Termination of the given JBC could be proven:

(0) Obligation:

JBC Problem based on JBC Program:
Manifest-Version: 1.0 Created-By: 1.6.0_16 (Sun Microsystems Inc.) Main-Class: PastaC11

/**
 * Example taken from "A Term Rewriting Approach to the Automated Termination
 * Analysis of Imperative Programs" (http://www.cs.unm.edu/~spf/papers/2009-02.pdf)
 * and converted to Java.
 */

public class PastaC11 {
    public static void main(String[] args) {
        Random.args = args;
        int x = Random.random();
        int y = Random.random();

        while (true) {
			if (x >= 0) {
				x--;
				y = Random.random();
			} else if (y >= 0) {
				y--;
			} else {
				break;
			}
        }
    } 
}


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:
PastaC11.main([Ljava/lang/String;)V: Graph of 252 nodes with 1 SCC.

(3) TerminationGraphToSCCProof (SOUND transformation)

Splitted TerminationGraph to 1 SCCs.

(4) Obligation:

SCC of termination graph based on JBC Program.
SCC contains nodes from the following methods: PastaC11.main([Ljava/lang/String;)V
SCC calls the following helper methods:
Performed SCC analyses: UsedFieldsAnalysis

(5) SCCToIDPv1Proof (SOUND transformation)

Transformed FIGraph SCCs to IDPs. Log:

Generated 33 rules for P and 0 rules for R.

P rules:
933_0_main_LT(EOS(STATIC_933), matching1, i347, matching2) → 935_0_main_LT(EOS(STATIC_935), -1, i347, -1) | &&(=(matching1, -1), =(matching2, -1))
933_0_main_LT(EOS(STATIC_933), i353, i347, i353) → 936_0_main_LT(EOS(STATIC_936), i353, i347, i353)
935_0_main_LT(EOS(STATIC_935), matching1, i347, matching2) → 938_0_main_Load(EOS(STATIC_938), -1, i347) | &&(&&(<(-1, 0), =(matching1, -1)), =(matching2, -1))
938_0_main_Load(EOS(STATIC_938), matching1, i347) → 943_0_main_LT(EOS(STATIC_943), -1, i347, i347) | =(matching1, -1)
943_0_main_LT(EOS(STATIC_943), matching1, i360, i360) → 948_0_main_LT(EOS(STATIC_948), -1, i360, i360) | =(matching1, -1)
948_0_main_LT(EOS(STATIC_948), matching1, i360, i360) → 954_0_main_Inc(EOS(STATIC_954), -1, i360) | &&(>=(i360, 0), =(matching1, -1))
954_0_main_Inc(EOS(STATIC_954), matching1, i360) → 957_0_main_JMP(EOS(STATIC_957), -1, +(i360, -1)) | &&(>=(i360, 0), =(matching1, -1))
957_0_main_JMP(EOS(STATIC_957), matching1, i361) → 963_0_main_Load(EOS(STATIC_963), -1, i361) | =(matching1, -1)
963_0_main_Load(EOS(STATIC_963), matching1, i361) → 929_0_main_Load(EOS(STATIC_929), -1, i361) | =(matching1, -1)
929_0_main_Load(EOS(STATIC_929), i346, i347) → 933_0_main_LT(EOS(STATIC_933), i346, i347, i346)
936_0_main_LT(EOS(STATIC_936), i353, i347, i353) → 940_0_main_Inc(EOS(STATIC_940), i353) | >=(i353, 0)
940_0_main_Inc(EOS(STATIC_940), i353) → 945_0_main_InvokeMethod(EOS(STATIC_945), +(i353, -1)) | >=(i353, 0)
945_0_main_InvokeMethod(EOS(STATIC_945), i357) → 950_0_random_FieldAccess(EOS(STATIC_950), i357)
950_0_random_FieldAccess(EOS(STATIC_950), i357) → 960_0_random_FieldAccess(EOS(STATIC_960), i357)
960_0_random_FieldAccess(EOS(STATIC_960), i357) → 970_0_random_ArrayAccess(EOS(STATIC_970), i357)
970_0_random_ArrayAccess(EOS(STATIC_970), i357) → 972_0_random_ArrayAccess(EOS(STATIC_972), i357)
972_0_random_ArrayAccess(EOS(STATIC_972), i357) → 976_0_random_Store(EOS(STATIC_976), i357, o242)
976_0_random_Store(EOS(STATIC_976), i357, o242) → 981_0_random_FieldAccess(EOS(STATIC_981), i357, o242)
981_0_random_FieldAccess(EOS(STATIC_981), i357, o242) → 984_0_random_ConstantStackPush(EOS(STATIC_984), i357, o242)
984_0_random_ConstantStackPush(EOS(STATIC_984), i357, o242) → 989_0_random_IntArithmetic(EOS(STATIC_989), i357, o242)
989_0_random_IntArithmetic(EOS(STATIC_989), i357, o242) → 994_0_random_FieldAccess(EOS(STATIC_994), i357, o242)
994_0_random_FieldAccess(EOS(STATIC_994), i357, o242) → 997_0_random_Load(EOS(STATIC_997), i357, o242)
997_0_random_Load(EOS(STATIC_997), i357, o242) → 1009_0_random_InvokeMethod(EOS(STATIC_1009), i357, o242)
1009_0_random_InvokeMethod(EOS(STATIC_1009), i357, java.lang.Object(o262sub)) → 1012_0_random_InvokeMethod(EOS(STATIC_1012), i357, java.lang.Object(o262sub))
1012_0_random_InvokeMethod(EOS(STATIC_1012), i357, java.lang.Object(o262sub)) → 1016_0_length_Load(EOS(STATIC_1016), i357, java.lang.Object(o262sub), java.lang.Object(o262sub))
1016_0_length_Load(EOS(STATIC_1016), i357, java.lang.Object(o262sub), java.lang.Object(o262sub)) → 1030_0_length_FieldAccess(EOS(STATIC_1030), i357, java.lang.Object(o262sub), java.lang.Object(o262sub))
1030_0_length_FieldAccess(EOS(STATIC_1030), i357, java.lang.Object(java.lang.String(o271sub, i408)), java.lang.Object(java.lang.String(o271sub, i408))) → 1033_0_length_FieldAccess(EOS(STATIC_1033), i357, java.lang.Object(java.lang.String(o271sub, i408)), java.lang.Object(java.lang.String(o271sub, i408))) | &&(>=(i408, 0), >=(i409, 0))
1033_0_length_FieldAccess(EOS(STATIC_1033), i357, java.lang.Object(java.lang.String(o271sub, i408)), java.lang.Object(java.lang.String(o271sub, i408))) → 1041_0_length_Return(EOS(STATIC_1041), i357, java.lang.Object(java.lang.String(o271sub, i408)), i408)
1041_0_length_Return(EOS(STATIC_1041), i357, java.lang.Object(java.lang.String(o271sub, i408)), i408) → 1047_0_random_Return(EOS(STATIC_1047), i357, i408)
1047_0_random_Return(EOS(STATIC_1047), i357, i408) → 1050_0_main_Store(EOS(STATIC_1050), i357, i408)
1050_0_main_Store(EOS(STATIC_1050), i357, i408) → 1059_0_main_JMP(EOS(STATIC_1059), i357, i408)
1059_0_main_JMP(EOS(STATIC_1059), i357, i408) → 1067_0_main_Load(EOS(STATIC_1067), i357, i408)
1067_0_main_Load(EOS(STATIC_1067), i357, i408) → 929_0_main_Load(EOS(STATIC_929), i357, i408)
R rules:

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

P rules:
933_0_main_LT(EOS(STATIC_933), -1, x1, -1) → 933_0_main_LT(EOS(STATIC_933), -1, +(x1, -1), -1) | >(+(x1, 1), 0)
933_0_main_LT(EOS(STATIC_933), x0, x1, x0) → 933_0_main_LT(EOS(STATIC_933), +(x0, -1), x2, +(x0, -1)) | &&(>(+(x2, 1), 0), >(+(x0, 1), 0))
R rules:

Filtered ground terms:

933_0_main_LT(x1, x2, x3, x4) → 933_0_main_LT(x2, x3, x4)
EOS(x1) → EOS
Cond_933_0_main_LT1(x1, x2, x3, x4, x5, x6) → Cond_933_0_main_LT1(x1, x3, x4, x5, x6)
Cond_933_0_main_LT(x1, x2, x3, x4, x5) → Cond_933_0_main_LT(x1, x4)

Filtered duplicate args:

933_0_main_LT(x1, x2, x3) → 933_0_main_LT(x2, x3)
Cond_933_0_main_LT1(x1, x2, x3, x4, x5) → Cond_933_0_main_LT1(x1, x3, x4, x5)

Filtered unneeded arguments:

Cond_933_0_main_LT1(x1, x2, x3, x4) → Cond_933_0_main_LT1(x1, x3, x4)

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

P rules:
933_0_main_LT(x1, -1) → 933_0_main_LT(+(x1, -1), -1) | >(x1, -1)
933_0_main_LT(x1, x0) → 933_0_main_LT(x2, +(x0, -1)) | &&(>(x2, -1), >(x0, -1))
R rules:

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

P rules:
933_0_MAIN_LT(x1, -1) → COND_933_0_MAIN_LT(>(x1, -1), x1, -1)
COND_933_0_MAIN_LT(TRUE, x1, -1) → 933_0_MAIN_LT(+(x1, -1), -1)
933_0_MAIN_LT(x1, x0) → COND_933_0_MAIN_LT1(&&(>(x2, -1), >(x0, -1)), x1, x0, x2)
COND_933_0_MAIN_LT1(TRUE, x1, x0, x2) → 933_0_MAIN_LT(x2, +(x0, -1))
R rules:

(6) Obligation:

IDP problem:
The following function symbols are pre-defined:

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

The following domains are used:

Integer, Boolean

R is empty.

The integer pair graph contains the following rules and edges:
(0): 933_0_MAIN_LT(x1[0], -1) → COND_933_0_MAIN_LT(x1[0] > -1, x1[0], -1)
(1): COND_933_0_MAIN_LT(TRUE, x1[1], -1) → 933_0_MAIN_LT(x1[1] + -1, -1)
(2): 933_0_MAIN_LT(x1[2], x0[2]) → COND_933_0_MAIN_LT1(x2[2] > -1 && x0[2] > -1, x1[2], x0[2], x2[2])
(3): COND_933_0_MAIN_LT1(TRUE, x1[3], x0[3], x2[3]) → 933_0_MAIN_LT(x2[3], x0[3] + -1)

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

(1) -> (0), if x1[1] + -1 →^* x1[0]

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

(2) -> (3), if (x2[2] > -1 && x0[2] > -1 ∧x1[2] →^* x1[3]∧x0[2] →^* x0[3]∧x2[2] →^* x2[3])

(3) -> (0), if (x2[3] →^* x1[0]∧x0[3] + -1 →^* -1)

(3) -> (2), if (x2[3] →^* x1[2]∧x0[3] + -1 →^* x0[2])

The set Q is empty.

(7) IDPNonInfProof (SOUND transformation)

Used the following options for this NonInfProof:
IDPGPoloSolver: Range: [(-1,2)] IsNat: false Interpretation Shape Heuristic: aprove.DPFramework.IDPProblem.Processors.nonInf.poly.IdpDefaultShapeHeuristic@e1f8dfe 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 933_0_MAIN_LT(x1, -1) → COND_933_0_MAIN_LT(>(x1, -1), x1, -1) the following chains were created:

• We consider the chain 933_0_MAIN_LT(x1[0], -1) → COND_933_0_MAIN_LT(>(x1[0], -1), x1[0], -1), COND_933_0_MAIN_LT(TRUE, x1[1], -1) → 933_0_MAIN_LT(+(x1[1], -1), -1) which results in the following constraint:
  

  (1)    (>(x1[0], -1)=TRUE∧x1[0]=x1[1] ⇒ 933_0_MAIN_LT(x1[0], -1)≥NonInfC∧933_0_MAIN_LT(x1[0], -1)≥COND_933_0_MAIN_LT(>(x1[0], -1), x1[0], -1)∧(U^Increasing(COND_933_0_MAIN_LT(>(x1[0], -1), x1[0], -1)), ≥))

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

  (2)    (>(x1[0], -1)=TRUE ⇒ 933_0_MAIN_LT(x1[0], -1)≥NonInfC∧933_0_MAIN_LT(x1[0], -1)≥COND_933_0_MAIN_LT(>(x1[0], -1), x1[0], -1)∧(U^Increasing(COND_933_0_MAIN_LT(>(x1[0], -1), x1[0], -1)), ≥))

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

  (3)    (x1[0] ≥ 0 ⇒ (U^Increasing(COND_933_0_MAIN_LT(>(x1[0], -1), x1[0], -1)), ≥)∧[(-2)bni_17 + (-1)Bound*bni_17] ≥ 0∧[(-1)bso_18] ≥ 0)

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

  (4)    (x1[0] ≥ 0 ⇒ (U^Increasing(COND_933_0_MAIN_LT(>(x1[0], -1), x1[0], -1)), ≥)∧[(-2)bni_17 + (-1)Bound*bni_17] ≥ 0∧[(-1)bso_18] ≥ 0)

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

  (5)    (x1[0] ≥ 0 ⇒ (U^Increasing(COND_933_0_MAIN_LT(>(x1[0], -1), x1[0], -1)), ≥)∧[(-2)bni_17 + (-1)Bound*bni_17] ≥ 0∧[(-1)bso_18] ≥ 0)

  
  
  

For Pair COND_933_0_MAIN_LT(TRUE, x1, -1) → 933_0_MAIN_LT(+(x1, -1), -1) the following chains were created:

• We consider the chain 933_0_MAIN_LT(x1[0], -1) → COND_933_0_MAIN_LT(>(x1[0], -1), x1[0], -1), COND_933_0_MAIN_LT(TRUE, x1[1], -1) → 933_0_MAIN_LT(+(x1[1], -1), -1), 933_0_MAIN_LT(x1[0], -1) → COND_933_0_MAIN_LT(>(x1[0], -1), x1[0], -1) which results in the following constraint:
  

  (6)    (>(x1[0], -1)=TRUE∧x1[0]=x1[1]∧+(x1[1], -1)=x1[0]1 ⇒ COND_933_0_MAIN_LT(TRUE, x1[1], -1)≥NonInfC∧COND_933_0_MAIN_LT(TRUE, x1[1], -1)≥933_0_MAIN_LT(+(x1[1], -1), -1)∧(U^Increasing(933_0_MAIN_LT(+(x1[1], -1), -1)), ≥))

  
  
  We simplified constraint (6) using rules (III), (IV) which results in the following new constraint:
  

  (7)    (>(x1[0], -1)=TRUE ⇒ COND_933_0_MAIN_LT(TRUE, x1[0], -1)≥NonInfC∧COND_933_0_MAIN_LT(TRUE, x1[0], -1)≥933_0_MAIN_LT(+(x1[0], -1), -1)∧(U^Increasing(933_0_MAIN_LT(+(x1[1], -1), -1)), ≥))

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

  (8)    (x1[0] ≥ 0 ⇒ (U^Increasing(933_0_MAIN_LT(+(x1[1], -1), -1)), ≥)∧[(-2)bni_19 + (-1)Bound*bni_19] ≥ 0∧[(-1)bso_20] ≥ 0)

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

  (9)    (x1[0] ≥ 0 ⇒ (U^Increasing(933_0_MAIN_LT(+(x1[1], -1), -1)), ≥)∧[(-2)bni_19 + (-1)Bound*bni_19] ≥ 0∧[(-1)bso_20] ≥ 0)

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

  (10)    (x1[0] ≥ 0 ⇒ (U^Increasing(933_0_MAIN_LT(+(x1[1], -1), -1)), ≥)∧[(-2)bni_19 + (-1)Bound*bni_19] ≥ 0∧[(-1)bso_20] ≥ 0)

  
  
  
• We consider the chain 933_0_MAIN_LT(x1[0], -1) → COND_933_0_MAIN_LT(>(x1[0], -1), x1[0], -1), COND_933_0_MAIN_LT(TRUE, x1[1], -1) → 933_0_MAIN_LT(+(x1[1], -1), -1), 933_0_MAIN_LT(x1[2], x0[2]) → COND_933_0_MAIN_LT1(&&(>(x2[2], -1), >(x0[2], -1)), x1[2], x0[2], x2[2]) which results in the following constraint:
  

  (11)    (>(x1[0], -1)=TRUE∧x1[0]=x1[1]∧+(x1[1], -1)=x1[2]∧-1=x0[2] ⇒ COND_933_0_MAIN_LT(TRUE, x1[1], -1)≥NonInfC∧COND_933_0_MAIN_LT(TRUE, x1[1], -1)≥933_0_MAIN_LT(+(x1[1], -1), -1)∧(U^Increasing(933_0_MAIN_LT(+(x1[1], -1), -1)), ≥))

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

  (12)    (>(x1[0], -1)=TRUE ⇒ COND_933_0_MAIN_LT(TRUE, x1[0], -1)≥NonInfC∧COND_933_0_MAIN_LT(TRUE, x1[0], -1)≥933_0_MAIN_LT(+(x1[0], -1), -1)∧(U^Increasing(933_0_MAIN_LT(+(x1[1], -1), -1)), ≥))

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

  (13)    (x1[0] ≥ 0 ⇒ (U^Increasing(933_0_MAIN_LT(+(x1[1], -1), -1)), ≥)∧[(-2)bni_19 + (-1)Bound*bni_19] ≥ 0∧[(-1)bso_20] ≥ 0)

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

  (14)    (x1[0] ≥ 0 ⇒ (U^Increasing(933_0_MAIN_LT(+(x1[1], -1), -1)), ≥)∧[(-2)bni_19 + (-1)Bound*bni_19] ≥ 0∧[(-1)bso_20] ≥ 0)

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

  (15)    (x1[0] ≥ 0 ⇒ (U^Increasing(933_0_MAIN_LT(+(x1[1], -1), -1)), ≥)∧[(-2)bni_19 + (-1)Bound*bni_19] ≥ 0∧[(-1)bso_20] ≥ 0)

  
  
  

For Pair 933_0_MAIN_LT(x1, x0) → COND_933_0_MAIN_LT1(&&(>(x2, -1), >(x0, -1)), x1, x0, x2) the following chains were created:

• We consider the chain COND_933_0_MAIN_LT(TRUE, x1[1], -1) → 933_0_MAIN_LT(+(x1[1], -1), -1), 933_0_MAIN_LT(x1[2], x0[2]) → COND_933_0_MAIN_LT1(&&(>(x2[2], -1), >(x0[2], -1)), x1[2], x0[2], x2[2]), COND_933_0_MAIN_LT1(TRUE, x1[3], x0[3], x2[3]) → 933_0_MAIN_LT(x2[3], +(x0[3], -1)) which results in the following constraint:
  

  (16)    (+(x1[1], -1)=x1[2]∧-1=x0[2]∧&&(>(x2[2], -1), >(x0[2], -1))=TRUE∧x1[2]=x1[3]∧x0[2]=x0[3]∧x2[2]=x2[3] ⇒ 933_0_MAIN_LT(x1[2], x0[2])≥NonInfC∧933_0_MAIN_LT(x1[2], x0[2])≥COND_933_0_MAIN_LT1(&&(>(x2[2], -1), >(x0[2], -1)), x1[2], x0[2], x2[2])∧(U^Increasing(COND_933_0_MAIN_LT1(&&(>(x2[2], -1), >(x0[2], -1)), x1[2], x0[2], x2[2])), ≥))

  
  
  We solved constraint (16) using rules (I), (II), (III), (IV), (IDP_CONSTANT_FOLD), (IDP_BOOLEAN).
• We consider the chain COND_933_0_MAIN_LT1(TRUE, x1[3], x0[3], x2[3]) → 933_0_MAIN_LT(x2[3], +(x0[3], -1)), 933_0_MAIN_LT(x1[2], x0[2]) → COND_933_0_MAIN_LT1(&&(>(x2[2], -1), >(x0[2], -1)), x1[2], x0[2], x2[2]), COND_933_0_MAIN_LT1(TRUE, x1[3], x0[3], x2[3]) → 933_0_MAIN_LT(x2[3], +(x0[3], -1)) which results in the following constraint:
  

  (17)    (x2[3]=x1[2]∧+(x0[3], -1)=x0[2]∧&&(>(x2[2], -1), >(x0[2], -1))=TRUE∧x1[2]=x1[3]1∧x0[2]=x0[3]1∧x2[2]=x2[3]1 ⇒ 933_0_MAIN_LT(x1[2], x0[2])≥NonInfC∧933_0_MAIN_LT(x1[2], x0[2])≥COND_933_0_MAIN_LT1(&&(>(x2[2], -1), >(x0[2], -1)), x1[2], x0[2], x2[2])∧(U^Increasing(COND_933_0_MAIN_LT1(&&(>(x2[2], -1), >(x0[2], -1)), x1[2], x0[2], x2[2])), ≥))

  
  
  We simplified constraint (17) using rules (III), (IV), (IDP_BOOLEAN) which results in the following new constraint:
  

  (18)    (>(x2[2], -1)=TRUE∧>(+(x0[3], -1), -1)=TRUE ⇒ 933_0_MAIN_LT(x2[3], +(x0[3], -1))≥NonInfC∧933_0_MAIN_LT(x2[3], +(x0[3], -1))≥COND_933_0_MAIN_LT1(&&(>(x2[2], -1), >(+(x0[3], -1), -1)), x2[3], +(x0[3], -1), x2[2])∧(U^Increasing(COND_933_0_MAIN_LT1(&&(>(x2[2], -1), >(x0[2], -1)), x1[2], x0[2], x2[2])), ≥))

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

  (19)    (x2[2] ≥ 0∧x0[3] + [-1] ≥ 0 ⇒ (U^Increasing(COND_933_0_MAIN_LT1(&&(>(x2[2], -1), >(x0[2], -1)), x1[2], x0[2], x2[2])), ≥)∧[(-2)bni_21 + (-1)Bound*bni_21] + [(2)bni_21]x0[3] ≥ 0∧[(-1)bso_22] ≥ 0)

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

  (20)    (x2[2] ≥ 0∧x0[3] + [-1] ≥ 0 ⇒ (U^Increasing(COND_933_0_MAIN_LT1(&&(>(x2[2], -1), >(x0[2], -1)), x1[2], x0[2], x2[2])), ≥)∧[(-2)bni_21 + (-1)Bound*bni_21] + [(2)bni_21]x0[3] ≥ 0∧[(-1)bso_22] ≥ 0)

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

  (21)    (x2[2] ≥ 0∧x0[3] + [-1] ≥ 0 ⇒ (U^Increasing(COND_933_0_MAIN_LT1(&&(>(x2[2], -1), >(x0[2], -1)), x1[2], x0[2], x2[2])), ≥)∧[(-2)bni_21 + (-1)Bound*bni_21] + [(2)bni_21]x0[3] ≥ 0∧[(-1)bso_22] ≥ 0)

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

  (22)    (x2[2] ≥ 0∧x0[3] + [-1] ≥ 0 ⇒ (U^Increasing(COND_933_0_MAIN_LT1(&&(>(x2[2], -1), >(x0[2], -1)), x1[2], x0[2], x2[2])), ≥)∧0 = 0∧[(-2)bni_21 + (-1)Bound*bni_21] + [(2)bni_21]x0[3] ≥ 0∧0 = 0∧[(-1)bso_22] ≥ 0)

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

  (23)    (x2[2] ≥ 0∧x0[3] ≥ 0 ⇒ (U^Increasing(COND_933_0_MAIN_LT1(&&(>(x2[2], -1), >(x0[2], -1)), x1[2], x0[2], x2[2])), ≥)∧0 = 0∧[(-1)Bound*bni_21] + [(2)bni_21]x0[3] ≥ 0∧0 = 0∧[(-1)bso_22] ≥ 0)

  
  
  

For Pair COND_933_0_MAIN_LT1(TRUE, x1, x0, x2) → 933_0_MAIN_LT(x2, +(x0, -1)) the following chains were created:

• We consider the chain 933_0_MAIN_LT(x1[2], x0[2]) → COND_933_0_MAIN_LT1(&&(>(x2[2], -1), >(x0[2], -1)), x1[2], x0[2], x2[2]), COND_933_0_MAIN_LT1(TRUE, x1[3], x0[3], x2[3]) → 933_0_MAIN_LT(x2[3], +(x0[3], -1)), 933_0_MAIN_LT(x1[0], -1) → COND_933_0_MAIN_LT(>(x1[0], -1), x1[0], -1) which results in the following constraint:
  

  (24)    (&&(>(x2[2], -1), >(x0[2], -1))=TRUE∧x1[2]=x1[3]∧x0[2]=x0[3]∧x2[2]=x2[3]∧x2[3]=x1[0]∧+(x0[3], -1)=-1 ⇒ COND_933_0_MAIN_LT1(TRUE, x1[3], x0[3], x2[3])≥NonInfC∧COND_933_0_MAIN_LT1(TRUE, x1[3], x0[3], x2[3])≥933_0_MAIN_LT(x2[3], +(x0[3], -1))∧(U^Increasing(933_0_MAIN_LT(x2[3], +(x0[3], -1))), ≥))

  
  
  We simplified constraint (24) using rules (III), (IV), (IDP_BOOLEAN) which results in the following new constraint:
  

  (25)    (+(x0[2], -1)=-1∧>(x2[2], -1)=TRUE∧>(x0[2], -1)=TRUE ⇒ COND_933_0_MAIN_LT1(TRUE, x1[2], x0[2], x2[2])≥NonInfC∧COND_933_0_MAIN_LT1(TRUE, x1[2], x0[2], x2[2])≥933_0_MAIN_LT(x2[2], +(x0[2], -1))∧(U^Increasing(933_0_MAIN_LT(x2[3], +(x0[3], -1))), ≥))

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

  (26)    (x0[2] ≥ 0∧x2[2] ≥ 0∧x0[2] ≥ 0 ⇒ (U^Increasing(933_0_MAIN_LT(x2[3], +(x0[3], -1))), ≥)∧[(-1)bni_23 + (-1)Bound*bni_23] + [(2)bni_23]x0[2] ≥ 0∧[1 + (-1)bso_24] ≥ 0)

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

  (27)    (x0[2] ≥ 0∧x2[2] ≥ 0∧x0[2] ≥ 0 ⇒ (U^Increasing(933_0_MAIN_LT(x2[3], +(x0[3], -1))), ≥)∧[(-1)bni_23 + (-1)Bound*bni_23] + [(2)bni_23]x0[2] ≥ 0∧[1 + (-1)bso_24] ≥ 0)

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

  (28)    (x0[2] ≥ 0∧x2[2] ≥ 0∧x0[2] ≥ 0 ⇒ (U^Increasing(933_0_MAIN_LT(x2[3], +(x0[3], -1))), ≥)∧[(-1)bni_23 + (-1)Bound*bni_23] + [(2)bni_23]x0[2] ≥ 0∧[1 + (-1)bso_24] ≥ 0)

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

  (29)    (x0[2] ≥ 0∧x2[2] ≥ 0∧x0[2] ≥ 0 ⇒ (U^Increasing(933_0_MAIN_LT(x2[3], +(x0[3], -1))), ≥)∧0 = 0∧[(-1)bni_23 + (-1)Bound*bni_23] + [(2)bni_23]x0[2] ≥ 0∧0 = 0∧[1 + (-1)bso_24] ≥ 0)

  
  
  
• We consider the chain 933_0_MAIN_LT(x1[2], x0[2]) → COND_933_0_MAIN_LT1(&&(>(x2[2], -1), >(x0[2], -1)), x1[2], x0[2], x2[2]), COND_933_0_MAIN_LT1(TRUE, x1[3], x0[3], x2[3]) → 933_0_MAIN_LT(x2[3], +(x0[3], -1)), 933_0_MAIN_LT(x1[2], x0[2]) → COND_933_0_MAIN_LT1(&&(>(x2[2], -1), >(x0[2], -1)), x1[2], x0[2], x2[2]) which results in the following constraint:
  

  (30)    (&&(>(x2[2], -1), >(x0[2], -1))=TRUE∧x1[2]=x1[3]∧x0[2]=x0[3]∧x2[2]=x2[3]∧x2[3]=x1[2]1∧+(x0[3], -1)=x0[2]1 ⇒ COND_933_0_MAIN_LT1(TRUE, x1[3], x0[3], x2[3])≥NonInfC∧COND_933_0_MAIN_LT1(TRUE, x1[3], x0[3], x2[3])≥933_0_MAIN_LT(x2[3], +(x0[3], -1))∧(U^Increasing(933_0_MAIN_LT(x2[3], +(x0[3], -1))), ≥))

  
  
  We simplified constraint (30) using rules (III), (IV), (IDP_BOOLEAN) which results in the following new constraint:
  

  (31)    (>(x2[2], -1)=TRUE∧>(x0[2], -1)=TRUE ⇒ COND_933_0_MAIN_LT1(TRUE, x1[2], x0[2], x2[2])≥NonInfC∧COND_933_0_MAIN_LT1(TRUE, x1[2], x0[2], x2[2])≥933_0_MAIN_LT(x2[2], +(x0[2], -1))∧(U^Increasing(933_0_MAIN_LT(x2[3], +(x0[3], -1))), ≥))

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

  (32)    (x2[2] ≥ 0∧x0[2] ≥ 0 ⇒ (U^Increasing(933_0_MAIN_LT(x2[3], +(x0[3], -1))), ≥)∧[(-1)bni_23 + (-1)Bound*bni_23] + [(2)bni_23]x0[2] ≥ 0∧[1 + (-1)bso_24] ≥ 0)

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

  (33)    (x2[2] ≥ 0∧x0[2] ≥ 0 ⇒ (U^Increasing(933_0_MAIN_LT(x2[3], +(x0[3], -1))), ≥)∧[(-1)bni_23 + (-1)Bound*bni_23] + [(2)bni_23]x0[2] ≥ 0∧[1 + (-1)bso_24] ≥ 0)

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

  (34)    (x2[2] ≥ 0∧x0[2] ≥ 0 ⇒ (U^Increasing(933_0_MAIN_LT(x2[3], +(x0[3], -1))), ≥)∧[(-1)bni_23 + (-1)Bound*bni_23] + [(2)bni_23]x0[2] ≥ 0∧[1 + (-1)bso_24] ≥ 0)

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

  (35)    (x2[2] ≥ 0∧x0[2] ≥ 0 ⇒ (U^Increasing(933_0_MAIN_LT(x2[3], +(x0[3], -1))), ≥)∧0 = 0∧[(-1)bni_23 + (-1)Bound*bni_23] + [(2)bni_23]x0[2] ≥ 0∧0 = 0∧[1 + (-1)bso_24] ≥ 0)

  
  
  

To summarize, we get the following constraints P_≥ for the following pairs.

• 933_0_MAIN_LT(x1, -1) → COND_933_0_MAIN_LT(>(x1, -1), x1, -1)
  
  • (x1[0] ≥ 0 ⇒ (U^Increasing(COND_933_0_MAIN_LT(>(x1[0], -1), x1[0], -1)), ≥)∧[(-2)bni_17 + (-1)Bound*bni_17] ≥ 0∧[(-1)bso_18] ≥ 0)
    
  
  
• COND_933_0_MAIN_LT(TRUE, x1, -1) → 933_0_MAIN_LT(+(x1, -1), -1)
  
  • (x1[0] ≥ 0 ⇒ (U^Increasing(933_0_MAIN_LT(+(x1[1], -1), -1)), ≥)∧[(-2)bni_19 + (-1)Bound*bni_19] ≥ 0∧[(-1)bso_20] ≥ 0)
    
  • (x1[0] ≥ 0 ⇒ (U^Increasing(933_0_MAIN_LT(+(x1[1], -1), -1)), ≥)∧[(-2)bni_19 + (-1)Bound*bni_19] ≥ 0∧[(-1)bso_20] ≥ 0)
    
  
  
• 933_0_MAIN_LT(x1, x0) → COND_933_0_MAIN_LT1(&&(>(x2, -1), >(x0, -1)), x1, x0, x2)
  
  • (x2[2] ≥ 0∧x0[3] ≥ 0 ⇒ (U^Increasing(COND_933_0_MAIN_LT1(&&(>(x2[2], -1), >(x0[2], -1)), x1[2], x0[2], x2[2])), ≥)∧0 = 0∧[(-1)Bound*bni_21] + [(2)bni_21]x0[3] ≥ 0∧0 = 0∧[(-1)bso_22] ≥ 0)
    
  
  
• COND_933_0_MAIN_LT1(TRUE, x1, x0, x2) → 933_0_MAIN_LT(x2, +(x0, -1))
  
  • (x0[2] ≥ 0∧x2[2] ≥ 0∧x0[2] ≥ 0 ⇒ (U^Increasing(933_0_MAIN_LT(x2[3], +(x0[3], -1))), ≥)∧0 = 0∧[(-1)bni_23 + (-1)Bound*bni_23] + [(2)bni_23]x0[2] ≥ 0∧0 = 0∧[1 + (-1)bso_24] ≥ 0)
    
  • (x2[2] ≥ 0∧x0[2] ≥ 0 ⇒ (U^Increasing(933_0_MAIN_LT(x2[3], +(x0[3], -1))), ≥)∧0 = 0∧[(-1)bni_23 + (-1)Bound*bni_23] + [(2)bni_23]x0[2] ≥ 0∧0 = 0∧[1 + (-1)bso_24] ≥ 0)
    
  
  

The constraints for P_> respective P_bound 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 P_bound.
Using the following integer polynomial ordering the resulting constraints can be solved
Polynomial interpretation over integers[POLO]:

POL(TRUE) = 0   
POL(FALSE) = 0   
POL(933_0_MAIN_LT(x₁, x₂)) = [2]x₂   
POL(-1) = [-1]   
POL(COND_933_0_MAIN_LT(x₁, x₂, x₃)) = [2]x₃   
POL(>(x₁, x₂)) = [-1]   
POL(+(x₁, x₂)) = x₁ + x₂   
POL(COND_933_0_MAIN_LT1(x₁, x₂, x₃, x₄)) = [-1] + [2]x₃ + [-1]x₁   
POL(&&(x₁, x₂)) = [-1]   

The following pairs are in P_>:

COND_933_0_MAIN_LT1(TRUE, x1[3], x0[3], x2[3]) → 933_0_MAIN_LT(x2[3], +(x0[3], -1))

The following pairs are in P_bound:

933_0_MAIN_LT(x1[0], -1) → COND_933_0_MAIN_LT(>(x1[0], -1), x1[0], -1)
COND_933_0_MAIN_LT(TRUE, x1[1], -1) → 933_0_MAIN_LT(+(x1[1], -1), -1)
933_0_MAIN_LT(x1[2], x0[2]) → COND_933_0_MAIN_LT1(&&(>(x2[2], -1), >(x0[2], -1)), x1[2], x0[2], x2[2])
COND_933_0_MAIN_LT1(TRUE, x1[3], x0[3], x2[3]) → 933_0_MAIN_LT(x2[3], +(x0[3], -1))

The following pairs are in P_≥:

933_0_MAIN_LT(x1[0], -1) → COND_933_0_MAIN_LT(>(x1[0], -1), x1[0], -1)
COND_933_0_MAIN_LT(TRUE, x1[1], -1) → 933_0_MAIN_LT(+(x1[1], -1), -1)
933_0_MAIN_LT(x1[2], x0[2]) → COND_933_0_MAIN_LT1(&&(>(x2[2], -1), >(x0[2], -1)), x1[2], x0[2], x2[2])

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

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

(8) Obligation:

IDP problem:
The following function symbols are pre-defined:

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

The following domains are used:

Integer, Boolean

R is empty.

The integer pair graph contains the following rules and edges:
(0): 933_0_MAIN_LT(x1[0], -1) → COND_933_0_MAIN_LT(x1[0] > -1, x1[0], -1)
(1): COND_933_0_MAIN_LT(TRUE, x1[1], -1) → 933_0_MAIN_LT(x1[1] + -1, -1)
(2): 933_0_MAIN_LT(x1[2], x0[2]) → COND_933_0_MAIN_LT1(x2[2] > -1 && x0[2] > -1, x1[2], x0[2], x2[2])

(1) -> (0), if x1[1] + -1 →^* x1[0]

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

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

The set Q is empty.

(9) IDependencyGraphProof (EQUIVALENT transformation)

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

(10) Obligation:

IDP problem:
The following function symbols are pre-defined:

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

The following domains are used:

Integer

R is empty.

The integer pair graph contains the following rules and edges:
(1): COND_933_0_MAIN_LT(TRUE, x1[1], -1) → 933_0_MAIN_LT(x1[1] + -1, -1)
(0): 933_0_MAIN_LT(x1[0], -1) → COND_933_0_MAIN_LT(x1[0] > -1, x1[0], -1)

(1) -> (0), if x1[1] + -1 →^* x1[0]

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

The set Q is empty.

(11) IDPNonInfProof (SOUND transformation)

Used the following options for this NonInfProof:
IDPGPoloSolver: Range: [(-1,2)] IsNat: false Interpretation Shape Heuristic: aprove.DPFramework.IDPProblem.Processors.nonInf.poly.IdpDefaultShapeHeuristic@e1f8dfe 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_933_0_MAIN_LT(TRUE, x1[1], -1) → 933_0_MAIN_LT(+(x1[1], -1), -1) the following chains were created:

• We consider the chain 933_0_MAIN_LT(x1[0], -1) → COND_933_0_MAIN_LT(>(x1[0], -1), x1[0], -1), COND_933_0_MAIN_LT(TRUE, x1[1], -1) → 933_0_MAIN_LT(+(x1[1], -1), -1), 933_0_MAIN_LT(x1[0], -1) → COND_933_0_MAIN_LT(>(x1[0], -1), x1[0], -1) which results in the following constraint:
  

  (1)    (>(x1[0], -1)=TRUE∧x1[0]=x1[1]∧+(x1[1], -1)=x1[0]1 ⇒ COND_933_0_MAIN_LT(TRUE, x1[1], -1)≥NonInfC∧COND_933_0_MAIN_LT(TRUE, x1[1], -1)≥933_0_MAIN_LT(+(x1[1], -1), -1)∧(U^Increasing(933_0_MAIN_LT(+(x1[1], -1), -1)), ≥))

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

  (2)    (>(x1[0], -1)=TRUE ⇒ COND_933_0_MAIN_LT(TRUE, x1[0], -1)≥NonInfC∧COND_933_0_MAIN_LT(TRUE, x1[0], -1)≥933_0_MAIN_LT(+(x1[0], -1), -1)∧(U^Increasing(933_0_MAIN_LT(+(x1[1], -1), -1)), ≥))

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

  (3)    (x1[0] ≥ 0 ⇒ (U^Increasing(933_0_MAIN_LT(+(x1[1], -1), -1)), ≥)∧[(-1)Bound*bni_11] + [bni_11]x1[0] ≥ 0∧[1 + (-1)bso_12] ≥ 0)

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

  (4)    (x1[0] ≥ 0 ⇒ (U^Increasing(933_0_MAIN_LT(+(x1[1], -1), -1)), ≥)∧[(-1)Bound*bni_11] + [bni_11]x1[0] ≥ 0∧[1 + (-1)bso_12] ≥ 0)

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

  (5)    (x1[0] ≥ 0 ⇒ (U^Increasing(933_0_MAIN_LT(+(x1[1], -1), -1)), ≥)∧[(-1)Bound*bni_11] + [bni_11]x1[0] ≥ 0∧[1 + (-1)bso_12] ≥ 0)

  
  
  

For Pair 933_0_MAIN_LT(x1[0], -1) → COND_933_0_MAIN_LT(>(x1[0], -1), x1[0], -1) the following chains were created:

• We consider the chain 933_0_MAIN_LT(x1[0], -1) → COND_933_0_MAIN_LT(>(x1[0], -1), x1[0], -1), COND_933_0_MAIN_LT(TRUE, x1[1], -1) → 933_0_MAIN_LT(+(x1[1], -1), -1) which results in the following constraint:
  

  (6)    (>(x1[0], -1)=TRUE∧x1[0]=x1[1] ⇒ 933_0_MAIN_LT(x1[0], -1)≥NonInfC∧933_0_MAIN_LT(x1[0], -1)≥COND_933_0_MAIN_LT(>(x1[0], -1), x1[0], -1)∧(U^Increasing(COND_933_0_MAIN_LT(>(x1[0], -1), x1[0], -1)), ≥))

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

  (7)    (>(x1[0], -1)=TRUE ⇒ 933_0_MAIN_LT(x1[0], -1)≥NonInfC∧933_0_MAIN_LT(x1[0], -1)≥COND_933_0_MAIN_LT(>(x1[0], -1), x1[0], -1)∧(U^Increasing(COND_933_0_MAIN_LT(>(x1[0], -1), x1[0], -1)), ≥))

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

  (8)    (x1[0] ≥ 0 ⇒ (U^Increasing(COND_933_0_MAIN_LT(>(x1[0], -1), x1[0], -1)), ≥)∧[(-1)Bound*bni_13] + [bni_13]x1[0] ≥ 0∧[(-1)bso_14] ≥ 0)

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

  (9)    (x1[0] ≥ 0 ⇒ (U^Increasing(COND_933_0_MAIN_LT(>(x1[0], -1), x1[0], -1)), ≥)∧[(-1)Bound*bni_13] + [bni_13]x1[0] ≥ 0∧[(-1)bso_14] ≥ 0)

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

  (10)    (x1[0] ≥ 0 ⇒ (U^Increasing(COND_933_0_MAIN_LT(>(x1[0], -1), x1[0], -1)), ≥)∧[(-1)Bound*bni_13] + [bni_13]x1[0] ≥ 0∧[(-1)bso_14] ≥ 0)

  
  
  

To summarize, we get the following constraints P_≥ for the following pairs.

• COND_933_0_MAIN_LT(TRUE, x1[1], -1) → 933_0_MAIN_LT(+(x1[1], -1), -1)
  
  • (x1[0] ≥ 0 ⇒ (U^Increasing(933_0_MAIN_LT(+(x1[1], -1), -1)), ≥)∧[(-1)Bound*bni_11] + [bni_11]x1[0] ≥ 0∧[1 + (-1)bso_12] ≥ 0)
    
  
  
• 933_0_MAIN_LT(x1[0], -1) → COND_933_0_MAIN_LT(>(x1[0], -1), x1[0], -1)
  
  • (x1[0] ≥ 0 ⇒ (U^Increasing(COND_933_0_MAIN_LT(>(x1[0], -1), x1[0], -1)), ≥)∧[(-1)Bound*bni_13] + [bni_13]x1[0] ≥ 0∧[(-1)bso_14] ≥ 0)
    
  
  

The constraints for P_> respective P_bound 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 P_bound.
Using the following integer polynomial ordering the resulting constraints can be solved
Polynomial interpretation over integers[POLO]:

POL(TRUE) = 0   
POL(FALSE) = 0   
POL(COND_933_0_MAIN_LT(x₁, x₂, x₃)) = [-1] + [-1]x₃ + x₂   
POL(-1) = [-1]   
POL(933_0_MAIN_LT(x₁, x₂)) = [-1] + [-1]x₂ + x₁   
POL(+(x₁, x₂)) = x₁ + x₂   
POL(>(x₁, x₂)) = [-1]   

The following pairs are in P_>:

COND_933_0_MAIN_LT(TRUE, x1[1], -1) → 933_0_MAIN_LT(+(x1[1], -1), -1)

The following pairs are in P_bound:

COND_933_0_MAIN_LT(TRUE, x1[1], -1) → 933_0_MAIN_LT(+(x1[1], -1), -1)
933_0_MAIN_LT(x1[0], -1) → COND_933_0_MAIN_LT(>(x1[0], -1), x1[0], -1)

The following pairs are in P_≥:

933_0_MAIN_LT(x1[0], -1) → COND_933_0_MAIN_LT(>(x1[0], -1), x1[0], -1)

There are no usable rules.

(12) Obligation:

IDP problem:
The following function symbols are pre-defined:

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

The following domains are used:

Integer

R is empty.

The integer pair graph contains the following rules and edges:
(0): 933_0_MAIN_LT(x1[0], -1) → COND_933_0_MAIN_LT(x1[0] > -1, x1[0], -1)


The set Q is empty.

(13) IDependencyGraphProof (EQUIVALENT transformation)

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

(14) TRUE