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: PastaB8

/**
 * 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 PastaB8 {
    public static void main(String[] args) {
        Random.args = args;
        int x = Random.random();

        if (x > 0) {
            while (x != 0) {
                if (x % 2 == 0) {
                    x = x/2;
                } else {
                    x--;
                }
            }
        }
    }
}


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:
PastaB8.main([Ljava/lang/String;)V: Graph of 124 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: PastaB8.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 19 rules for P and 0 rules for R.

P rules:
246_0_main_EQ(EOS(STATIC_246), i34, i34) → 252_0_main_EQ(EOS(STATIC_252), i34, i34)
252_0_main_EQ(EOS(STATIC_252), i34, i34) → 258_0_main_Load(EOS(STATIC_258), i34) | >(i34, 0)
258_0_main_Load(EOS(STATIC_258), i34) → 266_0_main_ConstantStackPush(EOS(STATIC_266), i34, i34)
266_0_main_ConstantStackPush(EOS(STATIC_266), i34, i34) → 274_0_main_IntArithmetic(EOS(STATIC_274), i34, i34, 2)
274_0_main_IntArithmetic(EOS(STATIC_274), i34, i34, matching1) → 280_0_main_NE(EOS(STATIC_280), i34, %(i34, 2)) | =(matching1, 2)
280_0_main_NE(EOS(STATIC_280), i34, matching1) → 284_0_main_NE(EOS(STATIC_284), i34, 1) | =(matching1, 1)
280_0_main_NE(EOS(STATIC_280), i34, matching1) → 286_0_main_NE(EOS(STATIC_286), i34, 0) | =(matching1, 0)
284_0_main_NE(EOS(STATIC_284), i34, matching1) → 288_0_main_Inc(EOS(STATIC_288), i34) | &&(>(1, 0), =(matching1, 1))
288_0_main_Inc(EOS(STATIC_288), i34) → 292_0_main_JMP(EOS(STATIC_292), +(i34, -1)) | >(i34, 0)
292_0_main_JMP(EOS(STATIC_292), i41) → 298_0_main_Load(EOS(STATIC_298), i41)
298_0_main_Load(EOS(STATIC_298), i41) → 236_0_main_Load(EOS(STATIC_236), i41)
236_0_main_Load(EOS(STATIC_236), i29) → 246_0_main_EQ(EOS(STATIC_246), i29, i29)
286_0_main_NE(EOS(STATIC_286), i34, matching1) → 290_0_main_Load(EOS(STATIC_290), i34) | =(matching1, 0)
290_0_main_Load(EOS(STATIC_290), i34) → 294_0_main_ConstantStackPush(EOS(STATIC_294), i34)
294_0_main_ConstantStackPush(EOS(STATIC_294), i34) → 300_0_main_IntArithmetic(EOS(STATIC_300), i34, 2)
300_0_main_IntArithmetic(EOS(STATIC_300), i34, matching1) → 302_0_main_Store(EOS(STATIC_302), /(i34, 2)) | &&(>=(i34, 1), =(matching1, 2))
302_0_main_Store(EOS(STATIC_302), i44) → 304_0_main_JMP(EOS(STATIC_304), i44)
304_0_main_JMP(EOS(STATIC_304), i44) → 309_0_main_Load(EOS(STATIC_309), i44)
309_0_main_Load(EOS(STATIC_309), i44) → 236_0_main_Load(EOS(STATIC_236), i44)
R rules:

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

P rules:
246_0_main_EQ(EOS(STATIC_246), x0, x0) → 246_0_main_EQ(EOS(STATIC_246), +(x0, -1), +(x0, -1)) | &&(>(x0, 0), =(1, %(x0, 2)))
246_0_main_EQ(EOS(STATIC_246), x0, x0) → 246_0_main_EQ(EOS(STATIC_246), /(x0, 2), /(x0, 2)) | &&(>(+(x0, 1), 1), =(0, %(x0, 2)))
R rules:

Filtered ground terms:

246_0_main_EQ(x1, x2, x3) → 246_0_main_EQ(x2, x3)
EOS(x1) → EOS
Cond_246_0_main_EQ1(x1, x2, x3, x4) → Cond_246_0_main_EQ1(x1, x3, x4)
Cond_246_0_main_EQ(x1, x2, x3, x4) → Cond_246_0_main_EQ(x1, x3, x4)

Filtered duplicate args:

246_0_main_EQ(x1, x2) → 246_0_main_EQ(x2)
Cond_246_0_main_EQ(x1, x2, x3) → Cond_246_0_main_EQ(x1, x3)
Cond_246_0_main_EQ1(x1, x2, x3) → Cond_246_0_main_EQ1(x1, x3)

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

P rules:
246_0_main_EQ(x0) → 246_0_main_EQ(+(x0, -1)) | &&(>(x0, 0), =(1, %(x0, 2)))
246_0_main_EQ(x0) → 246_0_main_EQ(/(x0, 2)) | &&(>(x0, 0), =(0, %(x0, 2)))
R rules:

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

P rules:
246_0_MAIN_EQ(x0) → COND_246_0_MAIN_EQ(&&(>(x0, 0), =(1, %(x0, 2))), x0)
COND_246_0_MAIN_EQ(TRUE, x0) → 246_0_MAIN_EQ(+(x0, -1))
246_0_MAIN_EQ(x0) → COND_246_0_MAIN_EQ1(&&(>(x0, 0), =(0, %(x0, 2))), x0)
COND_246_0_MAIN_EQ1(TRUE, x0) → 246_0_MAIN_EQ(/(x0, 2))
R rules:

(6) Obligation:

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

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

The following domains are used:

Boolean, Integer

R is empty.

The integer pair graph contains the following rules and edges:
(0): 246_0_MAIN_EQ(x0[0]) → COND_246_0_MAIN_EQ(x0[0] > 0 && 1 = x0[0] % 2, x0[0])
(1): COND_246_0_MAIN_EQ(TRUE, x0[1]) → 246_0_MAIN_EQ(x0[1] + -1)
(2): 246_0_MAIN_EQ(x0[2]) → COND_246_0_MAIN_EQ1(x0[2] > 0 && 0 = x0[2] % 2, x0[2])
(3): COND_246_0_MAIN_EQ1(TRUE, x0[3]) → 246_0_MAIN_EQ(x0[3] / 2)

(0) -> (1), if (x0[0] > 0 && 1 = x0[0] % 2 ∧x0[0] →^* x0[1])

(1) -> (0), if (x0[1] + -1 →^* x0[0])

(1) -> (2), if (x0[1] + -1 →^* x0[2])

(2) -> (3), if (x0[2] > 0 && 0 = x0[2] % 2 ∧x0[2] →^* x0[3])

(3) -> (0), if (x0[3] / 2 →^* x0[0])

(3) -> (2), if (x0[3] / 2 →^* 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@36b37b66 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 246_0_MAIN_EQ(x0) → COND_246_0_MAIN_EQ(&&(>(x0, 0), =(1, %(x0, 2))), x0) the following chains were created:

• We consider the chain 246_0_MAIN_EQ(x0[0]) → COND_246_0_MAIN_EQ(&&(>(x0[0], 0), =(1, %(x0[0], 2))), x0[0]), COND_246_0_MAIN_EQ(TRUE, x0[1]) → 246_0_MAIN_EQ(+(x0[1], -1)) which results in the following constraint:
  

  (1)    (&&(>(x0[0], 0), =(1, %(x0[0], 2)))=TRUE∧x0[0]=x0[1] ⇒ 246_0_MAIN_EQ(x0[0])≥NonInfC∧246_0_MAIN_EQ(x0[0])≥COND_246_0_MAIN_EQ(&&(>(x0[0], 0), =(1, %(x0[0], 2))), x0[0])∧(U^Increasing(COND_246_0_MAIN_EQ(&&(>(x0[0], 0), =(1, %(x0[0], 2))), x0[0])), ≥))

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

  (2)    (>(x0[0], 0)=TRUE∧>=(1, %(x0[0], 2))=TRUE∧<=(1, %(x0[0], 2))=TRUE ⇒ 246_0_MAIN_EQ(x0[0])≥NonInfC∧246_0_MAIN_EQ(x0[0])≥COND_246_0_MAIN_EQ(&&(>(x0[0], 0), =(1, %(x0[0], 2))), x0[0])∧(U^Increasing(COND_246_0_MAIN_EQ(&&(>(x0[0], 0), =(1, %(x0[0], 2))), x0[0])), ≥))

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

  (3)    (x0[0] + [-1] ≥ 0∧[1] + [-1]min{[2], [-2]} ≥ 0∧max{[2], [-2]} + [-1] ≥ 0 ⇒ (U^Increasing(COND_246_0_MAIN_EQ(&&(>(x0[0], 0), =(1, %(x0[0], 2))), x0[0])), ≥)∧[(-1)Bound*bni_14] + [bni_14]x0[0] ≥ 0∧[(-1)bso_15] ≥ 0)

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

  (4)    (x0[0] + [-1] ≥ 0∧[1] + [-1]min{[2], [-2]} ≥ 0∧max{[2], [-2]} + [-1] ≥ 0 ⇒ (U^Increasing(COND_246_0_MAIN_EQ(&&(>(x0[0], 0), =(1, %(x0[0], 2))), x0[0])), ≥)∧[(-1)Bound*bni_14] + [bni_14]x0[0] ≥ 0∧[(-1)bso_15] ≥ 0)

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

  (5)    (x0[0] + [-1] ≥ 0∧[4] ≥ 0∧[3] ≥ 0∧[1] ≥ 0 ⇒ (U^Increasing(COND_246_0_MAIN_EQ(&&(>(x0[0], 0), =(1, %(x0[0], 2))), x0[0])), ≥)∧[(-1)Bound*bni_14] + [bni_14]x0[0] ≥ 0∧[(-1)bso_15] ≥ 0)

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

  (6)    (x0[0] ≥ 0∧[4] ≥ 0∧[3] ≥ 0∧[1] ≥ 0 ⇒ (U^Increasing(COND_246_0_MAIN_EQ(&&(>(x0[0], 0), =(1, %(x0[0], 2))), x0[0])), ≥)∧[(-1)Bound*bni_14 + bni_14] + [bni_14]x0[0] ≥ 0∧[(-1)bso_15] ≥ 0)

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

  (7)    (x0[0] ≥ 0∧[1] ≥ 0∧[1] ≥ 0∧[1] ≥ 0 ⇒ (U^Increasing(COND_246_0_MAIN_EQ(&&(>(x0[0], 0), =(1, %(x0[0], 2))), x0[0])), ≥)∧[(-1)Bound*bni_14 + bni_14] + [bni_14]x0[0] ≥ 0∧[(-1)bso_15] ≥ 0)

  
  
  

For Pair COND_246_0_MAIN_EQ(TRUE, x0) → 246_0_MAIN_EQ(+(x0, -1)) the following chains were created:

• We consider the chain 246_0_MAIN_EQ(x0[0]) → COND_246_0_MAIN_EQ(&&(>(x0[0], 0), =(1, %(x0[0], 2))), x0[0]), COND_246_0_MAIN_EQ(TRUE, x0[1]) → 246_0_MAIN_EQ(+(x0[1], -1)), 246_0_MAIN_EQ(x0[0]) → COND_246_0_MAIN_EQ(&&(>(x0[0], 0), =(1, %(x0[0], 2))), x0[0]) which results in the following constraint:
  

  (8)    (&&(>(x0[0], 0), =(1, %(x0[0], 2)))=TRUE∧x0[0]=x0[1]∧+(x0[1], -1)=x0[0]1 ⇒ COND_246_0_MAIN_EQ(TRUE, x0[1])≥NonInfC∧COND_246_0_MAIN_EQ(TRUE, x0[1])≥246_0_MAIN_EQ(+(x0[1], -1))∧(U^Increasing(246_0_MAIN_EQ(+(x0[1], -1))), ≥))

  
  
  We simplified constraint (8) using rules (III), (IV), (IDP_BOOLEAN) which results in the following new constraint:
  

  (9)    (>(x0[0], 0)=TRUE∧>=(1, %(x0[0], 2))=TRUE∧<=(1, %(x0[0], 2))=TRUE ⇒ COND_246_0_MAIN_EQ(TRUE, x0[0])≥NonInfC∧COND_246_0_MAIN_EQ(TRUE, x0[0])≥246_0_MAIN_EQ(+(x0[0], -1))∧(U^Increasing(246_0_MAIN_EQ(+(x0[1], -1))), ≥))

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

  (10)    (x0[0] + [-1] ≥ 0∧[1] + [-1]min{[2], [-2]} ≥ 0∧max{[2], [-2]} + [-1] ≥ 0 ⇒ (U^Increasing(246_0_MAIN_EQ(+(x0[1], -1))), ≥)∧[(-1)bni_16 + (-1)Bound*bni_16] + [bni_16]x0[0] ≥ 0∧[(-1)bso_17] ≥ 0)

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

  (11)    (x0[0] + [-1] ≥ 0∧[1] + [-1]min{[2], [-2]} ≥ 0∧max{[2], [-2]} + [-1] ≥ 0 ⇒ (U^Increasing(246_0_MAIN_EQ(+(x0[1], -1))), ≥)∧[(-1)bni_16 + (-1)Bound*bni_16] + [bni_16]x0[0] ≥ 0∧[(-1)bso_17] ≥ 0)

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

  (12)    (x0[0] + [-1] ≥ 0∧[4] ≥ 0∧[3] ≥ 0∧[1] ≥ 0 ⇒ (U^Increasing(246_0_MAIN_EQ(+(x0[1], -1))), ≥)∧[(-1)bni_16 + (-1)Bound*bni_16] + [bni_16]x0[0] ≥ 0∧[(-1)bso_17] ≥ 0)

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

  (13)    (x0[0] ≥ 0∧[4] ≥ 0∧[3] ≥ 0∧[1] ≥ 0 ⇒ (U^Increasing(246_0_MAIN_EQ(+(x0[1], -1))), ≥)∧[(-1)Bound*bni_16] + [bni_16]x0[0] ≥ 0∧[(-1)bso_17] ≥ 0)

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

  (14)    (x0[0] ≥ 0∧[1] ≥ 0∧[1] ≥ 0∧[1] ≥ 0 ⇒ (U^Increasing(246_0_MAIN_EQ(+(x0[1], -1))), ≥)∧[(-1)Bound*bni_16] + [bni_16]x0[0] ≥ 0∧[(-1)bso_17] ≥ 0)

  
  
  
• We consider the chain 246_0_MAIN_EQ(x0[0]) → COND_246_0_MAIN_EQ(&&(>(x0[0], 0), =(1, %(x0[0], 2))), x0[0]), COND_246_0_MAIN_EQ(TRUE, x0[1]) → 246_0_MAIN_EQ(+(x0[1], -1)), 246_0_MAIN_EQ(x0[2]) → COND_246_0_MAIN_EQ1(&&(>(x0[2], 0), =(0, %(x0[2], 2))), x0[2]) which results in the following constraint:
  

  (15)    (&&(>(x0[0], 0), =(1, %(x0[0], 2)))=TRUE∧x0[0]=x0[1]∧+(x0[1], -1)=x0[2] ⇒ COND_246_0_MAIN_EQ(TRUE, x0[1])≥NonInfC∧COND_246_0_MAIN_EQ(TRUE, x0[1])≥246_0_MAIN_EQ(+(x0[1], -1))∧(U^Increasing(246_0_MAIN_EQ(+(x0[1], -1))), ≥))

  
  
  We simplified constraint (15) using rules (III), (IV), (IDP_BOOLEAN) which results in the following new constraint:
  

  (16)    (>(x0[0], 0)=TRUE∧>=(1, %(x0[0], 2))=TRUE∧<=(1, %(x0[0], 2))=TRUE ⇒ COND_246_0_MAIN_EQ(TRUE, x0[0])≥NonInfC∧COND_246_0_MAIN_EQ(TRUE, x0[0])≥246_0_MAIN_EQ(+(x0[0], -1))∧(U^Increasing(246_0_MAIN_EQ(+(x0[1], -1))), ≥))

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

  (17)    (x0[0] + [-1] ≥ 0∧[1] + [-1]min{[2], [-2]} ≥ 0∧max{[2], [-2]} + [-1] ≥ 0 ⇒ (U^Increasing(246_0_MAIN_EQ(+(x0[1], -1))), ≥)∧[(-1)bni_16 + (-1)Bound*bni_16] + [bni_16]x0[0] ≥ 0∧[(-1)bso_17] ≥ 0)

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

  (18)    (x0[0] + [-1] ≥ 0∧[1] + [-1]min{[2], [-2]} ≥ 0∧max{[2], [-2]} + [-1] ≥ 0 ⇒ (U^Increasing(246_0_MAIN_EQ(+(x0[1], -1))), ≥)∧[(-1)bni_16 + (-1)Bound*bni_16] + [bni_16]x0[0] ≥ 0∧[(-1)bso_17] ≥ 0)

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

  (19)    (x0[0] + [-1] ≥ 0∧[4] ≥ 0∧[3] ≥ 0∧[1] ≥ 0 ⇒ (U^Increasing(246_0_MAIN_EQ(+(x0[1], -1))), ≥)∧[(-1)bni_16 + (-1)Bound*bni_16] + [bni_16]x0[0] ≥ 0∧[(-1)bso_17] ≥ 0)

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

  (20)    (x0[0] ≥ 0∧[4] ≥ 0∧[3] ≥ 0∧[1] ≥ 0 ⇒ (U^Increasing(246_0_MAIN_EQ(+(x0[1], -1))), ≥)∧[(-1)Bound*bni_16] + [bni_16]x0[0] ≥ 0∧[(-1)bso_17] ≥ 0)

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

  (21)    (x0[0] ≥ 0∧[1] ≥ 0∧[1] ≥ 0∧[1] ≥ 0 ⇒ (U^Increasing(246_0_MAIN_EQ(+(x0[1], -1))), ≥)∧[(-1)Bound*bni_16] + [bni_16]x0[0] ≥ 0∧[(-1)bso_17] ≥ 0)

  
  
  

For Pair 246_0_MAIN_EQ(x0) → COND_246_0_MAIN_EQ1(&&(>(x0, 0), =(0, %(x0, 2))), x0) the following chains were created:

• We consider the chain 246_0_MAIN_EQ(x0[2]) → COND_246_0_MAIN_EQ1(&&(>(x0[2], 0), =(0, %(x0[2], 2))), x0[2]), COND_246_0_MAIN_EQ1(TRUE, x0[3]) → 246_0_MAIN_EQ(/(x0[3], 2)) which results in the following constraint:
  

  (22)    (&&(>(x0[2], 0), =(0, %(x0[2], 2)))=TRUE∧x0[2]=x0[3] ⇒ 246_0_MAIN_EQ(x0[2])≥NonInfC∧246_0_MAIN_EQ(x0[2])≥COND_246_0_MAIN_EQ1(&&(>(x0[2], 0), =(0, %(x0[2], 2))), x0[2])∧(U^Increasing(COND_246_0_MAIN_EQ1(&&(>(x0[2], 0), =(0, %(x0[2], 2))), x0[2])), ≥))

  
  
  We simplified constraint (22) using rules (IV), (IDP_BOOLEAN) which results in the following new constraint:
  

  (23)    (>(x0[2], 0)=TRUE∧>=(0, %(x0[2], 2))=TRUE∧<=(0, %(x0[2], 2))=TRUE ⇒ 246_0_MAIN_EQ(x0[2])≥NonInfC∧246_0_MAIN_EQ(x0[2])≥COND_246_0_MAIN_EQ1(&&(>(x0[2], 0), =(0, %(x0[2], 2))), x0[2])∧(U^Increasing(COND_246_0_MAIN_EQ1(&&(>(x0[2], 0), =(0, %(x0[2], 2))), x0[2])), ≥))

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

  (24)    (x0[2] + [-1] ≥ 0∧[-1]min{[2], [-2]} ≥ 0∧max{[2], [-2]} ≥ 0 ⇒ (U^Increasing(COND_246_0_MAIN_EQ1(&&(>(x0[2], 0), =(0, %(x0[2], 2))), x0[2])), ≥)∧[(-1)Bound*bni_18] + [bni_18]x0[2] ≥ 0∧[1 + (-1)bso_19] ≥ 0)

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

  (25)    (x0[2] + [-1] ≥ 0∧[-1]min{[2], [-2]} ≥ 0∧max{[2], [-2]} ≥ 0 ⇒ (U^Increasing(COND_246_0_MAIN_EQ1(&&(>(x0[2], 0), =(0, %(x0[2], 2))), x0[2])), ≥)∧[(-1)Bound*bni_18] + [bni_18]x0[2] ≥ 0∧[1 + (-1)bso_19] ≥ 0)

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

  (26)    (x0[2] + [-1] ≥ 0∧[4] ≥ 0∧[2] ≥ 0∧[2] ≥ 0 ⇒ (U^Increasing(COND_246_0_MAIN_EQ1(&&(>(x0[2], 0), =(0, %(x0[2], 2))), x0[2])), ≥)∧[(-1)Bound*bni_18] + [bni_18]x0[2] ≥ 0∧[1 + (-1)bso_19] ≥ 0)

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

  (27)    (x0[2] ≥ 0∧[4] ≥ 0∧[2] ≥ 0∧[2] ≥ 0 ⇒ (U^Increasing(COND_246_0_MAIN_EQ1(&&(>(x0[2], 0), =(0, %(x0[2], 2))), x0[2])), ≥)∧[(-1)Bound*bni_18 + bni_18] + [bni_18]x0[2] ≥ 0∧[1 + (-1)bso_19] ≥ 0)

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

  (28)    (x0[2] ≥ 0∧[1] ≥ 0∧[1] ≥ 0∧[1] ≥ 0 ⇒ (U^Increasing(COND_246_0_MAIN_EQ1(&&(>(x0[2], 0), =(0, %(x0[2], 2))), x0[2])), ≥)∧[(-1)Bound*bni_18 + bni_18] + [bni_18]x0[2] ≥ 0∧[1 + (-1)bso_19] ≥ 0)

  
  
  

For Pair COND_246_0_MAIN_EQ1(TRUE, x0) → 246_0_MAIN_EQ(/(x0, 2)) the following chains were created:

• We consider the chain 246_0_MAIN_EQ(x0[2]) → COND_246_0_MAIN_EQ1(&&(>(x0[2], 0), =(0, %(x0[2], 2))), x0[2]), COND_246_0_MAIN_EQ1(TRUE, x0[3]) → 246_0_MAIN_EQ(/(x0[3], 2)) which results in the following constraint:
  

  (29)    (&&(>(x0[2], 0), =(0, %(x0[2], 2)))=TRUE∧x0[2]=x0[3] ⇒ COND_246_0_MAIN_EQ1(TRUE, x0[3])≥NonInfC∧COND_246_0_MAIN_EQ1(TRUE, x0[3])≥246_0_MAIN_EQ(/(x0[3], 2))∧(U^Increasing(246_0_MAIN_EQ(/(x0[3], 2))), ≥))

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

  (30)    (>(x0[2], 0)=TRUE∧>=(0, %(x0[2], 2))=TRUE∧<=(0, %(x0[2], 2))=TRUE ⇒ COND_246_0_MAIN_EQ1(TRUE, x0[2])≥NonInfC∧COND_246_0_MAIN_EQ1(TRUE, x0[2])≥246_0_MAIN_EQ(/(x0[2], 2))∧(U^Increasing(246_0_MAIN_EQ(/(x0[3], 2))), ≥))

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

  (31)    (x0[2] + [-1] ≥ 0∧[-1]min{[2], [-2]} ≥ 0∧max{[2], [-2]} ≥ 0 ⇒ (U^Increasing(246_0_MAIN_EQ(/(x0[3], 2))), ≥)∧[(-1)bni_20 + (-1)Bound*bni_20] + [bni_20]x0[2] ≥ 0∧[(-1)bso_24] + x0[2] + [-1]max{x0[2], [-1]x0[2]} ≥ 0)

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

  (32)    (x0[2] + [-1] ≥ 0∧[-1]min{[2], [-2]} ≥ 0∧max{[2], [-2]} ≥ 0 ⇒ (U^Increasing(246_0_MAIN_EQ(/(x0[3], 2))), ≥)∧[(-1)bni_20 + (-1)Bound*bni_20] + [bni_20]x0[2] ≥ 0∧[(-1)bso_24] + x0[2] + [-1]max{x0[2], [-1]x0[2]} ≥ 0)

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

  (33)    (x0[2] + [-1] ≥ 0∧[4] ≥ 0∧[2] ≥ 0∧[2] ≥ 0∧[2]x0[2] ≥ 0 ⇒ (U^Increasing(246_0_MAIN_EQ(/(x0[3], 2))), ≥)∧[(-1)bni_20 + (-1)Bound*bni_20] + [bni_20]x0[2] ≥ 0∧[(-1)bso_24] ≥ 0)

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

  (34)    (x0[2] ≥ 0∧[4] ≥ 0∧[2] ≥ 0∧[2] ≥ 0∧[2] + [2]x0[2] ≥ 0 ⇒ (U^Increasing(246_0_MAIN_EQ(/(x0[3], 2))), ≥)∧[(-1)Bound*bni_20] + [bni_20]x0[2] ≥ 0∧[(-1)bso_24] ≥ 0)

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

  (35)    (x0[2] ≥ 0∧[1] ≥ 0∧[1] ≥ 0∧[1] ≥ 0∧[1] + x0[2] ≥ 0 ⇒ (U^Increasing(246_0_MAIN_EQ(/(x0[3], 2))), ≥)∧[(-1)Bound*bni_20] + [bni_20]x0[2] ≥ 0∧[(-1)bso_24] ≥ 0)

  
  
  

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

• 246_0_MAIN_EQ(x0) → COND_246_0_MAIN_EQ(&&(>(x0, 0), =(1, %(x0, 2))), x0)
  
  • (x0[0] ≥ 0∧[1] ≥ 0∧[1] ≥ 0∧[1] ≥ 0 ⇒ (U^Increasing(COND_246_0_MAIN_EQ(&&(>(x0[0], 0), =(1, %(x0[0], 2))), x0[0])), ≥)∧[(-1)Bound*bni_14 + bni_14] + [bni_14]x0[0] ≥ 0∧[(-1)bso_15] ≥ 0)
    
  
  
• COND_246_0_MAIN_EQ(TRUE, x0) → 246_0_MAIN_EQ(+(x0, -1))
  
  • (x0[0] ≥ 0∧[1] ≥ 0∧[1] ≥ 0∧[1] ≥ 0 ⇒ (U^Increasing(246_0_MAIN_EQ(+(x0[1], -1))), ≥)∧[(-1)Bound*bni_16] + [bni_16]x0[0] ≥ 0∧[(-1)bso_17] ≥ 0)
    
  • (x0[0] ≥ 0∧[1] ≥ 0∧[1] ≥ 0∧[1] ≥ 0 ⇒ (U^Increasing(246_0_MAIN_EQ(+(x0[1], -1))), ≥)∧[(-1)Bound*bni_16] + [bni_16]x0[0] ≥ 0∧[(-1)bso_17] ≥ 0)
    
  
  
• 246_0_MAIN_EQ(x0) → COND_246_0_MAIN_EQ1(&&(>(x0, 0), =(0, %(x0, 2))), x0)
  
  • (x0[2] ≥ 0∧[1] ≥ 0∧[1] ≥ 0∧[1] ≥ 0 ⇒ (U^Increasing(COND_246_0_MAIN_EQ1(&&(>(x0[2], 0), =(0, %(x0[2], 2))), x0[2])), ≥)∧[(-1)Bound*bni_18 + bni_18] + [bni_18]x0[2] ≥ 0∧[1 + (-1)bso_19] ≥ 0)
    
  
  
• COND_246_0_MAIN_EQ1(TRUE, x0) → 246_0_MAIN_EQ(/(x0, 2))
  
  • (x0[2] ≥ 0∧[1] ≥ 0∧[1] ≥ 0∧[1] ≥ 0∧[1] + x0[2] ≥ 0 ⇒ (U^Increasing(246_0_MAIN_EQ(/(x0[3], 2))), ≥)∧[(-1)Bound*bni_20] + [bni_20]x0[2] ≥ 0∧[(-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) = [1]   
POL(246_0_MAIN_EQ(x₁)) = x₁   
POL(COND_246_0_MAIN_EQ(x₁, x₂)) = [-1] + x₂ + [-1]x₁   
POL(&&(x₁, x₂)) = [-1]   
POL(>(x₁, x₂)) = [-1]   
POL(0) = 0   
POL(=(x₁, x₂)) = [-1]   
POL(1) = [1]   
POL(2) = [2]   
POL(+(x₁, x₂)) = x₁ + x₂   
POL(-1) = [-1]   
POL(COND_246_0_MAIN_EQ1(x₁, x₂)) = [-1] + x₂   

Polynomial Interpretations with Context Sensitive Arithemetic Replacement
POL(Term^CSAR-Mode @ Context)

POL(%(x₁, 2)^-1 @ {}) = min{x₂, [-1]x₂}   
POL(%(x₁, 2)¹ @ {}) = max{x₂, [-1]x₂}   
POL(/(x₁, 2)¹ @ {246_0_MAIN_EQ_1/0}) = max{x₁, [-1]x₁} + [-1]   

The following pairs are in P_>:

246_0_MAIN_EQ(x0[2]) → COND_246_0_MAIN_EQ1(&&(>(x0[2], 0), =(0, %(x0[2], 2))), x0[2])

The following pairs are in P_bound:

246_0_MAIN_EQ(x0[0]) → COND_246_0_MAIN_EQ(&&(>(x0[0], 0), =(1, %(x0[0], 2))), x0[0])
COND_246_0_MAIN_EQ(TRUE, x0[1]) → 246_0_MAIN_EQ(+(x0[1], -1))
246_0_MAIN_EQ(x0[2]) → COND_246_0_MAIN_EQ1(&&(>(x0[2], 0), =(0, %(x0[2], 2))), x0[2])
COND_246_0_MAIN_EQ1(TRUE, x0[3]) → 246_0_MAIN_EQ(/(x0[3], 2))

The following pairs are in P_≥:

246_0_MAIN_EQ(x0[0]) → COND_246_0_MAIN_EQ(&&(>(x0[0], 0), =(1, %(x0[0], 2))), x0[0])
COND_246_0_MAIN_EQ(TRUE, x0[1]) → 246_0_MAIN_EQ(+(x0[1], -1))
COND_246_0_MAIN_EQ1(TRUE, x0[3]) → 246_0_MAIN_EQ(/(x0[3], 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
/¹ →

(8) Obligation:

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

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

The following domains are used:

Boolean, Integer

R is empty.

The integer pair graph contains the following rules and edges:
(0): 246_0_MAIN_EQ(x0[0]) → COND_246_0_MAIN_EQ(x0[0] > 0 && 1 = x0[0] % 2, x0[0])
(1): COND_246_0_MAIN_EQ(TRUE, x0[1]) → 246_0_MAIN_EQ(x0[1] + -1)
(3): COND_246_0_MAIN_EQ1(TRUE, x0[3]) → 246_0_MAIN_EQ(x0[3] / 2)

(1) -> (0), if (x0[1] + -1 →^* x0[0])

(3) -> (0), if (x0[3] / 2 →^* x0[0])

(0) -> (1), if (x0[0] > 0 && 1 = x0[0] % 2 ∧x0[0] →^* x0[1])

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, Boolean

R is empty.

The integer pair graph contains the following rules and edges:
(1): COND_246_0_MAIN_EQ(TRUE, x0[1]) → 246_0_MAIN_EQ(x0[1] + -1)
(0): 246_0_MAIN_EQ(x0[0]) → COND_246_0_MAIN_EQ(x0[0] > 0 && 1 = x0[0] % 2, x0[0])

(1) -> (0), if (x0[1] + -1 →^* x0[0])

(0) -> (1), if (x0[0] > 0 && 1 = x0[0] % 2 ∧x0[0] →^* x0[1])

The set Q is empty.

(11) IDPNonInfProof (SOUND transformation)

Used the following options for this NonInfProof:
IDPGPoloSolver: Range: [(-1,2)] IsNat: false Interpretation Shape Heuristic: aprove.DPFramework.IDPProblem.Processors.nonInf.poly.IdpDefaultShapeHeuristic@36b37b66 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_246_0_MAIN_EQ(TRUE, x0[1]) → 246_0_MAIN_EQ(+(x0[1], -1)) the following chains were created:

• We consider the chain 246_0_MAIN_EQ(x0[0]) → COND_246_0_MAIN_EQ(&&(>(x0[0], 0), =(1, %(x0[0], 2))), x0[0]), COND_246_0_MAIN_EQ(TRUE, x0[1]) → 246_0_MAIN_EQ(+(x0[1], -1)), 246_0_MAIN_EQ(x0[0]) → COND_246_0_MAIN_EQ(&&(>(x0[0], 0), =(1, %(x0[0], 2))), x0[0]) which results in the following constraint:
  

  (1)    (&&(>(x0[0], 0), =(1, %(x0[0], 2)))=TRUE∧x0[0]=x0[1]∧+(x0[1], -1)=x0[0]1 ⇒ COND_246_0_MAIN_EQ(TRUE, x0[1])≥NonInfC∧COND_246_0_MAIN_EQ(TRUE, x0[1])≥246_0_MAIN_EQ(+(x0[1], -1))∧(U^Increasing(246_0_MAIN_EQ(+(x0[1], -1))), ≥))

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

  (2)    (>(x0[0], 0)=TRUE∧>=(1, %(x0[0], 2))=TRUE∧<=(1, %(x0[0], 2))=TRUE ⇒ COND_246_0_MAIN_EQ(TRUE, x0[0])≥NonInfC∧COND_246_0_MAIN_EQ(TRUE, x0[0])≥246_0_MAIN_EQ(+(x0[0], -1))∧(U^Increasing(246_0_MAIN_EQ(+(x0[1], -1))), ≥))

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

  (3)    (x0[0] + [-1] ≥ 0∧[1] + [-1]min{[2], [-2]} ≥ 0∧max{[2], [-2]} + [-1] ≥ 0 ⇒ (U^Increasing(246_0_MAIN_EQ(+(x0[1], -1))), ≥)∧[(-1)bni_11 + (-1)Bound*bni_11] + [bni_11]x0[0] ≥ 0∧[1 + (-1)bso_12] ≥ 0)

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

  (4)    (x0[0] + [-1] ≥ 0∧[1] + [-1]min{[2], [-2]} ≥ 0∧max{[2], [-2]} + [-1] ≥ 0 ⇒ (U^Increasing(246_0_MAIN_EQ(+(x0[1], -1))), ≥)∧[(-1)bni_11 + (-1)Bound*bni_11] + [bni_11]x0[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)    (x0[0] + [-1] ≥ 0∧[4] ≥ 0∧[3] ≥ 0∧[1] ≥ 0 ⇒ (U^Increasing(246_0_MAIN_EQ(+(x0[1], -1))), ≥)∧[(-1)bni_11 + (-1)Bound*bni_11] + [bni_11]x0[0] ≥ 0∧[1 + (-1)bso_12] ≥ 0)

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

  (6)    (x0[0] ≥ 0∧[4] ≥ 0∧[3] ≥ 0∧[1] ≥ 0 ⇒ (U^Increasing(246_0_MAIN_EQ(+(x0[1], -1))), ≥)∧[(-1)Bound*bni_11] + [bni_11]x0[0] ≥ 0∧[1 + (-1)bso_12] ≥ 0)

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

  (7)    (x0[0] ≥ 0∧[1] ≥ 0∧[1] ≥ 0∧[1] ≥ 0 ⇒ (U^Increasing(246_0_MAIN_EQ(+(x0[1], -1))), ≥)∧[(-1)Bound*bni_11] + [bni_11]x0[0] ≥ 0∧[1 + (-1)bso_12] ≥ 0)

  
  
  

For Pair 246_0_MAIN_EQ(x0[0]) → COND_246_0_MAIN_EQ(&&(>(x0[0], 0), =(1, %(x0[0], 2))), x0[0]) the following chains were created:

• We consider the chain 246_0_MAIN_EQ(x0[0]) → COND_246_0_MAIN_EQ(&&(>(x0[0], 0), =(1, %(x0[0], 2))), x0[0]), COND_246_0_MAIN_EQ(TRUE, x0[1]) → 246_0_MAIN_EQ(+(x0[1], -1)) which results in the following constraint:
  

  (8)    (&&(>(x0[0], 0), =(1, %(x0[0], 2)))=TRUE∧x0[0]=x0[1] ⇒ 246_0_MAIN_EQ(x0[0])≥NonInfC∧246_0_MAIN_EQ(x0[0])≥COND_246_0_MAIN_EQ(&&(>(x0[0], 0), =(1, %(x0[0], 2))), x0[0])∧(U^Increasing(COND_246_0_MAIN_EQ(&&(>(x0[0], 0), =(1, %(x0[0], 2))), x0[0])), ≥))

  
  
  We simplified constraint (8) using rules (IV), (IDP_BOOLEAN) which results in the following new constraint:
  

  (9)    (>(x0[0], 0)=TRUE∧>=(1, %(x0[0], 2))=TRUE∧<=(1, %(x0[0], 2))=TRUE ⇒ 246_0_MAIN_EQ(x0[0])≥NonInfC∧246_0_MAIN_EQ(x0[0])≥COND_246_0_MAIN_EQ(&&(>(x0[0], 0), =(1, %(x0[0], 2))), x0[0])∧(U^Increasing(COND_246_0_MAIN_EQ(&&(>(x0[0], 0), =(1, %(x0[0], 2))), x0[0])), ≥))

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

  (10)    (x0[0] + [-1] ≥ 0∧[1] + [-1]min{[2], [-2]} ≥ 0∧max{[2], [-2]} + [-1] ≥ 0 ⇒ (U^Increasing(COND_246_0_MAIN_EQ(&&(>(x0[0], 0), =(1, %(x0[0], 2))), x0[0])), ≥)∧[(-1)bni_13 + (-1)Bound*bni_13] + [bni_13]x0[0] ≥ 0∧[(-1)bso_14] ≥ 0)

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

  (11)    (x0[0] + [-1] ≥ 0∧[1] + [-1]min{[2], [-2]} ≥ 0∧max{[2], [-2]} + [-1] ≥ 0 ⇒ (U^Increasing(COND_246_0_MAIN_EQ(&&(>(x0[0], 0), =(1, %(x0[0], 2))), x0[0])), ≥)∧[(-1)bni_13 + (-1)Bound*bni_13] + [bni_13]x0[0] ≥ 0∧[(-1)bso_14] ≥ 0)

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

  (12)    (x0[0] + [-1] ≥ 0∧[4] ≥ 0∧[3] ≥ 0∧[1] ≥ 0 ⇒ (U^Increasing(COND_246_0_MAIN_EQ(&&(>(x0[0], 0), =(1, %(x0[0], 2))), x0[0])), ≥)∧[(-1)bni_13 + (-1)Bound*bni_13] + [bni_13]x0[0] ≥ 0∧[(-1)bso_14] ≥ 0)

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

  (13)    (x0[0] ≥ 0∧[4] ≥ 0∧[3] ≥ 0∧[1] ≥ 0 ⇒ (U^Increasing(COND_246_0_MAIN_EQ(&&(>(x0[0], 0), =(1, %(x0[0], 2))), x0[0])), ≥)∧[(-1)Bound*bni_13] + [bni_13]x0[0] ≥ 0∧[(-1)bso_14] ≥ 0)

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

  (14)    (x0[0] ≥ 0∧[1] ≥ 0∧[1] ≥ 0∧[1] ≥ 0 ⇒ (U^Increasing(COND_246_0_MAIN_EQ(&&(>(x0[0], 0), =(1, %(x0[0], 2))), x0[0])), ≥)∧[(-1)Bound*bni_13] + [bni_13]x0[0] ≥ 0∧[(-1)bso_14] ≥ 0)

  
  
  

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

• COND_246_0_MAIN_EQ(TRUE, x0[1]) → 246_0_MAIN_EQ(+(x0[1], -1))
  
  • (x0[0] ≥ 0∧[1] ≥ 0∧[1] ≥ 0∧[1] ≥ 0 ⇒ (U^Increasing(246_0_MAIN_EQ(+(x0[1], -1))), ≥)∧[(-1)Bound*bni_11] + [bni_11]x0[0] ≥ 0∧[1 + (-1)bso_12] ≥ 0)
    
  
  
• 246_0_MAIN_EQ(x0[0]) → COND_246_0_MAIN_EQ(&&(>(x0[0], 0), =(1, %(x0[0], 2))), x0[0])
  
  • (x0[0] ≥ 0∧[1] ≥ 0∧[1] ≥ 0∧[1] ≥ 0 ⇒ (U^Increasing(COND_246_0_MAIN_EQ(&&(>(x0[0], 0), =(1, %(x0[0], 2))), x0[0])), ≥)∧[(-1)Bound*bni_13] + [bni_13]x0[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_246_0_MAIN_EQ(x₁, x₂)) = [-1] + x₂ + [-1]x₁   
POL(246_0_MAIN_EQ(x₁)) = [-1] + x₁   
POL(+(x₁, x₂)) = x₁ + x₂   
POL(-1) = [-1]   
POL(&&(x₁, x₂)) = 0   
POL(>(x₁, x₂)) = [-1]   
POL(0) = 0   
POL(=(x₁, x₂)) = [-1]   
POL(1) = [1]   
POL(2) = [2]   

Polynomial Interpretations with Context Sensitive Arithemetic Replacement
POL(Term^CSAR-Mode @ Context)

POL(%(x₁, 2)^-1 @ {}) = min{x₂, [-1]x₂}   
POL(%(x₁, 2)¹ @ {}) = max{x₂, [-1]x₂}   

The following pairs are in P_>:

COND_246_0_MAIN_EQ(TRUE, x0[1]) → 246_0_MAIN_EQ(+(x0[1], -1))

The following pairs are in P_bound:

COND_246_0_MAIN_EQ(TRUE, x0[1]) → 246_0_MAIN_EQ(+(x0[1], -1))
246_0_MAIN_EQ(x0[0]) → COND_246_0_MAIN_EQ(&&(>(x0[0], 0), =(1, %(x0[0], 2))), x0[0])

The following pairs are in P_≥:

246_0_MAIN_EQ(x0[0]) → COND_246_0_MAIN_EQ(&&(>(x0[0], 0), =(1, %(x0[0], 2))), x0[0])

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

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

(12) Obligation:

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

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

The following domains are used:

Boolean, Integer

R is empty.

The integer pair graph contains the following rules and edges:
(0): 246_0_MAIN_EQ(x0[0]) → COND_246_0_MAIN_EQ(x0[0] > 0 && 1 = x0[0] % 2, x0[0])


The set Q is empty.

(13) IDependencyGraphProof (EQUIVALENT transformation)

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

(14) TRUE