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: PastaB17

/**
 * 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 PastaB17 {
    public static void main(String[] args) {
        Random.args = args;
        int x = Random.random();
        int y = Random.random();
        int z = Random.random();

        while (x > z) {
            while (y > z) {
                y--;
            }
            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:
PastaB17.main([Ljava/lang/String;)V: Graph of 255 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: PastaB17.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 16 rules for P and 0 rules for R.

P rules:
889_0_main_Load(EOS(STATIC_889), i160, i161, i89, i160) → 891_0_main_LE(EOS(STATIC_891), i160, i161, i89, i160, i89)
891_0_main_LE(EOS(STATIC_891), i160, i161, i89, i160, i89) → 894_0_main_LE(EOS(STATIC_894), i160, i161, i89, i160, i89)
894_0_main_LE(EOS(STATIC_894), i160, i161, i89, i160, i89) → 898_0_main_Load(EOS(STATIC_898), i160, i161, i89) | >(i160, i89)
898_0_main_Load(EOS(STATIC_898), i160, i161, i89) → 901_0_main_Load(EOS(STATIC_901), i160, i161, i89, i161)
901_0_main_Load(EOS(STATIC_901), i160, i161, i89, i161) → 903_0_main_LE(EOS(STATIC_903), i160, i161, i89, i161, i89)
903_0_main_LE(EOS(STATIC_903), i160, i161, i89, i161, i89) → 905_0_main_LE(EOS(STATIC_905), i160, i161, i89, i161, i89)
903_0_main_LE(EOS(STATIC_903), i160, i161, i89, i161, i89) → 907_0_main_LE(EOS(STATIC_907), i160, i161, i89, i161, i89)
905_0_main_LE(EOS(STATIC_905), i160, i161, i89, i161, i89) → 908_0_main_Inc(EOS(STATIC_908), i160, i161, i89) | <=(i161, i89)
908_0_main_Inc(EOS(STATIC_908), i160, i161, i89) → 912_0_main_JMP(EOS(STATIC_912), +(i160, -1), i161, i89)
912_0_main_JMP(EOS(STATIC_912), i165, i161, i89) → 918_0_main_Load(EOS(STATIC_918), i165, i161, i89)
918_0_main_Load(EOS(STATIC_918), i165, i161, i89) → 886_0_main_Load(EOS(STATIC_886), i165, i161, i89)
886_0_main_Load(EOS(STATIC_886), i160, i161, i89) → 889_0_main_Load(EOS(STATIC_889), i160, i161, i89, i160)
907_0_main_LE(EOS(STATIC_907), i160, i161, i89, i161, i89) → 910_0_main_Inc(EOS(STATIC_910), i160, i161, i89) | >(i161, i89)
910_0_main_Inc(EOS(STATIC_910), i160, i161, i89) → 914_0_main_JMP(EOS(STATIC_914), i160, +(i161, -1), i89)
914_0_main_JMP(EOS(STATIC_914), i160, i167, i89) → 921_0_main_Load(EOS(STATIC_921), i160, i167, i89)
921_0_main_Load(EOS(STATIC_921), i160, i167, i89) → 898_0_main_Load(EOS(STATIC_898), i160, i167, i89)
R rules:

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

P rules:
903_0_main_LE(EOS(STATIC_903), x0, x1, x2, x1, x2) → 903_0_main_LE(EOS(STATIC_903), +(x0, -1), x1, x2, x1, x2) | &&(>=(x2, x1), <(x2, +(x0, -1)))
903_0_main_LE(EOS(STATIC_903), x0, x1, x2, x1, x2) → 903_0_main_LE(EOS(STATIC_903), x0, +(x1, -1), x2, +(x1, -1), x2) | <(x2, x1)
R rules:

Filtered ground terms:

903_0_main_LE(x1, x2, x3, x4, x5, x6) → 903_0_main_LE(x2, x3, x4, x5, x6)
EOS(x1) → EOS
Cond_903_0_main_LE1(x1, x2, x3, x4, x5, x6, x7) → Cond_903_0_main_LE1(x1, x3, x4, x5, x6, x7)
Cond_903_0_main_LE(x1, x2, x3, x4, x5, x6, x7) → Cond_903_0_main_LE(x1, x3, x4, x5, x6, x7)

Filtered duplicate args:

903_0_main_LE(x1, x2, x3, x4, x5) → 903_0_main_LE(x1, x4, x5)
Cond_903_0_main_LE(x1, x2, x3, x4, x5, x6) → Cond_903_0_main_LE(x1, x2, x5, x6)
Cond_903_0_main_LE1(x1, x2, x3, x4, x5, x6) → Cond_903_0_main_LE1(x1, x2, x5, x6)

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

P rules:
903_0_main_LE(x0, x1, x2) → 903_0_main_LE(+(x0, -1), x1, x2) | &&(>=(x2, x1), <(x2, +(x0, -1)))
903_0_main_LE(x0, x1, x2) → 903_0_main_LE(x0, +(x1, -1), x2) | <(x2, x1)
R rules:

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

P rules:
903_0_MAIN_LE(x0, x1, x2) → COND_903_0_MAIN_LE(&&(>=(x2, x1), <(x2, +(x0, -1))), x0, x1, x2)
COND_903_0_MAIN_LE(TRUE, x0, x1, x2) → 903_0_MAIN_LE(+(x0, -1), x1, x2)
903_0_MAIN_LE(x0, x1, x2) → COND_903_0_MAIN_LE1(<(x2, x1), x0, x1, x2)
COND_903_0_MAIN_LE1(TRUE, x0, x1, x2) → 903_0_MAIN_LE(x0, +(x1, -1), x2)
R rules:

(6) Obligation:

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

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

The following domains are used:

Boolean, Integer

R is empty.

The integer pair graph contains the following rules and edges:
(0): 903_0_MAIN_LE(x0[0], x1[0], x2[0]) → COND_903_0_MAIN_LE(x2[0] >= x1[0] && x2[0] < x0[0] + -1, x0[0], x1[0], x2[0])
(1): COND_903_0_MAIN_LE(TRUE, x0[1], x1[1], x2[1]) → 903_0_MAIN_LE(x0[1] + -1, x1[1], x2[1])
(2): 903_0_MAIN_LE(x0[2], x1[2], x2[2]) → COND_903_0_MAIN_LE1(x2[2] < x1[2], x0[2], x1[2], x2[2])
(3): COND_903_0_MAIN_LE1(TRUE, x0[3], x1[3], x2[3]) → 903_0_MAIN_LE(x0[3], x1[3] + -1, x2[3])

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

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

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

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

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

(3) -> (2), if (x0[3] →^* x0[2]∧x1[3] + -1 →^* x1[2]∧x2[3] →^* x2[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.IdpCand1ShapeHeuristic@5f16e402 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 903_0_MAIN_LE(x0, x1, x2) → COND_903_0_MAIN_LE(&&(>=(x2, x1), <(x2, +(x0, -1))), x0, x1, x2) the following chains were created:

• We consider the chain 903_0_MAIN_LE(x0[0], x1[0], x2[0]) → COND_903_0_MAIN_LE(&&(>=(x2[0], x1[0]), <(x2[0], +(x0[0], -1))), x0[0], x1[0], x2[0]), COND_903_0_MAIN_LE(TRUE, x0[1], x1[1], x2[1]) → 903_0_MAIN_LE(+(x0[1], -1), x1[1], x2[1]) which results in the following constraint:
  

  (1)    (&&(>=(x2[0], x1[0]), <(x2[0], +(x0[0], -1)))=TRUE∧x0[0]=x0[1]∧x1[0]=x1[1]∧x2[0]=x2[1] ⇒ 903_0_MAIN_LE(x0[0], x1[0], x2[0])≥NonInfC∧903_0_MAIN_LE(x0[0], x1[0], x2[0])≥COND_903_0_MAIN_LE(&&(>=(x2[0], x1[0]), <(x2[0], +(x0[0], -1))), x0[0], x1[0], x2[0])∧(U^Increasing(COND_903_0_MAIN_LE(&&(>=(x2[0], x1[0]), <(x2[0], +(x0[0], -1))), x0[0], x1[0], x2[0])), ≥))

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

  (2)    (>=(x2[0], x1[0])=TRUE∧<(x2[0], +(x0[0], -1))=TRUE ⇒ 903_0_MAIN_LE(x0[0], x1[0], x2[0])≥NonInfC∧903_0_MAIN_LE(x0[0], x1[0], x2[0])≥COND_903_0_MAIN_LE(&&(>=(x2[0], x1[0]), <(x2[0], +(x0[0], -1))), x0[0], x1[0], x2[0])∧(U^Increasing(COND_903_0_MAIN_LE(&&(>=(x2[0], x1[0]), <(x2[0], +(x0[0], -1))), x0[0], x1[0], x2[0])), ≥))

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

  (3)    (x2[0] + [-1]x1[0] ≥ 0∧x0[0] + [-2] + [-1]x2[0] ≥ 0 ⇒ (U^Increasing(COND_903_0_MAIN_LE(&&(>=(x2[0], x1[0]), <(x2[0], +(x0[0], -1))), x0[0], x1[0], x2[0])), ≥)∧[(-1)bni_12 + (-1)Bound*bni_12] + [(-1)bni_12]x2[0] + [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)    (x2[0] + [-1]x1[0] ≥ 0∧x0[0] + [-2] + [-1]x2[0] ≥ 0 ⇒ (U^Increasing(COND_903_0_MAIN_LE(&&(>=(x2[0], x1[0]), <(x2[0], +(x0[0], -1))), x0[0], x1[0], x2[0])), ≥)∧[(-1)bni_12 + (-1)Bound*bni_12] + [(-1)bni_12]x2[0] + [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)    (x2[0] + [-1]x1[0] ≥ 0∧x0[0] + [-2] + [-1]x2[0] ≥ 0 ⇒ (U^Increasing(COND_903_0_MAIN_LE(&&(>=(x2[0], x1[0]), <(x2[0], +(x0[0], -1))), x0[0], x1[0], x2[0])), ≥)∧[(-1)bni_12 + (-1)Bound*bni_12] + [(-1)bni_12]x2[0] + [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)    (x2[0] ≥ 0∧x0[0] + [-2] + [-1]x1[0] + [-1]x2[0] ≥ 0 ⇒ (U^Increasing(COND_903_0_MAIN_LE(&&(>=(x2[0], x1[0]), <(x2[0], +(x0[0], -1))), x0[0], x1[0], x2[0])), ≥)∧[(-1)bni_12 + (-1)Bound*bni_12] + [(-1)bni_12]x2[0] ≥ 0∧[(-1)bso_13] ≥ 0)

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

  (7)    (x2[0] ≥ 0∧x1[0] ≥ 0 ⇒ (U^Increasing(COND_903_0_MAIN_LE(&&(>=(x2[0], x1[0]), <(x2[0], +(x0[0], -1))), x0[0], x1[0], x2[0])), ≥)∧[(-1)bni_12 + (-1)Bound*bni_12] + [(-1)bni_12]x2[0] ≥ 0∧[(-1)bso_13] ≥ 0)

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

  (8)    (x2[0] ≥ 0∧x1[0] ≥ 0∧x0[0] ≥ 0 ⇒ (U^Increasing(COND_903_0_MAIN_LE(&&(>=(x2[0], x1[0]), <(x2[0], +(x0[0], -1))), x0[0], x1[0], x2[0])), ≥)∧[(-1)bni_12 + (-1)Bound*bni_12] + [(-1)bni_12]x2[0] ≥ 0∧[(-1)bso_13] ≥ 0)

  
  

  (9)    (x2[0] ≥ 0∧x1[0] ≥ 0∧x0[0] ≥ 0 ⇒ (U^Increasing(COND_903_0_MAIN_LE(&&(>=(x2[0], x1[0]), <(x2[0], +(x0[0], -1))), x0[0], x1[0], x2[0])), ≥)∧[(-1)bni_12 + (-1)Bound*bni_12] + [(-1)bni_12]x2[0] ≥ 0∧[(-1)bso_13] ≥ 0)

  
  
  

For Pair COND_903_0_MAIN_LE(TRUE, x0, x1, x2) → 903_0_MAIN_LE(+(x0, -1), x1, x2) the following chains were created:

• We consider the chain COND_903_0_MAIN_LE(TRUE, x0[1], x1[1], x2[1]) → 903_0_MAIN_LE(+(x0[1], -1), x1[1], x2[1]) which results in the following constraint:
  

  (10)    (COND_903_0_MAIN_LE(TRUE, x0[1], x1[1], x2[1])≥NonInfC∧COND_903_0_MAIN_LE(TRUE, x0[1], x1[1], x2[1])≥903_0_MAIN_LE(+(x0[1], -1), x1[1], x2[1])∧(U^Increasing(903_0_MAIN_LE(+(x0[1], -1), x1[1], x2[1])), ≥))

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

  (11)    ((U^Increasing(903_0_MAIN_LE(+(x0[1], -1), x1[1], x2[1])), ≥)∧[bni_14] = 0∧[(-1)bso_15] ≥ 0)

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

  (12)    ((U^Increasing(903_0_MAIN_LE(+(x0[1], -1), x1[1], x2[1])), ≥)∧[bni_14] = 0∧[(-1)bso_15] ≥ 0)

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

  (13)    ((U^Increasing(903_0_MAIN_LE(+(x0[1], -1), x1[1], x2[1])), ≥)∧[bni_14] = 0∧[(-1)bso_15] ≥ 0)

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

  (14)    ((U^Increasing(903_0_MAIN_LE(+(x0[1], -1), x1[1], x2[1])), ≥)∧[bni_14] = 0∧0 = 0∧0 = 0∧0 = 0∧[(-1)bso_15] ≥ 0)

  
  
  

For Pair 903_0_MAIN_LE(x0, x1, x2) → COND_903_0_MAIN_LE1(<(x2, x1), x0, x1, x2) the following chains were created:

• We consider the chain 903_0_MAIN_LE(x0[2], x1[2], x2[2]) → COND_903_0_MAIN_LE1(<(x2[2], x1[2]), x0[2], x1[2], x2[2]), COND_903_0_MAIN_LE1(TRUE, x0[3], x1[3], x2[3]) → 903_0_MAIN_LE(x0[3], +(x1[3], -1), x2[3]) which results in the following constraint:
  

  (15)    (<(x2[2], x1[2])=TRUE∧x0[2]=x0[3]∧x1[2]=x1[3]∧x2[2]=x2[3] ⇒ 903_0_MAIN_LE(x0[2], x1[2], x2[2])≥NonInfC∧903_0_MAIN_LE(x0[2], x1[2], x2[2])≥COND_903_0_MAIN_LE1(<(x2[2], x1[2]), x0[2], x1[2], x2[2])∧(U^Increasing(COND_903_0_MAIN_LE1(<(x2[2], x1[2]), x0[2], x1[2], x2[2])), ≥))

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

  (16)    (<(x2[2], x1[2])=TRUE ⇒ 903_0_MAIN_LE(x0[2], x1[2], x2[2])≥NonInfC∧903_0_MAIN_LE(x0[2], x1[2], x2[2])≥COND_903_0_MAIN_LE1(<(x2[2], x1[2]), x0[2], x1[2], x2[2])∧(U^Increasing(COND_903_0_MAIN_LE1(<(x2[2], x1[2]), x0[2], x1[2], x2[2])), ≥))

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

  (17)    (x1[2] + [-1] + [-1]x2[2] ≥ 0 ⇒ (U^Increasing(COND_903_0_MAIN_LE1(<(x2[2], x1[2]), x0[2], x1[2], x2[2])), ≥)∧[(-1)bni_16 + (-1)Bound*bni_16] + [(-1)bni_16]x2[2] + [bni_16]x1[2] ≥ 0∧[(-1)bso_17] ≥ 0)

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

  (18)    (x1[2] + [-1] + [-1]x2[2] ≥ 0 ⇒ (U^Increasing(COND_903_0_MAIN_LE1(<(x2[2], x1[2]), x0[2], x1[2], x2[2])), ≥)∧[(-1)bni_16 + (-1)Bound*bni_16] + [(-1)bni_16]x2[2] + [bni_16]x1[2] ≥ 0∧[(-1)bso_17] ≥ 0)

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

  (19)    (x1[2] + [-1] + [-1]x2[2] ≥ 0 ⇒ (U^Increasing(COND_903_0_MAIN_LE1(<(x2[2], x1[2]), x0[2], x1[2], x2[2])), ≥)∧[(-1)bni_16 + (-1)Bound*bni_16] + [(-1)bni_16]x2[2] + [bni_16]x1[2] ≥ 0∧[(-1)bso_17] ≥ 0)

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

  (20)    (x1[2] + [-1] + [-1]x2[2] ≥ 0 ⇒ (U^Increasing(COND_903_0_MAIN_LE1(<(x2[2], x1[2]), x0[2], x1[2], x2[2])), ≥)∧0 = 0∧[(-1)bni_16 + (-1)Bound*bni_16] + [(-1)bni_16]x2[2] + [bni_16]x1[2] ≥ 0∧0 = 0∧[(-1)bso_17] ≥ 0)

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

  (21)    (x1[2] ≥ 0 ⇒ (U^Increasing(COND_903_0_MAIN_LE1(<(x2[2], x1[2]), x0[2], x1[2], x2[2])), ≥)∧0 = 0∧[(-1)Bound*bni_16] + [bni_16]x1[2] ≥ 0∧0 = 0∧[(-1)bso_17] ≥ 0)

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

  (22)    (x1[2] ≥ 0∧x2[2] ≥ 0 ⇒ (U^Increasing(COND_903_0_MAIN_LE1(<(x2[2], x1[2]), x0[2], x1[2], x2[2])), ≥)∧0 = 0∧[(-1)Bound*bni_16] + [bni_16]x1[2] ≥ 0∧0 = 0∧[(-1)bso_17] ≥ 0)

  
  

  (23)    (x1[2] ≥ 0∧x2[2] ≥ 0 ⇒ (U^Increasing(COND_903_0_MAIN_LE1(<(x2[2], x1[2]), x0[2], x1[2], x2[2])), ≥)∧0 = 0∧[(-1)Bound*bni_16] + [bni_16]x1[2] ≥ 0∧0 = 0∧[(-1)bso_17] ≥ 0)

  
  
  

For Pair COND_903_0_MAIN_LE1(TRUE, x0, x1, x2) → 903_0_MAIN_LE(x0, +(x1, -1), x2) the following chains were created:

• We consider the chain COND_903_0_MAIN_LE1(TRUE, x0[3], x1[3], x2[3]) → 903_0_MAIN_LE(x0[3], +(x1[3], -1), x2[3]) which results in the following constraint:
  

  (24)    (COND_903_0_MAIN_LE1(TRUE, x0[3], x1[3], x2[3])≥NonInfC∧COND_903_0_MAIN_LE1(TRUE, x0[3], x1[3], x2[3])≥903_0_MAIN_LE(x0[3], +(x1[3], -1), x2[3])∧(U^Increasing(903_0_MAIN_LE(x0[3], +(x1[3], -1), x2[3])), ≥))

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

  (25)    ((U^Increasing(903_0_MAIN_LE(x0[3], +(x1[3], -1), x2[3])), ≥)∧[bni_18] = 0∧[1 + (-1)bso_19] ≥ 0)

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

  (26)    ((U^Increasing(903_0_MAIN_LE(x0[3], +(x1[3], -1), x2[3])), ≥)∧[bni_18] = 0∧[1 + (-1)bso_19] ≥ 0)

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

  (27)    ((U^Increasing(903_0_MAIN_LE(x0[3], +(x1[3], -1), x2[3])), ≥)∧[bni_18] = 0∧[1 + (-1)bso_19] ≥ 0)

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

  (28)    ((U^Increasing(903_0_MAIN_LE(x0[3], +(x1[3], -1), x2[3])), ≥)∧[bni_18] = 0∧0 = 0∧0 = 0∧0 = 0∧[1 + (-1)bso_19] ≥ 0)

  
  
  

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

• 903_0_MAIN_LE(x0, x1, x2) → COND_903_0_MAIN_LE(&&(>=(x2, x1), <(x2, +(x0, -1))), x0, x1, x2)
  
  • (x2[0] ≥ 0∧x1[0] ≥ 0∧x0[0] ≥ 0 ⇒ (U^Increasing(COND_903_0_MAIN_LE(&&(>=(x2[0], x1[0]), <(x2[0], +(x0[0], -1))), x0[0], x1[0], x2[0])), ≥)∧[(-1)bni_12 + (-1)Bound*bni_12] + [(-1)bni_12]x2[0] ≥ 0∧[(-1)bso_13] ≥ 0)
    
  • (x2[0] ≥ 0∧x1[0] ≥ 0∧x0[0] ≥ 0 ⇒ (U^Increasing(COND_903_0_MAIN_LE(&&(>=(x2[0], x1[0]), <(x2[0], +(x0[0], -1))), x0[0], x1[0], x2[0])), ≥)∧[(-1)bni_12 + (-1)Bound*bni_12] + [(-1)bni_12]x2[0] ≥ 0∧[(-1)bso_13] ≥ 0)
    
  
  
• COND_903_0_MAIN_LE(TRUE, x0, x1, x2) → 903_0_MAIN_LE(+(x0, -1), x1, x2)
  
  • ((U^Increasing(903_0_MAIN_LE(+(x0[1], -1), x1[1], x2[1])), ≥)∧[bni_14] = 0∧0 = 0∧0 = 0∧0 = 0∧[(-1)bso_15] ≥ 0)
    
  
  
• 903_0_MAIN_LE(x0, x1, x2) → COND_903_0_MAIN_LE1(<(x2, x1), x0, x1, x2)
  
  • (x1[2] ≥ 0∧x2[2] ≥ 0 ⇒ (U^Increasing(COND_903_0_MAIN_LE1(<(x2[2], x1[2]), x0[2], x1[2], x2[2])), ≥)∧0 = 0∧[(-1)Bound*bni_16] + [bni_16]x1[2] ≥ 0∧0 = 0∧[(-1)bso_17] ≥ 0)
    
  • (x1[2] ≥ 0∧x2[2] ≥ 0 ⇒ (U^Increasing(COND_903_0_MAIN_LE1(<(x2[2], x1[2]), x0[2], x1[2], x2[2])), ≥)∧0 = 0∧[(-1)Bound*bni_16] + [bni_16]x1[2] ≥ 0∧0 = 0∧[(-1)bso_17] ≥ 0)
    
  
  
• COND_903_0_MAIN_LE1(TRUE, x0, x1, x2) → 903_0_MAIN_LE(x0, +(x1, -1), x2)
  
  • ((U^Increasing(903_0_MAIN_LE(x0[3], +(x1[3], -1), x2[3])), ≥)∧[bni_18] = 0∧0 = 0∧0 = 0∧0 = 0∧[1 + (-1)bso_19] ≥ 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(903_0_MAIN_LE(x₁, x₂, x₃)) = [-1] + [-1]x₃ + x₂   
POL(COND_903_0_MAIN_LE(x₁, x₂, x₃, x₄)) = [-1] + [-1]x₄ + x₃   
POL(&&(x₁, x₂)) = [-1]   
POL(>=(x₁, x₂)) = [-1]   
POL(<(x₁, x₂)) = [-1]   
POL(+(x₁, x₂)) = x₁ + x₂   
POL(-1) = [-1]   
POL(COND_903_0_MAIN_LE1(x₁, x₂, x₃, x₄)) = [-1] + [-1]x₄ + x₃   

The following pairs are in P_>:

COND_903_0_MAIN_LE1(TRUE, x0[3], x1[3], x2[3]) → 903_0_MAIN_LE(x0[3], +(x1[3], -1), x2[3])

The following pairs are in P_bound:

903_0_MAIN_LE(x0[2], x1[2], x2[2]) → COND_903_0_MAIN_LE1(<(x2[2], x1[2]), x0[2], x1[2], x2[2])

The following pairs are in P_≥:

903_0_MAIN_LE(x0[0], x1[0], x2[0]) → COND_903_0_MAIN_LE(&&(>=(x2[0], x1[0]), <(x2[0], +(x0[0], -1))), x0[0], x1[0], x2[0])
COND_903_0_MAIN_LE(TRUE, x0[1], x1[1], x2[1]) → 903_0_MAIN_LE(+(x0[1], -1), x1[1], x2[1])
903_0_MAIN_LE(x0[2], x1[2], x2[2]) → COND_903_0_MAIN_LE1(<(x2[2], x1[2]), x0[2], x1[2], x2[2])

There are no usable rules.

(8) Complex Obligation (AND)

(9) Obligation:

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

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

The following domains are used:

Boolean, Integer

R is empty.

The integer pair graph contains the following rules and edges:
(0): 903_0_MAIN_LE(x0[0], x1[0], x2[0]) → COND_903_0_MAIN_LE(x2[0] >= x1[0] && x2[0] < x0[0] + -1, x0[0], x1[0], x2[0])
(1): COND_903_0_MAIN_LE(TRUE, x0[1], x1[1], x2[1]) → 903_0_MAIN_LE(x0[1] + -1, x1[1], x2[1])
(2): 903_0_MAIN_LE(x0[2], x1[2], x2[2]) → COND_903_0_MAIN_LE1(x2[2] < x1[2], x0[2], x1[2], x2[2])

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

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

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

The set Q is empty.

(10) IDependencyGraphProof (EQUIVALENT transformation)

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

(11) 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_903_0_MAIN_LE(TRUE, x0[1], x1[1], x2[1]) → 903_0_MAIN_LE(x0[1] + -1, x1[1], x2[1])
(0): 903_0_MAIN_LE(x0[0], x1[0], x2[0]) → COND_903_0_MAIN_LE(x2[0] >= x1[0] && x2[0] < x0[0] + -1, x0[0], x1[0], x2[0])

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

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

The set Q is empty.

(12) 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@5f16e402 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 COND_903_0_MAIN_LE(TRUE, x0[1], x1[1], x2[1]) → 903_0_MAIN_LE(+(x0[1], -1), x1[1], x2[1]) the following chains were created:

• We consider the chain COND_903_0_MAIN_LE(TRUE, x0[1], x1[1], x2[1]) → 903_0_MAIN_LE(+(x0[1], -1), x1[1], x2[1]) which results in the following constraint:
  

  (1)    (COND_903_0_MAIN_LE(TRUE, x0[1], x1[1], x2[1])≥NonInfC∧COND_903_0_MAIN_LE(TRUE, x0[1], x1[1], x2[1])≥903_0_MAIN_LE(+(x0[1], -1), x1[1], x2[1])∧(U^Increasing(903_0_MAIN_LE(+(x0[1], -1), x1[1], x2[1])), ≥))

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

  (2)    ((U^Increasing(903_0_MAIN_LE(+(x0[1], -1), x1[1], x2[1])), ≥)∧[bni_8] = 0∧[1 + (-1)bso_9] ≥ 0)

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

  (3)    ((U^Increasing(903_0_MAIN_LE(+(x0[1], -1), x1[1], x2[1])), ≥)∧[bni_8] = 0∧[1 + (-1)bso_9] ≥ 0)

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

  (4)    ((U^Increasing(903_0_MAIN_LE(+(x0[1], -1), x1[1], x2[1])), ≥)∧[bni_8] = 0∧[1 + (-1)bso_9] ≥ 0)

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

  (5)    ((U^Increasing(903_0_MAIN_LE(+(x0[1], -1), x1[1], x2[1])), ≥)∧[bni_8] = 0∧0 = 0∧0 = 0∧0 = 0∧[1 + (-1)bso_9] ≥ 0)

  
  
  

For Pair 903_0_MAIN_LE(x0[0], x1[0], x2[0]) → COND_903_0_MAIN_LE(&&(>=(x2[0], x1[0]), <(x2[0], +(x0[0], -1))), x0[0], x1[0], x2[0]) the following chains were created:

• We consider the chain 903_0_MAIN_LE(x0[0], x1[0], x2[0]) → COND_903_0_MAIN_LE(&&(>=(x2[0], x1[0]), <(x2[0], +(x0[0], -1))), x0[0], x1[0], x2[0]), COND_903_0_MAIN_LE(TRUE, x0[1], x1[1], x2[1]) → 903_0_MAIN_LE(+(x0[1], -1), x1[1], x2[1]) which results in the following constraint:
  

  (6)    (&&(>=(x2[0], x1[0]), <(x2[0], +(x0[0], -1)))=TRUE∧x0[0]=x0[1]∧x1[0]=x1[1]∧x2[0]=x2[1] ⇒ 903_0_MAIN_LE(x0[0], x1[0], x2[0])≥NonInfC∧903_0_MAIN_LE(x0[0], x1[0], x2[0])≥COND_903_0_MAIN_LE(&&(>=(x2[0], x1[0]), <(x2[0], +(x0[0], -1))), x0[0], x1[0], x2[0])∧(U^Increasing(COND_903_0_MAIN_LE(&&(>=(x2[0], x1[0]), <(x2[0], +(x0[0], -1))), x0[0], x1[0], x2[0])), ≥))

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

  (7)    (>=(x2[0], x1[0])=TRUE∧<(x2[0], +(x0[0], -1))=TRUE ⇒ 903_0_MAIN_LE(x0[0], x1[0], x2[0])≥NonInfC∧903_0_MAIN_LE(x0[0], x1[0], x2[0])≥COND_903_0_MAIN_LE(&&(>=(x2[0], x1[0]), <(x2[0], +(x0[0], -1))), x0[0], x1[0], x2[0])∧(U^Increasing(COND_903_0_MAIN_LE(&&(>=(x2[0], x1[0]), <(x2[0], +(x0[0], -1))), x0[0], x1[0], x2[0])), ≥))

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

  (8)    (x2[0] + [-1]x1[0] ≥ 0∧x0[0] + [-2] + [-1]x2[0] ≥ 0 ⇒ (U^Increasing(COND_903_0_MAIN_LE(&&(>=(x2[0], x1[0]), <(x2[0], +(x0[0], -1))), x0[0], x1[0], x2[0])), ≥)∧[(-1)Bound*bni_10] + [bni_10]x0[0] + [(-1)bni_10]x1[0] ≥ 0∧[(-1)bso_11] ≥ 0)

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

  (9)    (x2[0] + [-1]x1[0] ≥ 0∧x0[0] + [-2] + [-1]x2[0] ≥ 0 ⇒ (U^Increasing(COND_903_0_MAIN_LE(&&(>=(x2[0], x1[0]), <(x2[0], +(x0[0], -1))), x0[0], x1[0], x2[0])), ≥)∧[(-1)Bound*bni_10] + [bni_10]x0[0] + [(-1)bni_10]x1[0] ≥ 0∧[(-1)bso_11] ≥ 0)

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

  (10)    (x2[0] + [-1]x1[0] ≥ 0∧x0[0] + [-2] + [-1]x2[0] ≥ 0 ⇒ (U^Increasing(COND_903_0_MAIN_LE(&&(>=(x2[0], x1[0]), <(x2[0], +(x0[0], -1))), x0[0], x1[0], x2[0])), ≥)∧[(-1)Bound*bni_10] + [bni_10]x0[0] + [(-1)bni_10]x1[0] ≥ 0∧[(-1)bso_11] ≥ 0)

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

  (11)    (x2[0] ≥ 0∧x0[0] + [-2] + [-1]x1[0] + [-1]x2[0] ≥ 0 ⇒ (U^Increasing(COND_903_0_MAIN_LE(&&(>=(x2[0], x1[0]), <(x2[0], +(x0[0], -1))), x0[0], x1[0], x2[0])), ≥)∧[(-1)Bound*bni_10] + [bni_10]x0[0] + [(-1)bni_10]x1[0] ≥ 0∧[(-1)bso_11] ≥ 0)

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

  (12)    (x2[0] ≥ 0∧x1[0] ≥ 0 ⇒ (U^Increasing(COND_903_0_MAIN_LE(&&(>=(x2[0], x1[0]), <(x2[0], +(x0[0], -1))), x0[0], x1[0], x2[0])), ≥)∧[(-1)Bound*bni_10 + (2)bni_10] + [bni_10]x2[0] + [bni_10]x1[0] ≥ 0∧[(-1)bso_11] ≥ 0)

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

  (13)    (x2[0] ≥ 0∧x1[0] ≥ 0∧x0[0] ≥ 0 ⇒ (U^Increasing(COND_903_0_MAIN_LE(&&(>=(x2[0], x1[0]), <(x2[0], +(x0[0], -1))), x0[0], x1[0], x2[0])), ≥)∧[(-1)Bound*bni_10 + (2)bni_10] + [bni_10]x2[0] + [bni_10]x1[0] ≥ 0∧[(-1)bso_11] ≥ 0)

  
  

  (14)    (x2[0] ≥ 0∧x1[0] ≥ 0∧x0[0] ≥ 0 ⇒ (U^Increasing(COND_903_0_MAIN_LE(&&(>=(x2[0], x1[0]), <(x2[0], +(x0[0], -1))), x0[0], x1[0], x2[0])), ≥)∧[(-1)Bound*bni_10 + (2)bni_10] + [bni_10]x2[0] + [bni_10]x1[0] ≥ 0∧[(-1)bso_11] ≥ 0)

  
  
  

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

• COND_903_0_MAIN_LE(TRUE, x0[1], x1[1], x2[1]) → 903_0_MAIN_LE(+(x0[1], -1), x1[1], x2[1])
  
  • ((U^Increasing(903_0_MAIN_LE(+(x0[1], -1), x1[1], x2[1])), ≥)∧[bni_8] = 0∧0 = 0∧0 = 0∧0 = 0∧[1 + (-1)bso_9] ≥ 0)
    
  
  
• 903_0_MAIN_LE(x0[0], x1[0], x2[0]) → COND_903_0_MAIN_LE(&&(>=(x2[0], x1[0]), <(x2[0], +(x0[0], -1))), x0[0], x1[0], x2[0])
  
  • (x2[0] ≥ 0∧x1[0] ≥ 0∧x0[0] ≥ 0 ⇒ (U^Increasing(COND_903_0_MAIN_LE(&&(>=(x2[0], x1[0]), <(x2[0], +(x0[0], -1))), x0[0], x1[0], x2[0])), ≥)∧[(-1)Bound*bni_10 + (2)bni_10] + [bni_10]x2[0] + [bni_10]x1[0] ≥ 0∧[(-1)bso_11] ≥ 0)
    
  • (x2[0] ≥ 0∧x1[0] ≥ 0∧x0[0] ≥ 0 ⇒ (U^Increasing(COND_903_0_MAIN_LE(&&(>=(x2[0], x1[0]), <(x2[0], +(x0[0], -1))), x0[0], x1[0], x2[0])), ≥)∧[(-1)Bound*bni_10 + (2)bni_10] + [bni_10]x2[0] + [bni_10]x1[0] ≥ 0∧[(-1)bso_11] ≥ 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_903_0_MAIN_LE(x₁, x₂, x₃, x₄)) = [-1]x₃ + x₂   
POL(903_0_MAIN_LE(x₁, x₂, x₃)) = x₁ + [-1]x₂   
POL(+(x₁, x₂)) = x₁ + x₂   
POL(-1) = [-1]   
POL(&&(x₁, x₂)) = [-1]   
POL(>=(x₁, x₂)) = [1]   
POL(<(x₁, x₂)) = [-1]   

The following pairs are in P_>:

COND_903_0_MAIN_LE(TRUE, x0[1], x1[1], x2[1]) → 903_0_MAIN_LE(+(x0[1], -1), x1[1], x2[1])

The following pairs are in P_bound:

903_0_MAIN_LE(x0[0], x1[0], x2[0]) → COND_903_0_MAIN_LE(&&(>=(x2[0], x1[0]), <(x2[0], +(x0[0], -1))), x0[0], x1[0], x2[0])

The following pairs are in P_≥:

903_0_MAIN_LE(x0[0], x1[0], x2[0]) → COND_903_0_MAIN_LE(&&(>=(x2[0], x1[0]), <(x2[0], +(x0[0], -1))), x0[0], x1[0], x2[0])

There are no usable rules.

(13) Complex Obligation (AND)

(14) 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): 903_0_MAIN_LE(x0[0], x1[0], x2[0]) → COND_903_0_MAIN_LE(x2[0] >= x1[0] && x2[0] < x0[0] + -1, x0[0], x1[0], x2[0])


The set Q is empty.

(15) IDependencyGraphProof (EQUIVALENT transformation)

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

(16) TRUE

(17) 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_903_0_MAIN_LE(TRUE, x0[1], x1[1], x2[1]) → 903_0_MAIN_LE(x0[1] + -1, x1[1], x2[1])


The set Q is empty.

(18) IDependencyGraphProof (EQUIVALENT transformation)

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

(19) TRUE

(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): 903_0_MAIN_LE(x0[0], x1[0], x2[0]) → COND_903_0_MAIN_LE(x2[0] >= x1[0] && x2[0] < x0[0] + -1, x0[0], x1[0], x2[0])
(1): COND_903_0_MAIN_LE(TRUE, x0[1], x1[1], x2[1]) → 903_0_MAIN_LE(x0[1] + -1, x1[1], x2[1])
(3): COND_903_0_MAIN_LE1(TRUE, x0[3], x1[3], x2[3]) → 903_0_MAIN_LE(x0[3], x1[3] + -1, x2[3])

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

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

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

The set Q is empty.

(21) IDependencyGraphProof (EQUIVALENT transformation)

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

(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, Boolean

R is empty.

The integer pair graph contains the following rules and edges:
(1): COND_903_0_MAIN_LE(TRUE, x0[1], x1[1], x2[1]) → 903_0_MAIN_LE(x0[1] + -1, x1[1], x2[1])
(0): 903_0_MAIN_LE(x0[0], x1[0], x2[0]) → COND_903_0_MAIN_LE(x2[0] >= x1[0] && x2[0] < x0[0] + -1, x0[0], x1[0], x2[0])

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

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

The set Q is empty.

(23) 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@5f16e402 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 COND_903_0_MAIN_LE(TRUE, x0[1], x1[1], x2[1]) → 903_0_MAIN_LE(+(x0[1], -1), x1[1], x2[1]) the following chains were created:

• We consider the chain COND_903_0_MAIN_LE(TRUE, x0[1], x1[1], x2[1]) → 903_0_MAIN_LE(+(x0[1], -1), x1[1], x2[1]) which results in the following constraint:
  

  (1)    (COND_903_0_MAIN_LE(TRUE, x0[1], x1[1], x2[1])≥NonInfC∧COND_903_0_MAIN_LE(TRUE, x0[1], x1[1], x2[1])≥903_0_MAIN_LE(+(x0[1], -1), x1[1], x2[1])∧(U^Increasing(903_0_MAIN_LE(+(x0[1], -1), x1[1], x2[1])), ≥))

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

  (2)    ((U^Increasing(903_0_MAIN_LE(+(x0[1], -1), x1[1], x2[1])), ≥)∧[bni_11] = 0∧[(-1)bso_12] ≥ 0)

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

  (3)    ((U^Increasing(903_0_MAIN_LE(+(x0[1], -1), x1[1], x2[1])), ≥)∧[bni_11] = 0∧[(-1)bso_12] ≥ 0)

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

  (4)    ((U^Increasing(903_0_MAIN_LE(+(x0[1], -1), x1[1], x2[1])), ≥)∧[bni_11] = 0∧[(-1)bso_12] ≥ 0)

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

  (5)    ((U^Increasing(903_0_MAIN_LE(+(x0[1], -1), x1[1], x2[1])), ≥)∧[bni_11] = 0∧0 = 0∧0 = 0∧0 = 0∧[(-1)bso_12] ≥ 0)

  
  
  

For Pair 903_0_MAIN_LE(x0[0], x1[0], x2[0]) → COND_903_0_MAIN_LE(&&(>=(x2[0], x1[0]), <(x2[0], +(x0[0], -1))), x0[0], x1[0], x2[0]) the following chains were created:

• We consider the chain 903_0_MAIN_LE(x0[0], x1[0], x2[0]) → COND_903_0_MAIN_LE(&&(>=(x2[0], x1[0]), <(x2[0], +(x0[0], -1))), x0[0], x1[0], x2[0]), COND_903_0_MAIN_LE(TRUE, x0[1], x1[1], x2[1]) → 903_0_MAIN_LE(+(x0[1], -1), x1[1], x2[1]) which results in the following constraint:
  

  (6)    (&&(>=(x2[0], x1[0]), <(x2[0], +(x0[0], -1)))=TRUE∧x0[0]=x0[1]∧x1[0]=x1[1]∧x2[0]=x2[1] ⇒ 903_0_MAIN_LE(x0[0], x1[0], x2[0])≥NonInfC∧903_0_MAIN_LE(x0[0], x1[0], x2[0])≥COND_903_0_MAIN_LE(&&(>=(x2[0], x1[0]), <(x2[0], +(x0[0], -1))), x0[0], x1[0], x2[0])∧(U^Increasing(COND_903_0_MAIN_LE(&&(>=(x2[0], x1[0]), <(x2[0], +(x0[0], -1))), x0[0], x1[0], x2[0])), ≥))

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

  (7)    (>=(x2[0], x1[0])=TRUE∧<(x2[0], +(x0[0], -1))=TRUE ⇒ 903_0_MAIN_LE(x0[0], x1[0], x2[0])≥NonInfC∧903_0_MAIN_LE(x0[0], x1[0], x2[0])≥COND_903_0_MAIN_LE(&&(>=(x2[0], x1[0]), <(x2[0], +(x0[0], -1))), x0[0], x1[0], x2[0])∧(U^Increasing(COND_903_0_MAIN_LE(&&(>=(x2[0], x1[0]), <(x2[0], +(x0[0], -1))), x0[0], x1[0], x2[0])), ≥))

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

  (8)    (x2[0] + [-1]x1[0] ≥ 0∧x0[0] + [-2] + [-1]x2[0] ≥ 0 ⇒ (U^Increasing(COND_903_0_MAIN_LE(&&(>=(x2[0], x1[0]), <(x2[0], +(x0[0], -1))), x0[0], x1[0], x2[0])), ≥)∧[bni_13 + (-1)Bound*bni_13] + [(2)bni_13]x0[0] + [(-1)bni_13]x2[0] + [(-1)bni_13]x1[0] ≥ 0∧[2 + (-1)bso_14] ≥ 0)

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

  (9)    (x2[0] + [-1]x1[0] ≥ 0∧x0[0] + [-2] + [-1]x2[0] ≥ 0 ⇒ (U^Increasing(COND_903_0_MAIN_LE(&&(>=(x2[0], x1[0]), <(x2[0], +(x0[0], -1))), x0[0], x1[0], x2[0])), ≥)∧[bni_13 + (-1)Bound*bni_13] + [(2)bni_13]x0[0] + [(-1)bni_13]x2[0] + [(-1)bni_13]x1[0] ≥ 0∧[2 + (-1)bso_14] ≥ 0)

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

  (10)    (x2[0] + [-1]x1[0] ≥ 0∧x0[0] + [-2] + [-1]x2[0] ≥ 0 ⇒ (U^Increasing(COND_903_0_MAIN_LE(&&(>=(x2[0], x1[0]), <(x2[0], +(x0[0], -1))), x0[0], x1[0], x2[0])), ≥)∧[bni_13 + (-1)Bound*bni_13] + [(2)bni_13]x0[0] + [(-1)bni_13]x2[0] + [(-1)bni_13]x1[0] ≥ 0∧[2 + (-1)bso_14] ≥ 0)

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

  (11)    (x2[0] ≥ 0∧x0[0] + [-2] + [-1]x1[0] + [-1]x2[0] ≥ 0 ⇒ (U^Increasing(COND_903_0_MAIN_LE(&&(>=(x2[0], x1[0]), <(x2[0], +(x0[0], -1))), x0[0], x1[0], x2[0])), ≥)∧[bni_13 + (-1)Bound*bni_13] + [(2)bni_13]x0[0] + [(-2)bni_13]x1[0] + [(-1)bni_13]x2[0] ≥ 0∧[2 + (-1)bso_14] ≥ 0)

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

  (12)    (x2[0] ≥ 0∧x1[0] ≥ 0 ⇒ (U^Increasing(COND_903_0_MAIN_LE(&&(>=(x2[0], x1[0]), <(x2[0], +(x0[0], -1))), x0[0], x1[0], x2[0])), ≥)∧[(5)bni_13 + (-1)Bound*bni_13] + [bni_13]x2[0] + [(2)bni_13]x1[0] ≥ 0∧[2 + (-1)bso_14] ≥ 0)

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

  (13)    (x2[0] ≥ 0∧x1[0] ≥ 0∧x0[0] ≥ 0 ⇒ (U^Increasing(COND_903_0_MAIN_LE(&&(>=(x2[0], x1[0]), <(x2[0], +(x0[0], -1))), x0[0], x1[0], x2[0])), ≥)∧[(5)bni_13 + (-1)Bound*bni_13] + [bni_13]x2[0] + [(2)bni_13]x1[0] ≥ 0∧[2 + (-1)bso_14] ≥ 0)

  
  

  (14)    (x2[0] ≥ 0∧x1[0] ≥ 0∧x0[0] ≥ 0 ⇒ (U^Increasing(COND_903_0_MAIN_LE(&&(>=(x2[0], x1[0]), <(x2[0], +(x0[0], -1))), x0[0], x1[0], x2[0])), ≥)∧[(5)bni_13 + (-1)Bound*bni_13] + [bni_13]x2[0] + [(2)bni_13]x1[0] ≥ 0∧[2 + (-1)bso_14] ≥ 0)

  
  
  

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

• COND_903_0_MAIN_LE(TRUE, x0[1], x1[1], x2[1]) → 903_0_MAIN_LE(+(x0[1], -1), x1[1], x2[1])
  
  • ((U^Increasing(903_0_MAIN_LE(+(x0[1], -1), x1[1], x2[1])), ≥)∧[bni_11] = 0∧0 = 0∧0 = 0∧0 = 0∧[(-1)bso_12] ≥ 0)
    
  
  
• 903_0_MAIN_LE(x0[0], x1[0], x2[0]) → COND_903_0_MAIN_LE(&&(>=(x2[0], x1[0]), <(x2[0], +(x0[0], -1))), x0[0], x1[0], x2[0])
  
  • (x2[0] ≥ 0∧x1[0] ≥ 0∧x0[0] ≥ 0 ⇒ (U^Increasing(COND_903_0_MAIN_LE(&&(>=(x2[0], x1[0]), <(x2[0], +(x0[0], -1))), x0[0], x1[0], x2[0])), ≥)∧[(5)bni_13 + (-1)Bound*bni_13] + [bni_13]x2[0] + [(2)bni_13]x1[0] ≥ 0∧[2 + (-1)bso_14] ≥ 0)
    
  • (x2[0] ≥ 0∧x1[0] ≥ 0∧x0[0] ≥ 0 ⇒ (U^Increasing(COND_903_0_MAIN_LE(&&(>=(x2[0], x1[0]), <(x2[0], +(x0[0], -1))), x0[0], x1[0], x2[0])), ≥)∧[(5)bni_13 + (-1)Bound*bni_13] + [bni_13]x2[0] + [(2)bni_13]x1[0] ≥ 0∧[2 + (-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_903_0_MAIN_LE(x₁, x₂, x₃, x₄)) = [-1] + [-1]x₄ + [-1]x₃ + [2]x₂   
POL(903_0_MAIN_LE(x₁, x₂, x₃)) = [1] + [2]x₁ + [-1]x₃ + [-1]x₂   
POL(+(x₁, x₂)) = x₁ + x₂   
POL(-1) = [-1]   
POL(&&(x₁, x₂)) = [-1]   
POL(>=(x₁, x₂)) = [-1]   
POL(<(x₁, x₂)) = [-1]   

The following pairs are in P_>:

903_0_MAIN_LE(x0[0], x1[0], x2[0]) → COND_903_0_MAIN_LE(&&(>=(x2[0], x1[0]), <(x2[0], +(x0[0], -1))), x0[0], x1[0], x2[0])

The following pairs are in P_bound:

903_0_MAIN_LE(x0[0], x1[0], x2[0]) → COND_903_0_MAIN_LE(&&(>=(x2[0], x1[0]), <(x2[0], +(x0[0], -1))), x0[0], x1[0], x2[0])

The following pairs are in P_≥:

COND_903_0_MAIN_LE(TRUE, x0[1], x1[1], x2[1]) → 903_0_MAIN_LE(+(x0[1], -1), x1[1], x2[1])

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:
(1): COND_903_0_MAIN_LE(TRUE, x0[1], x1[1], x2[1]) → 903_0_MAIN_LE(x0[1] + -1, x1[1], x2[1])


The set Q is empty.

(25) IDependencyGraphProof (EQUIVALENT transformation)

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

(26) TRUE