIC_2422), java.lang.Object(IntList(EOC, x0, x1)), x2, java.lang.Object(IntList(EOC, x0, x1)), x3, java.lang.Object(IntList(EOC, x0, x1)), x3, 1) → 2422_0_nth_LE(EOS(STATIC_2422), x0, x5, x0, x5, x0, x5, 1) | &&(&&(>(+(x5, 1), 0), <=(x3, 1)), >(x1, 0))
2422_0_nth_LE(EOS(STATIC_2422), java.lang.Object(IntList(EOC, x0, x1)), x2, java.lang.Object(IntList(EOC, x0, x1)), x3, NULL, x3, x4) → 2422_0_nth_LE(EOS(STATIC_2422), x0, x5, x0, x5, x0, x5, 1) | FALSE
2422_0_nth_LE(EOS(STATIC_2422), x0, x1, x0, x2, java.lang.Object(IntList(EOC, x3, x4)), x2, 1) → 2422_0_nth_LE(EOS(STATIC_2422), x0, x1, x0, +(x2, -1), x3, +(x2, -1), 1) | >(x2, 1)
2422_0_nth_LE(EOS(STATIC_2422), java.lang.Object(IntList(EOC, x0, x1)), x2, java.lang.Object(IntList(EOC, x0, x1)), x3, java.lang.Object(IntList(EOC, x0, x1)), x3, 1) → 2422_0_nth_LE(EOS(STATIC_2422), java.lang.Object(IntList(EOC, x0, x1)), x2, java.lang.Object(IntList(EOC, x0, x1)), +(x3, -1), x0, +(x3, -1), 1) | >(x3, 1)
R rules:

Filtered ground terms:

2422_0_nth_LE(x1, x2, x3, x4, x5, x6, x7, x8) → 2422_0_nth_LE(x2, x3, x4, x5, x6, x7)
IntList(x1, x2, x3) → IntList(x2, x3)
EOS(x1) → EOS
Cond_2422_0_nth_LE3(x1, x2, x3, x4, x5, x6, x7, x8, x9) → Cond_2422_0_nth_LE3(x1, x3, x4, x5, x6, x7, x8)
Cond_2422_0_nth_LE2(x1, x2, x3, x4, x5, x6, x7, x8, x9) → Cond_2422_0_nth_LE2(x1, x3, x4, x5, x6, x7, x8)
Cond_2422_0_nth_LE1(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10) → Cond_2422_0_nth_LE1(x1, x3, x4, x5, x6, x7, x8, x10)
Cond_2422_0_nth_LE(x1, x2, x3, x4, x5, x6, x7, x8, x9, x10) → Cond_2422_0_nth_LE(x1, x3, x4, x5, x6, x7, x8, x10)

Filtered duplicate args:

2422_0_nth_LE(x1, x2, x3, x4, x5, x6) → 2422_0_nth_LE(x2, x3, x5, x6)
Cond_2422_0_nth_LE(x1, x2, x3, x4, x5, x6, x7, x8) → Cond_2422_0_nth_LE(x1, x3, x4, x6, x7, x8)
Cond_2422_0_nth_LE1(x1, x2, x3, x4, x5, x6, x7, x8) → Cond_2422_0_nth_LE1(x1, x3, x6, x7, x8)
Cond_2422_0_nth_LE2(x1, x2, x3, x4, x5, x6, x7) → Cond_2422_0_nth_LE2(x1, x3, x4, x6, x7)
Cond_2422_0_nth_LE3(x1, x2, x3, x4, x5, x6, x7) → Cond_2422_0_nth_LE3(x1, x3, x6, x7)

Filtered unneeded arguments:

2422_0_nth_LE(x1, x2, x3, x4) → 2422_0_nth_LE(x2, x3, x4)
Cond_2422_0_nth_LE(x1, x2, x3, x4, x5, x6) → Cond_2422_0_nth_LE(x1, x3, x6)
Cond_2422_0_nth_LE1(x1, x2, x3, x4, x5) → Cond_2422_0_nth_LE1(x1, x3, x5)
Cond_2422_0_nth_LE2(x1, x2, x3, x4, x5) → Cond_2422_0_nth_LE2(x1, x3, x4, x5)
Cond_2422_0_nth_LE3(x1, x2, x3, x4) → Cond_2422_0_nth_LE3(x1, x3, x4)

Combined rules. Obtained 4 conditional rules for P and 0 conditional rules for R.

P rules:
2422_0_nth_LE(java.lang.Object(IntList(x0, x1)), java.lang.Object(IntList(x4, x5)), x3) → 2422_0_nth_LE(x0, x0, x7) | &&(&&(>(x7, -1), >(x5, 0)), <=(x3, 1))
2422_0_nth_LE(java.lang.Object(IntList(x0, x1)), java.lang.Object(IntList(x0, x1)), x3) → 2422_0_nth_LE(x0, x0, x5) | &&(&&(>(x5, -1), <=(x3, 1)), >(x1, 0))
2422_0_nth_LE(x0, java.lang.Object(IntList(x3, x4)), x2) → 2422_0_nth_LE(x0, x3, +(x2, -1)) | >(x2, 1)
2422_0_nth_LE(java.lang.Object(IntList(x0, x1)), java.lang.Object(IntList(x0, x1)), x3) → 2422_0_nth_LE(java.lang.Object(IntList(x0, x1)), x0, +(x3, -1)) | >(x3, 1)
R rules:

Finished conversion. Obtained 8 rules for P and 0 rules for R. System has predefined symbols.

P rules:
2422_0_NTH_LE(java.lang.Object(IntList(x0, x1)), java.lang.Object(IntList(x4, x5)), x3) → COND_2422_0_NTH_LE(&&(&&(>(x7, -1), >(x5, 0)), <=(x3, 1)), java.lang.Object(IntList(x0, x1)), java.lang.Object(IntList(x4, x5)), x3, x7)
COND_2422_0_NTH_LE(TRUE, java.lang.Object(IntList(x0, x1)), java.lang.Object(IntList(x4, x5)), x3, x7) → 2422_0_NTH_LE(x0, x0, x7)
2422_0_NTH_LE(java.lang.Object(IntList(x0, x1)), java.lang.Object(IntList(x0, x1)), x3) → COND_2422_0_NTH_LE1(&&(&&(>(x5, -1), <=(x3, 1)), >(x1, 0)), java.lang.Object(IntList(x0, x1)), java.lang.Object(IntList(x0, x1)), x3, x5)
COND_2422_0_NTH_LE1(TRUE, java.lang.Object(IntList(x0, x1)), java.lang.Object(IntList(x0, x1)), x3, x5) → 2422_0_NTH_LE(x0, x0, x5)
2422_0_NTH_LE(x0, java.lang.Object(IntList(x3, x4)), x2) → COND_2422_0_NTH_LE2(>(x2, 1), x0, java.lang.Object(IntList(x3, x4)), x2)
COND_2422_0_NTH_LE2(TRUE, x0, java.lang.Object(IntList(x3, x4)), x2) → 2422_0_NTH_LE(x0, x3, +(x2, -1))
2422_0_NTH_LE(java.lang.Object(IntList(x0, x1)), java.lang.Object(IntList(x0, x1)), x3) → COND_2422_0_NTH_LE3(>(x3, 1), java.lang.Object(IntList(x0, x1)), java.lang.Object(IntList(x0, x1)), x3)
COND_2422_0_NTH_LE3(TRUE, java.lang.Object(IntList(x0, x1)), java.lang.Object(IntList(x0, x1)), x3) → 2422_0_NTH_LE(java.lang.Object(IntList(x0, x1)), x0, +(x3, -1))
R rules:

(18) 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): 2422_0_NTH_LE(java.lang.Object(IntList(x0[0], x1[0])), java.lang.Object(IntList(x4[0], x5[0])), x3[0]) → COND_2422_0_NTH_LE(x7[0] > -1 && x5[0] > 0 && x3[0] <= 1, java.lang.Object(IntList(x0[0], x1[0])), java.lang.Object(IntList(x4[0], x5[0])), x3[0], x7[0])
(1): COND_2422_0_NTH_LE(TRUE, java.lang.Object(IntList(x0[1], x1[1])), java.lang.Object(IntList(x4[1], x5[1])), x3[1], x7[1]) → 2422_0_NTH_LE(x0[1], x0[1], x7[1])
(2): 2422_0_NTH_LE(java.lang.Object(IntList(x0[2], x1[2])), java.lang.Object(IntList(x0[2], x1[2])), x3[2]) → COND_2422_0_NTH_LE1(x5[2] > -1 && x3[2] <= 1 && x1[2] > 0, java.lang.Object(IntList(x0[2], x1[2])), java.lang.Object(IntList(x0[2], x1[2])), x3[2], x5[2])
(3): COND_2422_0_NTH_LE1(TRUE, java.lang.Object(IntList(x0[3], x1[3])), java.lang.Object(IntList(x0[3], x1[3])), x3[3], x5[3]) → 2422_0_NTH_LE(x0[3], x0[3], x5[3])
(4): 2422_0_NTH_LE(x0[4], java.lang.Object(IntList(x3[4], x4[4])), x2[4]) → COND_2422_0_NTH_LE2(x2[4] > 1, x0[4], java.lang.Object(IntList(x3[4], x4[4])), x2[4])
(5): COND_2422_0_NTH_LE2(TRUE, x0[5], java.lang.Object(IntList(x3[5], x4[5])), x2[5]) → 2422_0_NTH_LE(x0[5], x3[5], x2[5] + -1)
(6): 2422_0_NTH_LE(java.lang.Object(IntList(x0[6], x1[6])), java.lang.Object(IntList(x0[6], x1[6])), x3[6]) → COND_2422_0_NTH_LE3(x3[6] > 1, java.lang.Object(IntList(x0[6], x1[6])), java.lang.Object(IntList(x0[6], x1[6])), x3[6])
(7): COND_2422_0_NTH_LE3(TRUE, java.lang.Object(IntList(x0[7], x1[7])), java.lang.Object(IntList(x0[7], x1[7])), x3[7]) → 2422_0_NTH_LE(java.lang.Object(IntList(x0[7], x1[7])), x0[7], x3[7] + -1)

(0) -> (1), if (x7[0] > -1 && x5[0] > 0 && x3[0] <= 1java.lang.Object(IntList(x0[0], x1[0])) →* java.lang.Object(IntList(x0[1], x1[1]))∧java.lang.Object(IntList(x4[0], x5[0])) →* java.lang.Object(IntList(x4[1], x5[1]))∧x3[0]* x3[1]x7[0]* x7[1])

(1) -> (0), if (x0[1]* java.lang.Object(IntList(x0[0], x1[0]))∧x0[1]* java.lang.Object(IntList(x4[0], x5[0]))∧x7[1]* x3[0])

(1) -> (2), if (x0[1]* java.lang.Object(IntList(x0[2], x1[2]))∧x7[1]* x3[2])

(1) -> (4), if (x0[1]* x0[4]x0[1]* java.lang.Object(IntList(x3[4], x4[4]))∧x7[1]* x2[4])

(1) -> (6), if (x0[1]* java.lang.Object(IntList(x0[6], x1[6]))∧x7[1]* x3[6])

(2) -> (3), if (x5[2] > -1 && x3[2] <= 1 && x1[2] > 0java.lang.Object(IntList(x0[2], x1[2])) →* java.lang.Object(IntList(x0[3], x1[3]))∧x3[2]* x3[3]x5[2]* x5[3])

(3) -> (0), if (x0[3]* java.lang.Object(IntList(x0[0], x1[0]))∧x0[3]* java.lang.Object(IntList(x4[0], x5[0]))∧x5[3]* x3[0])

(3) -> (2), if (x0[3]* java.lang.Object(IntList(x0[2], x1[2]))∧x5[3]* x3[2])

(3) -> (4), if (x0[3]* x0[4]x0[3]* java.lang.Object(IntList(x3[4], x4[4]))∧x5[3]* x2[4])

(3) -> (6), if (x0[3]* java.lang.Object(IntList(x0[6], x1[6]))∧x5[3]* x3[6])

(4) -> (5), if (x2[4] > 1x0[4]* x0[5]java.lang.Object(IntList(x3[4], x4[4])) →* java.lang.Object(IntList(x3[5], x4[5]))∧x2[4]* x2[5])

(5) -> (0), if (x0[5]* java.lang.Object(IntList(x0[0], x1[0]))∧x3[5]* java.lang.Object(IntList(x4[0], x5[0]))∧x2[5] + -1* x3[0])

(5) -> (2), if (x0[5]* java.lang.Object(IntList(x0[2], x1[2]))∧x3[5]* java.lang.Object(IntList(x0[2], x1[2]))∧x2[5] + -1* x3[2])

(5) -> (4), if (x0[5]* x0[4]x3[5]* java.lang.Object(IntList(x3[4], x4[4]))∧x2[5] + -1* x2[4])

(5) -> (6), if (x0[5]* java.lang.Object(IntList(x0[6], x1[6]))∧x3[5]* java.lang.Object(IntList(x0[6], x1[6]))∧x2[5] + -1* x3[6])

(6) -> (7), if (x3[6] > 1java.lang.Object(IntList(x0[6], x1[6])) →* java.lang.Object(IntList(x0[7], x1[7]))∧x3[6]* x3[7])

(7) -> (0), if (java.lang.Object(IntList(x0[7], x1[7])) →* java.lang.Object(IntList(x0[0], x1[0]))∧x0[7]* java.lang.Object(IntList(x4[0], x5[0]))∧x3[7] + -1* x3[0])

(7) -> (2), if (java.lang.Object(IntList(x0[7], x1[7])) →* java.lang.Object(IntList(x0[2], x1[2]))∧x0[7]* java.lang.Object(IntList(x0[2], x1[2]))∧x3[7] + -1* x3[2])

(7) -> (4), if (java.lang.Object(IntList(x0[7], x1[7])) →* x0[4]x0[7]* java.lang.Object(IntList(x3[4], x4[4]))∧x3[7] + -1* x2[4])

(7) -> (6), if (java.lang.Object(IntList(x0[7], x1[7])) →* java.lang.Object(IntList(x0[6], x1[6]))∧x0[7]* java.lang.Object(IntList(x0[6], x1[6]))∧x3[7] + -1* x3[6])

The set Q is empty.

(19) IDPNonInfProof (SOUND transformation)

Used the following options for this NonInfProof:
IDPGPoloSolver: Range: [(-1,2)] IsNat: true Interpretation Shape Heuristic: aprove.DPFramework.IDPProblem.Processors.nonInf.poly.IdpDefaultShapeHeuristic@3e48983b 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 2422_0_NTH_LE(java.lang.Object(IntList(x0, x1)), java.lang.Object(IntList(x4, x5)), x3) → COND_2422_0_NTH_LE(&&(&&(>(x7, -1), >(x5, 0)), <=(x3, 1)), java.lang.Object(IntList(x0, x1)), java.lang.Object(IntList(x4, x5)), x3, x7) the following chains were created:
• We consider the chain COND_2422_0_NTH_LE(TRUE, java.lang.Object(IntList(x0[1], x1[1])), java.lang.Object(IntList(x4[1], x5[1])), x3[1], x7[1]) → 2422_0_NTH_LE(x0[1], x0[1], x7[1]), 2422_0_NTH_LE(java.lang.Object(IntList(x0[0], x1[0])), java.lang.Object(IntList(x4[0], x5[0])), x3[0]) → COND_2422_0_NTH_LE(&&(&&(>(x7[0], -1), >(x5[0], 0)), <=(x3[0], 1)), java.lang.Object(IntList(x0[0], x1[0])), java.lang.Object(IntList(x4[0], x5[0])), x3[0], x7[0]), COND_2422_0_NTH_LE(TRUE, java.lang.Object(IntList(x0[1], x1[1])), java.lang.Object(IntList(x4[1], x5[1])), x3[1], x7[1]) → 2422_0_NTH_LE(x0[1], x0[1], x7[1]) which results in the following constraint:

(1)    (x0[1]=java.lang.Object(IntList(x0[0], x1[0]))∧x0[1]=java.lang.Object(IntList(x4[0], x5[0]))∧x7[1]=x3[0]&&(&&(>(x7[0], -1), >(x5[0], 0)), <=(x3[0], 1))=TRUEjava.lang.Object(IntList(x0[0], x1[0]))=java.lang.Object(IntList(x0[1]1, x1[1]1))∧java.lang.Object(IntList(x4[0], x5[0]))=java.lang.Object(IntList(x4[1]1, x5[1]1))∧x3[0]=x3[1]1x7[0]=x7[1]12422_0_NTH_LE(java.lang.Object(IntList(x0[0], x1[0])), java.lang.Object(IntList(x4[0], x5[0])), x3[0])≥NonInfC∧2422_0_NTH_LE(java.lang.Object(IntList(x0[0], x1[0])), java.lang.Object(IntList(x4[0], x5[0])), x3[0])≥COND_2422_0_NTH_LE(&&(&&(>(x7[0], -1), >(x5[0], 0)), <=(x3[0], 1)), java.lang.Object(IntList(x0[0], x1[0])), java.lang.Object(IntList(x4[0], x5[0])), x3[0], x7[0])∧(UIncreasing(COND_2422_0_NTH_LE(&&(&&(>(x7[0], -1), >(x5[0], 0)), <=(x3[0], 1)), java.lang.Object(IntList(x0[0], x1[0])), java.lang.Object(IntList(x4[0], x5[0])), x3[0], x7[0])), ≥))

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

(2)    (&&(&&(>(x7[0], -1), >(x5[0], 0)), <=(x3[0], 1))=TRUE2422_0_NTH_LE(java.lang.Object(IntList(x0[0], x5[0])), java.lang.Object(IntList(x0[0], x5[0])), x3[0])≥NonInfC∧2422_0_NTH_LE(java.lang.Object(IntList(x0[0], x5[0])), java.lang.Object(IntList(x0[0], x5[0])), x3[0])≥COND_2422_0_NTH_LE(&&(&&(>(x7[0], -1), >(x5[0], 0)), <=(x3[0], 1)), java.lang.Object(IntList(x0[0], x5[0])), java.lang.Object(IntList(x0[0], x5[0])), x3[0], x7[0])∧(UIncreasing(COND_2422_0_NTH_LE(&&(&&(>(x7[0], -1), >(x5[0], 0)), <=(x3[0], 1)), java.lang.Object(IntList(x0[0], x1[0])), java.lang.Object(IntList(x4[0], x5[0])), x3[0], x7[0])), ≥))

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

(3)    (0 ≥ 0 ⇒ (UIncreasing(COND_2422_0_NTH_LE(&&(&&(>(x7[0], -1), >(x5[0], 0)), <=(x3[0], 1)), java.lang.Object(IntList(x0[0], x1[0])), java.lang.Object(IntList(x4[0], x5[0])), x3[0], x7[0])), ≥)∧[(-1)Bound*bni_32] + [bni_32]x0[0] ≥ 0∧[(-1)bso_33] ≥ 0)

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

(4)    (0 ≥ 0 ⇒ (UIncreasing(COND_2422_0_NTH_LE(&&(&&(>(x7[0], -1), >(x5[0], 0)), <=(x3[0], 1)), java.lang.Object(IntList(x0[0], x1[0])), java.lang.Object(IntList(x4[0], x5[0])), x3[0], x7[0])), ≥)∧[(-1)Bound*bni_32] + [bni_32]x0[0] ≥ 0∧[(-1)bso_33] ≥ 0)

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

(5)    (0 ≥ 0 ⇒ (UIncreasing(COND_2422_0_NTH_LE(&&(&&(>(x7[0], -1), >(x5[0], 0)), <=(x3[0], 1)), java.lang.Object(IntList(x0[0], x1[0])), java.lang.Object(IntList(x4[0], x5[0])), x3[0], x7[0])), ≥)∧[(-1)Bound*bni_32] + [bni_32]x0[0] ≥ 0∧[(-1)bso_33] ≥ 0)

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

(6)    (0 ≥ 0 ⇒ (UIncreasing(COND_2422_0_NTH_LE(&&(&&(>(x7[0], -1), >(x5[0], 0)), <=(x3[0], 1)), java.lang.Object(IntList(x0[0], x1[0])), java.lang.Object(IntList(x4[0], x5[0])), x3[0], x7[0])), ≥)∧0 ≥ 0∧0 ≥ 0∧[bni_32] ≥ 0∧[(-1)Bound*bni_32] ≥ 0∧0 ≥ 0∧0 ≥ 0∧0 ≥ 0∧0 ≥ 0∧[(-1)bso_33] ≥ 0)

• We consider the chain COND_2422_0_NTH_LE1(TRUE, java.lang.Object(IntList(x0[3], x1[3])), java.lang.Object(IntList(x0[3], x1[3])), x3[3], x5[3]) → 2422_0_NTH_LE(x0[3], x0[3], x5[3]), 2422_0_NTH_LE(java.lang.Object(IntList(x0[0], x1[0])), java.lang.Object(IntList(x4[0], x5[0])), x3[0]) → COND_2422_0_NTH_LE(&&(&&(>(x7[0], -1), >(x5[0], 0)), <=(x3[0], 1)), java.lang.Object(IntList(x0[0], x1[0])), java.lang.Object(IntList(x4[0], x5[0])), x3[0], x7[0]), COND_2422_0_NTH_LE(TRUE, java.lang.Object(IntList(x0[1], x1[1])), java.lang.Object(IntList(x4[1], x5[1])), x3[1], x7[1]) → 2422_0_NTH_LE(x0[1], x0[1], x7[1]) which results in the following constraint:

(7)    (x0[3]=java.lang.Object(IntList(x0[0], x1[0]))∧x0[3]=java.lang.Object(IntList(x4[0], x5[0]))∧x5[3]=x3[0]&&(&&(>(x7[0], -1), >(x5[0], 0)), <=(x3[0], 1))=TRUEjava.lang.Object(IntList(x0[0], x1[0]))=java.lang.Object(IntList(x0[1], x1[1]))∧java.lang.Object(IntList(x4[0], x5[0]))=java.lang.Object(IntList(x4[1], x5[1]))∧x3[0]=x3[1]x7[0]=x7[1]2422_0_NTH_LE(java.lang.Object(IntList(x0[0], x1[0])), java.lang.Object(IntList(x4[0], x5[0])), x3[0])≥NonInfC∧2422_0_NTH_LE(java.lang.Object(IntList(x0[0], x1[0])), java.lang.Object(IntList(x4[0], x5[0])), x3[0])≥COND_2422_0_NTH_LE(&&(&&(>(x7[0], -1), >(x5[0], 0)), <=(x3[0], 1)), java.lang.Object(IntList(x0[0], x1[0])), java.lang.Object(IntList(x4[0], x5[0])), x3[0], x7[0])∧(UIncreasing(COND_2422_0_NTH_LE(&&(&&(>(x7[0], -1), >(x5[0], 0)), <=(x3[0], 1)), java.lang.Object(IntList(x0[0], x1[0])), java.lang.Object(IntList(x4[0], x5[0])), x3[0], x7[0])), ≥))

We simplified constraint (7) using rules (I), (II), (III), (IV) which results in the following new constraint:

(8)    (&&(&&(>(x7[0], -1), >(x5[0], 0)), <=(x3[0], 1))=TRUE2422_0_NTH_LE(java.lang.Object(IntList(x0[0], x5[0])), java.lang.Object(IntList(x0[0], x5[0])), x3[0])≥NonInfC∧2422_0_NTH_LE(java.lang.Object(IntList(x0[0], x5[0])), java.lang.Object(IntList(x0[0], x5[0])), x3[0])≥COND_2422_0_NTH_LE(&&(&&(>(x7[0], -1), >(x5[0], 0)), <=(x3[0], 1)), java.lang.Object(IntList(x0[0], x5[0])), java.lang.Object(IntList(x0[0], x5[0])), x3[0], x7[0])∧(UIncreasing(COND_2422_0_NTH_LE(&&(&&(>(x7[0], -1), >(x5[0], 0)), <=(x3[0], 1)), java.lang.Object(IntList(x0[0], x1[0])), java.lang.Object(IntList(x4[0], x5[0])), x3[0], x7[0])), ≥))

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

(9)    (0 ≥ 0 ⇒ (UIncreasing(COND_2422_0_NTH_LE(&&(&&(>(x7[0], -1), >(x5[0], 0)), <=(x3[0], 1)), java.lang.Object(IntList(x0[0], x1[0])), java.lang.Object(IntList(x4[0], x5[0])), x3[0], x7[0])), ≥)∧[(-1)Bound*bni_32] + [bni_32]x0[0] ≥ 0∧[(-1)bso_33] ≥ 0)

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

(10)    (0 ≥ 0 ⇒ (UIncreasing(COND_2422_0_NTH_LE(&&(&&(>(x7[0], -1), >(x5[0], 0)), <=(x3[0], 1)), java.lang.Object(IntList(x0[0], x1[0])), java.lang.Object(IntList(x4[0], x5[0])), x3[0], x7[0])), ≥)∧[(-1)Bound*bni_32] + [bni_32]x0[0] ≥ 0∧[(-1)bso_33] ≥ 0)

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

(11)    (0 ≥ 0 ⇒ (UIncreasing(COND_2422_0_NTH_LE(&&(&&(>(x7[0], -1), >(x5[0], 0)), <=(x3[0], 1)), java.lang.Object(IntList(x0[0], x1[0])), java.lang.Object(IntList(x4[0], x5[0])), x3[0], x7[0])), ≥)∧[(-1)Bound*bni_32] + [bni_32]x0[0] ≥ 0∧[(-1)bso_33] ≥ 0)

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

(12)    (0 ≥ 0 ⇒ (UIncreasing(COND_2422_0_NTH_LE(&&(&&(>(x7[0], -1), >(x5[0], 0)), <=(x3[0], 1)), java.lang.Object(IntList(x0[0], x1[0])), java.lang.Object(IntList(x4[0], x5[0])), x3[0], x7[0])), ≥)∧0 ≥ 0∧0 ≥ 0∧[bni_32] ≥ 0∧[(-1)Bound*bni_32] ≥ 0∧0 ≥ 0∧0 ≥ 0∧0 ≥ 0∧0 ≥ 0∧[(-1)bso_33] ≥ 0)

• We consider the chain COND_2422_0_NTH_LE2(TRUE, x0[5], java.lang.Object(IntList(x3[5], x4[5])), x2[5]) → 2422_0_NTH_LE(x0[5], x3[5], +(x2[5], -1)), 2422_0_NTH_LE(java.lang.Object(IntList(x0[0], x1[0])), java.lang.Object(IntList(x4[0], x5[0])), x3[0]) → COND_2422_0_NTH_LE(&&(&&(>(x7[0], -1), >(x5[0], 0)), <=(x3[0], 1)), java.lang.Object(IntList(x0[0], x1[0])), java.lang.Object(IntList(x4[0], x5[0])), x3[0], x7[0]), COND_2422_0_NTH_LE(TRUE, java.lang.Object(IntList(x0[1], x1[1])), java.lang.Object(IntList(x4[1], x5[1])), x3[1], x7[1]) → 2422_0_NTH_LE(x0[1], x0[1], x7[1]) which results in the following constraint:

(13)    (x0[5]=java.lang.Object(IntList(x0[0], x1[0]))∧x3[5]=java.lang.Object(IntList(x4[0], x5[0]))∧+(x2[5], -1)=x3[0]&&(&&(>(x7[0], -1), >(x5[0], 0)), <=(x3[0], 1))=TRUEjava.lang.Object(IntList(x0[0], x1[0]))=java.lang.Object(IntList(x0[1], x1[1]))∧java.lang.Object(IntList(x4[0], x5[0]))=java.lang.Object(IntList(x4[1], x5[1]))∧x3[0]=x3[1]x7[0]=x7[1]2422_0_NTH_LE(java.lang.Object(IntList(x0[0], x1[0])), java.lang.Object(IntList(x4[0], x5[0])), x3[0])≥NonInfC∧2422_0_NTH_LE(java.lang.Object(IntList(x0[0], x1[0])), java.lang.Object(IntList(x4[0], x5[0])), x3[0])≥COND_2422_0_NTH_LE(&&(&&(>(x7[0], -1), >(x5[0], 0)), <=(x3[0], 1)), java.lang.Object(IntList(x0[0], x1[0])), java.lang.Object(IntList(x4[0], x5[0])), x3[0], x7[0])∧(UIncreasing(COND_2422_0_NTH_LE(&&(&&(>(x7[0], -1), >(x5[0], 0)), <=(x3[0], 1)), java.lang.Object(IntList(x0[0], x1[0])), java.lang.Object(IntList(x4[0], x5[0])), x3[0], x7[0])), ≥))

We simplified constraint (13) using rules (I), (II), (III), (IV) which results in the following new constraint:

(14)    (&&(&&(>(x7[0], -1), >(x5[0], 0)), <=(+(x2[5], -1), 1))=TRUE2422_0_NTH_LE(java.lang.Object(IntList(x0[0], x1[0])), java.lang.Object(IntList(x4[0], x5[0])), +(x2[5], -1))≥NonInfC∧2422_0_NTH_LE(java.lang.Object(IntList(x0[0], x1[0])), java.lang.Object(IntList(x4[0], x5[0])), +(x2[5], -1))≥COND_2422_0_NTH_LE(&&(&&(>(x7[0], -1), >(x5[0], 0)), <=(+(x2[5], -1), 1)), java.lang.Object(IntList(x0[0], x1[0])), java.lang.Object(IntList(x4[0], x5[0])), +(x2[5], -1), x7[0])∧(UIncreasing(COND_2422_0_NTH_LE(&&(&&(>(x7[0], -1), >(x5[0], 0)), <=(x3[0], 1)), java.lang.Object(IntList(x0[0], x1[0])), java.lang.Object(IntList(x4[0], x5[0])), x3[0], x7[0])), ≥))

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

(15)    (0 ≥ 0 ⇒ (UIncreasing(COND_2422_0_NTH_LE(&&(&&(>(x7[0], -1), >(x5[0], 0)), <=(x3[0], 1)), java.lang.Object(IntList(x0[0], x1[0])), java.lang.Object(IntList(x4[0], x5[0])), x3[0], x7[0])), ≥)∧[(-1)Bound*bni_32] + [bni_32]x0[0] ≥ 0∧[(-1)bso_33] ≥ 0)

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

(16)    (0 ≥ 0 ⇒ (UIncreasing(COND_2422_0_NTH_LE(&&(&&(>(x7[0], -1), >(x5[0], 0)), <=(x3[0], 1)), java.lang.Object(IntList(x0[0], x1[0])), java.lang.Object(IntList(x4[0], x5[0])), x3[0], x7[0])), ≥)∧[(-1)Bound*bni_32] + [bni_32]x0[0] ≥ 0∧[(-1)bso_33] ≥ 0)

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

(17)    (0 ≥ 0 ⇒ (UIncreasing(COND_2422_0_NTH_LE(&&(&&(>(x7[0], -1), >(x5[0], 0)), <=(x3[0], 1)), java.lang.Object(IntList(x0[0], x1[0])), java.lang.Object(IntList(x4[0], x5[0])), x3[0], x7[0])), ≥)∧[(-1)Bound*bni_32] + [bni_32]x0[0] ≥ 0∧[(-1)bso_33] ≥ 0)

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

(18)    (0 ≥ 0 ⇒ (UIncreasing(COND_2422_0_NTH_LE(&&(&&(>(x7[0], -1), >(x5[0], 0)), <=(x3[0], 1)), java.lang.Object(IntList(x0[0], x1[0])), java.lang.Object(IntList(x4[0], x5[0])), x3[0], x7[0])), ≥)∧0 ≥ 0∧0 ≥ 0∧0 ≥ 0∧[bni_32] ≥ 0∧[(-1)Bound*bni_32] ≥ 0∧0 ≥ 0∧0 ≥ 0∧0 ≥ 0∧0 ≥ 0∧0 ≥ 0∧[(-1)bso_33] ≥ 0)

• We consider the chain COND_2422_0_NTH_LE3(TRUE, java.lang.Object(IntList(x0[7], x1[7])), java.lang.Object(IntList(x0[7], x1[7])), x3[7]) → 2422_0_NTH_LE(java.lang.Object(IntList(x0[7], x1[7])), x0[7], +(x3[7], -1)), 2422_0_NTH_LE(java.lang.Object(IntList(x0[0], x1[0])), java.lang.Object(IntList(x4[0], x5[0])), x3[0]) → COND_2422_0_NTH_LE(&&(&&(>(x7[0], -1), >(x5[0], 0)), <=(x3[0], 1)), java.lang.Object(IntList(x0[0], x1[0])), java.lang.Object(IntList(x4[0], x5[0])), x3[0], x7[0]), COND_2422_0_NTH_LE(TRUE, java.lang.Object(IntList(x0[1], x1[1])), java.lang.Object(IntList(x4[1], x5[1])), x3[1], x7[1]) → 2422_0_NTH_LE(x0[1], x0[1], x7[1]) which results in the following constraint:

(19)    (java.lang.Object(IntList(x0[7], x1[7]))=java.lang.Object(IntList(x0[0], x1[0]))∧x0[7]=java.lang.Object(IntList(x4[0], x5[0]))∧+(x3[7], -1)=x3[0]&&(&&(>(x7[0], -1), >(x5[0], 0)), <=(x3[0], 1))=TRUEjava.lang.Object(IntList(x0[0], x1[0]))=java.lang.Object(IntList(x0[1], x1[1]))∧java.lang.Object(IntList(x4[0], x5[0]))=java.lang.Object(IntList(x4[1], x5[1]))∧x3[0]=x3[1]x7[0]=x7[1]2422_0_NTH_LE(java.lang.Object(IntList(x0[0], x1[0])), java.lang.Object(IntList(x4[0], x5[0])), x3[0])≥NonInfC∧2422_0_NTH_LE(java.lang.Object(IntList(x0[0], x1[0])), java.lang.Object(IntList(x4[0], x5[0])), x3[0])≥COND_2422_0_NTH_LE(&&(&&(>(x7[0], -1), >(x5[0], 0)), <=(x3[0], 1)), java.lang.Object(IntList(x0[0], x1[0])), java.lang.Object(IntList(x4[0], x5[0])), x3[0], x7[0])∧(UIncreasing(COND_2422_0_NTH_LE(&&(&&(>(x7[0], -1), >(x5[0], 0)), <=(x3[0], 1)), java.lang.Object(IntList(x0[0], x1[0])), java.lang.Object(IntList(x4[0], x5[0])), x3[0], x7[0])), ≥))

We simplified constraint (19) using rules (I), (II), (III), (IV) which results in the following new constraint:

(20)    (&&(&&(>(x7[0], -1), >(x5[0], 0)), <=(+(x3[7], -1), 1))=TRUE2422_0_NTH_LE(java.lang.Object(IntList(java.lang.Object(IntList(x4[0], x5[0])), x1[7])), java.lang.Object(IntList(x4[0], x5[0])), +(x3[7], -1))≥NonInfC∧2422_0_NTH_LE(java.lang.Object(IntList(java.lang.Object(IntList(x4[0], x5[0])), x1[7])), java.lang.Object(IntList(x4[0], x5[0])), +(x3[7], -1))≥COND_2422_0_NTH_LE(&&(&&(>(x7[0], -1), >(x5[0], 0)), <=(+(x3[7], -1), 1)), java.lang.Object(IntList(java.lang.Object(IntList(x4[0], x5[0])), x1[7])), java.lang.Object(IntList(x4[0], x5[0])), +(x3[7], -1), x7[0])∧(UIncreasing(COND_2422_0_NTH_LE(&&(&&(>(x7[0], -1), >(x5[0], 0)), <=(x3[0], 1)), java.lang.Object(IntList(x0[0], x1[0])), java.lang.Object(IntList(x4[0], x5[0])), x3[0], x7[0])), ≥))

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

(21)    (0 ≥ 0 ⇒ (UIncreasing(COND_2422_0_NTH_LE(&&(&&(>(x7[0], -1), >(x5[0], 0)), <=(x3[0], 1)), java.lang.Object(IntList(x0[0], x1[0])), java.lang.Object(IntList(x4[0], x5[0])), x3[0], x7[0])), ≥)∧[bni_32 + (-1)Bound*bni_32] + [bni_32]x4[0] ≥ 0∧[(-1)bso_33] ≥ 0)

We simplified constraint (21) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:

(22)    (0 ≥ 0 ⇒ (UIncreasing(COND_2422_0_NTH_LE(&&(&&(>(x7[0], -1), >(x5[0], 0)), <=(x3[0], 1)), java.lang.Object(IntList(x0[0], x1[0])), java.lang.Object(IntList(x4[0], x5[0])), x3[0], x7[0])), ≥)∧[bni_32 + (-1)Bound*bni_32] + [bni_32]x4[0] ≥ 0∧[(-1)bso_33] ≥ 0)

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

(23)    (0 ≥ 0 ⇒ (UIncreasing(COND_2422_0_NTH_LE(&&(&&(>(x7[0], -1), >(x5[0], 0)), <=(x3[0], 1)), java.lang.Object(IntList(x0[0], x1[0])), java.lang.Object(IntList(x4[0], x5[0])), x3[0], x7[0])), ≥)∧[bni_32 + (-1)Bound*bni_32] + [bni_32]x4[0] ≥ 0∧[(-1)bso_33] ≥ 0)

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

(24)    (0 ≥ 0 ⇒ (UIncreasing(COND_2422_0_NTH_LE(&&(&&(>(x7[0], -1), >(x5[0], 0)), <=(x3[0], 1)), java.lang.Object(IntList(x0[0], x1[0])), java.lang.Object(IntList(x4[0], x5[0])), x3[0], x7[0])), ≥)∧0 ≥ 0∧[bni_32] ≥ 0∧0 ≥ 0∧[bni_32 + (-1)Bound*bni_32] ≥ 0∧0 ≥ 0∧0 ≥ 0∧0 ≥ 0∧0 ≥ 0∧[(-1)bso_33] ≥ 0)

For Pair COND_2422_0_NTH_LE(TRUE, java.lang.Object(IntList(x0, x1)), java.lang.Object(IntList(x4, x5)), x3, x7) → 2422_0_NTH_LE(x0, x0, x7) the following chains were created:
• We consider the chain 2422_0_NTH_LE(java.lang.Object(IntList(x0[0], x1[0])), java.lang.Object(IntList(x4[0], x5[0])), x3[0]) → COND_2422_0_NTH_LE(&&(&&(>(x7[0], -1), >(x5[0], 0)), <=(x3[0], 1)), java.lang.Object(IntList(x0[0], x1[0])), java.lang.Object(IntList(x4[0], x5[0])), x3[0], x7[0]), COND_2422_0_NTH_LE(TRUE, java.lang.Object(IntList(x0[1], x1[1])), java.lang.Object(IntList(x4[1], x5[1])), x3[1], x7[1]) → 2422_0_NTH_LE(x0[1], x0[1], x7[1]) which results in the following constraint:

(25)    (&&(&&(>(x7[0], -1), >(x5[0], 0)), <=(x3[0], 1))=TRUEjava.lang.Object(IntList(x0[0], x1[0]))=java.lang.Object(IntList(x0[1], x1[1]))∧java.lang.Object(IntList(x4[0], x5[0]))=java.lang.Object(IntList(x4[1], x5[1]))∧x3[0]=x3[1]x7[0]=x7[1]COND_2422_0_NTH_LE(TRUE, java.lang.Object(IntList(x0[1], x1[1])), java.lang.Object(IntList(x4[1], x5[1])), x3[1], x7[1])≥NonInfC∧COND_2422_0_NTH_LE(TRUE, java.lang.Object(IntList(x0[1], x1[1])), java.lang.Object(IntList(x4[1], x5[1])), x3[1], x7[1])≥2422_0_NTH_LE(x0[1], x0[1], x7[1])∧(UIncreasing(2422_0_NTH_LE(x0[1], x0[1], x7[1])), ≥))

We simplified constraint (25) using rules (I), (II), (III) which results in the following new constraint:

(26)    (&&(&&(>(x7[0], -1), >(x5[0], 0)), <=(x3[0], 1))=TRUECOND_2422_0_NTH_LE(TRUE, java.lang.Object(IntList(x0[0], x1[0])), java.lang.Object(IntList(x4[0], x5[0])), x3[0], x7[0])≥NonInfC∧COND_2422_0_NTH_LE(TRUE, java.lang.Object(IntList(x0[0], x1[0])), java.lang.Object(IntList(x4[0], x5[0])), x3[0], x7[0])≥2422_0_NTH_LE(x0[0], x0[0], x7[0])∧(UIncreasing(2422_0_NTH_LE(x0[1], x0[1], x7[1])), ≥))

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

(27)    (0 ≥ 0 ⇒ (UIncreasing(2422_0_NTH_LE(x0[1], x0[1], x7[1])), ≥)∧[(-1)Bound*bni_34] + [bni_34]x0[0] ≥ 0∧[1 + (-1)bso_35] ≥ 0)

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

(28)    (0 ≥ 0 ⇒ (UIncreasing(2422_0_NTH_LE(x0[1], x0[1], x7[1])), ≥)∧[(-1)Bound*bni_34] + [bni_34]x0[0] ≥ 0∧[1 + (-1)bso_35] ≥ 0)

We simplified constraint (28) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:

(29)    (0 ≥ 0 ⇒ (UIncreasing(2422_0_NTH_LE(x0[1], x0[1], x7[1])), ≥)∧[(-1)Bound*bni_34] + [bni_34]x0[0] ≥ 0∧[1 + (-1)bso_35] ≥ 0)

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

(30)    (0 ≥ 0 ⇒ (UIncreasing(2422_0_NTH_LE(x0[1], x0[1], x7[1])), ≥)∧0 ≥ 0∧0 ≥ 0∧0 ≥ 0∧0 ≥ 0∧0 ≥ 0∧[bni_34] ≥ 0∧[(-1)Bound*bni_34] ≥ 0∧0 ≥ 0∧0 ≥ 0∧0 ≥ 0∧0 ≥ 0∧0 ≥ 0∧0 ≥ 0∧[1 + (-1)bso_35] ≥ 0)

For Pair 2422_0_NTH_LE(java.lang.Object(IntList(x0, x1)), java.lang.Object(IntList(x0, x1)), x3) → COND_2422_0_NTH_LE1(&&(&&(>(x5, -1), <=(x3, 1)), >(x1, 0)), java.lang.Object(IntList(x0, x1)), java.lang.Object(IntList(x0, x1)), x3, x5) the following chains were created:
• We consider the chain COND_2422_0_NTH_LE(TRUE, java.lang.Object(IntList(x0[1], x1[1])), java.lang.Object(IntList(x4[1], x5[1])), x3[1], x7[1]) → 2422_0_NTH_LE(x0[1], x0[1], x7[1]), 2422_0_NTH_LE(java.lang.Object(IntList(x0[2], x1[2])), java.lang.Object(IntList(x0[2], x1[2])), x3[2]) → COND_2422_0_NTH_LE1(&&(&&(>(x5[2], -1), <=(x3[2], 1)), >(x1[2], 0)), java.lang.Object(IntList(x0[2], x1[2])), java.lang.Object(IntList(x0[2], x1[2])), x3[2], x5[2]), COND_2422_0_NTH_LE1(TRUE, java.lang.Object(IntList(x0[3], x1[3])), java.lang.Object(IntList(x0[3], x1[3])), x3[3], x5[3]) → 2422_0_NTH_LE(x0[3], x0[3], x5[3]) which results in the following constraint:

(31)    (x0[1]=java.lang.Object(IntList(x0[2], x1[2]))∧x7[1]=x3[2]&&(&&(>(x5[2], -1), <=(x3[2], 1)), >(x1[2], 0))=TRUEjava.lang.Object(IntList(x0[2], x1[2]))=java.lang.Object(IntList(x0[3], x1[3]))∧x3[2]=x3[3]x5[2]=x5[3]2422_0_NTH_LE(java.lang.Object(IntList(x0[2], x1[2])), java.lang.Object(IntList(x0[2], x1[2])), x3[2])≥NonInfC∧2422_0_NTH_LE(java.lang.Object(IntList(x0[2], x1[2])), java.lang.Object(IntList(x0[2], x1[2])), x3[2])≥COND_2422_0_NTH_LE1(&&(&&(>(x5[2], -1), <=(x3[2], 1)), >(x1[2], 0)), java.lang.Object(IntList(x0[2], x1[2])), java.lang.Object(IntList(x0[2], x1[2])), x3[2], x5[2])∧(UIncreasing(COND_2422_0_NTH_LE1(&&(&&(>(x5[2], -1), <=(x3[2], 1)), >(x1[2], 0)), java.lang.Object(IntList(x0[2], x1[2])), java.lang.Object(IntList(x0[2], x1[2])), x3[2], x5[2])), ≥))

We simplified constraint (31) using rules (I), (II), (III), (IV) which results in the following new constraint:

(32)    (&&(&&(>(x5[2], -1), <=(x3[2], 1)), >(x1[2], 0))=TRUE2422_0_NTH_LE(java.lang.Object(IntList(x0[2], x1[2])), java.lang.Object(IntList(x0[2], x1[2])), x3[2])≥NonInfC∧2422_0_NTH_LE(java.lang.Object(IntList(x0[2], x1[2])), java.lang.Object(IntList(x0[2], x1[2])), x3[2])≥COND_2422_0_NTH_LE1(&&(&&(>(x5[2], -1), <=(x3[2], 1)), >(x1[2], 0)), java.lang.Object(IntList(x0[2], x1[2])), java.lang.Object(IntList(x0[2], x1[2])), x3[2], x5[2])∧(UIncreasing(COND_2422_0_NTH_LE1(&&(&&(>(x5[2], -1), <=(x3[2], 1)), >(x1[2], 0)), java.lang.Object(IntList(x0[2], x1[2])), java.lang.Object(IntList(x0[2], x1[2])), x3[2], x5[2])), ≥))

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

(33)    (0 ≥ 0 ⇒ (UIncreasing(COND_2422_0_NTH_LE1(&&(&&(>(x5[2], -1), <=(x3[2], 1)), >(x1[2], 0)), java.lang.Object(IntList(x0[2], x1[2])), java.lang.Object(IntList(x0[2], x1[2])), x3[2], x5[2])), ≥)∧[(-1)Bound*bni_36] + [bni_36]x0[2] ≥ 0∧[(-1)bso_37] ≥ 0)

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

(34)    (0 ≥ 0 ⇒ (UIncreasing(COND_2422_0_NTH_LE1(&&(&&(>(x5[2], -1), <=(x3[2], 1)), >(x1[2], 0)), java.lang.Object(IntList(x0[2], x1[2])), java.lang.Object(IntList(x0[2], x1[2])), x3[2], x5[2])), ≥)∧[(-1)Bound*bni_36] + [bni_36]x0[2] ≥ 0∧[(-1)bso_37] ≥ 0)

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

(35)    (0 ≥ 0 ⇒ (UIncreasing(COND_2422_0_NTH_LE1(&&(&&(>(x5[2], -1), <=(x3[2], 1)), >(x1[2], 0)), java.lang.Object(IntList(x0[2], x1[2])), java.lang.Object(IntList(x0[2], x1[2])), x3[2], x5[2])), ≥)∧[(-1)Bound*bni_36] + [bni_36]x0[2] ≥ 0∧[(-1)bso_37] ≥ 0)

We simplified constraint (35) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint:

(36)    (0 ≥ 0 ⇒ (UIncreasing(COND_2422_0_NTH_LE1(&&(&&(>(x5[2], -1), <=(x3[2], 1)), >(x1[2], 0)), java.lang.Object(IntList(x0[2], x1[2])), java.lang.Object(IntList(x0[2], x1[2])), x3[2], x5[2])), ≥)∧0 ≥ 0∧0 ≥ 0∧[bni_36] ≥ 0∧[(-1)Bound*bni_36] ≥ 0∧0 ≥ 0∧0 ≥ 0∧0 ≥ 0∧0 ≥ 0∧[(-1)bso_37] ≥ 0)

• We consider the chain COND_2422_0_NTH_LE1(TRUE, java.lang.Object(IntList(x0[3], x1[3])), java.lang.Object(IntList(x0[3], x1[3])), x3[3], x5[3]) → 2422_0_NTH_LE(x0[3], x0[3], x5[3]), 2422_0_NTH_LE(java.lang.Object(IntList(x0[2], x1[2])), java.lang.Object(IntList(x0[2], x1[2])), x3[2]) → COND_2422_0_NTH_LE1(&&(&&(>(x5[2], -1), <=(x3[2], 1)), >(x1[2], 0)), java.lang.Object(IntList(x0[2], x1[2])), java.lang.Object(IntList(x0[2], x1[2])), x3[2], x5[2]), COND_2422_0_NTH_LE1(TRUE, java.lang.Object(IntList(x0[3], x1[3])), java.lang.Object(IntList(x0[3], x1[3])), x3[3], x5[3]) → 2422_0_NTH_LE(x0[3], x0[3], x5[3]) which results in the following constraint:

(37)    (x0[3]=java.lang.Object(IntList(x0[2], x1[2]))∧x5[3]=x3[2]&&(&&(>(x5[2], -1), <=(x3[2], 1)), >(x1[2], 0))=TRUEjava.lang.Object(IntList(x0[2], x1[2]))=java.lang.Object(IntList(x0[3]1, x1[3]1))∧x3[2]=x3[3]1x5[2]=x5[3]12422_0_NTH_LE(java.lang.Object(IntList(x0[2], x1[2])), java.lang.Object(IntList(x0[2], x1[2])), x3[2])≥NonInfC∧2422_0_NTH_LE(java.lang.Object(IntList(x0[2], x1[2])), java.lang.Object(IntList(x0[2], x1[2])), x3[2])≥COND_2422_0_NTH_LE1(&&(&&(>(x5[2], -1), <=(x3[2], 1)), >(x1[2], 0)), java.lang.Object(IntList(x0[2], x1[2])), java.lang.Object(IntList(x0[2], x1[2])), x3[2], x5[2])∧(UIncreasing(COND_2422_0_NTH_LE1(&&(&&(>(x5[2], -1), <=(x3[2], 1)), >(x1[2], 0)), java.lang.Object(IntList(x0[2], x1[2])), java.lang.Object(IntList(x0[2], x1[2])), x3[2], x5[2])), ≥))

We simplified constraint (37) using rules (I), (II), (III), (IV) which results in the following new constraint:

(38)    (&&(&&(>(x5[2], -1), <=(x3[2], 1)), >(x1[2], 0))=TRUE2422_0_NTH_LE(java.lang.Object(IntList(x0[2], x1[2])), java.lang.Object(IntList(x0[2], x1[2])), x3[2])≥NonInfC∧2422_0_NTH_LE(java.lang.Object(IntList(x0[2], x1[2])), java.lang.Object(IntList(x0[2], x1[2])), x3[2])≥COND_2422_0_NTH_LE1(&&(&&(>(x5[2], -1), <=(x3[2], 1)), >(x1[2], 0)), java.lang.Object(IntList(x0[2], x1[2])), java.lang.Object(IntList(x0[2], x1[2])), x3[2], x5[2])∧(UIncreasing(COND_2422_0_NTH_LE1(&&(&&(>(x5[2], -1), <=(x3[2], 1)), >(x1[2], 0)), java.lang.Object(IntList(x0[2], x1[2])), java.lang.Object(IntList(x0[2], x1[2])), x3[2], x5[2])), ≥))

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

(39)    (0 ≥ 0 ⇒ (UIncreasing(COND_2422_0_NTH_LE1(&&(&&(>(x5[2], -1), <=(x3[2], 1)), >(x1[2], 0)), java.lang.Object(IntList(x0[2], x1[2])), java.lang.Object(IntList(x0[2], x1[2])), x3[2], x5[2])), ≥)∧[(-1)Bound*bni_36] + [bni_36]x0[2] ≥ 0∧[(-1)bso_37] ≥ 0)

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

(40)    (0 ≥ 0 ⇒ (UIncreasing(COND_2422_0_NTH_LE1(&&(&&(>(x5[2], -1), <=(x3[2], 1)), >(x1[2], 0)), java.lang.Object(IntList(x0[2], x1[2])), java.lang.Object(IntList(x0[2], x1[2])), x3[2], x5[2])), ≥)∧[(-1)Bound*bni_36] + [bni_36]x0[2] ≥ 0∧[(-1)bso_37] ≥ 0)

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

(41)    (0 ≥ 0 ⇒ (UIncreasing(COND_2422_0_NTH_LE1(&&(&&(>(x5[2], -1), <=(x3[2], 1)), >(x1[2], 0)), java.lang.Object(IntList(x0[2], x1[2])), java.lang.Object(IntList(x0[2], x1[2])), x3[2], x5[2])), ≥)∧[(-1)Bound*bni_36] + [bni_36]x0[2] ≥ 0∧[(-1)bso_37] ≥ 0)

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

(42)    (0 ≥ 0 ⇒ (UIncreasing(COND_2422_0_NTH_LE1(&&(&&(>(x5[2], -1), <=(x3[2], 1)), >(x1[2], 0)), java.lang.Object(IntList(x0[2], x1[2])), java.lang.Object(IntList(x0[2], x1[2])), x3[2], x5[2])), ≥)∧0 ≥ 0∧0 ≥ 0∧[bni_36] ≥ 0∧[(-1)Bound*bni_36] ≥ 0∧0 ≥ 0∧0 ≥ 0∧0 ≥ 0∧0 ≥ 0∧[(-1)bso_37] ≥ 0)

• We consider the chain COND_2422_0_NTH_LE2(TRUE, x0[5], java.lang.Object(IntList(x3[5], x4[5])), x2[5]) → 2422_0_NTH_LE(x0[5], x3[5], +(x2[5], -1)), 2422_0_NTH_LE(java.lang.Object(IntList(x0[2], x1[2])), java.lang.Object(IntList(x0[2], x1[2])), x3[2]) → COND_2422_0_NTH_LE1(&&(&&(>(x5[2], -1), <=(x3[2], 1)), >(x1[2], 0)), java.lang.Object(IntList(x0[2], x1[2])), java.lang.Object(IntList(x0[2], x1[2])), x3[2], x5[2]), COND_2422_0_NTH_LE1(TRUE, java.lang.Object(IntList(x0[3], x1[3])), java.lang.Object(IntList(x0[3], x1[3])), x3[3], x5[3]) → 2422_0_NTH_LE(x0[3], x0[3], x5[3]) which results in the following constraint:

(43)    (x0[5]=java.lang.Object(IntList(x0[2], x1[2]))∧x3[5]=java.lang.Object(IntList(x0[2], x1[2]))∧+(x2[5], -1)=x3[2]&&(&&(>(x5[2], -1), <=(x3[2], 1)), >(x1[2], 0))=TRUEjava.lang.Object(IntList(x0[2], x1[2]))=java.lang.Object(IntList(x0[3], x1[3]))∧x3[2]=x3[3]x5[2]=x5[3]2422_0_NTH_LE(java.lang.Object(IntList(x0[2], x1[2])), java.lang.Object(IntList(x0[2], x1[2])), x3[2])≥NonInfC∧2422_0_NTH_LE(java.lang.Object(IntList(x0[2], x1[2])), java.lang.Object(IntList(x0[2], x1[2])), x3[2])≥COND_2422_0_NTH_LE1(&&(&&(>(x5[2], -1), <=(x3[2], 1)), >(x1[2], 0)), java.lang.Object(IntList(x0[2], x1[2])), java.lang.Object(IntList(x0[2], x1[2])), x3[2], x5[2])∧(UIncreasing(COND_2422_0_NTH_LE1(&&(&&(>(x5[2], -1), <=(x3[2], 1)), >(x1[2], 0)), java.lang.Object(IntList(x0[2], x1[2])), java.lang.Object(IntList(x0[2], x1[2])), x3[2], x5[2])), ≥))

We simplified constraint (43) using rules (I), (II), (III), (IV) which results in the following new constraint:

(44)    (&&(&&(>(x5[2], -1), <=(+(x2[5], -1), 1)), >(x1[2], 0))=TRUE2422_0_NTH_LE(java.lang.Object(IntList(x0[2], x1[2])), java.lang.Object(IntList(x0[2], x1[2])), +(x2[5], -1))≥NonInfC∧2422_0_NTH_LE(java.lang.Object(IntList(x0[2], x1[2])), java.lang.Object(IntList(x0[2], x1[2])), +(x2[5], -1))≥COND_2422_0_NTH_LE1(&&(&&(>(x5[2], -1), <=(+(x2[5], -1), 1)), >(x1[2], 0)), java.lang.Object(IntList(x0[2], x1[2])), java.lang.Object(IntList(x0[2], x1[2])), +(x2[5], -1), x5[2])∧(UIncreasing(COND_2422_0_NTH_LE1(&&(&&(>(x5[2], -1), <=(x3[2], 1)), >(x1[2], 0)), java.lang.Object(IntList(x0[2], x1[2])), java.lang.Object(IntList(x0[2], x1[2])), x3[2], x5[2])), ≥))

We simplified constraint (44) using rule (POLY_CONSTRAINTS) which results in the following new constraint:

(45)    (0 ≥ 0 ⇒ (UIncreasing(COND_2422_0_NTH_LE1(&&(&&(>(x5[2], -1), <=(x3[2], 1)), >(x1[2], 0)), java.lang.Object(IntList(x0[2], x1[2])), java.lang.Object(IntList(x0[2], x1[2])), x3[2], x5[2])), ≥)∧[(-1)Bound*bni_36] + [bni_36]x0[2] ≥ 0∧[(-1)bso_37] ≥ 0)

We simplified constraint (45) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:

(46)    (0 ≥ 0 ⇒ (UIncreasing(COND_2422_0_NTH_LE1(&&(&&(>(x5[2], -1), <=(x3[2], 1)), >(x1[2], 0)), java.lang.Object(IntList(x0[2], x1[2])), java.lang.Object(IntList(x0[2], x1[2])), x3[2], x5[2])), ≥)∧[(-1)Bound*bni_36] + [bni_36]x0[2] ≥ 0∧[(-1)bso_37] ≥ 0)

We simplified constraint (46) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:

(47)    (0 ≥ 0 ⇒ (UIncreasing(COND_2422_0_NTH_LE1(&&(&&(>(x5[2], -1), <=(x3[2], 1)), >(x1[2], 0)), java.lang.Object(IntList(x0[2], x1[2])), java.lang.Object(IntList(x0[2], x1[2])), x3[2], x5[2])), ≥)∧[(-1)Bound*bni_36] + [bni_36]x0[2] ≥ 0∧[(-1)bso_37] ≥ 0)

We simplified constraint (47) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint:

(48)    (0 ≥ 0 ⇒ (UIncreasing(COND_2422_0_NTH_LE1(&&(&&(>(x5[2], -1), <=(x3[2], 1)), >(x1[2], 0)), java.lang.Object(IntList(x0[2], x1[2])), java.lang.Object(IntList(x0[2], x1[2])), x3[2], x5[2])), ≥)∧0 ≥ 0∧[bni_36] ≥ 0∧[(-1)Bound*bni_36] ≥ 0∧0 ≥ 0∧0 ≥ 0∧0 ≥ 0∧[(-1)bso_37] ≥ 0)

• We consider the chain COND_2422_0_NTH_LE3(TRUE, java.lang.Object(IntList(x0[7], x1[7])), java.lang.Object(IntList(x0[7], x1[7])), x3[7]) → 2422_0_NTH_LE(java.lang.Object(IntList(x0[7], x1[7])), x0[7], +(x3[7], -1)), 2422_0_NTH_LE(java.lang.Object(IntList(x0[2], x1[2])), java.lang.Object(IntList(x0[2], x1[2])), x3[2]) → COND_2422_0_NTH_LE1(&&(&&(>(x5[2], -1), <=(x3[2], 1)), >(x1[2], 0)), java.lang.Object(IntList(x0[2], x1[2])), java.lang.Object(IntList(x0[2], x1[2])), x3[2], x5[2]), COND_2422_0_NTH_LE1(TRUE, java.lang.Object(IntList(x0[3], x1[3])), java.lang.Object(IntList(x0[3], x1[3])), x3[3], x5[3]) → 2422_0_NTH_LE(x0[3], x0[3], x5[3]) which results in the following constraint:

(49)    (java.lang.Object(IntList(x0[7], x1[7]))=java.lang.Object(IntList(x0[2], x1[2]))∧x0[7]=java.lang.Object(IntList(x0[2], x1[2]))∧+(x3[7], -1)=x3[2]&&(&&(>(x5[2], -1), <=(x3[2], 1)), >(x1[2], 0))=TRUEjava.lang.Object(IntList(x0[2], x1[2]))=java.lang.Object(IntList(x0[3], x1[3]))∧x3[2]=x3[3]x5[2]=x5[3]2422_0_NTH_LE(java.lang.Object(IntList(x0[2], x1[2])), java.lang.Object(IntList(x0[2], x1[2])), x3[2])≥NonInfC∧2422_0_NTH_LE(java.lang.Object(IntList(x0[2], x1[2])), java.lang.Object(IntList(x0[2], x1[2])), x3[2])≥COND_2422_0_NTH_LE1(&&(&&(>(x5[2], -1), <=(x3[2], 1)), >(x1[2], 0)), java.lang.Object(IntList(x0[2], x1[2])), java.lang.Object(IntList(x0[2], x1[2])), x3[2], x5[2])∧(UIncreasing(COND_2422_0_NTH_LE1(&&(&&(>(x5[2], -1), <=(x3[2], 1)), >(x1[2], 0)), java.lang.Object(IntList(x0[2], x1[2])), java.lang.Object(IntList(x0[2], x1[2])), x3[2], x5[2])), ≥))

We simplified constraint (49) using rules (I), (II), (III), (IV) which results in the following new constraint:

(50)    (&&(&&(>(x5[2], -1), <=(+(x3[7], -1), 1)), >(x1[2], 0))=TRUE2422_0_NTH_LE(java.lang.Object(IntList(java.lang.Object(IntList(x0[7], x1[2])), x1[2])), java.lang.Object(IntList(java.lang.Object(IntList(x0[7], x1[2])), x1[2])), +(x3[7], -1))≥NonInfC∧2422_0_NTH_LE(java.lang.Object(IntList(java.lang.Object(IntList(x0[7], x1[2])), x1[2])), java.lang.Object(IntList(java.lang.Object(IntList(x0[7], x1[2])), x1[2])), +(x3[7], -1))≥COND_2422_0_NTH_LE1(&&(&&(>(x5[2], -1), <=(+(x3[7], -1), 1)), >(x1[2], 0)), java.lang.Object(IntList(java.lang.Object(IntList(x0[7], x1[2])), x1[2])), java.lang.Object(IntList(java.lang.Object(IntList(x0[7], x1[2])), x1[2])), +(x3[7], -1), x5[2])∧(UIncreasing(COND_2422_0_NTH_LE1(&&(&&(>(x5[2], -1), <=(x3[2], 1)), >(x1[2], 0)), java.lang.Object(IntList(x0[2], x1[2])), java.lang.Object(IntList(x0[2], x1[2])), x3[2], x5[2])), ≥))

We simplified constraint (50) using rule (POLY_CONSTRAINTS) which results in the following new constraint:

(51)    (0 ≥ 0 ⇒ (UIncreasing(COND_2422_0_NTH_LE1(&&(&&(>(x5[2], -1), <=(x3[2], 1)), >(x1[2], 0)), java.lang.Object(IntList(x0[2], x1[2])), java.lang.Object(IntList(x0[2], x1[2])), x3[2], x5[2])), ≥)∧[bni_36 + (-1)Bound*bni_36] + [bni_36]x0[7] ≥ 0∧[(-1)bso_37] ≥ 0)

We simplified constraint (51) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:

(52)    (0 ≥ 0 ⇒ (UIncreasing(COND_2422_0_NTH_LE1(&&(&&(>(x5[2], -1), <=(x3[2], 1)), >(x1[2], 0)), java.lang.Object(IntList(x0[2], x1[2])), java.lang.Object(IntList(x0[2], x1[2])), x3[2], x5[2])), ≥)∧[bni_36 + (-1)Bound*bni_36] + [bni_36]x0[7] ≥ 0∧[(-1)bso_37] ≥ 0)

We simplified constraint (52) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:

(53)    (0 ≥ 0 ⇒ (UIncreasing(COND_2422_0_NTH_LE1(&&(&&(>(x5[2], -1), <=(x3[2], 1)), >(x1[2], 0)), java.lang.Object(IntList(x0[2], x1[2])), java.lang.Object(IntList(x0[2], x1[2])), x3[2], x5[2])), ≥)∧[bni_36 + (-1)Bound*bni_36] + [bni_36]x0[7] ≥ 0∧[(-1)bso_37] ≥ 0)

We simplified constraint (53) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint:

(54)    (0 ≥ 0 ⇒ (UIncreasing(COND_2422_0_NTH_LE1(&&(&&(>(x5[2], -1), <=(x3[2], 1)), >(x1[2], 0)), java.lang.Object(IntList(x0[2], x1[2])), java.lang.Object(IntList(x0[2], x1[2])), x3[2], x5[2])), ≥)∧0 ≥ 0∧[bni_36] ≥ 0∧[bni_36 + (-1)Bound*bni_36] ≥ 0∧0 ≥ 0∧0 ≥ 0∧0 ≥ 0∧[(-1)bso_37] ≥ 0)

For Pair COND_2422_0_NTH_LE1(TRUE, java.lang.Object(IntList(x0, x1)), java.lang.Object(IntList(x0, x1)), x3, x5) → 2422_0_NTH_LE(x0, x0, x5) the following chains were created:
• We consider the chain 2422_0_NTH_LE(java.lang.Object(IntList(x0[2], x1[2])), java.lang.Object(IntList(x0[2], x1[2])), x3[2]) → COND_2422_0_NTH_LE1(&&(&&(>(x5[2], -1), <=(x3[2], 1)), >(x1[2], 0)), java.lang.Object(IntList(x0[2], x1[2])), java.lang.Object(IntList(x0[2], x1[2])), x3[2], x5[2]), COND_2422_0_NTH_LE1(TRUE, java.lang.Object(IntList(x0[3], x1[3])), java.lang.Object(IntList(x0[3], x1[3])), x3[3], x5[3]) → 2422_0_NTH_LE(x0[3], x0[3], x5[3]) which results in the following constraint:

(55)    (&&(&&(>(x5[2], -1), <=(x3[2], 1)), >(x1[2], 0))=TRUEjava.lang.Object(IntList(x0[2], x1[2]))=java.lang.Object(IntList(x0[3], x1[3]))∧x3[2]=x3[3]x5[2]=x5[3]COND_2422_0_NTH_LE1(TRUE, java.lang.Object(IntList(x0[3], x1[3])), java.lang.Object(IntList(x0[3], x1[3])), x3[3], x5[3])≥NonInfC∧COND_2422_0_NTH_LE1(TRUE, java.lang.Object(IntList(x0[3], x1[3])), java.lang.Object(IntList(x0[3], x1[3])), x3[3], x5[3])≥2422_0_NTH_LE(x0[3], x0[3], x5[3])∧(UIncreasing(2422_0_NTH_LE(x0[3], x0[3], x5[3])), ≥))

We simplified constraint (55) using rules (I), (II), (III) which results in the following new constraint:

(56)    (&&(&&(>(x5[2], -1), <=(x3[2], 1)), >(x1[2], 0))=TRUECOND_2422_0_NTH_LE1(TRUE, java.lang.Object(IntList(x0[2], x1[2])), java.lang.Object(IntList(x0[2], x1[2])), x3[2], x5[2])≥NonInfC∧COND_2422_0_NTH_LE1(TRUE, java.lang.Object(IntList(x0[2], x1[2])), java.lang.Object(IntList(x0[2], x1[2])), x3[2], x5[2])≥2422_0_NTH_LE(x0[2], x0[2], x5[2])∧(UIncreasing(2422_0_NTH_LE(x0[3], x0[3], x5[3])), ≥))

We simplified constraint (56) using rule (POLY_CONSTRAINTS) which results in the following new constraint:

(57)    (0 ≥ 0 ⇒ (UIncreasing(2422_0_NTH_LE(x0[3], x0[3], x5[3])), ≥)∧[(-1)Bound*bni_38] + [bni_38]x0[2] ≥ 0∧[1 + (-1)bso_39] ≥ 0)

We simplified constraint (57) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:

(58)    (0 ≥ 0 ⇒ (UIncreasing(2422_0_NTH_LE(x0[3], x0[3], x5[3])), ≥)∧[(-1)Bound*bni_38] + [bni_38]x0[2] ≥ 0∧[1 + (-1)bso_39] ≥ 0)

We simplified constraint (58) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:

(59)    (0 ≥ 0 ⇒ (UIncreasing(2422_0_NTH_LE(x0[3], x0[3], x5[3])), ≥)∧[(-1)Bound*bni_38] + [bni_38]x0[2] ≥ 0∧[1 + (-1)bso_39] ≥ 0)

We simplified constraint (59) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint:

(60)    (0 ≥ 0 ⇒ (UIncreasing(2422_0_NTH_LE(x0[3], x0[3], x5[3])), ≥)∧0 ≥ 0∧0 ≥ 0∧0 ≥ 0∧[bni_38] ≥ 0∧[(-1)Bound*bni_38] ≥ 0∧0 ≥ 0∧0 ≥ 0∧0 ≥ 0∧0 ≥ 0∧[1 + (-1)bso_39] ≥ 0)

For Pair 2422_0_NTH_LE(x0, java.lang.Object(IntList(x3, x4)), x2) → COND_2422_0_NTH_LE2(>(x2, 1), x0, java.lang.Object(IntList(x3, x4)), x2) the following chains were created:
• We consider the chain 2422_0_NTH_LE(x0[4], java.lang.Object(IntList(x3[4], x4[4])), x2[4]) → COND_2422_0_NTH_LE2(>(x2[4], 1), x0[4], java.lang.Object(IntList(x3[4], x4[4])), x2[4]), COND_2422_0_NTH_LE2(TRUE, x0[5], java.lang.Object(IntList(x3[5], x4[5])), x2[5]) → 2422_0_NTH_LE(x0[5], x3[5], +(x2[5], -1)) which results in the following constraint:

(61)    (>(x2[4], 1)=TRUEx0[4]=x0[5]java.lang.Object(IntList(x3[4], x4[4]))=java.lang.Object(IntList(x3[5], x4[5]))∧x2[4]=x2[5]2422_0_NTH_LE(x0[4], java.lang.Object(IntList(x3[4], x4[4])), x2[4])≥NonInfC∧2422_0_NTH_LE(x0[4], java.lang.Object(IntList(x3[4], x4[4])), x2[4])≥COND_2422_0_NTH_LE2(>(x2[4], 1), x0[4], java.lang.Object(IntList(x3[4], x4[4])), x2[4])∧(UIncreasing(COND_2422_0_NTH_LE2(>(x2[4], 1), x0[4], java.lang.Object(IntList(x3[4], x4[4])), x2[4])), ≥))

We simplified constraint (61) using rules (I), (II), (IV) which results in the following new constraint:

(62)    (>(x2[4], 1)=TRUE2422_0_NTH_LE(x0[4], java.lang.Object(IntList(x3[4], x4[4])), x2[4])≥NonInfC∧2422_0_NTH_LE(x0[4], java.lang.Object(IntList(x3[4], x4[4])), x2[4])≥COND_2422_0_NTH_LE2(>(x2[4], 1), x0[4], java.lang.Object(IntList(x3[4], x4[4])), x2[4])∧(UIncreasing(COND_2422_0_NTH_LE2(>(x2[4], 1), x0[4], java.lang.Object(IntList(x3[4], x4[4])), x2[4])), ≥))

We simplified constraint (62) using rule (POLY_CONSTRAINTS) which results in the following new constraint:

(63)    (0 ≥ 0 ⇒ (UIncreasing(COND_2422_0_NTH_LE2(>(x2[4], 1), x0[4], java.lang.Object(IntList(x3[4], x4[4])), x2[4])), ≥)∧[(-1)bni_40 + (-1)Bound*bni_40] + [bni_40]x0[4] ≥ 0∧[(-1)bso_41] ≥ 0)

We simplified constraint (63) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:

(64)    (0 ≥ 0 ⇒ (UIncreasing(COND_2422_0_NTH_LE2(>(x2[4], 1), x0[4], java.lang.Object(IntList(x3[4], x4[4])), x2[4])), ≥)∧[(-1)bni_40 + (-1)Bound*bni_40] + [bni_40]x0[4] ≥ 0∧[(-1)bso_41] ≥ 0)

We simplified constraint (64) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:

(65)    (0 ≥ 0 ⇒ (UIncreasing(COND_2422_0_NTH_LE2(>(x2[4], 1), x0[4], java.lang.Object(IntList(x3[4], x4[4])), x2[4])), ≥)∧[(-1)bni_40 + (-1)Bound*bni_40] + [bni_40]x0[4] ≥ 0∧[(-1)bso_41] ≥ 0)

We simplified constraint (65) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint:

(66)    (0 ≥ 0 ⇒ (UIncreasing(COND_2422_0_NTH_LE2(>(x2[4], 1), x0[4], java.lang.Object(IntList(x3[4], x4[4])), x2[4])), ≥)∧0 ≥ 0∧0 ≥ 0∧0 ≥ 0∧[bni_40] ≥ 0∧[(-1)bni_40 + (-1)Bound*bni_40] ≥ 0∧0 ≥ 0∧0 ≥ 0∧0 ≥ 0∧0 ≥ 0∧[(-1)bso_41] ≥ 0)

For Pair COND_2422_0_NTH_LE2(TRUE, x0, java.lang.Object(IntList(x3, x4)), x2) → 2422_0_NTH_LE(x0, x3, +(x2, -1)) the following chains were created:
• We consider the chain 2422_0_NTH_LE(x0[4], java.lang.Object(IntList(x3[4], x4[4])), x2[4]) → COND_2422_0_NTH_LE2(>(x2[4], 1), x0[4], java.lang.Object(IntList(x3[4], x4[4])), x2[4]), COND_2422_0_NTH_LE2(TRUE, x0[5], java.lang.Object(IntList(x3[5], x4[5])), x2[5]) → 2422_0_NTH_LE(x0[5], x3[5], +(x2[5], -1)) which results in the following constraint:

(67)    (>(x2[4], 1)=TRUEx0[4]=x0[5]java.lang.Object(IntList(x3[4], x4[4]))=java.lang.Object(IntList(x3[5], x4[5]))∧x2[4]=x2[5]COND_2422_0_NTH_LE2(TRUE, x0[5], java.lang.Object(IntList(x3[5], x4[5])), x2[5])≥NonInfC∧COND_2422_0_NTH_LE2(TRUE, x0[5], java.lang.Object(IntList(x3[5], x4[5])), x2[5])≥2422_0_NTH_LE(x0[5], x3[5], +(x2[5], -1))∧(UIncreasing(2422_0_NTH_LE(x0[5], x3[5], +(x2[5], -1))), ≥))

We simplified constraint (67) using rules (I), (II), (III) which results in the following new constraint:

(68)    (>(x2[4], 1)=TRUECOND_2422_0_NTH_LE2(TRUE, x0[4], java.lang.Object(IntList(x3[4], x4[4])), x2[4])≥NonInfC∧COND_2422_0_NTH_LE2(TRUE, x0[4], java.lang.Object(IntList(x3[4], x4[4])), x2[4])≥2422_0_NTH_LE(x0[4], x3[4], +(x2[4], -1))∧(UIncreasing(2422_0_NTH_LE(x0[5], x3[5], +(x2[5], -1))), ≥))

We simplified constraint (68) using rule (POLY_CONSTRAINTS) which results in the following new constraint:

(69)    (0 ≥ 0 ⇒ (UIncreasing(2422_0_NTH_LE(x0[5], x3[5], +(x2[5], -1))), ≥)∧[(-1)bni_42 + (-1)Bound*bni_42] + [bni_42]x0[4] ≥ 0∧[(-1)bso_43] ≥ 0)

We simplified constraint (69) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:

(70)    (0 ≥ 0 ⇒ (UIncreasing(2422_0_NTH_LE(x0[5], x3[5], +(x2[5], -1))), ≥)∧[(-1)bni_42 + (-1)Bound*bni_42] + [bni_42]x0[4] ≥ 0∧[(-1)bso_43] ≥ 0)

We simplified constraint (70) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:

(71)    (0 ≥ 0 ⇒ (UIncreasing(2422_0_NTH_LE(x0[5], x3[5], +(x2[5], -1))), ≥)∧[(-1)bni_42 + (-1)Bound*bni_42] + [bni_42]x0[4] ≥ 0∧[(-1)bso_43] ≥ 0)

We simplified constraint (71) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint:

(72)    (0 ≥ 0 ⇒ (UIncreasing(2422_0_NTH_LE(x0[5], x3[5], +(x2[5], -1))), ≥)∧0 ≥ 0∧0 ≥ 0∧0 ≥ 0∧[bni_42] ≥ 0∧[(-1)bni_42 + (-1)Bound*bni_42] ≥ 0∧0 ≥ 0∧0 ≥ 0∧0 ≥ 0∧0 ≥ 0∧[(-1)bso_43] ≥ 0)

For Pair 2422_0_NTH_LE(java.lang.Object(IntList(x0, x1)), java.lang.Object(IntList(x0, x1)), x3) → COND_2422_0_NTH_LE3(>(x3, 1), java.lang.Object(IntList(x0, x1)), java.lang.Object(IntList(x0, x1)), x3) the following chains were created:
• We consider the chain 2422_0_NTH_LE(java.lang.Object(IntList(x0[6], x1[6])), java.lang.Object(IntList(x0[6], x1[6])), x3[6]) → COND_2422_0_NTH_LE3(>(x3[6], 1), java.lang.Object(IntList(x0[6], x1[6])), java.lang.Object(IntList(x0[6], x1[6])), x3[6]), COND_2422_0_NTH_LE3(TRUE, java.lang.Object(IntList(x0[7], x1[7])), java.lang.Object(IntList(x0[7], x1[7])), x3[7]) → 2422_0_NTH_LE(java.lang.Object(IntList(x0[7], x1[7])), x0[7], +(x3[7], -1)) which results in the following constraint:

(73)    (>(x3[6], 1)=TRUEjava.lang.Object(IntList(x0[6], x1[6]))=java.lang.Object(IntList(x0[7], x1[7]))∧x3[6]=x3[7]2422_0_NTH_LE(java.lang.Object(IntList(x0[6], x1[6])), java.lang.Object(IntList(x0[6], x1[6])), x3[6])≥NonInfC∧2422_0_NTH_LE(java.lang.Object(IntList(x0[6], x1[6])), java.lang.Object(IntList(x0[6], x1[6])), x3[6])≥COND_2422_0_NTH_LE3(>(x3[6], 1), java.lang.Object(IntList(x0[6], x1[6])), java.lang.Object(IntList(x0[6], x1[6])), x3[6])∧(UIncreasing(COND_2422_0_NTH_LE3(>(x3[6], 1), java.lang.Object(IntList(x0[6], x1[6])), java.lang.Object(IntList(x0[6], x1[6])), x3[6])), ≥))

We simplified constraint (73) using rules (I), (II), (IV) which results in the following new constraint:

(74)    (>(x3[6], 1)=TRUE2422_0_NTH_LE(java.lang.Object(IntList(x0[6], x1[6])), java.lang.Object(IntList(x0[6], x1[6])), x3[6])≥NonInfC∧2422_0_NTH_LE(java.lang.Object(IntList(x0[6], x1[6])), java.lang.Object(IntList(x0[6], x1[6])), x3[6])≥COND_2422_0_NTH_LE3(>(x3[6], 1), java.lang.Object(IntList(x0[6], x1[6])), java.lang.Object(IntList(x0[6], x1[6])), x3[6])∧(UIncreasing(COND_2422_0_NTH_LE3(>(x3[6], 1), java.lang.Object(IntList(x0[6], x1[6])), java.lang.Object(IntList(x0[6], x1[6])), x3[6])), ≥))

We simplified constraint (74) using rule (POLY_CONSTRAINTS) which results in the following new constraint:

(75)    (0 ≥ 0 ⇒ (UIncreasing(COND_2422_0_NTH_LE3(>(x3[6], 1), java.lang.Object(IntList(x0[6], x1[6])), java.lang.Object(IntList(x0[6], x1[6])), x3[6])), ≥)∧[(-1)Bound*bni_44] + [bni_44]x0[6] ≥ 0∧[(-1)bso_45] ≥ 0)

We simplified constraint (75) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:

(76)    (0 ≥ 0 ⇒ (UIncreasing(COND_2422_0_NTH_LE3(>(x3[6], 1), java.lang.Object(IntList(x0[6], x1[6])), java.lang.Object(IntList(x0[6], x1[6])), x3[6])), ≥)∧[(-1)Bound*bni_44] + [bni_44]x0[6] ≥ 0∧[(-1)bso_45] ≥ 0)

We simplified constraint (76) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:

(77)    (0 ≥ 0 ⇒ (UIncreasing(COND_2422_0_NTH_LE3(>(x3[6], 1), java.lang.Object(IntList(x0[6], x1[6])), java.lang.Object(IntList(x0[6], x1[6])), x3[6])), ≥)∧[(-1)Bound*bni_44] + [bni_44]x0[6] ≥ 0∧[(-1)bso_45] ≥ 0)

We simplified constraint (77) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint:

(78)    (0 ≥ 0 ⇒ (UIncreasing(COND_2422_0_NTH_LE3(>(x3[6], 1), java.lang.Object(IntList(x0[6], x1[6])), java.lang.Object(IntList(x0[6], x1[6])), x3[6])), ≥)∧0 ≥ 0∧0 ≥ 0∧[bni_44] ≥ 0∧[(-1)Bound*bni_44] ≥ 0∧0 ≥ 0∧0 ≥ 0∧0 ≥ 0∧[(-1)bso_45] ≥ 0)

For Pair COND_2422_0_NTH_LE3(TRUE, java.lang.Object(IntList(x0, x1)), java.lang.Object(IntList(x0, x1)), x3) → 2422_0_NTH_LE(java.lang.Object(IntList(x0, x1)), x0, +(x3, -1)) the following chains were created:
• We consider the chain 2422_0_NTH_LE(java.lang.Object(IntList(x0[6], x1[6])), java.lang.Object(IntList(x0[6], x1[6])), x3[6]) → COND_2422_0_NTH_LE3(>(x3[6], 1), java.lang.Object(IntList(x0[6], x1[6])), java.lang.Object(IntList(x0[6], x1[6])), x3[6]), COND_2422_0_NTH_LE3(TRUE, java.lang.Object(IntList(x0[7], x1[7])), java.lang.Object(IntList(x0[7], x1[7])), x3[7]) → 2422_0_NTH_LE(java.lang.Object(IntList(x0[7], x1[7])), x0[7], +(x3[7], -1)) which results in the following constraint:

(79)    (>(x3[6], 1)=TRUEjava.lang.Object(IntList(x0[6], x1[6]))=java.lang.Object(IntList(x0[7], x1[7]))∧x3[6]=x3[7]COND_2422_0_NTH_LE3(TRUE, java.lang.Object(IntList(x0[7], x1[7])), java.lang.Object(IntList(x0[7], x1[7])), x3[7])≥NonInfC∧COND_2422_0_NTH_LE3(TRUE, java.lang.Object(IntList(x0[7], x1[7])), java.lang.Object(IntList(x0[7], x1[7])), x3[7])≥2422_0_NTH_LE(java.lang.Object(IntList(x0[7], x1[7])), x0[7], +(x3[7], -1))∧(UIncreasing(2422_0_NTH_LE(java.lang.Object(IntList(x0[7], x1[7])), x0[7], +(x3[7], -1))), ≥))

We simplified constraint (79) using rules (I), (II), (III) which results in the following new constraint:

(80)    (>(x3[6], 1)=TRUECOND_2422_0_NTH_LE3(TRUE, java.lang.Object(IntList(x0[6], x1[6])), java.lang.Object(IntList(x0[6], x1[6])), x3[6])≥NonInfC∧COND_2422_0_NTH_LE3(TRUE, java.lang.Object(IntList(x0[6], x1[6])), java.lang.Object(IntList(x0[6], x1[6])), x3[6])≥2422_0_NTH_LE(java.lang.Object(IntList(x0[6], x1[6])), x0[6], +(x3[6], -1))∧(UIncreasing(2422_0_NTH_LE(java.lang.Object(IntList(x0[7], x1[7])), x0[7], +(x3[7], -1))), ≥))

We simplified constraint (80) using rule (POLY_CONSTRAINTS) which results in the following new constraint:

(81)    (0 ≥ 0 ⇒ (UIncreasing(2422_0_NTH_LE(java.lang.Object(IntList(x0[7], x1[7])), x0[7], +(x3[7], -1))), ≥)∧[(-1)Bound*bni_46] + [bni_46]x0[6] ≥ 0∧[(-1)bso_47] ≥ 0)

We simplified constraint (81) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:

(82)    (0 ≥ 0 ⇒ (UIncreasing(2422_0_NTH_LE(java.lang.Object(IntList(x0[7], x1[7])), x0[7], +(x3[7], -1))), ≥)∧[(-1)Bound*bni_46] + [bni_46]x0[6] ≥ 0∧[(-1)bso_47] ≥ 0)

We simplified constraint (82) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:

(83)    (0 ≥ 0 ⇒ (UIncreasing(2422_0_NTH_LE(java.lang.Object(IntList(x0[7], x1[7])), x0[7], +(x3[7], -1))), ≥)∧[(-1)Bound*bni_46] + [bni_46]x0[6] ≥ 0∧[(-1)bso_47] ≥ 0)

We simplified constraint (83) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint:

(84)    (0 ≥ 0 ⇒ (UIncreasing(2422_0_NTH_LE(java.lang.Object(IntList(x0[7], x1[7])), x0[7], +(x3[7], -1))), ≥)∧0 ≥ 0∧0 ≥ 0∧[bni_46] ≥ 0∧[(-1)Bound*bni_46] ≥ 0∧0 ≥ 0∧0 ≥ 0∧0 ≥ 0∧[(-1)bso_47] ≥ 0)

To summarize, we get the following constraints P for the following pairs.
• 2422_0_NTH_LE(java.lang.Object(IntList(x0, x1)), java.lang.Object(IntList(x4, x5)), x3) → COND_2422_0_NTH_LE(&&(&&(>(x7, -1), >(x5, 0)), <=(x3, 1)), java.lang.Object(IntList(x0, x1)), java.lang.Object(IntList(x4, x5)), x3, x7)
• (0 ≥ 0 ⇒ (UIncreasing(COND_2422_0_NTH_LE(&&(&&(>(x7[0], -1), >(x5[0], 0)), <=(x3[0], 1)), java.lang.Object(IntList(x0[0], x1[0])), java.lang.Object(IntList(x4[0], x5[0])), x3[0], x7[0])), ≥)∧0 ≥ 0∧0 ≥ 0∧[bni_32] ≥ 0∧[(-1)Bound*bni_32] ≥ 0∧0 ≥ 0∧0 ≥ 0∧0 ≥ 0∧0 ≥ 0∧[(-1)bso_33] ≥ 0)
• (0 ≥ 0 ⇒ (UIncreasing(COND_2422_0_NTH_LE(&&(&&(>(x7[0], -1), >(x5[0], 0)), <=(x3[0], 1)), java.lang.Object(IntList(x0[0], x1[0])), java.lang.Object(IntList(x4[0], x5[0])), x3[0], x7[0])), ≥)∧0 ≥ 0∧0 ≥ 0∧[bni_32] ≥ 0∧[(-1)Bound*bni_32] ≥ 0∧0 ≥ 0∧0 ≥ 0∧0 ≥ 0∧0 ≥ 0∧[(-1)bso_33] ≥ 0)
• (0 ≥ 0 ⇒ (UIncreasing(COND_2422_0_NTH_LE(&&(&&(>(x7[0], -1), >(x5[0], 0)), <=(x3[0], 1)), java.lang.Object(IntList(x0[0], x1[0])), java.lang.Object(IntList(x4[0], x5[0])), x3[0], x7[0])), ≥)∧0 ≥ 0∧0 ≥ 0∧0 ≥ 0∧[bni_32] ≥ 0∧[(-1)Bound*bni_32] ≥ 0∧0 ≥ 0∧0 ≥ 0∧0 ≥ 0∧0 ≥ 0∧0 ≥ 0∧[(-1)bso_33] ≥ 0)
• (0 ≥ 0 ⇒ (UIncreasing(COND_2422_0_NTH_LE(&&(&&(>(x7[0], -1), >(x5[0], 0)), <=(x3[0], 1)), java.lang.Object(IntList(x0[0], x1[0])), java.lang.Object(IntList(x4[0], x5[0])), x3[0], x7[0])), ≥)∧0 ≥ 0∧[bni_32] ≥ 0∧0 ≥ 0∧[bni_32 + (-1)Bound*bni_32] ≥ 0∧0 ≥ 0∧0 ≥ 0∧0 ≥ 0∧0 ≥ 0∧[(-1)bso_33] ≥ 0)

• COND_2422_0_NTH_LE(TRUE, java.lang.Object(IntList(x0, x1)), java.lang.Object(IntList(x4, x5)), x3, x7) → 2422_0_NTH_LE(x0, x0, x7)
• (0 ≥ 0 ⇒ (UIncreasing(2422_0_NTH_LE(x0[1], x0[1], x7[1])), ≥)∧0 ≥ 0∧0 ≥ 0∧0 ≥ 0∧0 ≥ 0∧0 ≥ 0∧[bni_34] ≥ 0∧[(-1)Bound*bni_34] ≥ 0∧0 ≥ 0∧0 ≥ 0∧0 ≥ 0∧0 ≥ 0∧0 ≥ 0∧0 ≥ 0∧[1 + (-1)bso_35] ≥ 0)

• 2422_0_NTH_LE(java.lang.Object(IntList(x0, x1)), java.lang.Object(IntList(x0, x1)), x3) → COND_2422_0_NTH_LE1(&&(&&(>(x5, -1), <=(x3, 1)), >(x1, 0)), java.lang.Object(IntList(x0, x1)), java.lang.Object(IntList(x0, x1)), x3, x5)
• (0 ≥ 0 ⇒ (UIncreasing(COND_2422_0_NTH_LE1(&&(&&(>(x5[2], -1), <=(x3[2], 1)), >(x1[2], 0)), java.lang.Object(IntList(x0[2], x1[2])), java.lang.Object(IntList(x0[2], x1[2])), x3[2], x5[2])), ≥)∧0 ≥ 0∧0 ≥ 0∧[bni_36] ≥ 0∧[(-1)Bound*bni_36] ≥ 0∧0 ≥ 0∧0 ≥ 0∧0 ≥ 0∧0 ≥ 0∧[(-1)bso_37] ≥ 0)
• (0 ≥ 0 ⇒ (UIncreasing(COND_2422_0_NTH_LE1(&&(&&(>(x5[2], -1), <=(x3[2], 1)), >(x1[2], 0)), java.lang.Object(IntList(x0[2], x1[2])), java.lang.Object(IntList(x0[2], x1[2])), x3[2], x5[2])), ≥)∧0 ≥ 0∧0 ≥ 0∧[bni_36] ≥ 0∧[(-1)Bound*bni_36] ≥ 0∧0 ≥ 0∧0 ≥ 0∧0 ≥ 0∧0 ≥ 0∧[(-1)bso_37] ≥ 0)
• (0 ≥ 0 ⇒ (UIncreasing(COND_2422_0_NTH_LE1(&&(&&(>(x5[2], -1), <=(x3[2], 1)), >(x1[2], 0)), java.lang.Object(IntList(x0[2], x1[2])), java.lang.Object(IntList(x0[2], x1[2])), x3[2], x5[2])), ≥)∧0 ≥ 0∧[bni_36] ≥ 0∧[(-1)Bound*bni_36] ≥ 0∧0 ≥ 0∧0 ≥ 0∧0 ≥ 0∧[(-1)bso_37] ≥ 0)
• (0 ≥ 0 ⇒ (UIncreasing(COND_2422_0_NTH_LE1(&&(&&(>(x5[2], -1), <=(x3[2], 1)), >(x1[2], 0)), java.lang.Object(IntList(x0[2], x1[2])), java.lang.Object(IntList(x0[2], x1[2])), x3[2], x5[2])), ≥)∧0 ≥ 0∧[bni_36] ≥ 0∧[bni_36 + (-1)Bound*bni_36] ≥ 0∧0 ≥ 0∧0 ≥ 0∧0 ≥ 0∧[(-1)bso_37] ≥ 0)

• COND_2422_0_NTH_LE1(TRUE, java.lang.Object(IntList(x0, x1)), java.lang.Object(IntList(x0, x1)), x3, x5) → 2422_0_NTH_LE(x0, x0, x5)
• (0 ≥ 0 ⇒ (UIncreasing(2422_0_NTH_LE(x0[3], x0[3], x5[3])), ≥)∧0 ≥ 0∧0 ≥ 0∧0 ≥ 0∧[bni_38] ≥ 0∧[(-1)Bound*bni_38] ≥ 0∧0 ≥ 0∧0 ≥ 0∧0 ≥ 0∧0 ≥ 0∧[1 + (-1)bso_39] ≥ 0)

• 2422_0_NTH_LE(x0, java.lang.Object(IntList(x3, x4)), x2) → COND_2422_0_NTH_LE2(>(x2, 1), x0, java.lang.Object(IntList(x3, x4)), x2)
• (0 ≥ 0 ⇒ (UIncreasing(COND_2422_0_NTH_LE2(>(x2[4], 1), x0[4], java.lang.Object(IntList(x3[4], x4[4])), x2[4])), ≥)∧0 ≥ 0∧0 ≥ 0∧0 ≥ 0∧[bni_40] ≥ 0∧[(-1)bni_40 + (-1)Bound*bni_40] ≥ 0∧0 ≥ 0∧0 ≥ 0∧0 ≥ 0∧0 ≥ 0∧[(-1)bso_41] ≥ 0)

• COND_2422_0_NTH_LE2(TRUE, x0, java.lang.Object(IntList(x3, x4)), x2) → 2422_0_NTH_LE(x0, x3, +(x2, -1))
• (0 ≥ 0 ⇒ (UIncreasing(2422_0_NTH_LE(x0[5], x3[5], +(x2[5], -1))), ≥)∧0 ≥ 0∧0 ≥ 0∧0 ≥ 0∧[bni_42] ≥ 0∧[(-1)bni_42 + (-1)Bound*bni_42] ≥ 0∧0 ≥ 0∧0 ≥ 0∧0 ≥ 0∧0 ≥ 0∧[(-1)bso_43] ≥ 0)

• 2422_0_NTH_LE(java.lang.Object(IntList(x0, x1)), java.lang.Object(IntList(x0, x1)), x3) → COND_2422_0_NTH_LE3(>(x3, 1), java.lang.Object(IntList(x0, x1)), java.lang.Object(IntList(x0, x1)), x3)
• (0 ≥ 0 ⇒ (UIncreasing(COND_2422_0_NTH_LE3(>(x3[6], 1), java.lang.Object(IntList(x0[6], x1[6])), java.lang.Object(IntList(x0[6], x1[6])), x3[6])), ≥)∧0 ≥ 0∧0 ≥ 0∧[bni_44] ≥ 0∧[(-1)Bound*bni_44] ≥ 0∧0 ≥ 0∧0 ≥ 0∧0 ≥ 0∧[(-1)bso_45] ≥ 0)

• COND_2422_0_NTH_LE3(TRUE, java.lang.Object(IntList(x0, x1)), java.lang.Object(IntList(x0, x1)), x3) → 2422_0_NTH_LE(java.lang.Object(IntList(x0, x1)), x0, +(x3, -1))
• (0 ≥ 0 ⇒ (UIncreasing(2422_0_NTH_LE(java.lang.Object(IntList(x0[7], x1[7])), x0[7], +(x3[7], -1))), ≥)∧0 ≥ 0∧0 ≥ 0∧[bni_46] ≥ 0∧[(-1)Bound*bni_46] ≥ 0∧0 ≥ 0∧0 ≥ 0∧0 ≥ 0∧[(-1)bso_47] ≥ 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 with natural coefficients for non-tuple symbols [NONINF][POLO]:

POL(TRUE) = 0
POL(FALSE) = 0
POL(2422_0_NTH_LE(x1, x2, x3)) = [-1] + x1
POL(java.lang.Object(x1)) = [1] + x1
POL(IntList(x1, x2)) = x1
POL(COND_2422_0_NTH_LE(x1, x2, x3, x4, x5)) = [-1] + x2 + x1
POL(&&(x1, x2)) = 0
POL(>(x1, x2)) = 0
POL(-1) = 0
POL(0) = 0
POL(<=(x1, x2)) = 0
POL(1) = 0
POL(COND_2422_0_NTH_LE1(x1, x2, x3, x4, x5)) = [-1] + [-1]x3 + [2]x2 + [2]x1
POL(COND_2422_0_NTH_LE2(x1, x2, x3, x4)) = [-1] + x2
POL(+(x1, x2)) = 0
POL(COND_2422_0_NTH_LE3(x1, x2, x3, x4)) = [-1] + [2]x3 + [-1]x2

The following pairs are in P>:

COND_2422_0_NTH_LE(TRUE, java.lang.Object(IntList(x0[1], x1[1])), java.lang.Object(IntList(x4[1], x5[1])), x3[1], x7[1]) → 2422_0_NTH_LE(x0[1], x0[1], x7[1])
COND_2422_0_NTH_LE1(TRUE, java.lang.Object(IntList(x0[3], x1[3])), java.lang.Object(IntList(x0[3], x1[3])), x3[3], x5[3]) → 2422_0_NTH_LE(x0[3], x0[3], x5[3])

The following pairs are in Pbound:

2422_0_NTH_LE(java.lang.Object(IntList(x0[0], x1[0])), java.lang.Object(IntList(x4[0], x5[0])), x3[0]) → COND_2422_0_NTH_LE(&&(&&(>(x7[0], -1), >(x5[0], 0)), <=(x3[0], 1)), java.lang.Object(IntList(x0[0], x1[0])), java.lang.Object(IntList(x4[0], x5[0])), x3[0], x7[0])
COND_2422_0_NTH_LE(TRUE, java.lang.Object(IntList(x0[1], x1[1])), java.lang.Object(IntList(x4[1], x5[1])), x3[1], x7[1]) → 2422_0_NTH_LE(x0[1], x0[1], x7[1])
2422_0_NTH_LE(java.lang.Object(IntList(x0[2], x1[2])), java.lang.Object(IntList(x0[2], x1[2])), x3[2]) → COND_2422_0_NTH_LE1(&&(&&(>(x5[2], -1), <=(x3[2], 1)), >(x1[2], 0)), java.lang.Object(IntList(x0[2], x1[2])), java.lang.Object(IntList(x0[2], x1[2])), x3[2], x5[2])
COND_2422_0_NTH_LE1(TRUE, java.lang.Object(IntList(x0[3], x1[3])), java.lang.Object(IntList(x0[3], x1[3])), x3[3], x5[3]) → 2422_0_NTH_LE(x0[3], x0[3], x5[3])
2422_0_NTH_LE(x0[4], java.lang.Object(IntList(x3[4], x4[4])), x2[4]) → COND_2422_0_NTH_LE2(>(x2[4], 1), x0[4], java.lang.Object(IntList(x3[4], x4[4])), x2[4])
COND_2422_0_NTH_LE2(TRUE, x0[5], java.lang.Object(IntList(x3[5], x4[5])), x2[5]) → 2422_0_NTH_LE(x0[5], x3[5], +(x2[5], -1))
2422_0_NTH_LE(java.lang.Object(IntList(x0[6], x1[6])), java.lang.Object(IntList(x0[6], x1[6])), x3[6]) → COND_2422_0_NTH_LE3(>(x3[6], 1), java.lang.Object(IntList(x0[6], x1[6])), java.lang.Object(IntList(x0[6], x1[6])), x3[6])
COND_2422_0_NTH_LE3(TRUE, java.lang.Object(IntList(x0[7], x1[7])), java.lang.Object(IntList(x0[7], x1[7])), x3[7]) → 2422_0_NTH_LE(java.lang.Object(IntList(x0[7], x1[7])), x0[7], +(x3[7], -1))

The following pairs are in P:

2422_0_NTH_LE(java.lang.Object(IntList(x0[0], x1[0])), java.lang.Object(IntList(x4[0], x5[0])), x3[0]) → COND_2422_0_NTH_LE(&&(&&(>(x7[0], -1), >(x5[0], 0)), <=(x3[0], 1)), java.lang.Object(IntList(x0[0], x1[0])), java.lang.Object(IntList(x4[0], x5[0])), x3[0], x7[0])
2422_0_NTH_LE(java.lang.Object(IntList(x0[2], x1[2])), java.lang.Object(IntList(x0[2], x1[2])), x3[2]) → COND_2422_0_NTH_LE1(&&(&&(>(x5[2], -1), <=(x3[2], 1)), >(x1[2], 0)), java.lang.Object(IntList(x0[2], x1[2])), java.lang.Object(IntList(x0[2], x1[2])), x3[2], x5[2])
2422_0_NTH_LE(x0[4], java.lang.Object(IntList(x3[4], x4[4])), x2[4]) → COND_2422_0_NTH_LE2(>(x2[4], 1), x0[4], java.lang.Object(IntList(x3[4], x4[4])), x2[4])
COND_2422_0_NTH_LE2(TRUE, x0[5], java.lang.Object(IntList(x3[5], x4[5])), x2[5]) → 2422_0_NTH_LE(x0[5], x3[5], +(x2[5], -1))
2422_0_NTH_LE(java.lang.Object(IntList(x0[6], x1[6])), java.lang.Object(IntList(x0[6], x1[6])), x3[6]) → COND_2422_0_NTH_LE3(>(x3[6], 1), java.lang.Object(IntList(x0[6], x1[6])), java.lang.Object(IntList(x0[6], x1[6])), x3[6])
COND_2422_0_NTH_LE3(TRUE, java.lang.Object(IntList(x0[7], x1[7])), java.lang.Object(IntList(x0[7], x1[7])), x3[7]) → 2422_0_NTH_LE(java.lang.Object(IntList(x0[7], x1[7])), x0[7], +(x3[7], -1))

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

(20) 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): 2422_0_NTH_LE(java.lang.Object(IntList(x0[0], x1[0])), java.lang.Object(IntList(x4[0], x5[0])), x3[0]) → COND_2422_0_NTH_LE(x7[0] > -1 && x5[0] > 0 && x3[0] <= 1, java.lang.Object(IntList(x0[0], x1[0])), java.lang.Object(IntList(x4[0], x5[0])), x3[0], x7[0])
(2): 2422_0_NTH_LE(java.lang.Object(IntList(x0[2], x1[2])), java.lang.Object(IntList(x0[2], x1[2])), x3[2]) → COND_2422_0_NTH_LE1(x5[2] > -1 && x3[2] <= 1 && x1[2] > 0, java.lang.Object(IntList(x0[2], x1[2])), java.lang.Object(IntList(x0[2], x1[2])), x3[2], x5[2])
(4): 2422_0_NTH_LE(x0[4], java.lang.Object(IntList(x3[4], x4[4])), x2[4]) → COND_2422_0_NTH_LE2(x2[4] > 1, x0[4], java.lang.Object(IntList(x3[4], x4[4])), x2[4])
(5): COND_2422_0_NTH_LE2(TRUE, x0[5], java.lang.Object(IntList(x3[5], x4[5])), x2[5]) → 2422_0_NTH_LE(x0[5], x3[5], x2[5] + -1)
(6): 2422_0_NTH_LE(java.lang.Object(IntList(x0[6], x1[6])), java.lang.Object(IntList(x0[6], x1[6])), x3[6]) → COND_2422_0_NTH_LE3(x3[6] > 1, java.lang.Object(IntList(x0[6], x1[6])), java.lang.Object(IntList(x0[6], x1[6])), x3[6])
(7): COND_2422_0_NTH_LE3(TRUE, java.lang.Object(IntList(x0[7], x1[7])), java.lang.Object(IntList(x0[7], x1[7])), x3[7]) → 2422_0_NTH_LE(java.lang.Object(IntList(x0[7], x1[7])), x0[7], x3[7] + -1)

(5) -> (0), if (x0[5]* java.lang.Object(IntList(x0[0], x1[0]))∧x3[5]* java.lang.Object(IntList(x4[0], x5[0]))∧x2[5] + -1* x3[0])

(7) -> (0), if (java.lang.Object(IntList(x0[7], x1[7])) →* java.lang.Object(IntList(x0[0], x1[0]))∧x0[7]* java.lang.Object(IntList(x4[0], x5[0]))∧x3[7] + -1* x3[0])

(5) -> (2), if (x0[5]* java.lang.Object(IntList(x0[2], x1[2]))∧x3[5]* java.lang.Object(IntList(x0[2], x1[2]))∧x2[5] + -1* x3[2])

(7) -> (2), if (java.lang.Object(IntList(x0[7], x1[7])) →* java.lang.Object(IntList(x0[2], x1[2]))∧x0[7]* java.lang.Object(IntList(x0[2], x1[2]))∧x3[7] + -1* x3[2])

(5) -> (4), if (x0[5]* x0[4]x3[5]* java.lang.Object(IntList(x3[4], x4[4]))∧x2[5] + -1* x2[4])

(7) -> (4), if (java.lang.Object(IntList(x0[7], x1[7])) →* x0[4]x0[7]* java.lang.Object(IntList(x3[4], x4[4]))∧x3[7] + -1* x2[4])

(4) -> (5), if (x2[4] > 1x0[4]* x0[5]java.lang.Object(IntList(x3[4], x4[4])) →* java.lang.Object(IntList(x3[5], x4[5]))∧x2[4]* x2[5])

(5) -> (6), if (x0[5]* java.lang.Object(IntList(x0[6], x1[6]))∧x3[5]* java.lang.Object(IntList(x0[6], x1[6]))∧x2[5] + -1* x3[6])

(7) -> (6), if (java.lang.Object(IntList(x0[7], x1[7])) →* java.lang.Object(IntList(x0[6], x1[6]))∧x0[7]* java.lang.Object(IntList(x0[6], x1[6]))∧x3[7] + -1* x3[6])

(6) -> (7), if (x3[6] > 1java.lang.Object(IntList(x0[6], x1[6])) →* java.lang.Object(IntList(x0[7], x1[7]))∧x3[6]* x3[7])

The set Q is empty.

(21) IDependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 2 less nodes.

(22) 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:
(7): COND_2422_0_NTH_LE3(TRUE, java.lang.Object(IntList(x0[7], x1[7])), java.lang.Object(IntList(x0[7], x1[7])), x3[7]) → 2422_0_NTH_LE(java.lang.Object(IntList(x0[7], x1[7])), x0[7], x3[7] + -1)
(6): 2422_0_NTH_LE(java.lang.Object(IntList(x0[6], x1[6])), java.lang.Object(IntList(x0[6], x1[6])), x3[6]) → COND_2422_0_NTH_LE3(x3[6] > 1, java.lang.Object(IntList(x0[6], x1[6])), java.lang.Object(IntList(x0[6], x1[6])), x3[6])
(5): COND_2422_0_NTH_LE2(TRUE, x0[5], java.lang.Object(IntList(x3[5], x4[5])), x2[5]) → 2422_0_NTH_LE(x0[5], x3[5], x2[5] + -1)
(4): 2422_0_NTH_LE(x0[4], java.lang.Object(IntList(x3[4], x4[4])), x2[4]) → COND_2422_0_NTH_LE2(x2[4] > 1, x0[4], java.lang.Object(IntList(x3[4], x4[4])), x2[4])

(5) -> (4), if (x0[5]* x0[4]x3[5]* java.lang.Object(IntList(x3[4], x4[4]))∧x2[5] + -1* x2[4])

(7) -> (4), if (java.lang.Object(IntList(x0[7], x1[7])) →* x0[4]x0[7]* java.lang.Object(IntList(x3[4], x4[4]))∧x3[7] + -1* x2[4])

(4) -> (5), if (x2[4] > 1x0[4]* x0[5]java.lang.Object(IntList(x3[4], x4[4])) →* java.lang.Object(IntList(x3[5], x4[5]))∧x2[4]* x2[5])

(5) -> (6), if (x0[5]* java.lang.Object(IntList(x0[6], x1[6]))∧x3[5]* java.lang.Object(IntList(x0[6], x1[6]))∧x2[5] + -1* x3[6])

(7) -> (6), if (java.lang.Object(IntList(x0[7], x1[7])) →* java.lang.Object(IntList(x0[6], x1[6]))∧x0[7]* java.lang.Object(IntList(x0[6], x1[6]))∧x3[7] + -1* x3[6])

(6) -> (7), if (x3[6] > 1java.lang.Object(IntList(x0[6], x1[6])) →* java.lang.Object(IntList(x0[7], x1[7]))∧x3[6]* x3[7])

The set Q is empty.

(23) IDPNonInfProof (SOUND transformation)

Used the following options for this NonInfProof:
IDPGPoloSolver: Range: [(-1,2)] IsNat: true Interpretation Shape Heuristic: aprove.DPFramework.IDPProblem.Processors.nonInf.poly.IdpDefaultShapeHeuristic@3e48983b 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_2422_0_NTH_LE3(TRUE, java.lang.Object(IntList(x0[7], x1[7])), java.lang.Object(IntList(x0[7], x1[7])), x3[7]) → 2422_0_NTH_LE(java.lang.Object(IntList(x0[7], x1[7])), x0[7], +(x3[7], -1)) the following chains were created:
• We consider the chain 2422_0_NTH_LE(java.lang.Object(IntList(x0[6], x1[6])), java.lang.Object(IntList(x0[6], x1[6])), x3[6]) → COND_2422_0_NTH_LE3(>(x3[6], 1), java.lang.Object(IntList(x0[6], x1[6])), java.lang.Object(IntList(x0[6], x1[6])), x3[6]), COND_2422_0_NTH_LE3(TRUE, java.lang.Object(IntList(x0[7], x1[7])), java.lang.Object(IntList(x0[7], x1[7])), x3[7]) → 2422_0_NTH_LE(java.lang.Object(IntList(x0[7], x1[7])), x0[7], +(x3[7], -1)), 2422_0_NTH_LE(x0[4], java.lang.Object(IntList(x3[4], x4[4])), x2[4]) → COND_2422_0_NTH_LE2(>(x2[4], 1), x0[4], java.lang.Object(IntList(x3[4], x4[4])), x2[4]) which results in the following constraint:

(1)    (>(x3[6], 1)=TRUEjava.lang.Object(IntList(x0[6], x1[6]))=java.lang.Object(IntList(x0[7], x1[7]))∧x3[6]=x3[7]java.lang.Object(IntList(x0[7], x1[7]))=x0[4]x0[7]=java.lang.Object(IntList(x3[4], x4[4]))∧+(x3[7], -1)=x2[4]COND_2422_0_NTH_LE3(TRUE, java.lang.Object(IntList(x0[7], x1[7])), java.lang.Object(IntList(x0[7], x1[7])), x3[7])≥NonInfC∧COND_2422_0_NTH_LE3(TRUE, java.lang.Object(IntList(x0[7], x1[7])), java.lang.Object(IntList(x0[7], x1[7])), x3[7])≥2422_0_NTH_LE(java.lang.Object(IntList(x0[7], x1[7])), x0[7], +(x3[7], -1))∧(UIncreasing(2422_0_NTH_LE(java.lang.Object(IntList(x0[7], x1[7])), x0[7], +(x3[7], -1))), ≥))

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

(2)    (>(x3[6], 1)=TRUECOND_2422_0_NTH_LE3(TRUE, java.lang.Object(IntList(java.lang.Object(IntList(x3[4], x4[4])), x1[6])), java.lang.Object(IntList(java.lang.Object(IntList(x3[4], x4[4])), x1[6])), x3[6])≥NonInfC∧COND_2422_0_NTH_LE3(TRUE, java.lang.Object(IntList(java.lang.Object(IntList(x3[4], x4[4])), x1[6])), java.lang.Object(IntList(java.lang.Object(IntList(x3[4], x4[4])), x1[6])), x3[6])≥2422_0_NTH_LE(java.lang.Object(IntList(java.lang.Object(IntList(x3[4], x4[4])), x1[6])), java.lang.Object(IntList(x3[4], x4[4])), +(x3[6], -1))∧(UIncreasing(2422_0_NTH_LE(java.lang.Object(IntList(x0[7], x1[7])), x0[7], +(x3[7], -1))), ≥))

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

(3)    (0 ≥ 0 ⇒ (UIncreasing(2422_0_NTH_LE(java.lang.Object(IntList(x0[7], x1[7])), x0[7], +(x3[7], -1))), ≥)∧[(2)bni_20 + (-1)Bound*bni_20] ≥ 0∧[7 + (-1)bso_21] + [8]x3[4] ≥ 0)

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

(4)    (0 ≥ 0 ⇒ (UIncreasing(2422_0_NTH_LE(java.lang.Object(IntList(x0[7], x1[7])), x0[7], +(x3[7], -1))), ≥)∧[(2)bni_20 + (-1)Bound*bni_20] ≥ 0∧[7 + (-1)bso_21] + [8]x3[4] ≥ 0)

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

(5)    (0 ≥ 0 ⇒ (UIncreasing(2422_0_NTH_LE(java.lang.Object(IntList(x0[7], x1[7])), x0[7], +(x3[7], -1))), ≥)∧[(2)bni_20 + (-1)Bound*bni_20] ≥ 0∧[7 + (-1)bso_21] + [8]x3[4] ≥ 0)

We simplified constraint (5) using rules (IDP_UNRESTRICTED_VARS), (IDP_POLY_GCD) which results in the following new constraint:

(6)    (0 ≥ 0 ⇒ (UIncreasing(2422_0_NTH_LE(java.lang.Object(IntList(x0[7], x1[7])), x0[7], +(x3[7], -1))), ≥)∧0 ≥ 0∧0 ≥ 0∧0 ≥ 0∧0 ≥ 0∧[(2)bni_20 + (-1)Bound*bni_20] ≥ 0∧0 ≥ 0∧0 ≥ 0∧0 ≥ 0∧[7 + (-1)bso_21] ≥ 0∧[1] ≥ 0)

• We consider the chain 2422_0_NTH_LE(java.lang.Object(IntList(x0[6], x1[6])), java.lang.Object(IntList(x0[6], x1[6])), x3[6]) → COND_2422_0_NTH_LE3(>(x3[6], 1), java.lang.Object(IntList(x0[6], x1[6])), java.lang.Object(IntList(x0[6], x1[6])), x3[6]), COND_2422_0_NTH_LE3(TRUE, java.lang.Object(IntList(x0[7], x1[7])), java.lang.Object(IntList(x0[7], x1[7])), x3[7]) → 2422_0_NTH_LE(java.lang.Object(IntList(x0[7], x1[7])), x0[7], +(x3[7], -1)), 2422_0_NTH_LE(java.lang.Object(IntList(x0[6], x1[6])), java.lang.Object(IntList(x0[6], x1[6])), x3[6]) → COND_2422_0_NTH_LE3(>(x3[6], 1), java.lang.Object(IntList(x0[6], x1[6])), java.lang.Object(IntList(x0[6], x1[6])), x3[6]) which results in the following constraint:

(7)    (>(x3[6], 1)=TRUEjava.lang.Object(IntList(x0[6], x1[6]))=java.lang.Object(IntList(x0[7], x1[7]))∧x3[6]=x3[7]java.lang.Object(IntList(x0[7], x1[7]))=java.lang.Object(IntList(x0[6]1, x1[6]1))∧x0[7]=java.lang.Object(IntList(x0[6]1, x1[6]1))∧+(x3[7], -1)=x3[6]1COND_2422_0_NTH_LE3(TRUE, java.lang.Object(IntList(x0[7], x1[7])), java.lang.Object(IntList(x0[7], x1[7])), x3[7])≥NonInfC∧COND_2422_0_NTH_LE3(TRUE, java.lang.Object(IntList(x0[7], x1[7])), java.lang.Object(IntList(x0[7], x1[7])), x3[7])≥2422_0_NTH_LE(java.lang.Object(IntList(x0[7], x1[7])), x0[7], +(x3[7], -1))∧(UIncreasing(2422_0_NTH_LE(java.lang.Object(IntList(x0[7], x1[7])), x0[7], +(x3[7], -1))), ≥))

We simplified constraint (7) using rules (I), (II), (III), (IV) which results in the following new constraint:

(8)    (>(x3[6], 1)=TRUECOND_2422_0_NTH_LE3(TRUE, java.lang.Object(IntList(java.lang.Object(IntList(x0[6], x1[6])), x1[6])), java.lang.Object(IntList(java.lang.Object(IntList(x0[6], x1[6])), x1[6])), x3[6])≥NonInfC∧COND_2422_0_NTH_LE3(TRUE, java.lang.Object(IntList(java.lang.Object(IntList(x0[6], x1[6])), x1[6])), java.lang.Object(IntList(java.lang.Object(IntList(x0[6], x1[6])), x1[6])), x3[6])≥2422_0_NTH_LE(java.lang.Object(IntList(java.lang.Object(IntList(x0[6], x1[6])), x1[6])), java.lang.Object(IntList(x0[6], x1[6])), +(x3[6], -1))∧(UIncreasing(2422_0_NTH_LE(java.lang.Object(IntList(x0[7], x1[7])), x0[7], +(x3[7], -1))), ≥))

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

(9)    (0 ≥ 0 ⇒ (UIncreasing(2422_0_NTH_LE(java.lang.Object(IntList(x0[7], x1[7])), x0[7], +(x3[7], -1))), ≥)∧[(2)bni_20 + (-1)Bound*bni_20] ≥ 0∧[7 + (-1)bso_21] + [8]x0[6] ≥ 0)

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

(10)    (0 ≥ 0 ⇒ (UIncreasing(2422_0_NTH_LE(java.lang.Object(IntList(x0[7], x1[7])), x0[7], +(x3[7], -1))), ≥)∧[(2)bni_20 + (-1)Bound*bni_20] ≥ 0∧[7 + (-1)bso_21] + [8]x0[6] ≥ 0)

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

(11)    (0 ≥ 0 ⇒ (UIncreasing(2422_0_NTH_LE(java.lang.Object(IntList(x0[7], x1[7])), x0[7], +(x3[7], -1))), ≥)∧[(2)bni_20 + (-1)Bound*bni_20] ≥ 0∧[7 + (-1)bso_21] + [8]x0[6] ≥ 0)

We simplified constraint (11) using rules (IDP_UNRESTRICTED_VARS), (IDP_POLY_GCD) which results in the following new constraint:

(12)    (0 ≥ 0 ⇒ (UIncreasing(2422_0_NTH_LE(java.lang.Object(IntList(x0[7], x1[7])), x0[7], +(x3[7], -1))), ≥)∧0 ≥ 0∧0 ≥ 0∧0 ≥ 0∧[(2)bni_20 + (-1)Bound*bni_20] ≥ 0∧0 ≥ 0∧0 ≥ 0∧[7 + (-1)bso_21] ≥ 0∧[1] ≥ 0)

For Pair 2422_0_NTH_LE(java.lang.Object(IntList(x0[6], x1[6])), java.lang.Object(IntList(x0[6], x1[6])), x3[6]) → COND_2422_0_NTH_LE3(>(x3[6], 1), java.lang.Object(IntList(x0[6], x1[6])), java.lang.Object(IntList(x0[6], x1[6])), x3[6]) the following chains were created:
• We consider the chain 2422_0_NTH_LE(java.lang.Object(IntList(x0[6], x1[6])), java.lang.Object(IntList(x0[6], x1[6])), x3[6]) → COND_2422_0_NTH_LE3(>(x3[6], 1), java.lang.Object(IntList(x0[6], x1[6])), java.lang.Object(IntList(x0[6], x1[6])), x3[6]), COND_2422_0_NTH_LE3(TRUE, java.lang.Object(IntList(x0[7], x1[7])), java.lang.Object(IntList(x0[7], x1[7])), x3[7]) → 2422_0_NTH_LE(java.lang.Object(IntList(x0[7], x1[7])), x0[7], +(x3[7], -1)) which results in the following constraint:

(13)    (>(x3[6], 1)=TRUEjava.lang.Object(IntList(x0[6], x1[6]))=java.lang.Object(IntList(x0[7], x1[7]))∧x3[6]=x3[7]2422_0_NTH_LE(java.lang.Object(IntList(x0[6], x1[6])), java.lang.Object(IntList(x0[6], x1[6])), x3[6])≥NonInfC∧2422_0_NTH_LE(java.lang.Object(IntList(x0[6], x1[6])), java.lang.Object(IntList(x0[6], x1[6])), x3[6])≥COND_2422_0_NTH_LE3(>(x3[6], 1), java.lang.Object(IntList(x0[6], x1[6])), java.lang.Object(IntList(x0[6], x1[6])), x3[6])∧(UIncreasing(COND_2422_0_NTH_LE3(>(x3[6], 1), java.lang.Object(IntList(x0[6], x1[6])), java.lang.Object(IntList(x0[6], x1[6])), x3[6])), ≥))

We simplified constraint (13) using rules (I), (II), (IV) which results in the following new constraint:

(14)    (>(x3[6], 1)=TRUE2422_0_NTH_LE(java.lang.Object(IntList(x0[6], x1[6])), java.lang.Object(IntList(x0[6], x1[6])), x3[6])≥NonInfC∧2422_0_NTH_LE(java.lang.Object(IntList(x0[6], x1[6])), java.lang.Object(IntList(x0[6], x1[6])), x3[6])≥COND_2422_0_NTH_LE3(>(x3[6], 1), java.lang.Object(IntList(x0[6], x1[6])), java.lang.Object(IntList(x0[6], x1[6])), x3[6])∧(UIncreasing(COND_2422_0_NTH_LE3(>(x3[6], 1), java.lang.Object(IntList(x0[6], x1[6])), java.lang.Object(IntList(x0[6], x1[6])), x3[6])), ≥))

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

(15)    (0 ≥ 0 ⇒ (UIncreasing(COND_2422_0_NTH_LE3(>(x3[6], 1), java.lang.Object(IntList(x0[6], x1[6])), java.lang.Object(IntList(x0[6], x1[6])), x3[6])), ≥)∧[(3)bni_22 + (-1)Bound*bni_22] + [bni_22]x3[6] + [(4)bni_22]x0[6] ≥ 0∧[1 + (-1)bso_23] + x3[6] + [4]x0[6] ≥ 0)

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

(16)    (0 ≥ 0 ⇒ (UIncreasing(COND_2422_0_NTH_LE3(>(x3[6], 1), java.lang.Object(IntList(x0[6], x1[6])), java.lang.Object(IntList(x0[6], x1[6])), x3[6])), ≥)∧[(3)bni_22 + (-1)Bound*bni_22] + [bni_22]x3[6] + [(4)bni_22]x0[6] ≥ 0∧[1 + (-1)bso_23] + x3[6] + [4]x0[6] ≥ 0)

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

(17)    (0 ≥ 0 ⇒ (UIncreasing(COND_2422_0_NTH_LE3(>(x3[6], 1), java.lang.Object(IntList(x0[6], x1[6])), java.lang.Object(IntList(x0[6], x1[6])), x3[6])), ≥)∧[(3)bni_22 + (-1)Bound*bni_22] + [bni_22]x3[6] + [(4)bni_22]x0[6] ≥ 0∧[1 + (-1)bso_23] + x3[6] + [4]x0[6] ≥ 0)

We simplified constraint (17) using rules (IDP_UNRESTRICTED_VARS), (IDP_POLY_GCD) which results in the following new constraint:

(18)    (0 ≥ 0 ⇒ (UIncreasing(COND_2422_0_NTH_LE3(>(x3[6], 1), java.lang.Object(IntList(x0[6], x1[6])), java.lang.Object(IntList(x0[6], x1[6])), x3[6])), ≥)∧[bni_22] ≥ 0∧0 ≥ 0∧[(4)bni_22] ≥ 0∧[(3)bni_22 + (-1)Bound*bni_22] ≥ 0∧[1] ≥ 0∧0 ≥ 0∧[1 + (-1)bso_23] ≥ 0∧[1] ≥ 0)

For Pair COND_2422_0_NTH_LE2(TRUE, x0[5], java.lang.Object(IntList(x3[5], x4[5])), x2[5]) → 2422_0_NTH_LE(x0[5], x3[5], +(x2[5], -1)) the following chains were created:
• We consider the chain 2422_0_NTH_LE(x0[4], java.lang.Object(IntList(x3[4], x4[4])), x2[4]) → COND_2422_0_NTH_LE2(>(x2[4], 1), x0[4], java.lang.Object(IntList(x3[4], x4[4])), x2[4]), COND_2422_0_NTH_LE2(TRUE, x0[5], java.lang.Object(IntList(x3[5], x4[5])), x2[5]) → 2422_0_NTH_LE(x0[5], x3[5], +(x2[5], -1)), 2422_0_NTH_LE(x0[4], java.lang.Object(IntList(x3[4], x4[4])), x2[4]) → COND_2422_0_NTH_LE2(>(x2[4], 1), x0[4], java.lang.Object(IntList(x3[4], x4[4])), x2[4]) which results in the following constraint:

(19)    (>(x2[4], 1)=TRUEx0[4]=x0[5]java.lang.Object(IntList(x3[4], x4[4]))=java.lang.Object(IntList(x3[5], x4[5]))∧x2[4]=x2[5]x0[5]=x0[4]1x3[5]=java.lang.Object(IntList(x3[4]1, x4[4]1))∧+(x2[5], -1)=x2[4]1COND_2422_0_NTH_LE2(TRUE, x0[5], java.lang.Object(IntList(x3[5], x4[5])), x2[5])≥NonInfC∧COND_2422_0_NTH_LE2(TRUE, x0[5], java.lang.Object(IntList(x3[5], x4[5])), x2[5])≥2422_0_NTH_LE(x0[5], x3[5], +(x2[5], -1))∧(UIncreasing(2422_0_NTH_LE(x0[5], x3[5], +(x2[5], -1))), ≥))

We simplified constraint (19) using rules (I), (II), (III), (IV) which results in the following new constraint:

(20)    (>(x2[4], 1)=TRUECOND_2422_0_NTH_LE2(TRUE, x0[4], java.lang.Object(IntList(java.lang.Object(IntList(x3[4]1, x4[4]1)), x4[4])), x2[4])≥NonInfC∧COND_2422_0_NTH_LE2(TRUE, x0[4], java.lang.Object(IntList(java.lang.Object(IntList(x3[4]1, x4[4]1)), x4[4])), x2[4])≥2422_0_NTH_LE(x0[4], java.lang.Object(IntList(x3[4]1, x4[4]1)), +(x2[4], -1))∧(UIncreasing(2422_0_NTH_LE(x0[5], x3[5], +(x2[5], -1))), ≥))

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

(21)    (0 ≥ 0 ⇒ (UIncreasing(2422_0_NTH_LE(x0[5], x3[5], +(x2[5], -1))), ≥)∧[(9)bni_24 + (-1)Bound*bni_24] + [(16)bni_24]x3[4]1 + [(-1)bni_24]x0[4] ≥ 0∧[4 + (-1)bso_25] + [8]x3[4]1 ≥ 0)

We simplified constraint (21) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:

(22)    (0 ≥ 0 ⇒ (UIncreasing(2422_0_NTH_LE(x0[5], x3[5], +(x2[5], -1))), ≥)∧[(9)bni_24 + (-1)Bound*bni_24] + [(16)bni_24]x3[4]1 + [(-1)bni_24]x0[4] ≥ 0∧[4 + (-1)bso_25] + [8]x3[4]1 ≥ 0)

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

(23)    (0 ≥ 0 ⇒ (UIncreasing(2422_0_NTH_LE(x0[5], x3[5], +(x2[5], -1))), ≥)∧[(9)bni_24 + (-1)Bound*bni_24] + [(16)bni_24]x3[4]1 + [(-1)bni_24]x0[4] ≥ 0∧[4 + (-1)bso_25] + [8]x3[4]1 ≥ 0)

We simplified constraint (23) using rules (IDP_UNRESTRICTED_VARS), (IDP_POLY_GCD) which results in the following new constraint:

(24)    (0 ≥ 0 ⇒ (UIncreasing(2422_0_NTH_LE(x0[5], x3[5], +(x2[5], -1))), ≥)∧0 ≥ 0∧0 ≥ 0∧0 ≥ 0∧[(16)bni_24] ≥ 0∧[(-1)bni_24] ≥ 0∧[(9)bni_24 + (-1)Bound*bni_24] ≥ 0∧0 ≥ 0∧0 ≥ 0∧0 ≥ 0∧0 ≥ 0∧[4 + (-1)bso_25] ≥ 0∧[1] ≥ 0)

• We consider the chain 2422_0_NTH_LE(x0[4], java.lang.Object(IntList(x3[4], x4[4])), x2[4]) → COND_2422_0_NTH_LE2(>(x2[4], 1), x0[4], java.lang.Object(IntList(x3[4], x4[4])), x2[4]), COND_2422_0_NTH_LE2(TRUE, x0[5], java.lang.Object(IntList(x3[5], x4[5])), x2[5]) → 2422_0_NTH_LE(x0[5], x3[5], +(x2[5], -1)), 2422_0_NTH_LE(java.lang.Object(IntList(x0[6], x1[6])), java.lang.Object(IntList(x0[6], x1[6])), x3[6]) → COND_2422_0_NTH_LE3(>(x3[6], 1), java.lang.Object(IntList(x0[6], x1[6])), java.lang.Object(IntList(x0[6], x1[6])), x3[6]) which results in the following constraint:

(25)    (>(x2[4], 1)=TRUEx0[4]=x0[5]java.lang.Object(IntList(x3[4], x4[4]))=java.lang.Object(IntList(x3[5], x4[5]))∧x2[4]=x2[5]x0[5]=java.lang.Object(IntList(x0[6], x1[6]))∧x3[5]=java.lang.Object(IntList(x0[6], x1[6]))∧+(x2[5], -1)=x3[6]COND_2422_0_NTH_LE2(TRUE, x0[5], java.lang.Object(IntList(x3[5], x4[5])), x2[5])≥NonInfC∧COND_2422_0_NTH_LE2(TRUE, x0[5], java.lang.Object(IntList(x3[5], x4[5])), x2[5])≥2422_0_NTH_LE(x0[5], x3[5], +(x2[5], -1))∧(UIncreasing(2422_0_NTH_LE(x0[5], x3[5], +(x2[5], -1))), ≥))

We simplified constraint (25) using rules (I), (II), (III), (IV) which results in the following new constraint:

(26)    (>(x2[4], 1)=TRUECOND_2422_0_NTH_LE2(TRUE, java.lang.Object(IntList(x0[6], x1[6])), java.lang.Object(IntList(java.lang.Object(IntList(x0[6], x1[6])), x4[4])), x2[4])≥NonInfC∧COND_2422_0_NTH_LE2(TRUE, java.lang.Object(IntList(x0[6], x1[6])), java.lang.Object(IntList(java.lang.Object(IntList(x0[6], x1[6])), x4[4])), x2[4])≥2422_0_NTH_LE(java.lang.Object(IntList(x0[6], x1[6])), java.lang.Object(IntList(x0[6], x1[6])), +(x2[4], -1))∧(UIncreasing(2422_0_NTH_LE(x0[5], x3[5], +(x2[5], -1))), ≥))

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

(27)    (0 ≥ 0 ⇒ (UIncreasing(2422_0_NTH_LE(x0[5], x3[5], +(x2[5], -1))), ≥)∧[(7)bni_24 + (-1)Bound*bni_24] + [(12)bni_24]x0[6] ≥ 0∧[4 + (-1)bso_25] + [8]x0[6] ≥ 0)

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

(28)    (0 ≥ 0 ⇒ (UIncreasing(2422_0_NTH_LE(x0[5], x3[5], +(x2[5], -1))), ≥)∧[(7)bni_24 + (-1)Bound*bni_24] + [(12)bni_24]x0[6] ≥ 0∧[4 + (-1)bso_25] + [8]x0[6] ≥ 0)

We simplified constraint (28) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:

(29)    (0 ≥ 0 ⇒ (UIncreasing(2422_0_NTH_LE(x0[5], x3[5], +(x2[5], -1))), ≥)∧[(7)bni_24 + (-1)Bound*bni_24] + [(12)bni_24]x0[6] ≥ 0∧[4 + (-1)bso_25] + [8]x0[6] ≥ 0)

We simplified constraint (29) using rules (IDP_UNRESTRICTED_VARS), (IDP_POLY_GCD) which results in the following new constraint:

(30)    (0 ≥ 0 ⇒ (UIncreasing(2422_0_NTH_LE(x0[5], x3[5], +(x2[5], -1))), ≥)∧0 ≥ 0∧0 ≥ 0∧0 ≥ 0∧[(12)bni_24] ≥ 0∧[(7)bni_24 + (-1)Bound*bni_24] ≥ 0∧0 ≥ 0∧0 ≥ 0∧0 ≥ 0∧[4 + (-1)bso_25] ≥ 0∧[1] ≥ 0)

For Pair 2422_0_NTH_LE(x0[4], java.lang.Object(IntList(x3[4], x4[4])), x2[4]) → COND_2422_0_NTH_LE2(>(x2[4], 1), x0[4], java.lang.Object(IntList(x3[4], x4[4])), x2[4]) the following chains were created:
• We consider the chain 2422_0_NTH_LE(x0[4], java.lang.Object(IntList(x3[4], x4[4])), x2[4]) → COND_2422_0_NTH_LE2(>(x2[4], 1), x0[4], java.lang.Object(IntList(x3[4], x4[4])), x2[4]), COND_2422_0_NTH_LE2(TRUE, x0[5], java.lang.Object(IntList(x3[5], x4[5])), x2[5]) → 2422_0_NTH_LE(x0[5], x3[5], +(x2[5], -1)) which results in the following constraint:

(31)    (>(x2[4], 1)=TRUEx0[4]=x0[5]java.lang.Object(IntList(x3[4], x4[4]))=java.lang.Object(IntList(x3[5], x4[5]))∧x2[4]=x2[5]2422_0_NTH_LE(x0[4], java.lang.Object(IntList(x3[4], x4[4])), x2[4])≥NonInfC∧2422_0_NTH_LE(x0[4], java.lang.Object(IntList(x3[4], x4[4])), x2[4])≥COND_2422_0_NTH_LE2(>(x2[4], 1), x0[4], java.lang.Object(IntList(x3[4], x4[4])), x2[4])∧(UIncreasing(COND_2422_0_NTH_LE2(>(x2[4], 1), x0[4], java.lang.Object(IntList(x3[4], x4[4])), x2[4])), ≥))

We simplified constraint (31) using rules (I), (II), (IV) which results in the following new constraint:

(32)    (>(x2[4], 1)=TRUE2422_0_NTH_LE(x0[4], java.lang.Object(IntList(x3[4], x4[4])), x2[4])≥NonInfC∧2422_0_NTH_LE(x0[4], java.lang.Object(IntList(x3[4], x4[4])), x2[4])≥COND_2422_0_NTH_LE2(>(x2[4], 1), x0[4], java.lang.Object(IntList(x3[4], x4[4])), x2[4])∧(UIncreasing(COND_2422_0_NTH_LE2(>(x2[4], 1), x0[4], java.lang.Object(IntList(x3[4], x4[4])), x2[4])), ≥))

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

(33)    (0 ≥ 0 ⇒ (UIncreasing(COND_2422_0_NTH_LE2(>(x2[4], 1), x0[4], java.lang.Object(IntList(x3[4], x4[4])), x2[4])), ≥)∧[(5)bni_26 + (-1)Bound*bni_26] + [bni_26]x2[4] + [(8)bni_26]x3[4] + [(-1)bni_26]x0[4] ≥ 0∧[4 + (-1)bso_27] + x2[4] + [4]x3[4] ≥ 0)

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

(34)    (0 ≥ 0 ⇒ (UIncreasing(COND_2422_0_NTH_LE2(>(x2[4], 1), x0[4], java.lang.Object(IntList(x3[4], x4[4])), x2[4])), ≥)∧[(5)bni_26 + (-1)Bound*bni_26] + [bni_26]x2[4] + [(8)bni_26]x3[4] + [(-1)bni_26]x0[4] ≥ 0∧[4 + (-1)bso_27] + x2[4] + [4]x3[4] ≥ 0)

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

(35)    (0 ≥ 0 ⇒ (UIncreasing(COND_2422_0_NTH_LE2(>(x2[4], 1), x0[4], java.lang.Object(IntList(x3[4], x4[4])), x2[4])), ≥)∧[(5)bni_26 + (-1)Bound*bni_26] + [bni_26]x2[4] + [(8)bni_26]x3[4] + [(-1)bni_26]x0[4] ≥ 0∧[4 + (-1)bso_27] + x2[4] + [4]x3[4] ≥ 0)

We simplified constraint (35) using rules (IDP_UNRESTRICTED_VARS), (IDP_POLY_GCD) which results in the following new constraint:

(36)    (0 ≥ 0 ⇒ (UIncreasing(COND_2422_0_NTH_LE2(>(x2[4], 1), x0[4], java.lang.Object(IntList(x3[4], x4[4])), x2[4])), ≥)∧[bni_26] ≥ 0∧0 ≥ 0∧[(8)bni_26] ≥ 0∧[(-1)bni_26] ≥ 0∧[(5)bni_26 + (-1)Bound*bni_26] ≥ 0∧[1] ≥ 0∧0 ≥ 0∧0 ≥ 0∧[4 + (-1)bso_27] ≥ 0∧[1] ≥ 0)

To summarize, we get the following constraints P for the following pairs.
• COND_2422_0_NTH_LE3(TRUE, java.lang.Object(IntList(x0[7], x1[7])), java.lang.Object(IntList(x0[7], x1[7])), x3[7]) → 2422_0_NTH_LE(java.lang.Object(IntList(x0[7], x1[7])), x0[7], +(x3[7], -1))
• (0 ≥ 0 ⇒ (UIncreasing(2422_0_NTH_LE(java.lang.Object(IntList(x0[7], x1[7])), x0[7], +(x3[7], -1))), ≥)∧0 ≥ 0∧0 ≥ 0∧0 ≥ 0∧0 ≥ 0∧[(2)bni_20 + (-1)Bound*bni_20] ≥ 0∧0 ≥ 0∧0 ≥ 0∧0 ≥ 0∧[7 + (-1)bso_21] ≥ 0∧[1] ≥ 0)
• (0 ≥ 0 ⇒ (UIncreasing(2422_0_NTH_LE(java.lang.Object(IntList(x0[7], x1[7])), x0[7], +(x3[7], -1))), ≥)∧0 ≥ 0∧0 ≥ 0∧0 ≥ 0∧[(2)bni_20 + (-1)Bound*bni_20] ≥ 0∧0 ≥ 0∧0 ≥ 0∧[7 + (-1)bso_21] ≥ 0∧[1] ≥ 0)

• 2422_0_NTH_LE(java.lang.Object(IntList(x0[6], x1[6])), java.lang.Object(IntList(x0[6], x1[6])), x3[6]) → COND_2422_0_NTH_LE3(>(x3[6], 1), java.lang.Object(IntList(x0[6], x1[6])), java.lang.Object(IntList(x0[6], x1[6])), x3[6])
• (0 ≥ 0 ⇒ (UIncreasing(COND_2422_0_NTH_LE3(>(x3[6], 1), java.lang.Object(IntList(x0[6], x1[6])), java.lang.Object(IntList(x0[6], x1[6])), x3[6])), ≥)∧[bni_22] ≥ 0∧0 ≥ 0∧[(4)bni_22] ≥ 0∧[(3)bni_22 + (-1)Bound*bni_22] ≥ 0∧[1] ≥ 0∧0 ≥ 0∧[1 + (-1)bso_23] ≥ 0∧[1] ≥ 0)

• COND_2422_0_NTH_LE2(TRUE, x0[5], java.lang.Object(IntList(x3[5], x4[5])), x2[5]) → 2422_0_NTH_LE(x0[5], x3[5], +(x2[5], -1))
• (0 ≥ 0 ⇒ (UIncreasing(2422_0_NTH_LE(x0[5], x3[5], +(x2[5], -1))), ≥)∧0 ≥ 0∧0 ≥ 0∧0 ≥ 0∧[(16)bni_24] ≥ 0∧[(-1)bni_24] ≥ 0∧[(9)bni_24 + (-1)Bound*bni_24] ≥ 0∧0 ≥ 0∧0 ≥ 0∧0 ≥ 0∧0 ≥ 0∧[4 + (-1)bso_25] ≥ 0∧[1] ≥ 0)
• (0 ≥ 0 ⇒ (UIncreasing(2422_0_NTH_LE(x0[5], x3[5], +(x2[5], -1))), ≥)∧0 ≥ 0∧0 ≥ 0∧0 ≥ 0∧[(12)bni_24] ≥ 0∧[(7)bni_24 + (-1)Bound*bni_24] ≥ 0∧0 ≥ 0∧0 ≥ 0∧0 ≥ 0∧[4 + (-1)bso_25] ≥ 0∧[1] ≥ 0)

• 2422_0_NTH_LE(x0[4], java.lang.Object(IntList(x3[4], x4[4])), x2[4]) → COND_2422_0_NTH_LE2(>(x2[4], 1), x0[4], java.lang.Object(IntList(x3[4], x4[4])), x2[4])
• (0 ≥ 0 ⇒ (UIncreasing(COND_2422_0_NTH_LE2(>(x2[4], 1), x0[4], java.lang.Object(IntList(x3[4], x4[4])), x2[4])), ≥)∧[bni_26] ≥ 0∧0 ≥ 0∧[(8)bni_26] ≥ 0∧[(-1)bni_26] ≥ 0∧[(5)bni_26 + (-1)Bound*bni_26] ≥ 0∧[1] ≥ 0∧0 ≥ 0∧0 ≥ 0∧[4 + (-1)bso_27] ≥ 0∧[1] ≥ 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 with natural coefficients for non-tuple symbols [NONINF][POLO]:

POL(TRUE) = 0
POL(FALSE) = 0
POL(COND_2422_0_NTH_LE3(x1, x2, x3, x4)) = [2] + [-1]x3 + x2
POL(java.lang.Object(x1)) = [2] + [2]x1
POL(IntList(x1, x2)) = [2]x1
POL(2422_0_NTH_LE(x1, x2, x3)) = [1] + x3 + [2]x2 + [-1]x1
POL(+(x1, x2)) = 0
POL(-1) = 0
POL(>(x1, x2)) = 0
POL(1) = 0
POL(COND_2422_0_NTH_LE2(x1, x2, x3, x4)) = [-1] + x3 + [-1]x2

The following pairs are in P>:

COND_2422_0_NTH_LE3(TRUE, java.lang.Object(IntList(x0[7], x1[7])), java.lang.Object(IntList(x0[7], x1[7])), x3[7]) → 2422_0_NTH_LE(java.lang.Object(IntList(x0[7], x1[7])), x0[7], +(x3[7], -1))
2422_0_NTH_LE(java.lang.Object(IntList(x0[6], x1[6])), java.lang.Object(IntList(x0[6], x1[6])), x3[6]) → COND_2422_0_NTH_LE3(>(x3[6], 1), java.lang.Object(IntList(x0[6], x1[6])), java.lang.Object(IntList(x0[6], x1[6])), x3[6])
COND_2422_0_NTH_LE2(TRUE, x0[5], java.lang.Object(IntList(x3[5], x4[5])), x2[5]) → 2422_0_NTH_LE(x0[5], x3[5], +(x2[5], -1))
2422_0_NTH_LE(x0[4], java.lang.Object(IntList(x3[4], x4[4])), x2[4]) → COND_2422_0_NTH_LE2(>(x2[4], 1), x0[4], java.lang.Object(IntList(x3[4], x4[4])), x2[4])

The following pairs are in Pbound:

COND_2422_0_NTH_LE3(TRUE, java.lang.Object(IntList(x0[7], x1[7])), java.lang.Object(IntList(x0[7], x1[7])), x3[7]) → 2422_0_NTH_LE(java.lang.Object(IntList(x0[7], x1[7])), x0[7], +(x3[7], -1))
2422_0_NTH_LE(java.lang.Object(IntList(x0[6], x1[6])), java.lang.Object(IntList(x0[6], x1[6])), x3[6]) → COND_2422_0_NTH_LE3(>(x3[6], 1), java.lang.Object(IntList(x0[6], x1[6])), java.lang.Object(IntList(x0[6], x1[6])), x3[6])

The following pairs are in P:
none

There are no usable rules.

(24) 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:
(5): COND_2422_0_NTH_LE2(TRUE, x0[5], java.lang.Object(IntList(x3[5], x4[5])), x2[5]) → 2422_0_NTH_LE(x0[5], x3[5], x2[5] + -1)
(4): 2422_0_NTH_LE(x0[4], java.lang.Object(IntList(x3[4], x4[4])), x2[4]) → COND_2422_0_NTH_LE2(x2[4] > 1, x0[4], java.lang.Object(IntList(x3[4], x4[4])), x2[4])

(5) -> (4), if (x0[5]* x0[4]x3[5]* java.lang.Object(IntList(x3[4], x4[4]))∧x2[5] + -1* x2[4])

(4) -> (5), if (x2[4] > 1x0[4]* x0[5]java.lang.Object(IntList(x3[4], x4[4])) →* java.lang.Object(IntList(x3[5], x4[5]))∧x2[4]* x2[5])

The set Q is empty.

(25) IDPNonInfProof (SOUND transformation)

Used the following options for this NonInfProof:
IDPGPoloSolver: Range: [(-1,2)] IsNat: true Interpretation Shape Heuristic: aprove.DPFramework.IDPProblem.Processors.nonInf.poly.IdpDefaultShapeHeuristic@3e48983b 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_2422_0_NTH_LE2(TRUE, x0[5], java.lang.Object(IntList(x3[5], x4[5])), x2[5]) → 2422_0_NTH_LE(x0[5], x3[5], +(x2[5], -1)) the following chains were created:
• We consider the chain 2422_0_NTH_LE(x0[4], java.lang.Object(IntList(x3[4], x4[4])), x2[4]) → COND_2422_0_NTH_LE2(>(x2[4], 1), x0[4], java.lang.Object(IntList(x3[4], x4[4])), x2[4]), COND_2422_0_NTH_LE2(TRUE, x0[5], java.lang.Object(IntList(x3[5], x4[5])), x2[5]) → 2422_0_NTH_LE(x0[5], x3[5], +(x2[5], -1)), 2422_0_NTH_LE(x0[4], java.lang.Object(IntList(x3[4], x4[4])), x2[4]) → COND_2422_0_NTH_LE2(>(x2[4], 1), x0[4], java.lang.Object(IntList(x3[4], x4[4])), x2[4]) which results in the following constraint:

(1)    (>(x2[4], 1)=TRUEx0[4]=x0[5]java.lang.Object(IntList(x3[4], x4[4]))=java.lang.Object(IntList(x3[5], x4[5]))∧x2[4]=x2[5]x0[5]=x0[4]1x3[5]=java.lang.Object(IntList(x3[4]1, x4[4]1))∧+(x2[5], -1)=x2[4]1COND_2422_0_NTH_LE2(TRUE, x0[5], java.lang.Object(IntList(x3[5], x4[5])), x2[5])≥NonInfC∧COND_2422_0_NTH_LE2(TRUE, x0[5], java.lang.Object(IntList(x3[5], x4[5])), x2[5])≥2422_0_NTH_LE(x0[5], x3[5], +(x2[5], -1))∧(UIncreasing(2422_0_NTH_LE(x0[5], x3[5], +(x2[5], -1))), ≥))

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

(2)    (>(x2[4], 1)=TRUECOND_2422_0_NTH_LE2(TRUE, x0[4], java.lang.Object(IntList(java.lang.Object(IntList(x3[4]1, x4[4]1)), x4[4])), x2[4])≥NonInfC∧COND_2422_0_NTH_LE2(TRUE, x0[4], java.lang.Object(IntList(java.lang.Object(IntList(x3[4]1, x4[4]1)), x4[4])), x2[4])≥2422_0_NTH_LE(x0[4], java.lang.Object(IntList(x3[4]1, x4[4]1)), +(x2[4], -1))∧(UIncreasing(2422_0_NTH_LE(x0[5], x3[5], +(x2[5], -1))), ≥))

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

(3)    (0 ≥ 0 ⇒ (UIncreasing(2422_0_NTH_LE(x0[5], x3[5], +(x2[5], -1))), ≥)∧[(140)bni_15 + (-1)Bound*bni_15] + [bni_15]x2[4] + [(72)bni_15]x3[4]1 ≥ 0∧[119 + (-1)bso_16] + x2[4] + [60]x3[4]1 ≥ 0)

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

(4)    (0 ≥ 0 ⇒ (UIncreasing(2422_0_NTH_LE(x0[5], x3[5], +(x2[5], -1))), ≥)∧[(140)bni_15 + (-1)Bound*bni_15] + [bni_15]x2[4] + [(72)bni_15]x3[4]1 ≥ 0∧[119 + (-1)bso_16] + x2[4] + [60]x3[4]1 ≥ 0)

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

(5)    (0 ≥ 0 ⇒ (UIncreasing(2422_0_NTH_LE(x0[5], x3[5], +(x2[5], -1))), ≥)∧[(140)bni_15 + (-1)Bound*bni_15] + [bni_15]x2[4] + [(72)bni_15]x3[4]1 ≥ 0∧[119 + (-1)bso_16] + x2[4] + [60]x3[4]1 ≥ 0)

We simplified constraint (5) using rules (IDP_UNRESTRICTED_VARS), (IDP_POLY_GCD) which results in the following new constraint:

(6)    (0 ≥ 0 ⇒ (UIncreasing(2422_0_NTH_LE(x0[5], x3[5], +(x2[5], -1))), ≥)∧[bni_15] ≥ 0∧0 ≥ 0∧0 ≥ 0∧[(72)bni_15] ≥ 0∧0 ≥ 0∧[(140)bni_15 + (-1)Bound*bni_15] ≥ 0∧[1] ≥ 0∧0 ≥ 0∧0 ≥ 0∧0 ≥ 0∧[119 + (-1)bso_16] ≥ 0∧[1] ≥ 0)

For Pair 2422_0_NTH_LE(x0[4], java.lang.Object(IntList(x3[4], x4[4])), x2[4]) → COND_2422_0_NTH_LE2(>(x2[4], 1), x0[4], java.lang.Object(IntList(x3[4], x4[4])), x2[4]) the following chains were created:
• We consider the chain 2422_0_NTH_LE(x0[4], java.lang.Object(IntList(x3[4], x4[4])), x2[4]) → COND_2422_0_NTH_LE2(>(x2[4], 1), x0[4], java.lang.Object(IntList(x3[4], x4[4])), x2[4]), COND_2422_0_NTH_LE2(TRUE, x0[5], java.lang.Object(IntList(x3[5], x4[5])), x2[5]) → 2422_0_NTH_LE(x0[5], x3[5], +(x2[5], -1)) which results in the following constraint:

(7)    (>(x2[4], 1)=TRUEx0[4]=x0[5]java.lang.Object(IntList(x3[4], x4[4]))=java.lang.Object(IntList(x3[5], x4[5]))∧x2[4]=x2[5]2422_0_NTH_LE(x0[4], java.lang.Object(IntList(x3[4], x4[4])), x2[4])≥NonInfC∧2422_0_NTH_LE(x0[4], java.lang.Object(IntList(x3[4], x4[4])), x2[4])≥COND_2422_0_NTH_LE2(>(x2[4], 1), x0[4], java.lang.Object(IntList(x3[4], x4[4])), x2[4])∧(UIncreasing(COND_2422_0_NTH_LE2(>(x2[4], 1), x0[4], java.lang.Object(IntList(x3[4], x4[4])), x2[4])), ≥))

We simplified constraint (7) using rules (I), (II), (IV) which results in the following new constraint:

(8)    (>(x2[4], 1)=TRUE2422_0_NTH_LE(x0[4], java.lang.Object(IntList(x3[4], x4[4])), x2[4])≥NonInfC∧2422_0_NTH_LE(x0[4], java.lang.Object(IntList(x3[4], x4[4])), x2[4])≥COND_2422_0_NTH_LE2(>(x2[4], 1), x0[4], java.lang.Object(IntList(x3[4], x4[4])), x2[4])∧(UIncreasing(COND_2422_0_NTH_LE2(>(x2[4], 1), x0[4], java.lang.Object(IntList(x3[4], x4[4])), x2[4])), ≥))

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

(9)    (0 ≥ 0 ⇒ (UIncreasing(COND_2422_0_NTH_LE2(>(x2[4], 1), x0[4], java.lang.Object(IntList(x3[4], x4[4])), x2[4])), ≥)∧[(21)bni_17 + (-1)Bound*bni_17] + [bni_17]x2[4] + [(12)bni_17]x3[4] ≥ 0∧[1 + (-1)bso_18] ≥ 0)

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

(10)    (0 ≥ 0 ⇒ (UIncreasing(COND_2422_0_NTH_LE2(>(x2[4], 1), x0[4], java.lang.Object(IntList(x3[4], x4[4])), x2[4])), ≥)∧[(21)bni_17 + (-1)Bound*bni_17] + [bni_17]x2[4] + [(12)bni_17]x3[4] ≥ 0∧[1 + (-1)bso_18] ≥ 0)

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

(11)    (0 ≥ 0 ⇒ (UIncreasing(COND_2422_0_NTH_LE2(>(x2[4], 1), x0[4], java.lang.Object(IntList(x3[4], x4[4])), x2[4])), ≥)∧[(21)bni_17 + (-1)Bound*bni_17] + [bni_17]x2[4] + [(12)bni_17]x3[4] ≥ 0∧[1 + (-1)bso_18] ≥ 0)

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

(12)    (0 ≥ 0 ⇒ (UIncreasing(COND_2422_0_NTH_LE2(>(x2[4], 1), x0[4], java.lang.Object(IntList(x3[4], x4[4])), x2[4])), ≥)∧[bni_17] ≥ 0∧0 ≥ 0∧[(12)bni_17] ≥ 0∧0 ≥ 0∧[(21)bni_17 + (-1)Bound*bni_17] ≥ 0∧0 ≥ 0∧0 ≥ 0∧0 ≥ 0∧0 ≥ 0∧[1 + (-1)bso_18] ≥ 0)

To summarize, we get the following constraints P for the following pairs.
• COND_2422_0_NTH_LE2(TRUE, x0[5], java.lang.Object(IntList(x3[5], x4[5])), x2[5]) → 2422_0_NTH_LE(x0[5], x3[5], +(x2[5], -1))
• (0 ≥ 0 ⇒ (UIncreasing(2422_0_NTH_LE(x0[5], x3[5], +(x2[5], -1))), ≥)∧[bni_15] ≥ 0∧0 ≥ 0∧0 ≥ 0∧[(72)bni_15] ≥ 0∧0 ≥ 0∧[(140)bni_15 + (-1)Bound*bni_15] ≥ 0∧[1] ≥ 0∧0 ≥ 0∧0 ≥ 0∧0 ≥ 0∧[119 + (-1)bso_16] ≥ 0∧[1] ≥ 0)

• 2422_0_NTH_LE(x0[4], java.lang.Object(IntList(x3[4], x4[4])), x2[4]) → COND_2422_0_NTH_LE2(>(x2[4], 1), x0[4], java.lang.Object(IntList(x3[4], x4[4])), x2[4])
• (0 ≥ 0 ⇒ (UIncreasing(COND_2422_0_NTH_LE2(>(x2[4], 1), x0[4], java.lang.Object(IntList(x3[4], x4[4])), x2[4])), ≥)∧[bni_17] ≥ 0∧0 ≥ 0∧[(12)bni_17] ≥ 0∧0 ≥ 0∧[(21)bni_17 + (-1)Bound*bni_17] ≥ 0∧0 ≥ 0∧0 ≥ 0∧0 ≥ 0∧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 with natural coefficients for non-tuple symbols [NONINF][POLO]:

POL(TRUE) = 0
POL(FALSE) = 0
POL(COND_2422_0_NTH_LE2(x1, x2, x3, x4)) = x4 + [2]x3
POL(java.lang.Object(x1)) = [1] + [3]x1
POL(IntList(x1, x2)) = [3] + [2]x1
POL(2422_0_NTH_LE(x1, x2, x3)) = [1] + x3 + [2]x2
POL(+(x1, x2)) = 0
POL(-1) = 0
POL(>(x1, x2)) = 0
POL(1) = 0

The following pairs are in P>:

COND_2422_0_NTH_LE2(TRUE, x0[5], java.lang.Object(IntList(x3[5], x4[5])), x2[5]) → 2422_0_NTH_LE(x0[5], x3[5], +(x2[5], -1))
2422_0_NTH_LE(x0[4], java.lang.Object(IntList(x3[4], x4[4])), x2[4]) → COND_2422_0_NTH_LE2(>(x2[4], 1), x0[4], java.lang.Object(IntList(x3[4], x4[4])), x2[4])

The following pairs are in Pbound:

COND_2422_0_NTH_LE2(TRUE, x0[5], java.lang.Object(IntList(x3[5], x4[5])), x2[5]) → 2422_0_NTH_LE(x0[5], x3[5], +(x2[5], -1))
2422_0_NTH_LE(x0[4], java.lang.Object(IntList(x3[4], x4[4])), x2[4]) → COND_2422_0_NTH_LE2(>(x2[4], 1), x0[4], java.lang.Object(IntList(x3[4], x4[4])), x2[4])

The following pairs are in P:
none

There are no usable rules.

(26) 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:
none

R is empty.

The integer pair graph is empty.

The set Q is empty.

(27) IDependencyGraphProof (EQUIVALENT transformation)

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

(29) Obligation:

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

(30) SCCToIDPv1Proof (SOUND transformation)

Transformed FIGraph SCCs to IDPs. Log:

Generated 21 rules for P and 0 rules for R.

P rules:
1874_0_nth_ConstantStackPush(EOS(STATIC_1874), o740, o740, i482, o739, i482) → 1880_0_nth_LE(EOS(STATIC_1880), o740, o740, i482, o739, i482, 1)
1880_0_nth_LE(EOS(STATIC_1880), o740, o740, i494, o739, i494, matching1) → 1886_0_nth_LE(EOS(STATIC_1886), o740, o740, i494, o739, i494, 1) | =(matching1, 1)
1886_0_nth_LE(EOS(STATIC_1886), o740, o740, i494, o739, i494, matching1) → 1893_0_nth_Load(EOS(STATIC_1893), o740, o740, i494, o739) | &&(>(i494, 1), =(matching1, 1))
1893_0_nth_Load(EOS(STATIC_1893), o740, o740, i494, o739) → 1899_0_nth_NULL(EOS(STATIC_1899), o740, o740, i494, o739, o739)
1899_0_nth_NULL(EOS(STATIC_1899), o740, o740, i494, java.lang.Object(o764sub), java.lang.Object(o764sub)) → 1906_0_nth_NULL(EOS(STATIC_1906), o740, o740, i494, java.lang.Object(o764sub), java.lang.Object(o764sub))
1906_0_nth_NULL(EOS(STATIC_1906), o740, o740, i494, java.lang.Object(o764sub), java.lang.Object(o764sub)) → 1914_0_nth_Inc(EOS(STATIC_1914), o740, o740, i494, java.lang.Object(o764sub))
1914_0_nth_Inc(EOS(STATIC_1914), o740, o740, i494, java.lang.Object(o764sub)) → 1924_0_nth_Load(EOS(STATIC_1924), o740, o740, +(i494, -1), java.lang.Object(o764sub)) | >(i494, 0)
1924_0_nth_Load(EOS(STATIC_1924), o740, o740, i500, java.lang.Object(o764sub)) → 1935_0_nth_FieldAccess(EOS(STATIC_1935), o740, o740, i500, java.lang.Object(o764sub))
1935_0_nth_FieldAccess(EOS(STATIC_1935), o740, o740, i500, java.lang.Object(o764sub)) → 1946_0_nth_FieldAccess(EOS(STATIC_1946), o740, o740, i500, java.lang.Object(o764sub))
1935_0_nth_FieldAccess(EOS(STATIC_1935), java.lang.Object(o764sub), java.lang.Object(o764sub), i500, java.lang.Object(o764sub)) → 1947_0_nth_FieldAccess(EOS(STATIC_1947), java.lang.Object(o764sub), java.lang.Object(o764sub), i500, java.lang.Object(o764sub))
1946_0_nth_FieldAccess(EOS(STATIC_1946), o740, o740, i500, java.lang.Object(IntList(EOC, o787))) → 1954_0_nth_FieldAccess(EOS(STATIC_1954), o740, o740, i500, java.lang.Object(IntList(EOC, o787)))
1954_0_nth_FieldAccess(EOS(STATIC_1954), o740, o740, i500, java.lang.Object(IntList(EOC, o787))) → 1966_0_nth_Store(EOS(STATIC_1966), o740, o740, i500, o787)
1966_0_nth_Store(EOS(STATIC_1966), o740, o740, i500, o787) → 1979_0_nth_JMP(EOS(STATIC_1979), o740, o740, i500, o787)
1979_0_nth_JMP(EOS(STATIC_1979), o740, o740, i500, o787) → 1990_0_nth_Load(EOS(STATIC_1990), o740, o740, i500, o787)
1868_0_nth_Load(EOS(STATIC_1868), o740, o740, i482, o739) → 1874_0_nth_ConstantStackPush(EOS(STATIC_1874), o740, o740, i482, o739, i482)
1947_0_nth_FieldAccess(EOS(STATIC_1947), java.lang.Object(IntList(EOC, o789)), java.lang.Object(IntList(EOC, o789)), i500, java.lang.Object(IntList(EOC, o789))) → 1956_0_nth_FieldAccess(EOS(STATIC_1956), java.lang.Object(IntList(EOC, o789)), java.lang.Object(IntList(EOC, o789)), i500, java.lang.Object(IntList(EOC, o789)))
1956_0_nth_FieldAccess(EOS(STATIC_1956), java.lang.Object(IntList(EOC, o789)), java.lang.Object(IntList(EOC, o789)), i500, java.lang.Object(IntList(EOC, o789))) → 1968_0_nth_Store(EOS(STATIC_1968), java.lang.Object(IntList(EOC, o789)), java.lang.Object(IntList(EOC, o789)), i500, o789)
1968_0_nth_Store(EOS(STATIC_1968), java.lang.Object(IntList(EOC, o789)), java.lang.Object(IntList(EOC, o789)), i500, o789) → 1980_0_nth_JMP(EOS(STATIC_1980), java.lang.Object(IntList(EOC, o789)), java.lang.Object(IntList(EOC, o789)), i500, o789)
1980_0_nth_JMP(EOS(STATIC_1980), java.lang.Object(IntList(EOC, o789)), java.lang.Object(IntList(EOC, o789)), i500, o789) → 1993_0_nth_Load(EOS(STATIC_1993), java.lang.Object(IntList(EOC, o789)), java.lang.Object(IntList(EOC, o789)), i500, o789)
1993_0_nth_Load(EOS(STATIC_1993), java.lang.Object(IntList(EOC, o789)), java.lang.Object(IntList(EOC, o789)), i500, o789) → 1868_0_nth_Load(EOS(STATIC_1868), java.lang.Object(IntList(EOC, o789)), java.lang.Object(IntList(EOC, o789)), i500, o789)
R rules:

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

P rules:
1874_0_nth_ConstantStackPush(EOS(STATIC_1874), x0, x0, x1, java.lang.Object(IntList(EOC, x2)), x1) → 1874_0_nth_ConstantStackPush(EOS(STATIC_1874), x0, x0, +(x1, -1), x2, +(x1, -1)) | >(x1, 1)
1874_0_nth_ConstantStackPush(EOS(STATIC_1874), java.lang.Object(IntList(EOC, x0)), java.lang.Object(IntList(EOC, x0)), x1, java.lang.Object(IntList(EOC, x0)), x1) → 1874_0_nth_ConstantStackPush(EOS(STATIC_1874), java.lang.Object(IntList(EOC, x0)), java.lang.Object(IntList(EOC, x0)), +(x1, -1), x0, +(x1, -1)) | >(x1, 1)
R rules:

Filtered ground terms:

1874_0_nth_ConstantStackPush(x1, x2, x3, x4, x5, x6) → 1874_0_nth_ConstantStackPush(x2, x3, x4, x5, x6)
IntList(x1, x2) → IntList(x2)
EOS(x1) → EOS
Cond_1874_0_nth_ConstantStackPush1(x1, x2, x3, x4, x5, x6, x7) → Cond_1874_0_nth_ConstantStackPush1(x1, x3, x4, x5, x6, x7)
Cond_1874_0_nth_ConstantStackPush(x1, x2, x3, x4, x5, x6, x7) → Cond_1874_0_nth_ConstantStackPush(x1, x3, x4, x5, x6, x7)

Filtered duplicate args:

1874_0_nth_ConstantStackPush(x1, x2, x3, x4, x5) → 1874_0_nth_ConstantStackPush(x2, x4, x5)
Cond_1874_0_nth_ConstantStackPush(x1, x2, x3, x4, x5, x6) → Cond_1874_0_nth_ConstantStackPush(x1, x3, x5, x6)
Cond_1874_0_nth_ConstantStackPush1(x1, x2, x3, x4, x5, x6) → Cond_1874_0_nth_ConstantStackPush1(x1, x5, x6)

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

P rules:
1874_0_nth_ConstantStackPush(x0, java.lang.Object(IntList(x2)), x1) → 1874_0_nth_ConstantStackPush(x0, x2, +(x1, -1)) | >(x1, 1)
1874_0_nth_ConstantStackPush(java.lang.Object(IntList(x0)), java.lang.Object(IntList(x0)), x1) → 1874_0_nth_ConstantStackPush(java.lang.Object(IntList(x0)), x0, +(x1, -1)) | >(x1, 1)
R rules:

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

P rules:
1874_0_NTH_CONSTANTSTACKPUSH(x0, java.lang.Object(IntList(x2)), x1) → COND_1874_0_NTH_CONSTANTSTACKPUSH(>(x1, 1), x0, java.lang.Object(IntList(x2)), x1)
COND_1874_0_NTH_CONSTANTSTACKPUSH(TRUE, x0, java.lang.Object(IntList(x2)), x1) → 1874_0_NTH_CONSTANTSTACKPUSH(x0, x2, +(x1, -1))
1874_0_NTH_CONSTANTSTACKPUSH(java.lang.Object(IntList(x0)), java.lang.Object(IntList(x0)), x1) → COND_1874_0_NTH_CONSTANTSTACKPUSH1(>(x1, 1), java.lang.Object(IntList(x0)), java.lang.Object(IntList(x0)), x1)
COND_1874_0_NTH_CONSTANTSTACKPUSH1(TRUE, java.lang.Object(IntList(x0)), java.lang.Object(IntList(x0)), x1) → 1874_0_NTH_CONSTANTSTACKPUSH(java.lang.Object(IntList(x0)), x0, +(x1, -1))
R rules:

(31) 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:
(0): 1874_0_NTH_CONSTANTSTACKPUSH(x0[0], java.lang.Object(IntList(x2[0])), x1[0]) → COND_1874_0_NTH_CONSTANTSTACKPUSH(x1[0] > 1, x0[0], java.lang.Object(IntList(x2[0])), x1[0])
(1): COND_1874_0_NTH_CONSTANTSTACKPUSH(TRUE, x0[1], java.lang.Object(IntList(x2[1])), x1[1]) → 1874_0_NTH_CONSTANTSTACKPUSH(x0[1], x2[1], x1[1] + -1)
(2): 1874_0_NTH_CONSTANTSTACKPUSH(java.lang.Object(IntList(x0[2])), java.lang.Object(IntList(x0[2])), x1[2]) → COND_1874_0_NTH_CONSTANTSTACKPUSH1(x1[2] > 1, java.lang.Object(IntList(x0[2])), java.lang.Object(IntList(x0[2])), x1[2])
(3): COND_1874_0_NTH_CONSTANTSTACKPUSH1(TRUE, java.lang.Object(IntList(x0[3])), java.lang.Object(IntList(x0[3])), x1[3]) → 1874_0_NTH_CONSTANTSTACKPUSH(java.lang.Object(IntList(x0[3])), x0[3], x1[3] + -1)

(0) -> (1), if (x1[0] > 1x0[0]* x0[1]java.lang.Object(IntList(x2[0])) →* java.lang.Object(IntList(x2[1]))∧x1[0]* x1[1])

(1) -> (0), if (x0[1]* x0[0]x2[1]* java.lang.Object(IntList(x2[0]))∧x1[1] + -1* x1[0])

(1) -> (2), if (x0[1]* java.lang.Object(IntList(x0[2]))∧x2[1]* java.lang.Object(IntList(x0[2]))∧x1[1] + -1* x1[2])

(2) -> (3), if (x1[2] > 1java.lang.Object(IntList(x0[2])) →* java.lang.Object(IntList(x0[3]))∧x1[2]* x1[3])

(3) -> (0), if (java.lang.Object(IntList(x0[3])) →* x0[0]x0[3]* java.lang.Object(IntList(x2[0]))∧x1[3] + -1* x1[0])

(3) -> (2), if (java.lang.Object(IntList(x0[3])) →* java.lang.Object(IntList(x0[2]))∧x0[3]* java.lang.Object(IntList(x0[2]))∧x1[3] + -1* x1[2])

The set Q is empty.

(32) 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.IdpCand1ShapeHeuristic@21754bc Constraint Generator: NonInfConstraintGenerator: PathGenerator: MetricPathGenerator: Max Left Steps: 0 Max Right Steps: 0

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 1874_0_NTH_CONSTANTSTACKPUSH(x0, java.lang.Object(IntList(x2)), x1) → COND_1874_0_NTH_CONSTANTSTACKPUSH(>(x1, 1), x0, java.lang.Object(IntList(x2)), x1) the following chains were created:
• We consider the chain 1874_0_NTH_CONSTANTSTACKPUSH(x0[0], java.lang.Object(IntList(x2[0])), x1[0]) → COND_1874_0_NTH_CONSTANTSTACKPUSH(>(x1[0], 1), x0[0], java.lang.Object(IntList(x2[0])), x1[0]), COND_1874_0_NTH_CONSTANTSTACKPUSH(TRUE, x0[1], java.lang.Object(IntList(x2[1])), x1[1]) → 1874_0_NTH_CONSTANTSTACKPUSH(x0[1], x2[1], +(x1[1], -1)) which results in the following constraint:

(1)    (>(x1[0], 1)=TRUEx0[0]=x0[1]java.lang.Object(IntList(x2[0]))=java.lang.Object(IntList(x2[1]))∧x1[0]=x1[1]1874_0_NTH_CONSTANTSTACKPUSH(x0[0], java.lang.Object(IntList(x2[0])), x1[0])≥NonInfC∧1874_0_NTH_CONSTANTSTACKPUSH(x0[0], java.lang.Object(IntList(x2[0])), x1[0])≥COND_1874_0_NTH_CONSTANTSTACKPUSH(>(x1[0], 1), x0[0], java.lang.Object(IntList(x2[0])), x1[0])∧(UIncreasing(COND_1874_0_NTH_CONSTANTSTACKPUSH(>(x1[0], 1), x0[0], java.lang.Object(IntList(x2[0])), x1[0])), ≥))

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

(2)    (>(x1[0], 1)=TRUE1874_0_NTH_CONSTANTSTACKPUSH(x0[0], java.lang.Object(IntList(x2[0])), x1[0])≥NonInfC∧1874_0_NTH_CONSTANTSTACKPUSH(x0[0], java.lang.Object(IntList(x2[0])), x1[0])≥COND_1874_0_NTH_CONSTANTSTACKPUSH(>(x1[0], 1), x0[0], java.lang.Object(IntList(x2[0])), x1[0])∧(UIncreasing(COND_1874_0_NTH_CONSTANTSTACKPUSH(>(x1[0], 1), x0[0], java.lang.Object(IntList(x2[0])), x1[0])), ≥))

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

(3)    (x1[0] + [-2] ≥ 0 ⇒ (UIncreasing(COND_1874_0_NTH_CONSTANTSTACKPUSH(>(x1[0], 1), x0[0], java.lang.Object(IntList(x2[0])), x1[0])), ≥)∧[(-1)bni_12 + (-1)Bound*bni_12] + [(2)bni_12]x1[0] ≥ 0∧[(-1)bso_13] ≥ 0)

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

(4)    (x1[0] + [-2] ≥ 0 ⇒ (UIncreasing(COND_1874_0_NTH_CONSTANTSTACKPUSH(>(x1[0], 1), x0[0], java.lang.Object(IntList(x2[0])), x1[0])), ≥)∧[(-1)bni_12 + (-1)Bound*bni_12] + [(2)bni_12]x1[0] ≥ 0∧[(-1)bso_13] ≥ 0)

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

(5)    (x1[0] + [-2] ≥ 0 ⇒ (UIncreasing(COND_1874_0_NTH_CONSTANTSTACKPUSH(>(x1[0], 1), x0[0], java.lang.Object(IntList(x2[0])), x1[0])), ≥)∧[(-1)bni_12 + (-1)Bound*bni_12] + [(2)bni_12]x1[0] ≥ 0∧[(-1)bso_13] ≥ 0)

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

(6)    (x1[0] ≥ 0 ⇒ (UIncreasing(COND_1874_0_NTH_CONSTANTSTACKPUSH(>(x1[0], 1), x0[0], java.lang.Object(IntList(x2[0])), x1[0])), ≥)∧[(3)bni_12 + (-1)Bound*bni_12] + [(2)bni_12]x1[0] ≥ 0∧[(-1)bso_13] ≥ 0)

For Pair COND_1874_0_NTH_CONSTANTSTACKPUSH(TRUE, x0, java.lang.Object(IntList(x2)), x1) → 1874_0_NTH_CONSTANTSTACKPUSH(x0, x2, +(x1, -1)) the following chains were created:
• We consider the chain COND_1874_0_NTH_CONSTANTSTACKPUSH(TRUE, x0[1], java.lang.Object(IntList(x2[1])), x1[1]) → 1874_0_NTH_CONSTANTSTACKPUSH(x0[1], x2[1], +(x1[1], -1)) which results in the following constraint:

(7)    (COND_1874_0_NTH_CONSTANTSTACKPUSH(TRUE, x0[1], java.lang.Object(IntList(x2[1])), x1[1])≥NonInfC∧COND_1874_0_NTH_CONSTANTSTACKPUSH(TRUE, x0[1], java.lang.Object(IntList(x2[1])), x1[1])≥1874_0_NTH_CONSTANTSTACKPUSH(x0[1], x2[1], +(x1[1], -1))∧(UIncreasing(1874_0_NTH_CONSTANTSTACKPUSH(x0[1], x2[1], +(x1[1], -1))), ≥))

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

(8)    ((UIncreasing(1874_0_NTH_CONSTANTSTACKPUSH(x0[1], x2[1], +(x1[1], -1))), ≥)∧[bni_14] = 0∧[2 + (-1)bso_15] ≥ 0)

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

(9)    ((UIncreasing(1874_0_NTH_CONSTANTSTACKPUSH(x0[1], x2[1], +(x1[1], -1))), ≥)∧[bni_14] = 0∧[2 + (-1)bso_15] ≥ 0)

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

(10)    ((UIncreasing(1874_0_NTH_CONSTANTSTACKPUSH(x0[1], x2[1], +(x1[1], -1))), ≥)∧[bni_14] = 0∧[2 + (-1)bso_15] ≥ 0)

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

(11)    ((UIncreasing(1874_0_NTH_CONSTANTSTACKPUSH(x0[1], x2[1], +(x1[1], -1))), ≥)∧[bni_14] = 0∧0 = 0∧[2 + (-1)bso_15] ≥ 0)

For Pair 1874_0_NTH_CONSTANTSTACKPUSH(java.lang.Object(IntList(x0)), java.lang.Object(IntList(x0)), x1) → COND_1874_0_NTH_CONSTANTSTACKPUSH1(>(x1, 1), java.lang.Object(IntList(x0)), java.lang.Object(IntList(x0)), x1) the following chains were created:
• We consider the chain 1874_0_NTH_CONSTANTSTACKPUSH(java.lang.Object(IntList(x0[2])), java.lang.Object(IntList(x0[2])), x1[2]) → COND_1874_0_NTH_CONSTANTSTACKPUSH1(>(x1[2], 1), java.lang.Object(IntList(x0[2])), java.lang.Object(IntList(x0[2])), x1[2]), COND_1874_0_NTH_CONSTANTSTACKPUSH1(TRUE, java.lang.Object(IntList(x0[3])), java.lang.Object(IntList(x0[3])), x1[3]) → 1874_0_NTH_CONSTANTSTACKPUSH(java.lang.Object(IntList(x0[3])), x0[3], +(x1[3], -1)) which results in the following constraint:

(12)    (>(x1[2], 1)=TRUEjava.lang.Object(IntList(x0[2]))=java.lang.Object(IntList(x0[3]))∧x1[2]=x1[3]1874_0_NTH_CONSTANTSTACKPUSH(java.lang.Object(IntList(x0[2])), java.lang.Object(IntList(x0[2])), x1[2])≥NonInfC∧1874_0_NTH_CONSTANTSTACKPUSH(java.lang.Object(IntList(x0[2])), java.lang.Object(IntList(x0[2])), x1[2])≥COND_1874_0_NTH_CONSTANTSTACKPUSH1(>(x1[2], 1), java.lang.Object(IntList(x0[2])), java.lang.Object(IntList(x0[2])), x1[2])∧(UIncreasing(COND_1874_0_NTH_CONSTANTSTACKPUSH1(>(x1[2], 1), java.lang.Object(IntList(x0[2])), java.lang.Object(IntList(x0[2])), x1[2])), ≥))

We simplified constraint (12) using rules (I), (II), (IV) which results in the following new constraint:

(13)    (>(x1[2], 1)=TRUE1874_0_NTH_CONSTANTSTACKPUSH(java.lang.Object(IntList(x0[2])), java.lang.Object(IntList(x0[2])), x1[2])≥NonInfC∧1874_0_NTH_CONSTANTSTACKPUSH(java.lang.Object(IntList(x0[2])), java.lang.Object(IntList(x0[2])), x1[2])≥COND_1874_0_NTH_CONSTANTSTACKPUSH1(>(x1[2], 1), java.lang.Object(IntList(x0[2])), java.lang.Object(IntList(x0[2])), x1[2])∧(UIncreasing(COND_1874_0_NTH_CONSTANTSTACKPUSH1(>(x1[2], 1), java.lang.Object(IntList(x0[2])), java.lang.Object(IntList(x0[2])), x1[2])), ≥))

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

(14)    (x1[2] + [-2] ≥ 0 ⇒ (UIncreasing(COND_1874_0_NTH_CONSTANTSTACKPUSH1(>(x1[2], 1), java.lang.Object(IntList(x0[2])), java.lang.Object(IntList(x0[2])), x1[2])), ≥)∧[(-1)bni_16 + (-1)Bound*bni_16] + [(2)bni_16]x1[2] ≥ 0∧[(-1)bso_17] ≥ 0)

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

(15)    (x1[2] + [-2] ≥ 0 ⇒ (UIncreasing(COND_1874_0_NTH_CONSTANTSTACKPUSH1(>(x1[2], 1), java.lang.Object(IntList(x0[2])), java.lang.Object(IntList(x0[2])), x1[2])), ≥)∧[(-1)bni_16 + (-1)Bound*bni_16] + [(2)bni_16]x1[2] ≥ 0∧[(-1)bso_17] ≥ 0)

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

(16)    (x1[2] + [-2] ≥ 0 ⇒ (UIncreasing(COND_1874_0_NTH_CONSTANTSTACKPUSH1(>(x1[2], 1), java.lang.Object(IntList(x0[2])), java.lang.Object(IntList(x0[2])), x1[2])), ≥)∧[(-1)bni_16 + (-1)Bound*bni_16] + [(2)bni_16]x1[2] ≥ 0∧[(-1)bso_17] ≥ 0)

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

(17)    (x1[2] ≥ 0 ⇒ (UIncreasing(COND_1874_0_NTH_CONSTANTSTACKPUSH1(>(x1[2], 1), java.lang.Object(IntList(x0[2])), java.lang.Object(IntList(x0[2])), x1[2])), ≥)∧[(3)bni_16 + (-1)Bound*bni_16] + [(2)bni_16]x1[2] ≥ 0∧[(-1)bso_17] ≥ 0)

For Pair COND_1874_0_NTH_CONSTANTSTACKPUSH1(TRUE, java.lang.Object(IntList(x0)), java.lang.Object(IntList(x0)), x1) → 1874_0_NTH_CONSTANTSTACKPUSH(java.lang.Object(IntList(x0)), x0, +(x1, -1)) the following chains were created:
• We consider the chain COND_1874_0_NTH_CONSTANTSTACKPUSH1(TRUE, java.lang.Object(IntList(x0[3])), java.lang.Object(IntList(x0[3])), x1[3]) → 1874_0_NTH_CONSTANTSTACKPUSH(java.lang.Object(IntList(x0[3])), x0[3], +(x1[3], -1)) which results in the following constraint:

(18)    (COND_1874_0_NTH_CONSTANTSTACKPUSH1(TRUE, java.lang.Object(IntList(x0[3])), java.lang.Object(IntList(x0[3])), x1[3])≥NonInfC∧COND_1874_0_NTH_CONSTANTSTACKPUSH1(TRUE, java.lang.Object(IntList(x0[3])), java.lang.Object(IntList(x0[3])), x1[3])≥1874_0_NTH_CONSTANTSTACKPUSH(java.lang.Object(IntList(x0[3])), x0[3], +(x1[3], -1))∧(UIncreasing(1874_0_NTH_CONSTANTSTACKPUSH(java.lang.Object(IntList(x0[3])), x0[3], +(x1[3], -1))), ≥))

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

(19)    ((UIncreasing(1874_0_NTH_CONSTANTSTACKPUSH(java.lang.Object(IntList(x0[3])), x0[3], +(x1[3], -1))), ≥)∧[bni_18] = 0∧[2 + (-1)bso_19] ≥ 0)

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

(20)    ((UIncreasing(1874_0_NTH_CONSTANTSTACKPUSH(java.lang.Object(IntList(x0[3])), x0[3], +(x1[3], -1))), ≥)∧[bni_18] = 0∧[2 + (-1)bso_19] ≥ 0)

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

(21)    ((UIncreasing(1874_0_NTH_CONSTANTSTACKPUSH(java.lang.Object(IntList(x0[3])), x0[3], +(x1[3], -1))), ≥)∧[bni_18] = 0∧[2 + (-1)bso_19] ≥ 0)

We simplified constraint (21) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint:

(22)    ((UIncreasing(1874_0_NTH_CONSTANTSTACKPUSH(java.lang.Object(IntList(x0[3])), x0[3], +(x1[3], -1))), ≥)∧[bni_18] = 0∧0 = 0∧[2 + (-1)bso_19] ≥ 0)

To summarize, we get the following constraints P for the following pairs.
• 1874_0_NTH_CONSTANTSTACKPUSH(x0, java.lang.Object(IntList(x2)), x1) → COND_1874_0_NTH_CONSTANTSTACKPUSH(>(x1, 1), x0, java.lang.Object(IntList(x2)), x1)
• (x1[0] ≥ 0 ⇒ (UIncreasing(COND_1874_0_NTH_CONSTANTSTACKPUSH(>(x1[0], 1), x0[0], java.lang.Object(IntList(x2[0])), x1[0])), ≥)∧[(3)bni_12 + (-1)Bound*bni_12] + [(2)bni_12]x1[0] ≥ 0∧[(-1)bso_13] ≥ 0)

• COND_1874_0_NTH_CONSTANTSTACKPUSH(TRUE, x0, java.lang.Object(IntList(x2)), x1) → 1874_0_NTH_CONSTANTSTACKPUSH(x0, x2, +(x1, -1))
• ((UIncreasing(1874_0_NTH_CONSTANTSTACKPUSH(x0[1], x2[1], +(x1[1], -1))), ≥)∧[bni_14] = 0∧0 = 0∧[2 + (-1)bso_15] ≥ 0)

• 1874_0_NTH_CONSTANTSTACKPUSH(java.lang.Object(IntList(x0)), java.lang.Object(IntList(x0)), x1) → COND_1874_0_NTH_CONSTANTSTACKPUSH1(>(x1, 1), java.lang.Object(IntList(x0)), java.lang.Object(IntList(x0)), x1)
• (x1[2] ≥ 0 ⇒ (UIncreasing(COND_1874_0_NTH_CONSTANTSTACKPUSH1(>(x1[2], 1), java.lang.Object(IntList(x0[2])), java.lang.Object(IntList(x0[2])), x1[2])), ≥)∧[(3)bni_16 + (-1)Bound*bni_16] + [(2)bni_16]x1[2] ≥ 0∧[(-1)bso_17] ≥ 0)

• COND_1874_0_NTH_CONSTANTSTACKPUSH1(TRUE, java.lang.Object(IntList(x0)), java.lang.Object(IntList(x0)), x1) → 1874_0_NTH_CONSTANTSTACKPUSH(java.lang.Object(IntList(x0)), x0, +(x1, -1))
• ((UIncreasing(1874_0_NTH_CONSTANTSTACKPUSH(java.lang.Object(IntList(x0[3])), x0[3], +(x1[3], -1))), ≥)∧[bni_18] = 0∧0 = 0∧[2 + (-1)bso_19] ≥ 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(1874_0_NTH_CONSTANTSTACKPUSH(x1, x2, x3)) = [-1] + [2]x3
POL(java.lang.Object(x1)) = [-1]
POL(IntList(x1)) = [-1]
POL(COND_1874_0_NTH_CONSTANTSTACKPUSH(x1, x2, x3, x4)) = [-1] + [2]x4
POL(>(x1, x2)) = [-1]
POL(1) = [1]
POL(+(x1, x2)) = x1 + x2
POL(-1) = [-1]
POL(COND_1874_0_NTH_CONSTANTSTACKPUSH1(x1, x2, x3, x4)) = [-1] + [2]x4

The following pairs are in P>:

COND_1874_0_NTH_CONSTANTSTACKPUSH(TRUE, x0[1], java.lang.Object(IntList(x2[1])), x1[1]) → 1874_0_NTH_CONSTANTSTACKPUSH(x0[1], x2[1], +(x1[1], -1))
COND_1874_0_NTH_CONSTANTSTACKPUSH1(TRUE, java.lang.Object(IntList(x0[3])), java.lang.Object(IntList(x0[3])), x1[3]) → 1874_0_NTH_CONSTANTSTACKPUSH(java.lang.Object(IntList(x0[3])), x0[3], +(x1[3], -1))

The following pairs are in Pbound:

1874_0_NTH_CONSTANTSTACKPUSH(x0[0], java.lang.Object(IntList(x2[0])), x1[0]) → COND_1874_0_NTH_CONSTANTSTACKPUSH(>(x1[0], 1), x0[0], java.lang.Object(IntList(x2[0])), x1[0])
1874_0_NTH_CONSTANTSTACKPUSH(java.lang.Object(IntList(x0[2])), java.lang.Object(IntList(x0[2])), x1[2]) → COND_1874_0_NTH_CONSTANTSTACKPUSH1(>(x1[2], 1), java.lang.Object(IntList(x0[2])), java.lang.Object(IntList(x0[2])), x1[2])

The following pairs are in P:

1874_0_NTH_CONSTANTSTACKPUSH(x0[0], java.lang.Object(IntList(x2[0])), x1[0]) → COND_1874_0_NTH_CONSTANTSTACKPUSH(>(x1[0], 1), x0[0], java.lang.Object(IntList(x2[0])), x1[0])
1874_0_NTH_CONSTANTSTACKPUSH(java.lang.Object(IntList(x0[2])), java.lang.Object(IntList(x0[2])), x1[2]) → COND_1874_0_NTH_CONSTANTSTACKPUSH1(>(x1[2], 1), java.lang.Object(IntList(x0[2])), java.lang.Object(IntList(x0[2])), x1[2])

There are no usable rules.

(34) 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:
(0): 1874_0_NTH_CONSTANTSTACKPUSH(x0[0], java.lang.Object(IntList(x2[0])), x1[0]) → COND_1874_0_NTH_CONSTANTSTACKPUSH(x1[0] > 1, x0[0], java.lang.Object(IntList(x2[0])), x1[0])
(2): 1874_0_NTH_CONSTANTSTACKPUSH(java.lang.Object(IntList(x0[2])), java.lang.Object(IntList(x0[2])), x1[2]) → COND_1874_0_NTH_CONSTANTSTACKPUSH1(x1[2] > 1, java.lang.Object(IntList(x0[2])), java.lang.Object(IntList(x0[2])), x1[2])

The set Q is empty.

(35) IDependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 2 less nodes.

(37) 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_1874_0_NTH_CONSTANTSTACKPUSH(TRUE, x0[1], java.lang.Object(IntList(x2[1])), x1[1]) → 1874_0_NTH_CONSTANTSTACKPUSH(x0[1], x2[1], x1[1] + -1)
(3): COND_1874_0_NTH_CONSTANTSTACKPUSH1(TRUE, java.lang.Object(IntList(x0[3])), java.lang.Object(IntList(x0[3])), x1[3]) → 1874_0_NTH_CONSTANTSTACKPUSH(java.lang.Object(IntList(x0[3])), x0[3], x1[3] + -1)

The set Q is empty.

(38) IDependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 2 less nodes.