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_20 (Apple Inc.) Main-Class: Carre

class Curseur{
	private int X=0, Y=0, maxX, maxY;
	private boolean torique=false;

	public Curseur(int maxX, int maxY, boolean espaceTorique){
		super();
		this.maxX=maxX;
		this.maxY=maxY;
		this.torique=espaceTorique;		
	}

	public void centrer(){
		
			int cX=maxX/2;
			int cY=maxY/2;
			X=cX;
			Y=cY;
		
	}

	public void haut(){

			Y--;
			if(torique&&Y<0) Y=maxY-1;
	}

	public void bas(){
			Y++;
			if(torique&&Y==maxY) Y=0;
	}

	public void droite(){
			X++;
			if(torique&&X==maxX) X=0;
	}

	public void gauche(){
			X--;
			if(torique&&X<0) X=maxX-1;
	}

	public int getX(){
			return X;
	}

	public int getY(){
			return Y;
	}
	
	public void imprimer(){
	    //System.out.println("Curseur@["+getX()+","+getY()+"]");
	}
}

public class Carre {

	private Curseur curseur=null;
	private int cote=0;

	public Carre(int cote){
		if(cote>1&cote%2==1){
			this.cote=cote; 
		}else{
		    //System.out.println("Cette classe ne genere pas les carres magiques d\'ordre pair.");
		    //System.exit(0);
		}
		this.curseur=new Curseur(cote,cote,true);
	}
	
	private int [][] carre=null;

	public void init(){
		carre=new int[cote][cote];
		int n=0;
		for(int x=0;x<3;x++) for(int y=0;y<3;y++) carre[x][y]=0;
		curseur.centrer();
	}

	public void peupler(){
		curseur.bas();
		int nbre=1;
		int cpteur=1;
		while(cpteur<3){			
			if(!ajouter(nbre,curseur.getX(),curseur.getY())){
				curseur.bas();
				curseur.gauche();
				cpteur++;
			}else{
				cpteur=1;
				curseur.bas();
				curseur.droite();
				nbre++;
			}
		}
	}

	public Curseur curseur(){
		return curseur;
	}

	public int cote(){
		return cote;
	}

	public boolean ajouter(int nombre, int X, int Y){
		if(carre[X][Y]!=0) return false; 
		carre[X][Y]=nombre;
		return true;
	}

	public void imprimer(){
		for(int j=0;j<cote;j++){
			for(int i=0;i<cote;i++){
			    //System.out.print(carre[i][j]+"\t");
			}
			//System.out.println();
		}
	}

	public static void main(String args[]){
	    Random.args = args;
	    Carre carre=new Carre(2*Random.random()+1);
	    carre.init();
	    //carre.peupler();
	    carre.imprimer();
	}
    
}

public class Random {
  static String[] args;
  static int index = 0;

  public static int random() {
      if (index >= args.length)
	  return 0;

      String string = args[index];
      index++;
      return string.length();
  }
}

(1) JBC2FIG (SOUND transformation)

Constructed FIGraph.

(2) Obligation:

FIGraph based on JBC Program:
Carre.main([Ljava/lang/String;)V: Graph of 305 nodes with 2 SCCs.

Carre.<init>(I)V: Graph of 144 nodes with 0 SCCs.

(3) FIGtoITRSProof (SOUND transformation)

Transformed FIGraph SCCs to IDPs. Logs:

---

Log for SCC 0:

Generated 23 rules for P and 3 rules for R.

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

Filtered ground terms:

Carre(x1, x2) → Carre(x2)
3588_0_imprimer_GE(x1, x2, x3, x4, x5, x6) → 3588_0_imprimer_GE(x2, x3, x4, x5, x6)

Filtered duplicate args:

3588_0_imprimer_GE(x1, x2, x3, x4, x5) → 3588_0_imprimer_GE(x1, x2, x4, x5)

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

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

---

Log for SCC 1:

Generated 32 rules for P and 127 rules for R.

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

Filtered ground terms:

Carre(x1, x2) → Carre(x2)
2995_0_init_GE(x1, x2, x3, x4, x5, x6) → 2995_0_init_GE(x2, x3, x4, x5)

Filtered duplicate args:

2995_1_main_InvokeMethod(x1, x2, x3) → 2995_1_main_InvokeMethod(x1, x3)
2995_0_init_GE(x1, x2, x3, x4) → 2995_0_init_GE(x1, x2, x4)
Cond_2995_1_main_InvokeMethod1(x1, x2, x3, x4) → Cond_2995_1_main_InvokeMethod1(x1, x2, x4)
Cond_2995_1_main_InvokeMethod(x1, x2, x3, x4) → Cond_2995_1_main_InvokeMethod(x1, x2, x4)

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

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

(4) Complex Obligation (AND)

(5) 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): 3588_1_MAIN_INVOKEMETHOD(3588_0_imprimer_GE(java.lang.Object(Carre(x0[0])), x1[0], x2[0], x0[0]), java.lang.Object(Carre(x0[0]))) → COND_3588_1_MAIN_INVOKEMETHOD(x2[0] >= x0[0] && x1[0] >= 0 && x0[0] > x1[0] + 1, 3588_0_imprimer_GE(java.lang.Object(Carre(x0[0])), x1[0], x2[0], x0[0]), java.lang.Object(Carre(x0[0])))
(1): COND_3588_1_MAIN_INVOKEMETHOD(TRUE, 3588_0_imprimer_GE(java.lang.Object(Carre(x0[1])), x1[1], x2[1], x0[1]), java.lang.Object(Carre(x0[1]))) → 3588_1_MAIN_INVOKEMETHOD(3588_0_imprimer_GE(java.lang.Object(Carre(x0[1])), x1[1] + 1, 0, x0[1]), java.lang.Object(Carre(x0[1])))
(2): 3588_1_MAIN_INVOKEMETHOD(3588_0_imprimer_GE(java.lang.Object(Carre(x0[2])), x1[2], x2[2], x0[2]), java.lang.Object(Carre(x0[2]))) → COND_3588_1_MAIN_INVOKEMETHOD1(x2[2] >= 0 && x2[2] < x0[2], 3588_0_imprimer_GE(java.lang.Object(Carre(x0[2])), x1[2], x2[2], x0[2]), java.lang.Object(Carre(x0[2])))
(3): COND_3588_1_MAIN_INVOKEMETHOD1(TRUE, 3588_0_imprimer_GE(java.lang.Object(Carre(x0[3])), x1[3], x2[3], x0[3]), java.lang.Object(Carre(x0[3]))) → 3588_1_MAIN_INVOKEMETHOD(3588_0_imprimer_GE(java.lang.Object(Carre(x0[3])), x1[3], x2[3] + 1, x0[3]), java.lang.Object(Carre(x0[3])))

(0) -> (1), if ((x2[0] >= x0[0] && x1[0] >= 0 && x0[0] > x1[0] + 1 →^* TRUE)∧(3588_0_imprimer_GE(java.lang.Object(Carre(x0[0])), x1[0], x2[0], x0[0]) →^* 3588_0_imprimer_GE(java.lang.Object(Carre(x0[1])), x1[1], x2[1], x0[1]))∧(java.lang.Object(Carre(x0[0])) →^* java.lang.Object(Carre(x0[1]))))

(1) -> (0), if ((3588_0_imprimer_GE(java.lang.Object(Carre(x0[1])), x1[1] + 1, 0, x0[1]) →^* 3588_0_imprimer_GE(java.lang.Object(Carre(x0[0])), x1[0], x2[0], x0[0]))∧(java.lang.Object(Carre(x0[1])) →^* java.lang.Object(Carre(x0[0]))))

(1) -> (2), if ((3588_0_imprimer_GE(java.lang.Object(Carre(x0[1])), x1[1] + 1, 0, x0[1]) →^* 3588_0_imprimer_GE(java.lang.Object(Carre(x0[2])), x1[2], x2[2], x0[2]))∧(java.lang.Object(Carre(x0[1])) →^* java.lang.Object(Carre(x0[2]))))

(2) -> (3), if ((x2[2] >= 0 && x2[2] < x0[2] →^* TRUE)∧(3588_0_imprimer_GE(java.lang.Object(Carre(x0[2])), x1[2], x2[2], x0[2]) →^* 3588_0_imprimer_GE(java.lang.Object(Carre(x0[3])), x1[3], x2[3], x0[3]))∧(java.lang.Object(Carre(x0[2])) →^* java.lang.Object(Carre(x0[3]))))

(3) -> (0), if ((3588_0_imprimer_GE(java.lang.Object(Carre(x0[3])), x1[3], x2[3] + 1, x0[3]) →^* 3588_0_imprimer_GE(java.lang.Object(Carre(x0[0])), x1[0], x2[0], x0[0]))∧(java.lang.Object(Carre(x0[3])) →^* java.lang.Object(Carre(x0[0]))))

(3) -> (2), if ((3588_0_imprimer_GE(java.lang.Object(Carre(x0[3])), x1[3], x2[3] + 1, x0[3]) →^* 3588_0_imprimer_GE(java.lang.Object(Carre(x0[2])), x1[2], x2[2], x0[2]))∧(java.lang.Object(Carre(x0[3])) →^* java.lang.Object(Carre(x0[2]))))

The set Q is empty.

(6) IDPNonInfProof (SOUND transformation)

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 3588_1_MAIN_INVOKEMETHOD(3588_0_imprimer_GE(java.lang.Object(Carre(x0)), x1, x2, x0), java.lang.Object(Carre(x0))) → COND_3588_1_MAIN_INVOKEMETHOD(&&(&&(>=(x2, x0), >=(x1, 0)), >(x0, +(x1, 1))), 3588_0_imprimer_GE(java.lang.Object(Carre(x0)), x1, x2, x0), java.lang.Object(Carre(x0))) the following chains were created:

• We consider the chain 3588_1_MAIN_INVOKEMETHOD(3588_0_imprimer_GE(java.lang.Object(Carre(x0[0])), x1[0], x2[0], x0[0]), java.lang.Object(Carre(x0[0]))) → COND_3588_1_MAIN_INVOKEMETHOD(&&(&&(>=(x2[0], x0[0]), >=(x1[0], 0)), >(x0[0], +(x1[0], 1))), 3588_0_imprimer_GE(java.lang.Object(Carre(x0[0])), x1[0], x2[0], x0[0]), java.lang.Object(Carre(x0[0]))), COND_3588_1_MAIN_INVOKEMETHOD(TRUE, 3588_0_imprimer_GE(java.lang.Object(Carre(x0[1])), x1[1], x2[1], x0[1]), java.lang.Object(Carre(x0[1]))) → 3588_1_MAIN_INVOKEMETHOD(3588_0_imprimer_GE(java.lang.Object(Carre(x0[1])), +(x1[1], 1), 0, x0[1]), java.lang.Object(Carre(x0[1]))) which results in the following constraint:
  

  (1)    (&&(&&(>=(x2[0], x0[0]), >=(x1[0], 0)), >(x0[0], +(x1[0], 1)))=TRUE∧3588_0_imprimer_GE(java.lang.Object(Carre(x0[0])), x1[0], x2[0], x0[0])=3588_0_imprimer_GE(java.lang.Object(Carre(x0[1])), x1[1], x2[1], x0[1])∧java.lang.Object(Carre(x0[0]))=java.lang.Object(Carre(x0[1])) ⇒ 3588_1_MAIN_INVOKEMETHOD(3588_0_imprimer_GE(java.lang.Object(Carre(x0[0])), x1[0], x2[0], x0[0]), java.lang.Object(Carre(x0[0])))≥NonInfC∧3588_1_MAIN_INVOKEMETHOD(3588_0_imprimer_GE(java.lang.Object(Carre(x0[0])), x1[0], x2[0], x0[0]), java.lang.Object(Carre(x0[0])))≥COND_3588_1_MAIN_INVOKEMETHOD(&&(&&(>=(x2[0], x0[0]), >=(x1[0], 0)), >(x0[0], +(x1[0], 1))), 3588_0_imprimer_GE(java.lang.Object(Carre(x0[0])), x1[0], x2[0], x0[0]), java.lang.Object(Carre(x0[0])))∧(U^Increasing(COND_3588_1_MAIN_INVOKEMETHOD(&&(&&(>=(x2[0], x0[0]), >=(x1[0], 0)), >(x0[0], +(x1[0], 1))), 3588_0_imprimer_GE(java.lang.Object(Carre(x0[0])), x1[0], x2[0], x0[0]), java.lang.Object(Carre(x0[0])))), ≥))

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

  (2)    (>(x0[0], +(x1[0], 1))=TRUE∧>=(x2[0], x0[0])=TRUE∧>=(x1[0], 0)=TRUE ⇒ 3588_1_MAIN_INVOKEMETHOD(3588_0_imprimer_GE(java.lang.Object(Carre(x0[0])), x1[0], x2[0], x0[0]), java.lang.Object(Carre(x0[0])))≥NonInfC∧3588_1_MAIN_INVOKEMETHOD(3588_0_imprimer_GE(java.lang.Object(Carre(x0[0])), x1[0], x2[0], x0[0]), java.lang.Object(Carre(x0[0])))≥COND_3588_1_MAIN_INVOKEMETHOD(&&(&&(>=(x2[0], x0[0]), >=(x1[0], 0)), >(x0[0], +(x1[0], 1))), 3588_0_imprimer_GE(java.lang.Object(Carre(x0[0])), x1[0], x2[0], x0[0]), java.lang.Object(Carre(x0[0])))∧(U^Increasing(COND_3588_1_MAIN_INVOKEMETHOD(&&(&&(>=(x2[0], x0[0]), >=(x1[0], 0)), >(x0[0], +(x1[0], 1))), 3588_0_imprimer_GE(java.lang.Object(Carre(x0[0])), x1[0], x2[0], x0[0]), java.lang.Object(Carre(x0[0])))), ≥))

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

  (3)    (x0[0] + [-2] + [-1]x1[0] ≥ 0∧x2[0] + [-1]x0[0] ≥ 0∧x1[0] ≥ 0 ⇒ (U^Increasing(COND_3588_1_MAIN_INVOKEMETHOD(&&(&&(>=(x2[0], x0[0]), >=(x1[0], 0)), >(x0[0], +(x1[0], 1))), 3588_0_imprimer_GE(java.lang.Object(Carre(x0[0])), x1[0], x2[0], x0[0]), java.lang.Object(Carre(x0[0])))), ≥)∧[(-1)Bound*bni_17] + [bni_17]x0[0] + [(-1)bni_17]x1[0] ≥ 0∧[(-1)bso_18] ≥ 0)

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

  (4)    (x0[0] + [-2] + [-1]x1[0] ≥ 0∧x2[0] + [-1]x0[0] ≥ 0∧x1[0] ≥ 0 ⇒ (U^Increasing(COND_3588_1_MAIN_INVOKEMETHOD(&&(&&(>=(x2[0], x0[0]), >=(x1[0], 0)), >(x0[0], +(x1[0], 1))), 3588_0_imprimer_GE(java.lang.Object(Carre(x0[0])), x1[0], x2[0], x0[0]), java.lang.Object(Carre(x0[0])))), ≥)∧[(-1)Bound*bni_17] + [bni_17]x0[0] + [(-1)bni_17]x1[0] ≥ 0∧[(-1)bso_18] ≥ 0)

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

  (5)    (x0[0] + [-2] + [-1]x1[0] ≥ 0∧x2[0] + [-1]x0[0] ≥ 0∧x1[0] ≥ 0 ⇒ (U^Increasing(COND_3588_1_MAIN_INVOKEMETHOD(&&(&&(>=(x2[0], x0[0]), >=(x1[0], 0)), >(x0[0], +(x1[0], 1))), 3588_0_imprimer_GE(java.lang.Object(Carre(x0[0])), x1[0], x2[0], x0[0]), java.lang.Object(Carre(x0[0])))), ≥)∧[(-1)Bound*bni_17] + [bni_17]x0[0] + [(-1)bni_17]x1[0] ≥ 0∧[(-1)bso_18] ≥ 0)

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

  (6)    (x0[0] ≥ 0∧x2[0] + [-2] + [-1]x1[0] + [-1]x0[0] ≥ 0∧x1[0] ≥ 0 ⇒ (U^Increasing(COND_3588_1_MAIN_INVOKEMETHOD(&&(&&(>=(x2[0], x0[0]), >=(x1[0], 0)), >(x0[0], +(x1[0], 1))), 3588_0_imprimer_GE(java.lang.Object(Carre(x0[0])), x1[0], x2[0], x0[0]), java.lang.Object(Carre(x0[0])))), ≥)∧[(-1)Bound*bni_17 + (2)bni_17] + [bni_17]x0[0] ≥ 0∧[(-1)bso_18] ≥ 0)

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

  (7)    (x0[0] ≥ 0∧x2[0] ≥ 0∧x1[0] ≥ 0 ⇒ (U^Increasing(COND_3588_1_MAIN_INVOKEMETHOD(&&(&&(>=(x2[0], x0[0]), >=(x1[0], 0)), >(x0[0], +(x1[0], 1))), 3588_0_imprimer_GE(java.lang.Object(Carre(x0[0])), x1[0], x2[0], x0[0]), java.lang.Object(Carre(x0[0])))), ≥)∧[(-1)Bound*bni_17 + (2)bni_17] + [bni_17]x0[0] ≥ 0∧[(-1)bso_18] ≥ 0)

  
  
  

For Pair COND_3588_1_MAIN_INVOKEMETHOD(TRUE, 3588_0_imprimer_GE(java.lang.Object(Carre(x0)), x1, x2, x0), java.lang.Object(Carre(x0))) → 3588_1_MAIN_INVOKEMETHOD(3588_0_imprimer_GE(java.lang.Object(Carre(x0)), +(x1, 1), 0, x0), java.lang.Object(Carre(x0))) the following chains were created:

• We consider the chain COND_3588_1_MAIN_INVOKEMETHOD(TRUE, 3588_0_imprimer_GE(java.lang.Object(Carre(x0[1])), x1[1], x2[1], x0[1]), java.lang.Object(Carre(x0[1]))) → 3588_1_MAIN_INVOKEMETHOD(3588_0_imprimer_GE(java.lang.Object(Carre(x0[1])), +(x1[1], 1), 0, x0[1]), java.lang.Object(Carre(x0[1]))) which results in the following constraint:
  

  (8)    (COND_3588_1_MAIN_INVOKEMETHOD(TRUE, 3588_0_imprimer_GE(java.lang.Object(Carre(x0[1])), x1[1], x2[1], x0[1]), java.lang.Object(Carre(x0[1])))≥NonInfC∧COND_3588_1_MAIN_INVOKEMETHOD(TRUE, 3588_0_imprimer_GE(java.lang.Object(Carre(x0[1])), x1[1], x2[1], x0[1]), java.lang.Object(Carre(x0[1])))≥3588_1_MAIN_INVOKEMETHOD(3588_0_imprimer_GE(java.lang.Object(Carre(x0[1])), +(x1[1], 1), 0, x0[1]), java.lang.Object(Carre(x0[1])))∧(U^Increasing(3588_1_MAIN_INVOKEMETHOD(3588_0_imprimer_GE(java.lang.Object(Carre(x0[1])), +(x1[1], 1), 0, x0[1]), java.lang.Object(Carre(x0[1])))), ≥))

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

  (9)    ((U^Increasing(3588_1_MAIN_INVOKEMETHOD(3588_0_imprimer_GE(java.lang.Object(Carre(x0[1])), +(x1[1], 1), 0, x0[1]), java.lang.Object(Carre(x0[1])))), ≥)∧[1 + (-1)bso_20] ≥ 0)

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

  (10)    ((U^Increasing(3588_1_MAIN_INVOKEMETHOD(3588_0_imprimer_GE(java.lang.Object(Carre(x0[1])), +(x1[1], 1), 0, x0[1]), java.lang.Object(Carre(x0[1])))), ≥)∧[1 + (-1)bso_20] ≥ 0)

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

  (11)    ((U^Increasing(3588_1_MAIN_INVOKEMETHOD(3588_0_imprimer_GE(java.lang.Object(Carre(x0[1])), +(x1[1], 1), 0, x0[1]), java.lang.Object(Carre(x0[1])))), ≥)∧[1 + (-1)bso_20] ≥ 0)

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

  (12)    ((U^Increasing(3588_1_MAIN_INVOKEMETHOD(3588_0_imprimer_GE(java.lang.Object(Carre(x0[1])), +(x1[1], 1), 0, x0[1]), java.lang.Object(Carre(x0[1])))), ≥)∧0 = 0∧0 = 0∧0 = 0∧[1 + (-1)bso_20] ≥ 0)

  
  
  

For Pair 3588_1_MAIN_INVOKEMETHOD(3588_0_imprimer_GE(java.lang.Object(Carre(x0)), x1, x2, x0), java.lang.Object(Carre(x0))) → COND_3588_1_MAIN_INVOKEMETHOD1(&&(>=(x2, 0), <(x2, x0)), 3588_0_imprimer_GE(java.lang.Object(Carre(x0)), x1, x2, x0), java.lang.Object(Carre(x0))) the following chains were created:

• We consider the chain 3588_1_MAIN_INVOKEMETHOD(3588_0_imprimer_GE(java.lang.Object(Carre(x0[2])), x1[2], x2[2], x0[2]), java.lang.Object(Carre(x0[2]))) → COND_3588_1_MAIN_INVOKEMETHOD1(&&(>=(x2[2], 0), <(x2[2], x0[2])), 3588_0_imprimer_GE(java.lang.Object(Carre(x0[2])), x1[2], x2[2], x0[2]), java.lang.Object(Carre(x0[2]))), COND_3588_1_MAIN_INVOKEMETHOD1(TRUE, 3588_0_imprimer_GE(java.lang.Object(Carre(x0[3])), x1[3], x2[3], x0[3]), java.lang.Object(Carre(x0[3]))) → 3588_1_MAIN_INVOKEMETHOD(3588_0_imprimer_GE(java.lang.Object(Carre(x0[3])), x1[3], +(x2[3], 1), x0[3]), java.lang.Object(Carre(x0[3]))) which results in the following constraint:
  

  (13)    (&&(>=(x2[2], 0), <(x2[2], x0[2]))=TRUE∧3588_0_imprimer_GE(java.lang.Object(Carre(x0[2])), x1[2], x2[2], x0[2])=3588_0_imprimer_GE(java.lang.Object(Carre(x0[3])), x1[3], x2[3], x0[3])∧java.lang.Object(Carre(x0[2]))=java.lang.Object(Carre(x0[3])) ⇒ 3588_1_MAIN_INVOKEMETHOD(3588_0_imprimer_GE(java.lang.Object(Carre(x0[2])), x1[2], x2[2], x0[2]), java.lang.Object(Carre(x0[2])))≥NonInfC∧3588_1_MAIN_INVOKEMETHOD(3588_0_imprimer_GE(java.lang.Object(Carre(x0[2])), x1[2], x2[2], x0[2]), java.lang.Object(Carre(x0[2])))≥COND_3588_1_MAIN_INVOKEMETHOD1(&&(>=(x2[2], 0), <(x2[2], x0[2])), 3588_0_imprimer_GE(java.lang.Object(Carre(x0[2])), x1[2], x2[2], x0[2]), java.lang.Object(Carre(x0[2])))∧(U^Increasing(COND_3588_1_MAIN_INVOKEMETHOD1(&&(>=(x2[2], 0), <(x2[2], x0[2])), 3588_0_imprimer_GE(java.lang.Object(Carre(x0[2])), x1[2], x2[2], x0[2]), java.lang.Object(Carre(x0[2])))), ≥))

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

  (14)    (>=(x2[2], 0)=TRUE∧<(x2[2], x0[2])=TRUE ⇒ 3588_1_MAIN_INVOKEMETHOD(3588_0_imprimer_GE(java.lang.Object(Carre(x0[2])), x1[2], x2[2], x0[2]), java.lang.Object(Carre(x0[2])))≥NonInfC∧3588_1_MAIN_INVOKEMETHOD(3588_0_imprimer_GE(java.lang.Object(Carre(x0[2])), x1[2], x2[2], x0[2]), java.lang.Object(Carre(x0[2])))≥COND_3588_1_MAIN_INVOKEMETHOD1(&&(>=(x2[2], 0), <(x2[2], x0[2])), 3588_0_imprimer_GE(java.lang.Object(Carre(x0[2])), x1[2], x2[2], x0[2]), java.lang.Object(Carre(x0[2])))∧(U^Increasing(COND_3588_1_MAIN_INVOKEMETHOD1(&&(>=(x2[2], 0), <(x2[2], x0[2])), 3588_0_imprimer_GE(java.lang.Object(Carre(x0[2])), x1[2], x2[2], x0[2]), java.lang.Object(Carre(x0[2])))), ≥))

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

  (15)    (x2[2] ≥ 0∧x0[2] + [-1] + [-1]x2[2] ≥ 0 ⇒ (U^Increasing(COND_3588_1_MAIN_INVOKEMETHOD1(&&(>=(x2[2], 0), <(x2[2], x0[2])), 3588_0_imprimer_GE(java.lang.Object(Carre(x0[2])), x1[2], x2[2], x0[2]), java.lang.Object(Carre(x0[2])))), ≥)∧[(-1)Bound*bni_21] + [bni_21]x0[2] + [(-1)bni_21]x1[2] ≥ 0∧[(-1)bso_22] ≥ 0)

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

  (16)    (x2[2] ≥ 0∧x0[2] + [-1] + [-1]x2[2] ≥ 0 ⇒ (U^Increasing(COND_3588_1_MAIN_INVOKEMETHOD1(&&(>=(x2[2], 0), <(x2[2], x0[2])), 3588_0_imprimer_GE(java.lang.Object(Carre(x0[2])), x1[2], x2[2], x0[2]), java.lang.Object(Carre(x0[2])))), ≥)∧[(-1)Bound*bni_21] + [bni_21]x0[2] + [(-1)bni_21]x1[2] ≥ 0∧[(-1)bso_22] ≥ 0)

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

  (17)    (x2[2] ≥ 0∧x0[2] + [-1] + [-1]x2[2] ≥ 0 ⇒ (U^Increasing(COND_3588_1_MAIN_INVOKEMETHOD1(&&(>=(x2[2], 0), <(x2[2], x0[2])), 3588_0_imprimer_GE(java.lang.Object(Carre(x0[2])), x1[2], x2[2], x0[2]), java.lang.Object(Carre(x0[2])))), ≥)∧[(-1)Bound*bni_21] + [bni_21]x0[2] + [(-1)bni_21]x1[2] ≥ 0∧[(-1)bso_22] ≥ 0)

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

  (18)    (x2[2] ≥ 0∧x0[2] + [-1] + [-1]x2[2] ≥ 0 ⇒ (U^Increasing(COND_3588_1_MAIN_INVOKEMETHOD1(&&(>=(x2[2], 0), <(x2[2], x0[2])), 3588_0_imprimer_GE(java.lang.Object(Carre(x0[2])), x1[2], x2[2], x0[2]), java.lang.Object(Carre(x0[2])))), ≥)∧[(-1)bni_21] = 0∧[(-1)Bound*bni_21] + [bni_21]x0[2] ≥ 0∧0 = 0∧[(-1)bso_22] ≥ 0)

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

  (19)    (x2[2] ≥ 0∧x0[2] ≥ 0 ⇒ (U^Increasing(COND_3588_1_MAIN_INVOKEMETHOD1(&&(>=(x2[2], 0), <(x2[2], x0[2])), 3588_0_imprimer_GE(java.lang.Object(Carre(x0[2])), x1[2], x2[2], x0[2]), java.lang.Object(Carre(x0[2])))), ≥)∧[(-1)bni_21] = 0∧[(-1)Bound*bni_21 + bni_21] + [bni_21]x2[2] + [bni_21]x0[2] ≥ 0∧0 = 0∧[(-1)bso_22] ≥ 0)

  
  
  

For Pair COND_3588_1_MAIN_INVOKEMETHOD1(TRUE, 3588_0_imprimer_GE(java.lang.Object(Carre(x0)), x1, x2, x0), java.lang.Object(Carre(x0))) → 3588_1_MAIN_INVOKEMETHOD(3588_0_imprimer_GE(java.lang.Object(Carre(x0)), x1, +(x2, 1), x0), java.lang.Object(Carre(x0))) the following chains were created:

• We consider the chain COND_3588_1_MAIN_INVOKEMETHOD1(TRUE, 3588_0_imprimer_GE(java.lang.Object(Carre(x0[3])), x1[3], x2[3], x0[3]), java.lang.Object(Carre(x0[3]))) → 3588_1_MAIN_INVOKEMETHOD(3588_0_imprimer_GE(java.lang.Object(Carre(x0[3])), x1[3], +(x2[3], 1), x0[3]), java.lang.Object(Carre(x0[3]))) which results in the following constraint:
  

  (20)    (COND_3588_1_MAIN_INVOKEMETHOD1(TRUE, 3588_0_imprimer_GE(java.lang.Object(Carre(x0[3])), x1[3], x2[3], x0[3]), java.lang.Object(Carre(x0[3])))≥NonInfC∧COND_3588_1_MAIN_INVOKEMETHOD1(TRUE, 3588_0_imprimer_GE(java.lang.Object(Carre(x0[3])), x1[3], x2[3], x0[3]), java.lang.Object(Carre(x0[3])))≥3588_1_MAIN_INVOKEMETHOD(3588_0_imprimer_GE(java.lang.Object(Carre(x0[3])), x1[3], +(x2[3], 1), x0[3]), java.lang.Object(Carre(x0[3])))∧(U^Increasing(3588_1_MAIN_INVOKEMETHOD(3588_0_imprimer_GE(java.lang.Object(Carre(x0[3])), x1[3], +(x2[3], 1), x0[3]), java.lang.Object(Carre(x0[3])))), ≥))

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

  (21)    ((U^Increasing(3588_1_MAIN_INVOKEMETHOD(3588_0_imprimer_GE(java.lang.Object(Carre(x0[3])), x1[3], +(x2[3], 1), x0[3]), java.lang.Object(Carre(x0[3])))), ≥)∧[(-1)bso_24] ≥ 0)

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

  (22)    ((U^Increasing(3588_1_MAIN_INVOKEMETHOD(3588_0_imprimer_GE(java.lang.Object(Carre(x0[3])), x1[3], +(x2[3], 1), x0[3]), java.lang.Object(Carre(x0[3])))), ≥)∧[(-1)bso_24] ≥ 0)

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

  (23)    ((U^Increasing(3588_1_MAIN_INVOKEMETHOD(3588_0_imprimer_GE(java.lang.Object(Carre(x0[3])), x1[3], +(x2[3], 1), x0[3]), java.lang.Object(Carre(x0[3])))), ≥)∧[(-1)bso_24] ≥ 0)

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

  (24)    ((U^Increasing(3588_1_MAIN_INVOKEMETHOD(3588_0_imprimer_GE(java.lang.Object(Carre(x0[3])), x1[3], +(x2[3], 1), x0[3]), java.lang.Object(Carre(x0[3])))), ≥)∧0 = 0∧0 = 0∧0 = 0∧[(-1)bso_24] ≥ 0)

  
  
  

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

• 3588_1_MAIN_INVOKEMETHOD(3588_0_imprimer_GE(java.lang.Object(Carre(x0)), x1, x2, x0), java.lang.Object(Carre(x0))) → COND_3588_1_MAIN_INVOKEMETHOD(&&(&&(>=(x2, x0), >=(x1, 0)), >(x0, +(x1, 1))), 3588_0_imprimer_GE(java.lang.Object(Carre(x0)), x1, x2, x0), java.lang.Object(Carre(x0)))
  
  • (x0[0] ≥ 0∧x2[0] ≥ 0∧x1[0] ≥ 0 ⇒ (U^Increasing(COND_3588_1_MAIN_INVOKEMETHOD(&&(&&(>=(x2[0], x0[0]), >=(x1[0], 0)), >(x0[0], +(x1[0], 1))), 3588_0_imprimer_GE(java.lang.Object(Carre(x0[0])), x1[0], x2[0], x0[0]), java.lang.Object(Carre(x0[0])))), ≥)∧[(-1)Bound*bni_17 + (2)bni_17] + [bni_17]x0[0] ≥ 0∧[(-1)bso_18] ≥ 0)
    
  
  
• COND_3588_1_MAIN_INVOKEMETHOD(TRUE, 3588_0_imprimer_GE(java.lang.Object(Carre(x0)), x1, x2, x0), java.lang.Object(Carre(x0))) → 3588_1_MAIN_INVOKEMETHOD(3588_0_imprimer_GE(java.lang.Object(Carre(x0)), +(x1, 1), 0, x0), java.lang.Object(Carre(x0)))
  
  • ((U^Increasing(3588_1_MAIN_INVOKEMETHOD(3588_0_imprimer_GE(java.lang.Object(Carre(x0[1])), +(x1[1], 1), 0, x0[1]), java.lang.Object(Carre(x0[1])))), ≥)∧0 = 0∧0 = 0∧0 = 0∧[1 + (-1)bso_20] ≥ 0)
    
  
  
• 3588_1_MAIN_INVOKEMETHOD(3588_0_imprimer_GE(java.lang.Object(Carre(x0)), x1, x2, x0), java.lang.Object(Carre(x0))) → COND_3588_1_MAIN_INVOKEMETHOD1(&&(>=(x2, 0), <(x2, x0)), 3588_0_imprimer_GE(java.lang.Object(Carre(x0)), x1, x2, x0), java.lang.Object(Carre(x0)))
  
  • (x2[2] ≥ 0∧x0[2] ≥ 0 ⇒ (U^Increasing(COND_3588_1_MAIN_INVOKEMETHOD1(&&(>=(x2[2], 0), <(x2[2], x0[2])), 3588_0_imprimer_GE(java.lang.Object(Carre(x0[2])), x1[2], x2[2], x0[2]), java.lang.Object(Carre(x0[2])))), ≥)∧[(-1)bni_21] = 0∧[(-1)Bound*bni_21 + bni_21] + [bni_21]x2[2] + [bni_21]x0[2] ≥ 0∧0 = 0∧[(-1)bso_22] ≥ 0)
    
  
  
• COND_3588_1_MAIN_INVOKEMETHOD1(TRUE, 3588_0_imprimer_GE(java.lang.Object(Carre(x0)), x1, x2, x0), java.lang.Object(Carre(x0))) → 3588_1_MAIN_INVOKEMETHOD(3588_0_imprimer_GE(java.lang.Object(Carre(x0)), x1, +(x2, 1), x0), java.lang.Object(Carre(x0)))
  
  • ((U^Increasing(3588_1_MAIN_INVOKEMETHOD(3588_0_imprimer_GE(java.lang.Object(Carre(x0[3])), x1[3], +(x2[3], 1), x0[3]), java.lang.Object(Carre(x0[3])))), ≥)∧0 = 0∧0 = 0∧0 = 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) = 0   
POL(3588_1_MAIN_INVOKEMETHOD(x₁, x₂)) = [-1] + [-1]x₂ + [-1]x₁   
POL(3588_0_imprimer_GE(x₁, x₂, x₃, x₄)) = [-1] + [-1]x₄ + x₂ + [-1]x₁   
POL(java.lang.Object(x₁)) = x₁   
POL(Carre(x₁)) = x₁   
POL(COND_3588_1_MAIN_INVOKEMETHOD(x₁, x₂, x₃)) = [-1] + [-1]x₃ + [-1]x₂   
POL(&&(x₁, x₂)) = [-1]   
POL(>=(x₁, x₂)) = [-1]   
POL(0) = 0   
POL(>(x₁, x₂)) = [-1]   
POL(+(x₁, x₂)) = x₁ + x₂   
POL(1) = [1]   
POL(COND_3588_1_MAIN_INVOKEMETHOD1(x₁, x₂, x₃)) = [-1] + [-1]x₃ + [-1]x₂   
POL(<(x₁, x₂)) = [-1]   

The following pairs are in P_>:

COND_3588_1_MAIN_INVOKEMETHOD(TRUE, 3588_0_imprimer_GE(java.lang.Object(Carre(x0[1])), x1[1], x2[1], x0[1]), java.lang.Object(Carre(x0[1]))) → 3588_1_MAIN_INVOKEMETHOD(3588_0_imprimer_GE(java.lang.Object(Carre(x0[1])), +(x1[1], 1), 0, x0[1]), java.lang.Object(Carre(x0[1])))

The following pairs are in P_bound:

3588_1_MAIN_INVOKEMETHOD(3588_0_imprimer_GE(java.lang.Object(Carre(x0[0])), x1[0], x2[0], x0[0]), java.lang.Object(Carre(x0[0]))) → COND_3588_1_MAIN_INVOKEMETHOD(&&(&&(>=(x2[0], x0[0]), >=(x1[0], 0)), >(x0[0], +(x1[0], 1))), 3588_0_imprimer_GE(java.lang.Object(Carre(x0[0])), x1[0], x2[0], x0[0]), java.lang.Object(Carre(x0[0])))

The following pairs are in P_≥:

3588_1_MAIN_INVOKEMETHOD(3588_0_imprimer_GE(java.lang.Object(Carre(x0[0])), x1[0], x2[0], x0[0]), java.lang.Object(Carre(x0[0]))) → COND_3588_1_MAIN_INVOKEMETHOD(&&(&&(>=(x2[0], x0[0]), >=(x1[0], 0)), >(x0[0], +(x1[0], 1))), 3588_0_imprimer_GE(java.lang.Object(Carre(x0[0])), x1[0], x2[0], x0[0]), java.lang.Object(Carre(x0[0])))
3588_1_MAIN_INVOKEMETHOD(3588_0_imprimer_GE(java.lang.Object(Carre(x0[2])), x1[2], x2[2], x0[2]), java.lang.Object(Carre(x0[2]))) → COND_3588_1_MAIN_INVOKEMETHOD1(&&(>=(x2[2], 0), <(x2[2], x0[2])), 3588_0_imprimer_GE(java.lang.Object(Carre(x0[2])), x1[2], x2[2], x0[2]), java.lang.Object(Carre(x0[2])))
COND_3588_1_MAIN_INVOKEMETHOD1(TRUE, 3588_0_imprimer_GE(java.lang.Object(Carre(x0[3])), x1[3], x2[3], x0[3]), java.lang.Object(Carre(x0[3]))) → 3588_1_MAIN_INVOKEMETHOD(3588_0_imprimer_GE(java.lang.Object(Carre(x0[3])), x1[3], +(x2[3], 1), x0[3]), java.lang.Object(Carre(x0[3])))

There are no usable rules.

(7) Complex Obligation (AND)

(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): 3588_1_MAIN_INVOKEMETHOD(3588_0_imprimer_GE(java.lang.Object(Carre(x0[0])), x1[0], x2[0], x0[0]), java.lang.Object(Carre(x0[0]))) → COND_3588_1_MAIN_INVOKEMETHOD(x2[0] >= x0[0] && x1[0] >= 0 && x0[0] > x1[0] + 1, 3588_0_imprimer_GE(java.lang.Object(Carre(x0[0])), x1[0], x2[0], x0[0]), java.lang.Object(Carre(x0[0])))
(2): 3588_1_MAIN_INVOKEMETHOD(3588_0_imprimer_GE(java.lang.Object(Carre(x0[2])), x1[2], x2[2], x0[2]), java.lang.Object(Carre(x0[2]))) → COND_3588_1_MAIN_INVOKEMETHOD1(x2[2] >= 0 && x2[2] < x0[2], 3588_0_imprimer_GE(java.lang.Object(Carre(x0[2])), x1[2], x2[2], x0[2]), java.lang.Object(Carre(x0[2])))
(3): COND_3588_1_MAIN_INVOKEMETHOD1(TRUE, 3588_0_imprimer_GE(java.lang.Object(Carre(x0[3])), x1[3], x2[3], x0[3]), java.lang.Object(Carre(x0[3]))) → 3588_1_MAIN_INVOKEMETHOD(3588_0_imprimer_GE(java.lang.Object(Carre(x0[3])), x1[3], x2[3] + 1, x0[3]), java.lang.Object(Carre(x0[3])))

(3) -> (0), if ((3588_0_imprimer_GE(java.lang.Object(Carre(x0[3])), x1[3], x2[3] + 1, x0[3]) →^* 3588_0_imprimer_GE(java.lang.Object(Carre(x0[0])), x1[0], x2[0], x0[0]))∧(java.lang.Object(Carre(x0[3])) →^* java.lang.Object(Carre(x0[0]))))

(3) -> (2), if ((3588_0_imprimer_GE(java.lang.Object(Carre(x0[3])), x1[3], x2[3] + 1, x0[3]) →^* 3588_0_imprimer_GE(java.lang.Object(Carre(x0[2])), x1[2], x2[2], x0[2]))∧(java.lang.Object(Carre(x0[3])) →^* java.lang.Object(Carre(x0[2]))))

(2) -> (3), if ((x2[2] >= 0 && x2[2] < x0[2] →^* TRUE)∧(3588_0_imprimer_GE(java.lang.Object(Carre(x0[2])), x1[2], x2[2], x0[2]) →^* 3588_0_imprimer_GE(java.lang.Object(Carre(x0[3])), x1[3], x2[3], x0[3]))∧(java.lang.Object(Carre(x0[2])) →^* java.lang.Object(Carre(x0[3]))))

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:
(3): COND_3588_1_MAIN_INVOKEMETHOD1(TRUE, 3588_0_imprimer_GE(java.lang.Object(Carre(x0[3])), x1[3], x2[3], x0[3]), java.lang.Object(Carre(x0[3]))) → 3588_1_MAIN_INVOKEMETHOD(3588_0_imprimer_GE(java.lang.Object(Carre(x0[3])), x1[3], x2[3] + 1, x0[3]), java.lang.Object(Carre(x0[3])))
(2): 3588_1_MAIN_INVOKEMETHOD(3588_0_imprimer_GE(java.lang.Object(Carre(x0[2])), x1[2], x2[2], x0[2]), java.lang.Object(Carre(x0[2]))) → COND_3588_1_MAIN_INVOKEMETHOD1(x2[2] >= 0 && x2[2] < x0[2], 3588_0_imprimer_GE(java.lang.Object(Carre(x0[2])), x1[2], x2[2], x0[2]), java.lang.Object(Carre(x0[2])))

(3) -> (2), if ((3588_0_imprimer_GE(java.lang.Object(Carre(x0[3])), x1[3], x2[3] + 1, x0[3]) →^* 3588_0_imprimer_GE(java.lang.Object(Carre(x0[2])), x1[2], x2[2], x0[2]))∧(java.lang.Object(Carre(x0[3])) →^* java.lang.Object(Carre(x0[2]))))

(2) -> (3), if ((x2[2] >= 0 && x2[2] < x0[2] →^* TRUE)∧(3588_0_imprimer_GE(java.lang.Object(Carre(x0[2])), x1[2], x2[2], x0[2]) →^* 3588_0_imprimer_GE(java.lang.Object(Carre(x0[3])), x1[3], x2[3], x0[3]))∧(java.lang.Object(Carre(x0[2])) →^* java.lang.Object(Carre(x0[3]))))

The set Q is empty.

(11) IDPNonInfProof (SOUND transformation)

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_3588_1_MAIN_INVOKEMETHOD1(TRUE, 3588_0_imprimer_GE(java.lang.Object(Carre(x0[3])), x1[3], x2[3], x0[3]), java.lang.Object(Carre(x0[3]))) → 3588_1_MAIN_INVOKEMETHOD(3588_0_imprimer_GE(java.lang.Object(Carre(x0[3])), x1[3], +(x2[3], 1), x0[3]), java.lang.Object(Carre(x0[3]))) the following chains were created:

• We consider the chain COND_3588_1_MAIN_INVOKEMETHOD1(TRUE, 3588_0_imprimer_GE(java.lang.Object(Carre(x0[3])), x1[3], x2[3], x0[3]), java.lang.Object(Carre(x0[3]))) → 3588_1_MAIN_INVOKEMETHOD(3588_0_imprimer_GE(java.lang.Object(Carre(x0[3])), x1[3], +(x2[3], 1), x0[3]), java.lang.Object(Carre(x0[3]))) which results in the following constraint:
  

  (1)    (COND_3588_1_MAIN_INVOKEMETHOD1(TRUE, 3588_0_imprimer_GE(java.lang.Object(Carre(x0[3])), x1[3], x2[3], x0[3]), java.lang.Object(Carre(x0[3])))≥NonInfC∧COND_3588_1_MAIN_INVOKEMETHOD1(TRUE, 3588_0_imprimer_GE(java.lang.Object(Carre(x0[3])), x1[3], x2[3], x0[3]), java.lang.Object(Carre(x0[3])))≥3588_1_MAIN_INVOKEMETHOD(3588_0_imprimer_GE(java.lang.Object(Carre(x0[3])), x1[3], +(x2[3], 1), x0[3]), java.lang.Object(Carre(x0[3])))∧(U^Increasing(3588_1_MAIN_INVOKEMETHOD(3588_0_imprimer_GE(java.lang.Object(Carre(x0[3])), x1[3], +(x2[3], 1), x0[3]), java.lang.Object(Carre(x0[3])))), ≥))

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

  (2)    ((U^Increasing(3588_1_MAIN_INVOKEMETHOD(3588_0_imprimer_GE(java.lang.Object(Carre(x0[3])), x1[3], +(x2[3], 1), x0[3]), java.lang.Object(Carre(x0[3])))), ≥)∧[1 + (-1)bso_12] ≥ 0)

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

  (3)    ((U^Increasing(3588_1_MAIN_INVOKEMETHOD(3588_0_imprimer_GE(java.lang.Object(Carre(x0[3])), x1[3], +(x2[3], 1), x0[3]), java.lang.Object(Carre(x0[3])))), ≥)∧[1 + (-1)bso_12] ≥ 0)

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

  (4)    ((U^Increasing(3588_1_MAIN_INVOKEMETHOD(3588_0_imprimer_GE(java.lang.Object(Carre(x0[3])), x1[3], +(x2[3], 1), x0[3]), java.lang.Object(Carre(x0[3])))), ≥)∧[1 + (-1)bso_12] ≥ 0)

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

  (5)    ((U^Increasing(3588_1_MAIN_INVOKEMETHOD(3588_0_imprimer_GE(java.lang.Object(Carre(x0[3])), x1[3], +(x2[3], 1), x0[3]), java.lang.Object(Carre(x0[3])))), ≥)∧0 = 0∧0 = 0∧0 = 0∧[1 + (-1)bso_12] ≥ 0)

  
  
  

For Pair 3588_1_MAIN_INVOKEMETHOD(3588_0_imprimer_GE(java.lang.Object(Carre(x0[2])), x1[2], x2[2], x0[2]), java.lang.Object(Carre(x0[2]))) → COND_3588_1_MAIN_INVOKEMETHOD1(&&(>=(x2[2], 0), <(x2[2], x0[2])), 3588_0_imprimer_GE(java.lang.Object(Carre(x0[2])), x1[2], x2[2], x0[2]), java.lang.Object(Carre(x0[2]))) the following chains were created:

• We consider the chain 3588_1_MAIN_INVOKEMETHOD(3588_0_imprimer_GE(java.lang.Object(Carre(x0[2])), x1[2], x2[2], x0[2]), java.lang.Object(Carre(x0[2]))) → COND_3588_1_MAIN_INVOKEMETHOD1(&&(>=(x2[2], 0), <(x2[2], x0[2])), 3588_0_imprimer_GE(java.lang.Object(Carre(x0[2])), x1[2], x2[2], x0[2]), java.lang.Object(Carre(x0[2]))), COND_3588_1_MAIN_INVOKEMETHOD1(TRUE, 3588_0_imprimer_GE(java.lang.Object(Carre(x0[3])), x1[3], x2[3], x0[3]), java.lang.Object(Carre(x0[3]))) → 3588_1_MAIN_INVOKEMETHOD(3588_0_imprimer_GE(java.lang.Object(Carre(x0[3])), x1[3], +(x2[3], 1), x0[3]), java.lang.Object(Carre(x0[3]))) which results in the following constraint:
  

  (6)    (&&(>=(x2[2], 0), <(x2[2], x0[2]))=TRUE∧3588_0_imprimer_GE(java.lang.Object(Carre(x0[2])), x1[2], x2[2], x0[2])=3588_0_imprimer_GE(java.lang.Object(Carre(x0[3])), x1[3], x2[3], x0[3])∧java.lang.Object(Carre(x0[2]))=java.lang.Object(Carre(x0[3])) ⇒ 3588_1_MAIN_INVOKEMETHOD(3588_0_imprimer_GE(java.lang.Object(Carre(x0[2])), x1[2], x2[2], x0[2]), java.lang.Object(Carre(x0[2])))≥NonInfC∧3588_1_MAIN_INVOKEMETHOD(3588_0_imprimer_GE(java.lang.Object(Carre(x0[2])), x1[2], x2[2], x0[2]), java.lang.Object(Carre(x0[2])))≥COND_3588_1_MAIN_INVOKEMETHOD1(&&(>=(x2[2], 0), <(x2[2], x0[2])), 3588_0_imprimer_GE(java.lang.Object(Carre(x0[2])), x1[2], x2[2], x0[2]), java.lang.Object(Carre(x0[2])))∧(U^Increasing(COND_3588_1_MAIN_INVOKEMETHOD1(&&(>=(x2[2], 0), <(x2[2], x0[2])), 3588_0_imprimer_GE(java.lang.Object(Carre(x0[2])), x1[2], x2[2], x0[2]), java.lang.Object(Carre(x0[2])))), ≥))

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

  (7)    (>=(x2[2], 0)=TRUE∧<(x2[2], x0[2])=TRUE ⇒ 3588_1_MAIN_INVOKEMETHOD(3588_0_imprimer_GE(java.lang.Object(Carre(x0[2])), x1[2], x2[2], x0[2]), java.lang.Object(Carre(x0[2])))≥NonInfC∧3588_1_MAIN_INVOKEMETHOD(3588_0_imprimer_GE(java.lang.Object(Carre(x0[2])), x1[2], x2[2], x0[2]), java.lang.Object(Carre(x0[2])))≥COND_3588_1_MAIN_INVOKEMETHOD1(&&(>=(x2[2], 0), <(x2[2], x0[2])), 3588_0_imprimer_GE(java.lang.Object(Carre(x0[2])), x1[2], x2[2], x0[2]), java.lang.Object(Carre(x0[2])))∧(U^Increasing(COND_3588_1_MAIN_INVOKEMETHOD1(&&(>=(x2[2], 0), <(x2[2], x0[2])), 3588_0_imprimer_GE(java.lang.Object(Carre(x0[2])), x1[2], x2[2], x0[2]), java.lang.Object(Carre(x0[2])))), ≥))

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

  (8)    (x2[2] ≥ 0∧x0[2] + [-1] + [-1]x2[2] ≥ 0 ⇒ (U^Increasing(COND_3588_1_MAIN_INVOKEMETHOD1(&&(>=(x2[2], 0), <(x2[2], x0[2])), 3588_0_imprimer_GE(java.lang.Object(Carre(x0[2])), x1[2], x2[2], x0[2]), java.lang.Object(Carre(x0[2])))), ≥)∧[(3)bni_13 + (-1)Bound*bni_13] + [bni_13]x0[2] + [(-1)bni_13]x2[2] ≥ 0∧[(-1)bso_14] ≥ 0)

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

  (9)    (x2[2] ≥ 0∧x0[2] + [-1] + [-1]x2[2] ≥ 0 ⇒ (U^Increasing(COND_3588_1_MAIN_INVOKEMETHOD1(&&(>=(x2[2], 0), <(x2[2], x0[2])), 3588_0_imprimer_GE(java.lang.Object(Carre(x0[2])), x1[2], x2[2], x0[2]), java.lang.Object(Carre(x0[2])))), ≥)∧[(3)bni_13 + (-1)Bound*bni_13] + [bni_13]x0[2] + [(-1)bni_13]x2[2] ≥ 0∧[(-1)bso_14] ≥ 0)

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

  (10)    (x2[2] ≥ 0∧x0[2] + [-1] + [-1]x2[2] ≥ 0 ⇒ (U^Increasing(COND_3588_1_MAIN_INVOKEMETHOD1(&&(>=(x2[2], 0), <(x2[2], x0[2])), 3588_0_imprimer_GE(java.lang.Object(Carre(x0[2])), x1[2], x2[2], x0[2]), java.lang.Object(Carre(x0[2])))), ≥)∧[(3)bni_13 + (-1)Bound*bni_13] + [bni_13]x0[2] + [(-1)bni_13]x2[2] ≥ 0∧[(-1)bso_14] ≥ 0)

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

  (11)    (x2[2] ≥ 0∧x0[2] + [-1] + [-1]x2[2] ≥ 0 ⇒ (U^Increasing(COND_3588_1_MAIN_INVOKEMETHOD1(&&(>=(x2[2], 0), <(x2[2], x0[2])), 3588_0_imprimer_GE(java.lang.Object(Carre(x0[2])), x1[2], x2[2], x0[2]), java.lang.Object(Carre(x0[2])))), ≥)∧0 = 0∧[(3)bni_13 + (-1)Bound*bni_13] + [bni_13]x0[2] + [(-1)bni_13]x2[2] ≥ 0∧0 = 0∧[(-1)bso_14] ≥ 0)

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

  (12)    (x2[2] ≥ 0∧x0[2] ≥ 0 ⇒ (U^Increasing(COND_3588_1_MAIN_INVOKEMETHOD1(&&(>=(x2[2], 0), <(x2[2], x0[2])), 3588_0_imprimer_GE(java.lang.Object(Carre(x0[2])), x1[2], x2[2], x0[2]), java.lang.Object(Carre(x0[2])))), ≥)∧0 = 0∧[(4)bni_13 + (-1)Bound*bni_13] + [bni_13]x0[2] ≥ 0∧0 = 0∧[(-1)bso_14] ≥ 0)

  
  
  

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

• COND_3588_1_MAIN_INVOKEMETHOD1(TRUE, 3588_0_imprimer_GE(java.lang.Object(Carre(x0[3])), x1[3], x2[3], x0[3]), java.lang.Object(Carre(x0[3]))) → 3588_1_MAIN_INVOKEMETHOD(3588_0_imprimer_GE(java.lang.Object(Carre(x0[3])), x1[3], +(x2[3], 1), x0[3]), java.lang.Object(Carre(x0[3])))
  
  • ((U^Increasing(3588_1_MAIN_INVOKEMETHOD(3588_0_imprimer_GE(java.lang.Object(Carre(x0[3])), x1[3], +(x2[3], 1), x0[3]), java.lang.Object(Carre(x0[3])))), ≥)∧0 = 0∧0 = 0∧0 = 0∧[1 + (-1)bso_12] ≥ 0)
    
  
  
• 3588_1_MAIN_INVOKEMETHOD(3588_0_imprimer_GE(java.lang.Object(Carre(x0[2])), x1[2], x2[2], x0[2]), java.lang.Object(Carre(x0[2]))) → COND_3588_1_MAIN_INVOKEMETHOD1(&&(>=(x2[2], 0), <(x2[2], x0[2])), 3588_0_imprimer_GE(java.lang.Object(Carre(x0[2])), x1[2], x2[2], x0[2]), java.lang.Object(Carre(x0[2])))
  
  • (x2[2] ≥ 0∧x0[2] ≥ 0 ⇒ (U^Increasing(COND_3588_1_MAIN_INVOKEMETHOD1(&&(>=(x2[2], 0), <(x2[2], x0[2])), 3588_0_imprimer_GE(java.lang.Object(Carre(x0[2])), x1[2], x2[2], x0[2]), java.lang.Object(Carre(x0[2])))), ≥)∧0 = 0∧[(4)bni_13 + (-1)Bound*bni_13] + [bni_13]x0[2] ≥ 0∧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_3588_1_MAIN_INVOKEMETHOD1(x₁, x₂, x₃)) = [2] + [2]x₃ + [-1]x₂   
POL(3588_0_imprimer_GE(x₁, x₂, x₃, x₄)) = [-1] + [2]x₄ + x₃ + [-1]x₁   
POL(java.lang.Object(x₁)) = x₁   
POL(Carre(x₁)) = x₁   
POL(3588_1_MAIN_INVOKEMETHOD(x₁, x₂)) = [2] + [-1]x₁ + [2]x₂   
POL(+(x₁, x₂)) = x₁ + x₂   
POL(1) = [1]   
POL(&&(x₁, x₂)) = [1]   
POL(>=(x₁, x₂)) = 0   
POL(0) = 0   
POL(<(x₁, x₂)) = [-1]   

The following pairs are in P_>:

COND_3588_1_MAIN_INVOKEMETHOD1(TRUE, 3588_0_imprimer_GE(java.lang.Object(Carre(x0[3])), x1[3], x2[3], x0[3]), java.lang.Object(Carre(x0[3]))) → 3588_1_MAIN_INVOKEMETHOD(3588_0_imprimer_GE(java.lang.Object(Carre(x0[3])), x1[3], +(x2[3], 1), x0[3]), java.lang.Object(Carre(x0[3])))

The following pairs are in P_bound:

3588_1_MAIN_INVOKEMETHOD(3588_0_imprimer_GE(java.lang.Object(Carre(x0[2])), x1[2], x2[2], x0[2]), java.lang.Object(Carre(x0[2]))) → COND_3588_1_MAIN_INVOKEMETHOD1(&&(>=(x2[2], 0), <(x2[2], x0[2])), 3588_0_imprimer_GE(java.lang.Object(Carre(x0[2])), x1[2], x2[2], x0[2]), java.lang.Object(Carre(x0[2])))

The following pairs are in P_≥:

3588_1_MAIN_INVOKEMETHOD(3588_0_imprimer_GE(java.lang.Object(Carre(x0[2])), x1[2], x2[2], x0[2]), java.lang.Object(Carre(x0[2]))) → COND_3588_1_MAIN_INVOKEMETHOD1(&&(>=(x2[2], 0), <(x2[2], x0[2])), 3588_0_imprimer_GE(java.lang.Object(Carre(x0[2])), x1[2], x2[2], x0[2]), java.lang.Object(Carre(x0[2])))

There are no usable rules.

(12) Complex Obligation (AND)

(13) Obligation:

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

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

The following domains are used:

Boolean, Integer

R is empty.

The integer pair graph contains the following rules and edges:
(2): 3588_1_MAIN_INVOKEMETHOD(3588_0_imprimer_GE(java.lang.Object(Carre(x0[2])), x1[2], x2[2], x0[2]), java.lang.Object(Carre(x0[2]))) → COND_3588_1_MAIN_INVOKEMETHOD1(x2[2] >= 0 && x2[2] < x0[2], 3588_0_imprimer_GE(java.lang.Object(Carre(x0[2])), x1[2], x2[2], x0[2]), java.lang.Object(Carre(x0[2])))


The set Q is empty.

(14) IDependencyGraphProof (EQUIVALENT transformation)

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

(15) TRUE

(16) 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:
(3): COND_3588_1_MAIN_INVOKEMETHOD1(TRUE, 3588_0_imprimer_GE(java.lang.Object(Carre(x0[3])), x1[3], x2[3], x0[3]), java.lang.Object(Carre(x0[3]))) → 3588_1_MAIN_INVOKEMETHOD(3588_0_imprimer_GE(java.lang.Object(Carre(x0[3])), x1[3], x2[3] + 1, x0[3]), java.lang.Object(Carre(x0[3])))


The set Q is empty.

(17) IDependencyGraphProof (EQUIVALENT transformation)

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

(18) TRUE

(19) 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_3588_1_MAIN_INVOKEMETHOD(TRUE, 3588_0_imprimer_GE(java.lang.Object(Carre(x0[1])), x1[1], x2[1], x0[1]), java.lang.Object(Carre(x0[1]))) → 3588_1_MAIN_INVOKEMETHOD(3588_0_imprimer_GE(java.lang.Object(Carre(x0[1])), x1[1] + 1, 0, x0[1]), java.lang.Object(Carre(x0[1])))
(2): 3588_1_MAIN_INVOKEMETHOD(3588_0_imprimer_GE(java.lang.Object(Carre(x0[2])), x1[2], x2[2], x0[2]), java.lang.Object(Carre(x0[2]))) → COND_3588_1_MAIN_INVOKEMETHOD1(x2[2] >= 0 && x2[2] < x0[2], 3588_0_imprimer_GE(java.lang.Object(Carre(x0[2])), x1[2], x2[2], x0[2]), java.lang.Object(Carre(x0[2])))
(3): COND_3588_1_MAIN_INVOKEMETHOD1(TRUE, 3588_0_imprimer_GE(java.lang.Object(Carre(x0[3])), x1[3], x2[3], x0[3]), java.lang.Object(Carre(x0[3]))) → 3588_1_MAIN_INVOKEMETHOD(3588_0_imprimer_GE(java.lang.Object(Carre(x0[3])), x1[3], x2[3] + 1, x0[3]), java.lang.Object(Carre(x0[3])))

(1) -> (2), if ((3588_0_imprimer_GE(java.lang.Object(Carre(x0[1])), x1[1] + 1, 0, x0[1]) →^* 3588_0_imprimer_GE(java.lang.Object(Carre(x0[2])), x1[2], x2[2], x0[2]))∧(java.lang.Object(Carre(x0[1])) →^* java.lang.Object(Carre(x0[2]))))

(3) -> (2), if ((3588_0_imprimer_GE(java.lang.Object(Carre(x0[3])), x1[3], x2[3] + 1, x0[3]) →^* 3588_0_imprimer_GE(java.lang.Object(Carre(x0[2])), x1[2], x2[2], x0[2]))∧(java.lang.Object(Carre(x0[3])) →^* java.lang.Object(Carre(x0[2]))))

(2) -> (3), if ((x2[2] >= 0 && x2[2] < x0[2] →^* TRUE)∧(3588_0_imprimer_GE(java.lang.Object(Carre(x0[2])), x1[2], x2[2], x0[2]) →^* 3588_0_imprimer_GE(java.lang.Object(Carre(x0[3])), x1[3], x2[3], x0[3]))∧(java.lang.Object(Carre(x0[2])) →^* java.lang.Object(Carre(x0[3]))))

The set Q is empty.

(20) IDependencyGraphProof (EQUIVALENT transformation)

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

(21) 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:
(3): COND_3588_1_MAIN_INVOKEMETHOD1(TRUE, 3588_0_imprimer_GE(java.lang.Object(Carre(x0[3])), x1[3], x2[3], x0[3]), java.lang.Object(Carre(x0[3]))) → 3588_1_MAIN_INVOKEMETHOD(3588_0_imprimer_GE(java.lang.Object(Carre(x0[3])), x1[3], x2[3] + 1, x0[3]), java.lang.Object(Carre(x0[3])))
(2): 3588_1_MAIN_INVOKEMETHOD(3588_0_imprimer_GE(java.lang.Object(Carre(x0[2])), x1[2], x2[2], x0[2]), java.lang.Object(Carre(x0[2]))) → COND_3588_1_MAIN_INVOKEMETHOD1(x2[2] >= 0 && x2[2] < x0[2], 3588_0_imprimer_GE(java.lang.Object(Carre(x0[2])), x1[2], x2[2], x0[2]), java.lang.Object(Carre(x0[2])))

(3) -> (2), if ((3588_0_imprimer_GE(java.lang.Object(Carre(x0[3])), x1[3], x2[3] + 1, x0[3]) →^* 3588_0_imprimer_GE(java.lang.Object(Carre(x0[2])), x1[2], x2[2], x0[2]))∧(java.lang.Object(Carre(x0[3])) →^* java.lang.Object(Carre(x0[2]))))

(2) -> (3), if ((x2[2] >= 0 && x2[2] < x0[2] →^* TRUE)∧(3588_0_imprimer_GE(java.lang.Object(Carre(x0[2])), x1[2], x2[2], x0[2]) →^* 3588_0_imprimer_GE(java.lang.Object(Carre(x0[3])), x1[3], x2[3], x0[3]))∧(java.lang.Object(Carre(x0[2])) →^* java.lang.Object(Carre(x0[3]))))

The set Q is empty.

(22) IDPNonInfProof (SOUND transformation)

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_3588_1_MAIN_INVOKEMETHOD1(TRUE, 3588_0_imprimer_GE(java.lang.Object(Carre(x0[3])), x1[3], x2[3], x0[3]), java.lang.Object(Carre(x0[3]))) → 3588_1_MAIN_INVOKEMETHOD(3588_0_imprimer_GE(java.lang.Object(Carre(x0[3])), x1[3], +(x2[3], 1), x0[3]), java.lang.Object(Carre(x0[3]))) the following chains were created:

• We consider the chain COND_3588_1_MAIN_INVOKEMETHOD1(TRUE, 3588_0_imprimer_GE(java.lang.Object(Carre(x0[3])), x1[3], x2[3], x0[3]), java.lang.Object(Carre(x0[3]))) → 3588_1_MAIN_INVOKEMETHOD(3588_0_imprimer_GE(java.lang.Object(Carre(x0[3])), x1[3], +(x2[3], 1), x0[3]), java.lang.Object(Carre(x0[3]))) which results in the following constraint:
  

  (1)    (COND_3588_1_MAIN_INVOKEMETHOD1(TRUE, 3588_0_imprimer_GE(java.lang.Object(Carre(x0[3])), x1[3], x2[3], x0[3]), java.lang.Object(Carre(x0[3])))≥NonInfC∧COND_3588_1_MAIN_INVOKEMETHOD1(TRUE, 3588_0_imprimer_GE(java.lang.Object(Carre(x0[3])), x1[3], x2[3], x0[3]), java.lang.Object(Carre(x0[3])))≥3588_1_MAIN_INVOKEMETHOD(3588_0_imprimer_GE(java.lang.Object(Carre(x0[3])), x1[3], +(x2[3], 1), x0[3]), java.lang.Object(Carre(x0[3])))∧(U^Increasing(3588_1_MAIN_INVOKEMETHOD(3588_0_imprimer_GE(java.lang.Object(Carre(x0[3])), x1[3], +(x2[3], 1), x0[3]), java.lang.Object(Carre(x0[3])))), ≥))

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

  (2)    ((U^Increasing(3588_1_MAIN_INVOKEMETHOD(3588_0_imprimer_GE(java.lang.Object(Carre(x0[3])), x1[3], +(x2[3], 1), x0[3]), java.lang.Object(Carre(x0[3])))), ≥)∧[(-1)bso_10] ≥ 0)

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

  (3)    ((U^Increasing(3588_1_MAIN_INVOKEMETHOD(3588_0_imprimer_GE(java.lang.Object(Carre(x0[3])), x1[3], +(x2[3], 1), x0[3]), java.lang.Object(Carre(x0[3])))), ≥)∧[(-1)bso_10] ≥ 0)

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

  (4)    ((U^Increasing(3588_1_MAIN_INVOKEMETHOD(3588_0_imprimer_GE(java.lang.Object(Carre(x0[3])), x1[3], +(x2[3], 1), x0[3]), java.lang.Object(Carre(x0[3])))), ≥)∧[(-1)bso_10] ≥ 0)

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

  (5)    ((U^Increasing(3588_1_MAIN_INVOKEMETHOD(3588_0_imprimer_GE(java.lang.Object(Carre(x0[3])), x1[3], +(x2[3], 1), x0[3]), java.lang.Object(Carre(x0[3])))), ≥)∧0 = 0∧0 = 0∧0 = 0∧[(-1)bso_10] ≥ 0)

  
  
  

For Pair 3588_1_MAIN_INVOKEMETHOD(3588_0_imprimer_GE(java.lang.Object(Carre(x0[2])), x1[2], x2[2], x0[2]), java.lang.Object(Carre(x0[2]))) → COND_3588_1_MAIN_INVOKEMETHOD1(&&(>=(x2[2], 0), <(x2[2], x0[2])), 3588_0_imprimer_GE(java.lang.Object(Carre(x0[2])), x1[2], x2[2], x0[2]), java.lang.Object(Carre(x0[2]))) the following chains were created:

• We consider the chain 3588_1_MAIN_INVOKEMETHOD(3588_0_imprimer_GE(java.lang.Object(Carre(x0[2])), x1[2], x2[2], x0[2]), java.lang.Object(Carre(x0[2]))) → COND_3588_1_MAIN_INVOKEMETHOD1(&&(>=(x2[2], 0), <(x2[2], x0[2])), 3588_0_imprimer_GE(java.lang.Object(Carre(x0[2])), x1[2], x2[2], x0[2]), java.lang.Object(Carre(x0[2]))), COND_3588_1_MAIN_INVOKEMETHOD1(TRUE, 3588_0_imprimer_GE(java.lang.Object(Carre(x0[3])), x1[3], x2[3], x0[3]), java.lang.Object(Carre(x0[3]))) → 3588_1_MAIN_INVOKEMETHOD(3588_0_imprimer_GE(java.lang.Object(Carre(x0[3])), x1[3], +(x2[3], 1), x0[3]), java.lang.Object(Carre(x0[3]))) which results in the following constraint:
  

  (6)    (&&(>=(x2[2], 0), <(x2[2], x0[2]))=TRUE∧3588_0_imprimer_GE(java.lang.Object(Carre(x0[2])), x1[2], x2[2], x0[2])=3588_0_imprimer_GE(java.lang.Object(Carre(x0[3])), x1[3], x2[3], x0[3])∧java.lang.Object(Carre(x0[2]))=java.lang.Object(Carre(x0[3])) ⇒ 3588_1_MAIN_INVOKEMETHOD(3588_0_imprimer_GE(java.lang.Object(Carre(x0[2])), x1[2], x2[2], x0[2]), java.lang.Object(Carre(x0[2])))≥NonInfC∧3588_1_MAIN_INVOKEMETHOD(3588_0_imprimer_GE(java.lang.Object(Carre(x0[2])), x1[2], x2[2], x0[2]), java.lang.Object(Carre(x0[2])))≥COND_3588_1_MAIN_INVOKEMETHOD1(&&(>=(x2[2], 0), <(x2[2], x0[2])), 3588_0_imprimer_GE(java.lang.Object(Carre(x0[2])), x1[2], x2[2], x0[2]), java.lang.Object(Carre(x0[2])))∧(U^Increasing(COND_3588_1_MAIN_INVOKEMETHOD1(&&(>=(x2[2], 0), <(x2[2], x0[2])), 3588_0_imprimer_GE(java.lang.Object(Carre(x0[2])), x1[2], x2[2], x0[2]), java.lang.Object(Carre(x0[2])))), ≥))

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

  (7)    (>=(x2[2], 0)=TRUE∧<(x2[2], x0[2])=TRUE ⇒ 3588_1_MAIN_INVOKEMETHOD(3588_0_imprimer_GE(java.lang.Object(Carre(x0[2])), x1[2], x2[2], x0[2]), java.lang.Object(Carre(x0[2])))≥NonInfC∧3588_1_MAIN_INVOKEMETHOD(3588_0_imprimer_GE(java.lang.Object(Carre(x0[2])), x1[2], x2[2], x0[2]), java.lang.Object(Carre(x0[2])))≥COND_3588_1_MAIN_INVOKEMETHOD1(&&(>=(x2[2], 0), <(x2[2], x0[2])), 3588_0_imprimer_GE(java.lang.Object(Carre(x0[2])), x1[2], x2[2], x0[2]), java.lang.Object(Carre(x0[2])))∧(U^Increasing(COND_3588_1_MAIN_INVOKEMETHOD1(&&(>=(x2[2], 0), <(x2[2], x0[2])), 3588_0_imprimer_GE(java.lang.Object(Carre(x0[2])), x1[2], x2[2], x0[2]), java.lang.Object(Carre(x0[2])))), ≥))

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

  (8)    (x2[2] ≥ 0∧x0[2] + [-1] + [-1]x2[2] ≥ 0 ⇒ (U^Increasing(COND_3588_1_MAIN_INVOKEMETHOD1(&&(>=(x2[2], 0), <(x2[2], x0[2])), 3588_0_imprimer_GE(java.lang.Object(Carre(x0[2])), x1[2], x2[2], x0[2]), java.lang.Object(Carre(x0[2])))), ≥)∧[(3)bni_11 + (-1)Bound*bni_11] + [(2)bni_11]x0[2] + [(-1)bni_11]x2[2] ≥ 0∧[1 + (-1)bso_12] ≥ 0)

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

  (9)    (x2[2] ≥ 0∧x0[2] + [-1] + [-1]x2[2] ≥ 0 ⇒ (U^Increasing(COND_3588_1_MAIN_INVOKEMETHOD1(&&(>=(x2[2], 0), <(x2[2], x0[2])), 3588_0_imprimer_GE(java.lang.Object(Carre(x0[2])), x1[2], x2[2], x0[2]), java.lang.Object(Carre(x0[2])))), ≥)∧[(3)bni_11 + (-1)Bound*bni_11] + [(2)bni_11]x0[2] + [(-1)bni_11]x2[2] ≥ 0∧[1 + (-1)bso_12] ≥ 0)

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

  (10)    (x2[2] ≥ 0∧x0[2] + [-1] + [-1]x2[2] ≥ 0 ⇒ (U^Increasing(COND_3588_1_MAIN_INVOKEMETHOD1(&&(>=(x2[2], 0), <(x2[2], x0[2])), 3588_0_imprimer_GE(java.lang.Object(Carre(x0[2])), x1[2], x2[2], x0[2]), java.lang.Object(Carre(x0[2])))), ≥)∧[(3)bni_11 + (-1)Bound*bni_11] + [(2)bni_11]x0[2] + [(-1)bni_11]x2[2] ≥ 0∧[1 + (-1)bso_12] ≥ 0)

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

  (11)    (x2[2] ≥ 0∧x0[2] + [-1] + [-1]x2[2] ≥ 0 ⇒ (U^Increasing(COND_3588_1_MAIN_INVOKEMETHOD1(&&(>=(x2[2], 0), <(x2[2], x0[2])), 3588_0_imprimer_GE(java.lang.Object(Carre(x0[2])), x1[2], x2[2], x0[2]), java.lang.Object(Carre(x0[2])))), ≥)∧0 = 0∧[(3)bni_11 + (-1)Bound*bni_11] + [(2)bni_11]x0[2] + [(-1)bni_11]x2[2] ≥ 0∧0 = 0∧[1 + (-1)bso_12] ≥ 0)

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

  (12)    (x2[2] ≥ 0∧x0[2] ≥ 0 ⇒ (U^Increasing(COND_3588_1_MAIN_INVOKEMETHOD1(&&(>=(x2[2], 0), <(x2[2], x0[2])), 3588_0_imprimer_GE(java.lang.Object(Carre(x0[2])), x1[2], x2[2], x0[2]), java.lang.Object(Carre(x0[2])))), ≥)∧0 = 0∧[(5)bni_11 + (-1)Bound*bni_11] + [bni_11]x2[2] + [(2)bni_11]x0[2] ≥ 0∧0 = 0∧[1 + (-1)bso_12] ≥ 0)

  
  
  

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

• COND_3588_1_MAIN_INVOKEMETHOD1(TRUE, 3588_0_imprimer_GE(java.lang.Object(Carre(x0[3])), x1[3], x2[3], x0[3]), java.lang.Object(Carre(x0[3]))) → 3588_1_MAIN_INVOKEMETHOD(3588_0_imprimer_GE(java.lang.Object(Carre(x0[3])), x1[3], +(x2[3], 1), x0[3]), java.lang.Object(Carre(x0[3])))
  
  • ((U^Increasing(3588_1_MAIN_INVOKEMETHOD(3588_0_imprimer_GE(java.lang.Object(Carre(x0[3])), x1[3], +(x2[3], 1), x0[3]), java.lang.Object(Carre(x0[3])))), ≥)∧0 = 0∧0 = 0∧0 = 0∧[(-1)bso_10] ≥ 0)
    
  
  
• 3588_1_MAIN_INVOKEMETHOD(3588_0_imprimer_GE(java.lang.Object(Carre(x0[2])), x1[2], x2[2], x0[2]), java.lang.Object(Carre(x0[2]))) → COND_3588_1_MAIN_INVOKEMETHOD1(&&(>=(x2[2], 0), <(x2[2], x0[2])), 3588_0_imprimer_GE(java.lang.Object(Carre(x0[2])), x1[2], x2[2], x0[2]), java.lang.Object(Carre(x0[2])))
  
  • (x2[2] ≥ 0∧x0[2] ≥ 0 ⇒ (U^Increasing(COND_3588_1_MAIN_INVOKEMETHOD1(&&(>=(x2[2], 0), <(x2[2], x0[2])), 3588_0_imprimer_GE(java.lang.Object(Carre(x0[2])), x1[2], x2[2], x0[2]), java.lang.Object(Carre(x0[2])))), ≥)∧0 = 0∧[(5)bni_11 + (-1)Bound*bni_11] + [bni_11]x2[2] + [(2)bni_11]x0[2] ≥ 0∧0 = 0∧[1 + (-1)bso_12] ≥ 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_3588_1_MAIN_INVOKEMETHOD1(x₁, x₂, x₃)) = [2]x₃ + x₂   
POL(3588_0_imprimer_GE(x₁, x₂, x₃, x₄)) = [2] + x₄ + [-1]x₃ + [-1]x₁   
POL(java.lang.Object(x₁)) = x₁   
POL(Carre(x₁)) = x₁   
POL(3588_1_MAIN_INVOKEMETHOD(x₁, x₂)) = [1] + x₁ + [2]x₂   
POL(+(x₁, x₂)) = x₁ + x₂   
POL(1) = [1]   
POL(&&(x₁, x₂)) = [-1]   
POL(>=(x₁, x₂)) = [1]   
POL(0) = 0   
POL(<(x₁, x₂)) = [-1]   

The following pairs are in P_>:

3588_1_MAIN_INVOKEMETHOD(3588_0_imprimer_GE(java.lang.Object(Carre(x0[2])), x1[2], x2[2], x0[2]), java.lang.Object(Carre(x0[2]))) → COND_3588_1_MAIN_INVOKEMETHOD1(&&(>=(x2[2], 0), <(x2[2], x0[2])), 3588_0_imprimer_GE(java.lang.Object(Carre(x0[2])), x1[2], x2[2], x0[2]), java.lang.Object(Carre(x0[2])))

The following pairs are in P_bound:

3588_1_MAIN_INVOKEMETHOD(3588_0_imprimer_GE(java.lang.Object(Carre(x0[2])), x1[2], x2[2], x0[2]), java.lang.Object(Carre(x0[2]))) → COND_3588_1_MAIN_INVOKEMETHOD1(&&(>=(x2[2], 0), <(x2[2], x0[2])), 3588_0_imprimer_GE(java.lang.Object(Carre(x0[2])), x1[2], x2[2], x0[2]), java.lang.Object(Carre(x0[2])))

The following pairs are in P_≥:

COND_3588_1_MAIN_INVOKEMETHOD1(TRUE, 3588_0_imprimer_GE(java.lang.Object(Carre(x0[3])), x1[3], x2[3], x0[3]), java.lang.Object(Carre(x0[3]))) → 3588_1_MAIN_INVOKEMETHOD(3588_0_imprimer_GE(java.lang.Object(Carre(x0[3])), x1[3], +(x2[3], 1), x0[3]), java.lang.Object(Carre(x0[3])))

There are no usable rules.

(23) 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:
(3): COND_3588_1_MAIN_INVOKEMETHOD1(TRUE, 3588_0_imprimer_GE(java.lang.Object(Carre(x0[3])), x1[3], x2[3], x0[3]), java.lang.Object(Carre(x0[3]))) → 3588_1_MAIN_INVOKEMETHOD(3588_0_imprimer_GE(java.lang.Object(Carre(x0[3])), x1[3], x2[3] + 1, x0[3]), java.lang.Object(Carre(x0[3])))


The set Q is empty.

(24) IDependencyGraphProof (EQUIVALENT transformation)

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

(25) TRUE

(26) Obligation:

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

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

The following domains are used:

Boolean, Integer

R is empty.

The integer pair graph contains the following rules and edges:
(0): 2995_1_MAIN_INVOKEMETHOD(2995_0_init_GE(java.lang.Object(Carre(java.lang.Object(ARRAY(x0[0], x1[0])))), x2[0], x3[0]), java.lang.Object(Carre(java.lang.Object(ARRAY(x0[0], x1[0]))))) → COND_2995_1_MAIN_INVOKEMETHOD(x3[0] >= 0 && x3[0] <= 2 && x3[0] < 3 && x2[0] >= 0 && x2[0] <= 2, 2995_0_init_GE(java.lang.Object(Carre(java.lang.Object(ARRAY(x0[0], x1[0])))), x2[0], x3[0]), java.lang.Object(Carre(java.lang.Object(ARRAY(x0[0], x1[0])))))
(1): COND_2995_1_MAIN_INVOKEMETHOD(TRUE, 2995_0_init_GE(java.lang.Object(Carre(java.lang.Object(ARRAY(x0[1], x1[1])))), x2[1], x3[1]), java.lang.Object(Carre(java.lang.Object(ARRAY(x0[1], x1[1]))))) → 2995_1_MAIN_INVOKEMETHOD(2995_0_init_GE(java.lang.Object(Carre(java.lang.Object(ARRAY(x0[1], x1[1])))), x2[1], x3[1] + 1), java.lang.Object(Carre(java.lang.Object(ARRAY(x0[1], x1[1])))))
(2): 2995_1_MAIN_INVOKEMETHOD(2995_0_init_GE(java.lang.Object(Carre(java.lang.Object(ARRAY(x0[2], x1[2])))), x2[2], x3[2]), java.lang.Object(Carre(java.lang.Object(ARRAY(x0[2], x1[2]))))) → COND_2995_1_MAIN_INVOKEMETHOD1(x3[2] >= 3 && x2[2] >= 0 && 3 > x2[2] + 1, 2995_0_init_GE(java.lang.Object(Carre(java.lang.Object(ARRAY(x0[2], x1[2])))), x2[2], x3[2]), java.lang.Object(Carre(java.lang.Object(ARRAY(x0[2], x1[2])))))
(3): COND_2995_1_MAIN_INVOKEMETHOD1(TRUE, 2995_0_init_GE(java.lang.Object(Carre(java.lang.Object(ARRAY(x0[3], x1[3])))), x2[3], x3[3]), java.lang.Object(Carre(java.lang.Object(ARRAY(x0[3], x1[3]))))) → 2995_1_MAIN_INVOKEMETHOD(2995_0_init_GE(java.lang.Object(Carre(java.lang.Object(ARRAY(x0[3], x1[3])))), x2[3] + 1, 0), java.lang.Object(Carre(java.lang.Object(ARRAY(x0[3], x1[3])))))

(0) -> (1), if ((x3[0] >= 0 && x3[0] <= 2 && x3[0] < 3 && x2[0] >= 0 && x2[0] <= 2 →^* TRUE)∧(2995_0_init_GE(java.lang.Object(Carre(java.lang.Object(ARRAY(x0[0], x1[0])))), x2[0], x3[0]) →^* 2995_0_init_GE(java.lang.Object(Carre(java.lang.Object(ARRAY(x0[1], x1[1])))), x2[1], x3[1]))∧(java.lang.Object(Carre(java.lang.Object(ARRAY(x0[0], x1[0])))) →^* java.lang.Object(Carre(java.lang.Object(ARRAY(x0[1], x1[1]))))))

(1) -> (0), if ((2995_0_init_GE(java.lang.Object(Carre(java.lang.Object(ARRAY(x0[1], x1[1])))), x2[1], x3[1] + 1) →^* 2995_0_init_GE(java.lang.Object(Carre(java.lang.Object(ARRAY(x0[0], x1[0])))), x2[0], x3[0]))∧(java.lang.Object(Carre(java.lang.Object(ARRAY(x0[1], x1[1])))) →^* java.lang.Object(Carre(java.lang.Object(ARRAY(x0[0], x1[0]))))))

(1) -> (2), if ((2995_0_init_GE(java.lang.Object(Carre(java.lang.Object(ARRAY(x0[1], x1[1])))), x2[1], x3[1] + 1) →^* 2995_0_init_GE(java.lang.Object(Carre(java.lang.Object(ARRAY(x0[2], x1[2])))), x2[2], x3[2]))∧(java.lang.Object(Carre(java.lang.Object(ARRAY(x0[1], x1[1])))) →^* java.lang.Object(Carre(java.lang.Object(ARRAY(x0[2], x1[2]))))))

(2) -> (3), if ((x3[2] >= 3 && x2[2] >= 0 && 3 > x2[2] + 1 →^* TRUE)∧(2995_0_init_GE(java.lang.Object(Carre(java.lang.Object(ARRAY(x0[2], x1[2])))), x2[2], x3[2]) →^* 2995_0_init_GE(java.lang.Object(Carre(java.lang.Object(ARRAY(x0[3], x1[3])))), x2[3], x3[3]))∧(java.lang.Object(Carre(java.lang.Object(ARRAY(x0[2], x1[2])))) →^* java.lang.Object(Carre(java.lang.Object(ARRAY(x0[3], x1[3]))))))

(3) -> (0), if ((2995_0_init_GE(java.lang.Object(Carre(java.lang.Object(ARRAY(x0[3], x1[3])))), x2[3] + 1, 0) →^* 2995_0_init_GE(java.lang.Object(Carre(java.lang.Object(ARRAY(x0[0], x1[0])))), x2[0], x3[0]))∧(java.lang.Object(Carre(java.lang.Object(ARRAY(x0[3], x1[3])))) →^* java.lang.Object(Carre(java.lang.Object(ARRAY(x0[0], x1[0]))))))

(3) -> (2), if ((2995_0_init_GE(java.lang.Object(Carre(java.lang.Object(ARRAY(x0[3], x1[3])))), x2[3] + 1, 0) →^* 2995_0_init_GE(java.lang.Object(Carre(java.lang.Object(ARRAY(x0[2], x1[2])))), x2[2], x3[2]))∧(java.lang.Object(Carre(java.lang.Object(ARRAY(x0[3], x1[3])))) →^* java.lang.Object(Carre(java.lang.Object(ARRAY(x0[2], x1[2]))))))

The set Q is empty.

(27) IDPNonInfProof (SOUND transformation)

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 2995_1_MAIN_INVOKEMETHOD(2995_0_init_GE(java.lang.Object(Carre(java.lang.Object(ARRAY(x0, x1)))), x2, x3), java.lang.Object(Carre(java.lang.Object(ARRAY(x0, x1))))) → COND_2995_1_MAIN_INVOKEMETHOD(&&(&&(&&(&&(>=(x3, 0), <=(x3, 2)), <(x3, 3)), >=(x2, 0)), <=(x2, 2)), 2995_0_init_GE(java.lang.Object(Carre(java.lang.Object(ARRAY(x0, x1)))), x2, x3), java.lang.Object(Carre(java.lang.Object(ARRAY(x0, x1))))) the following chains were created:

• We consider the chain 2995_1_MAIN_INVOKEMETHOD(2995_0_init_GE(java.lang.Object(Carre(java.lang.Object(ARRAY(x0[0], x1[0])))), x2[0], x3[0]), java.lang.Object(Carre(java.lang.Object(ARRAY(x0[0], x1[0]))))) → COND_2995_1_MAIN_INVOKEMETHOD(&&(&&(&&(&&(>=(x3[0], 0), <=(x3[0], 2)), <(x3[0], 3)), >=(x2[0], 0)), <=(x2[0], 2)), 2995_0_init_GE(java.lang.Object(Carre(java.lang.Object(ARRAY(x0[0], x1[0])))), x2[0], x3[0]), java.lang.Object(Carre(java.lang.Object(ARRAY(x0[0], x1[0]))))), COND_2995_1_MAIN_INVOKEMETHOD(TRUE, 2995_0_init_GE(java.lang.Object(Carre(java.lang.Object(ARRAY(x0[1], x1[1])))), x2[1], x3[1]), java.lang.Object(Carre(java.lang.Object(ARRAY(x0[1], x1[1]))))) → 2995_1_MAIN_INVOKEMETHOD(2995_0_init_GE(java.lang.Object(Carre(java.lang.Object(ARRAY(x0[1], x1[1])))), x2[1], +(x3[1], 1)), java.lang.Object(Carre(java.lang.Object(ARRAY(x0[1], x1[1]))))) which results in the following constraint:
  

  (1)    (&&(&&(&&(&&(>=(x3[0], 0), <=(x3[0], 2)), <(x3[0], 3)), >=(x2[0], 0)), <=(x2[0], 2))=TRUE∧2995_0_init_GE(java.lang.Object(Carre(java.lang.Object(ARRAY(x0[0], x1[0])))), x2[0], x3[0])=2995_0_init_GE(java.lang.Object(Carre(java.lang.Object(ARRAY(x0[1], x1[1])))), x2[1], x3[1])∧java.lang.Object(Carre(java.lang.Object(ARRAY(x0[0], x1[0]))))=java.lang.Object(Carre(java.lang.Object(ARRAY(x0[1], x1[1])))) ⇒ 2995_1_MAIN_INVOKEMETHOD(2995_0_init_GE(java.lang.Object(Carre(java.lang.Object(ARRAY(x0[0], x1[0])))), x2[0], x3[0]), java.lang.Object(Carre(java.lang.Object(ARRAY(x0[0], x1[0])))))≥NonInfC∧2995_1_MAIN_INVOKEMETHOD(2995_0_init_GE(java.lang.Object(Carre(java.lang.Object(ARRAY(x0[0], x1[0])))), x2[0], x3[0]), java.lang.Object(Carre(java.lang.Object(ARRAY(x0[0], x1[0])))))≥COND_2995_1_MAIN_INVOKEMETHOD(&&(&&(&&(&&(>=(x3[0], 0), <=(x3[0], 2)), <(x3[0], 3)), >=(x2[0], 0)), <=(x2[0], 2)), 2995_0_init_GE(java.lang.Object(Carre(java.lang.Object(ARRAY(x0[0], x1[0])))), x2[0], x3[0]), java.lang.Object(Carre(java.lang.Object(ARRAY(x0[0], x1[0])))))∧(U^Increasing(COND_2995_1_MAIN_INVOKEMETHOD(&&(&&(&&(&&(>=(x3[0], 0), <=(x3[0], 2)), <(x3[0], 3)), >=(x2[0], 0)), <=(x2[0], 2)), 2995_0_init_GE(java.lang.Object(Carre(java.lang.Object(ARRAY(x0[0], x1[0])))), x2[0], x3[0]), java.lang.Object(Carre(java.lang.Object(ARRAY(x0[0], x1[0])))))), ≥))

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

  (2)    (<=(x2[0], 2)=TRUE∧>=(x2[0], 0)=TRUE∧<(x3[0], 3)=TRUE∧>=(x3[0], 0)=TRUE∧<=(x3[0], 2)=TRUE ⇒ 2995_1_MAIN_INVOKEMETHOD(2995_0_init_GE(java.lang.Object(Carre(java.lang.Object(ARRAY(x0[0], x1[0])))), x2[0], x3[0]), java.lang.Object(Carre(java.lang.Object(ARRAY(x0[0], x1[0])))))≥NonInfC∧2995_1_MAIN_INVOKEMETHOD(2995_0_init_GE(java.lang.Object(Carre(java.lang.Object(ARRAY(x0[0], x1[0])))), x2[0], x3[0]), java.lang.Object(Carre(java.lang.Object(ARRAY(x0[0], x1[0])))))≥COND_2995_1_MAIN_INVOKEMETHOD(&&(&&(&&(&&(>=(x3[0], 0), <=(x3[0], 2)), <(x3[0], 3)), >=(x2[0], 0)), <=(x2[0], 2)), 2995_0_init_GE(java.lang.Object(Carre(java.lang.Object(ARRAY(x0[0], x1[0])))), x2[0], x3[0]), java.lang.Object(Carre(java.lang.Object(ARRAY(x0[0], x1[0])))))∧(U^Increasing(COND_2995_1_MAIN_INVOKEMETHOD(&&(&&(&&(&&(>=(x3[0], 0), <=(x3[0], 2)), <(x3[0], 3)), >=(x2[0], 0)), <=(x2[0], 2)), 2995_0_init_GE(java.lang.Object(Carre(java.lang.Object(ARRAY(x0[0], x1[0])))), x2[0], x3[0]), java.lang.Object(Carre(java.lang.Object(ARRAY(x0[0], x1[0])))))), ≥))

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

  (3)    ([2] + [-1]x2[0] ≥ 0∧x2[0] ≥ 0∧[2] + [-1]x3[0] ≥ 0∧x3[0] ≥ 0∧[2] + [-1]x3[0] ≥ 0 ⇒ (U^Increasing(COND_2995_1_MAIN_INVOKEMETHOD(&&(&&(&&(&&(>=(x3[0], 0), <=(x3[0], 2)), <(x3[0], 3)), >=(x2[0], 0)), <=(x2[0], 2)), 2995_0_init_GE(java.lang.Object(Carre(java.lang.Object(ARRAY(x0[0], x1[0])))), x2[0], x3[0]), java.lang.Object(Carre(java.lang.Object(ARRAY(x0[0], x1[0])))))), ≥)∧[(-2)bni_20 + (-1)Bound*bni_20] + [(-1)bni_20]x2[0] ≥ 0∧[(-1)bso_21] ≥ 0)

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

  (4)    ([2] + [-1]x2[0] ≥ 0∧x2[0] ≥ 0∧[2] + [-1]x3[0] ≥ 0∧x3[0] ≥ 0∧[2] + [-1]x3[0] ≥ 0 ⇒ (U^Increasing(COND_2995_1_MAIN_INVOKEMETHOD(&&(&&(&&(&&(>=(x3[0], 0), <=(x3[0], 2)), <(x3[0], 3)), >=(x2[0], 0)), <=(x2[0], 2)), 2995_0_init_GE(java.lang.Object(Carre(java.lang.Object(ARRAY(x0[0], x1[0])))), x2[0], x3[0]), java.lang.Object(Carre(java.lang.Object(ARRAY(x0[0], x1[0])))))), ≥)∧[(-2)bni_20 + (-1)Bound*bni_20] + [(-1)bni_20]x2[0] ≥ 0∧[(-1)bso_21] ≥ 0)

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

  (5)    ([2] + [-1]x2[0] ≥ 0∧x2[0] ≥ 0∧[2] + [-1]x3[0] ≥ 0∧x3[0] ≥ 0∧[2] + [-1]x3[0] ≥ 0 ⇒ (U^Increasing(COND_2995_1_MAIN_INVOKEMETHOD(&&(&&(&&(&&(>=(x3[0], 0), <=(x3[0], 2)), <(x3[0], 3)), >=(x2[0], 0)), <=(x2[0], 2)), 2995_0_init_GE(java.lang.Object(Carre(java.lang.Object(ARRAY(x0[0], x1[0])))), x2[0], x3[0]), java.lang.Object(Carre(java.lang.Object(ARRAY(x0[0], x1[0])))))), ≥)∧[(-2)bni_20 + (-1)Bound*bni_20] + [(-1)bni_20]x2[0] ≥ 0∧[(-1)bso_21] ≥ 0)

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

  (6)    ([2] + [-1]x2[0] ≥ 0∧x2[0] ≥ 0∧[2] + [-1]x3[0] ≥ 0∧x3[0] ≥ 0∧[2] + [-1]x3[0] ≥ 0 ⇒ (U^Increasing(COND_2995_1_MAIN_INVOKEMETHOD(&&(&&(&&(&&(>=(x3[0], 0), <=(x3[0], 2)), <(x3[0], 3)), >=(x2[0], 0)), <=(x2[0], 2)), 2995_0_init_GE(java.lang.Object(Carre(java.lang.Object(ARRAY(x0[0], x1[0])))), x2[0], x3[0]), java.lang.Object(Carre(java.lang.Object(ARRAY(x0[0], x1[0])))))), ≥)∧0 = 0∧0 = 0∧[(-2)bni_20 + (-1)Bound*bni_20] + [(-1)bni_20]x2[0] ≥ 0∧0 = 0∧0 = 0∧[(-1)bso_21] ≥ 0)

  
  
  

For Pair COND_2995_1_MAIN_INVOKEMETHOD(TRUE, 2995_0_init_GE(java.lang.Object(Carre(java.lang.Object(ARRAY(x0, x1)))), x2, x3), java.lang.Object(Carre(java.lang.Object(ARRAY(x0, x1))))) → 2995_1_MAIN_INVOKEMETHOD(2995_0_init_GE(java.lang.Object(Carre(java.lang.Object(ARRAY(x0, x1)))), x2, +(x3, 1)), java.lang.Object(Carre(java.lang.Object(ARRAY(x0, x1))))) the following chains were created:

• We consider the chain COND_2995_1_MAIN_INVOKEMETHOD(TRUE, 2995_0_init_GE(java.lang.Object(Carre(java.lang.Object(ARRAY(x0[1], x1[1])))), x2[1], x3[1]), java.lang.Object(Carre(java.lang.Object(ARRAY(x0[1], x1[1]))))) → 2995_1_MAIN_INVOKEMETHOD(2995_0_init_GE(java.lang.Object(Carre(java.lang.Object(ARRAY(x0[1], x1[1])))), x2[1], +(x3[1], 1)), java.lang.Object(Carre(java.lang.Object(ARRAY(x0[1], x1[1]))))) which results in the following constraint:
  

  (7)    (COND_2995_1_MAIN_INVOKEMETHOD(TRUE, 2995_0_init_GE(java.lang.Object(Carre(java.lang.Object(ARRAY(x0[1], x1[1])))), x2[1], x3[1]), java.lang.Object(Carre(java.lang.Object(ARRAY(x0[1], x1[1])))))≥NonInfC∧COND_2995_1_MAIN_INVOKEMETHOD(TRUE, 2995_0_init_GE(java.lang.Object(Carre(java.lang.Object(ARRAY(x0[1], x1[1])))), x2[1], x3[1]), java.lang.Object(Carre(java.lang.Object(ARRAY(x0[1], x1[1])))))≥2995_1_MAIN_INVOKEMETHOD(2995_0_init_GE(java.lang.Object(Carre(java.lang.Object(ARRAY(x0[1], x1[1])))), x2[1], +(x3[1], 1)), java.lang.Object(Carre(java.lang.Object(ARRAY(x0[1], x1[1])))))∧(U^Increasing(2995_1_MAIN_INVOKEMETHOD(2995_0_init_GE(java.lang.Object(Carre(java.lang.Object(ARRAY(x0[1], x1[1])))), x2[1], +(x3[1], 1)), java.lang.Object(Carre(java.lang.Object(ARRAY(x0[1], x1[1])))))), ≥))

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

  (8)    ((U^Increasing(2995_1_MAIN_INVOKEMETHOD(2995_0_init_GE(java.lang.Object(Carre(java.lang.Object(ARRAY(x0[1], x1[1])))), x2[1], +(x3[1], 1)), java.lang.Object(Carre(java.lang.Object(ARRAY(x0[1], x1[1])))))), ≥)∧[(-1)bso_23] ≥ 0)

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

  (9)    ((U^Increasing(2995_1_MAIN_INVOKEMETHOD(2995_0_init_GE(java.lang.Object(Carre(java.lang.Object(ARRAY(x0[1], x1[1])))), x2[1], +(x3[1], 1)), java.lang.Object(Carre(java.lang.Object(ARRAY(x0[1], x1[1])))))), ≥)∧[(-1)bso_23] ≥ 0)

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

  (10)    ((U^Increasing(2995_1_MAIN_INVOKEMETHOD(2995_0_init_GE(java.lang.Object(Carre(java.lang.Object(ARRAY(x0[1], x1[1])))), x2[1], +(x3[1], 1)), java.lang.Object(Carre(java.lang.Object(ARRAY(x0[1], x1[1])))))), ≥)∧[(-1)bso_23] ≥ 0)

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

  (11)    ((U^Increasing(2995_1_MAIN_INVOKEMETHOD(2995_0_init_GE(java.lang.Object(Carre(java.lang.Object(ARRAY(x0[1], x1[1])))), x2[1], +(x3[1], 1)), java.lang.Object(Carre(java.lang.Object(ARRAY(x0[1], x1[1])))))), ≥)∧0 = 0∧0 = 0∧0 = 0∧0 = 0∧[(-1)bso_23] ≥ 0)

  
  
  

For Pair 2995_1_MAIN_INVOKEMETHOD(2995_0_init_GE(java.lang.Object(Carre(java.lang.Object(ARRAY(x0, x1)))), x2, x3), java.lang.Object(Carre(java.lang.Object(ARRAY(x0, x1))))) → COND_2995_1_MAIN_INVOKEMETHOD1(&&(&&(>=(x3, 3), >=(x2, 0)), >(3, +(x2, 1))), 2995_0_init_GE(java.lang.Object(Carre(java.lang.Object(ARRAY(x0, x1)))), x2, x3), java.lang.Object(Carre(java.lang.Object(ARRAY(x0, x1))))) the following chains were created:

• We consider the chain 2995_1_MAIN_INVOKEMETHOD(2995_0_init_GE(java.lang.Object(Carre(java.lang.Object(ARRAY(x0[2], x1[2])))), x2[2], x3[2]), java.lang.Object(Carre(java.lang.Object(ARRAY(x0[2], x1[2]))))) → COND_2995_1_MAIN_INVOKEMETHOD1(&&(&&(>=(x3[2], 3), >=(x2[2], 0)), >(3, +(x2[2], 1))), 2995_0_init_GE(java.lang.Object(Carre(java.lang.Object(ARRAY(x0[2], x1[2])))), x2[2], x3[2]), java.lang.Object(Carre(java.lang.Object(ARRAY(x0[2], x1[2]))))), COND_2995_1_MAIN_INVOKEMETHOD1(TRUE, 2995_0_init_GE(java.lang.Object(Carre(java.lang.Object(ARRAY(x0[3], x1[3])))), x2[3], x3[3]), java.lang.Object(Carre(java.lang.Object(ARRAY(x0[3], x1[3]))))) → 2995_1_MAIN_INVOKEMETHOD(2995_0_init_GE(java.lang.Object(Carre(java.lang.Object(ARRAY(x0[3], x1[3])))), +(x2[3], 1), 0), java.lang.Object(Carre(java.lang.Object(ARRAY(x0[3], x1[3]))))) which results in the following constraint:
  

  (12)    (&&(&&(>=(x3[2], 3), >=(x2[2], 0)), >(3, +(x2[2], 1)))=TRUE∧2995_0_init_GE(java.lang.Object(Carre(java.lang.Object(ARRAY(x0[2], x1[2])))), x2[2], x3[2])=2995_0_init_GE(java.lang.Object(Carre(java.lang.Object(ARRAY(x0[3], x1[3])))), x2[3], x3[3])∧java.lang.Object(Carre(java.lang.Object(ARRAY(x0[2], x1[2]))))=java.lang.Object(Carre(java.lang.Object(ARRAY(x0[3], x1[3])))) ⇒ 2995_1_MAIN_INVOKEMETHOD(2995_0_init_GE(java.lang.Object(Carre(java.lang.Object(ARRAY(x0[2], x1[2])))), x2[2], x3[2]), java.lang.Object(Carre(java.lang.Object(ARRAY(x0[2], x1[2])))))≥NonInfC∧2995_1_MAIN_INVOKEMETHOD(2995_0_init_GE(java.lang.Object(Carre(java.lang.Object(ARRAY(x0[2], x1[2])))), x2[2], x3[2]), java.lang.Object(Carre(java.lang.Object(ARRAY(x0[2], x1[2])))))≥COND_2995_1_MAIN_INVOKEMETHOD1(&&(&&(>=(x3[2], 3), >=(x2[2], 0)), >(3, +(x2[2], 1))), 2995_0_init_GE(java.lang.Object(Carre(java.lang.Object(ARRAY(x0[2], x1[2])))), x2[2], x3[2]), java.lang.Object(Carre(java.lang.Object(ARRAY(x0[2], x1[2])))))∧(U^Increasing(COND_2995_1_MAIN_INVOKEMETHOD1(&&(&&(>=(x3[2], 3), >=(x2[2], 0)), >(3, +(x2[2], 1))), 2995_0_init_GE(java.lang.Object(Carre(java.lang.Object(ARRAY(x0[2], x1[2])))), x2[2], x3[2]), java.lang.Object(Carre(java.lang.Object(ARRAY(x0[2], x1[2])))))), ≥))

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

  (13)    (>(3, +(x2[2], 1))=TRUE∧>=(x3[2], 3)=TRUE∧>=(x2[2], 0)=TRUE ⇒ 2995_1_MAIN_INVOKEMETHOD(2995_0_init_GE(java.lang.Object(Carre(java.lang.Object(ARRAY(x0[2], x1[2])))), x2[2], x3[2]), java.lang.Object(Carre(java.lang.Object(ARRAY(x0[2], x1[2])))))≥NonInfC∧2995_1_MAIN_INVOKEMETHOD(2995_0_init_GE(java.lang.Object(Carre(java.lang.Object(ARRAY(x0[2], x1[2])))), x2[2], x3[2]), java.lang.Object(Carre(java.lang.Object(ARRAY(x0[2], x1[2])))))≥COND_2995_1_MAIN_INVOKEMETHOD1(&&(&&(>=(x3[2], 3), >=(x2[2], 0)), >(3, +(x2[2], 1))), 2995_0_init_GE(java.lang.Object(Carre(java.lang.Object(ARRAY(x0[2], x1[2])))), x2[2], x3[2]), java.lang.Object(Carre(java.lang.Object(ARRAY(x0[2], x1[2])))))∧(U^Increasing(COND_2995_1_MAIN_INVOKEMETHOD1(&&(&&(>=(x3[2], 3), >=(x2[2], 0)), >(3, +(x2[2], 1))), 2995_0_init_GE(java.lang.Object(Carre(java.lang.Object(ARRAY(x0[2], x1[2])))), x2[2], x3[2]), java.lang.Object(Carre(java.lang.Object(ARRAY(x0[2], x1[2])))))), ≥))

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

  (14)    ([1] + [-1]x2[2] ≥ 0∧x3[2] + [-3] ≥ 0∧x2[2] ≥ 0 ⇒ (U^Increasing(COND_2995_1_MAIN_INVOKEMETHOD1(&&(&&(>=(x3[2], 3), >=(x2[2], 0)), >(3, +(x2[2], 1))), 2995_0_init_GE(java.lang.Object(Carre(java.lang.Object(ARRAY(x0[2], x1[2])))), x2[2], x3[2]), java.lang.Object(Carre(java.lang.Object(ARRAY(x0[2], x1[2])))))), ≥)∧[(-2)bni_24 + (-1)Bound*bni_24] + [(-1)bni_24]x2[2] ≥ 0∧[(-1)bso_25] ≥ 0)

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

  (15)    ([1] + [-1]x2[2] ≥ 0∧x3[2] + [-3] ≥ 0∧x2[2] ≥ 0 ⇒ (U^Increasing(COND_2995_1_MAIN_INVOKEMETHOD1(&&(&&(>=(x3[2], 3), >=(x2[2], 0)), >(3, +(x2[2], 1))), 2995_0_init_GE(java.lang.Object(Carre(java.lang.Object(ARRAY(x0[2], x1[2])))), x2[2], x3[2]), java.lang.Object(Carre(java.lang.Object(ARRAY(x0[2], x1[2])))))), ≥)∧[(-2)bni_24 + (-1)Bound*bni_24] + [(-1)bni_24]x2[2] ≥ 0∧[(-1)bso_25] ≥ 0)

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

  (16)    ([1] + [-1]x2[2] ≥ 0∧x3[2] + [-3] ≥ 0∧x2[2] ≥ 0 ⇒ (U^Increasing(COND_2995_1_MAIN_INVOKEMETHOD1(&&(&&(>=(x3[2], 3), >=(x2[2], 0)), >(3, +(x2[2], 1))), 2995_0_init_GE(java.lang.Object(Carre(java.lang.Object(ARRAY(x0[2], x1[2])))), x2[2], x3[2]), java.lang.Object(Carre(java.lang.Object(ARRAY(x0[2], x1[2])))))), ≥)∧[(-2)bni_24 + (-1)Bound*bni_24] + [(-1)bni_24]x2[2] ≥ 0∧[(-1)bso_25] ≥ 0)

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

  (17)    ([1] + [-1]x2[2] ≥ 0∧x3[2] + [-3] ≥ 0∧x2[2] ≥ 0 ⇒ (U^Increasing(COND_2995_1_MAIN_INVOKEMETHOD1(&&(&&(>=(x3[2], 3), >=(x2[2], 0)), >(3, +(x2[2], 1))), 2995_0_init_GE(java.lang.Object(Carre(java.lang.Object(ARRAY(x0[2], x1[2])))), x2[2], x3[2]), java.lang.Object(Carre(java.lang.Object(ARRAY(x0[2], x1[2])))))), ≥)∧0 = 0∧0 = 0∧[(-2)bni_24 + (-1)Bound*bni_24] + [(-1)bni_24]x2[2] ≥ 0∧0 = 0∧0 = 0∧[(-1)bso_25] ≥ 0)

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

  (18)    ([1] + [-1]x2[2] ≥ 0∧x3[2] ≥ 0∧x2[2] ≥ 0 ⇒ (U^Increasing(COND_2995_1_MAIN_INVOKEMETHOD1(&&(&&(>=(x3[2], 3), >=(x2[2], 0)), >(3, +(x2[2], 1))), 2995_0_init_GE(java.lang.Object(Carre(java.lang.Object(ARRAY(x0[2], x1[2])))), x2[2], x3[2]), java.lang.Object(Carre(java.lang.Object(ARRAY(x0[2], x1[2])))))), ≥)∧0 = 0∧0 = 0∧[(-2)bni_24 + (-1)Bound*bni_24] + [(-1)bni_24]x2[2] ≥ 0∧0 = 0∧0 = 0∧[(-1)bso_25] ≥ 0)

  
  
  

For Pair COND_2995_1_MAIN_INVOKEMETHOD1(TRUE, 2995_0_init_GE(java.lang.Object(Carre(java.lang.Object(ARRAY(x0, x1)))), x2, x3), java.lang.Object(Carre(java.lang.Object(ARRAY(x0, x1))))) → 2995_1_MAIN_INVOKEMETHOD(2995_0_init_GE(java.lang.Object(Carre(java.lang.Object(ARRAY(x0, x1)))), +(x2, 1), 0), java.lang.Object(Carre(java.lang.Object(ARRAY(x0, x1))))) the following chains were created:

• We consider the chain COND_2995_1_MAIN_INVOKEMETHOD1(TRUE, 2995_0_init_GE(java.lang.Object(Carre(java.lang.Object(ARRAY(x0[3], x1[3])))), x2[3], x3[3]), java.lang.Object(Carre(java.lang.Object(ARRAY(x0[3], x1[3]))))) → 2995_1_MAIN_INVOKEMETHOD(2995_0_init_GE(java.lang.Object(Carre(java.lang.Object(ARRAY(x0[3], x1[3])))), +(x2[3], 1), 0), java.lang.Object(Carre(java.lang.Object(ARRAY(x0[3], x1[3]))))) which results in the following constraint:
  

  (19)    (COND_2995_1_MAIN_INVOKEMETHOD1(TRUE, 2995_0_init_GE(java.lang.Object(Carre(java.lang.Object(ARRAY(x0[3], x1[3])))), x2[3], x3[3]), java.lang.Object(Carre(java.lang.Object(ARRAY(x0[3], x1[3])))))≥NonInfC∧COND_2995_1_MAIN_INVOKEMETHOD1(TRUE, 2995_0_init_GE(java.lang.Object(Carre(java.lang.Object(ARRAY(x0[3], x1[3])))), x2[3], x3[3]), java.lang.Object(Carre(java.lang.Object(ARRAY(x0[3], x1[3])))))≥2995_1_MAIN_INVOKEMETHOD(2995_0_init_GE(java.lang.Object(Carre(java.lang.Object(ARRAY(x0[3], x1[3])))), +(x2[3], 1), 0), java.lang.Object(Carre(java.lang.Object(ARRAY(x0[3], x1[3])))))∧(U^Increasing(2995_1_MAIN_INVOKEMETHOD(2995_0_init_GE(java.lang.Object(Carre(java.lang.Object(ARRAY(x0[3], x1[3])))), +(x2[3], 1), 0), java.lang.Object(Carre(java.lang.Object(ARRAY(x0[3], x1[3])))))), ≥))

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

  (20)    ((U^Increasing(2995_1_MAIN_INVOKEMETHOD(2995_0_init_GE(java.lang.Object(Carre(java.lang.Object(ARRAY(x0[3], x1[3])))), +(x2[3], 1), 0), java.lang.Object(Carre(java.lang.Object(ARRAY(x0[3], x1[3])))))), ≥)∧[1 + (-1)bso_27] ≥ 0)

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

  (21)    ((U^Increasing(2995_1_MAIN_INVOKEMETHOD(2995_0_init_GE(java.lang.Object(Carre(java.lang.Object(ARRAY(x0[3], x1[3])))), +(x2[3], 1), 0), java.lang.Object(Carre(java.lang.Object(ARRAY(x0[3], x1[3])))))), ≥)∧[1 + (-1)bso_27] ≥ 0)

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

  (22)    ((U^Increasing(2995_1_MAIN_INVOKEMETHOD(2995_0_init_GE(java.lang.Object(Carre(java.lang.Object(ARRAY(x0[3], x1[3])))), +(x2[3], 1), 0), java.lang.Object(Carre(java.lang.Object(ARRAY(x0[3], x1[3])))))), ≥)∧[1 + (-1)bso_27] ≥ 0)

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

  (23)    ((U^Increasing(2995_1_MAIN_INVOKEMETHOD(2995_0_init_GE(java.lang.Object(Carre(java.lang.Object(ARRAY(x0[3], x1[3])))), +(x2[3], 1), 0), java.lang.Object(Carre(java.lang.Object(ARRAY(x0[3], x1[3])))))), ≥)∧0 = 0∧0 = 0∧0 = 0∧0 = 0∧[1 + (-1)bso_27] ≥ 0)

  
  
  

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

• 2995_1_MAIN_INVOKEMETHOD(2995_0_init_GE(java.lang.Object(Carre(java.lang.Object(ARRAY(x0, x1)))), x2, x3), java.lang.Object(Carre(java.lang.Object(ARRAY(x0, x1))))) → COND_2995_1_MAIN_INVOKEMETHOD(&&(&&(&&(&&(>=(x3, 0), <=(x3, 2)), <(x3, 3)), >=(x2, 0)), <=(x2, 2)), 2995_0_init_GE(java.lang.Object(Carre(java.lang.Object(ARRAY(x0, x1)))), x2, x3), java.lang.Object(Carre(java.lang.Object(ARRAY(x0, x1)))))
  
  • ([2] + [-1]x2[0] ≥ 0∧x2[0] ≥ 0∧[2] + [-1]x3[0] ≥ 0∧x3[0] ≥ 0∧[2] + [-1]x3[0] ≥ 0 ⇒ (U^Increasing(COND_2995_1_MAIN_INVOKEMETHOD(&&(&&(&&(&&(>=(x3[0], 0), <=(x3[0], 2)), <(x3[0], 3)), >=(x2[0], 0)), <=(x2[0], 2)), 2995_0_init_GE(java.lang.Object(Carre(java.lang.Object(ARRAY(x0[0], x1[0])))), x2[0], x3[0]), java.lang.Object(Carre(java.lang.Object(ARRAY(x0[0], x1[0])))))), ≥)∧0 = 0∧0 = 0∧[(-2)bni_20 + (-1)Bound*bni_20] + [(-1)bni_20]x2[0] ≥ 0∧0 = 0∧0 = 0∧[(-1)bso_21] ≥ 0)
    
  
  
• COND_2995_1_MAIN_INVOKEMETHOD(TRUE, 2995_0_init_GE(java.lang.Object(Carre(java.lang.Object(ARRAY(x0, x1)))), x2, x3), java.lang.Object(Carre(java.lang.Object(ARRAY(x0, x1))))) → 2995_1_MAIN_INVOKEMETHOD(2995_0_init_GE(java.lang.Object(Carre(java.lang.Object(ARRAY(x0, x1)))), x2, +(x3, 1)), java.lang.Object(Carre(java.lang.Object(ARRAY(x0, x1)))))
  
  • ((U^Increasing(2995_1_MAIN_INVOKEMETHOD(2995_0_init_GE(java.lang.Object(Carre(java.lang.Object(ARRAY(x0[1], x1[1])))), x2[1], +(x3[1], 1)), java.lang.Object(Carre(java.lang.Object(ARRAY(x0[1], x1[1])))))), ≥)∧0 = 0∧0 = 0∧0 = 0∧0 = 0∧[(-1)bso_23] ≥ 0)
    
  
  
• 2995_1_MAIN_INVOKEMETHOD(2995_0_init_GE(java.lang.Object(Carre(java.lang.Object(ARRAY(x0, x1)))), x2, x3), java.lang.Object(Carre(java.lang.Object(ARRAY(x0, x1))))) → COND_2995_1_MAIN_INVOKEMETHOD1(&&(&&(>=(x3, 3), >=(x2, 0)), >(3, +(x2, 1))), 2995_0_init_GE(java.lang.Object(Carre(java.lang.Object(ARRAY(x0, x1)))), x2, x3), java.lang.Object(Carre(java.lang.Object(ARRAY(x0, x1)))))
  
  • ([1] + [-1]x2[2] ≥ 0∧x3[2] ≥ 0∧x2[2] ≥ 0 ⇒ (U^Increasing(COND_2995_1_MAIN_INVOKEMETHOD1(&&(&&(>=(x3[2], 3), >=(x2[2], 0)), >(3, +(x2[2], 1))), 2995_0_init_GE(java.lang.Object(Carre(java.lang.Object(ARRAY(x0[2], x1[2])))), x2[2], x3[2]), java.lang.Object(Carre(java.lang.Object(ARRAY(x0[2], x1[2])))))), ≥)∧0 = 0∧0 = 0∧[(-2)bni_24 + (-1)Bound*bni_24] + [(-1)bni_24]x2[2] ≥ 0∧0 = 0∧0 = 0∧[(-1)bso_25] ≥ 0)
    
  
  
• COND_2995_1_MAIN_INVOKEMETHOD1(TRUE, 2995_0_init_GE(java.lang.Object(Carre(java.lang.Object(ARRAY(x0, x1)))), x2, x3), java.lang.Object(Carre(java.lang.Object(ARRAY(x0, x1))))) → 2995_1_MAIN_INVOKEMETHOD(2995_0_init_GE(java.lang.Object(Carre(java.lang.Object(ARRAY(x0, x1)))), +(x2, 1), 0), java.lang.Object(Carre(java.lang.Object(ARRAY(x0, x1)))))
  
  • ((U^Increasing(2995_1_MAIN_INVOKEMETHOD(2995_0_init_GE(java.lang.Object(Carre(java.lang.Object(ARRAY(x0[3], x1[3])))), +(x2[3], 1), 0), java.lang.Object(Carre(java.lang.Object(ARRAY(x0[3], x1[3])))))), ≥)∧0 = 0∧0 = 0∧0 = 0∧0 = 0∧[1 + (-1)bso_27] ≥ 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(2995_1_MAIN_INVOKEMETHOD(x₁, x₂)) = [-1] + [-1]x₂ + [-1]x₁   
POL(2995_0_init_GE(x₁, x₂, x₃)) = [1] + x₂ + [-1]x₁   
POL(java.lang.Object(x₁)) = x₁   
POL(Carre(x₁)) = x₁   
POL(ARRAY(x₁, x₂)) = [-1] + [-1]x₂ + [-1]x₁   
POL(COND_2995_1_MAIN_INVOKEMETHOD(x₁, x₂, x₃)) = [-1] + [-1]x₃ + [-1]x₂   
POL(&&(x₁, x₂)) = [-1]   
POL(>=(x₁, x₂)) = [-1]   
POL(0) = 0   
POL(<=(x₁, x₂)) = [-1]   
POL(2) = [2]   
POL(<(x₁, x₂)) = [-1]   
POL(3) = [3]   
POL(+(x₁, x₂)) = x₁ + x₂   
POL(1) = [1]   
POL(COND_2995_1_MAIN_INVOKEMETHOD1(x₁, x₂, x₃)) = [-1] + [-1]x₃ + [-1]x₂   
POL(>(x₁, x₂)) = [-1]   

The following pairs are in P_>:

COND_2995_1_MAIN_INVOKEMETHOD1(TRUE, 2995_0_init_GE(java.lang.Object(Carre(java.lang.Object(ARRAY(x0[3], x1[3])))), x2[3], x3[3]), java.lang.Object(Carre(java.lang.Object(ARRAY(x0[3], x1[3]))))) → 2995_1_MAIN_INVOKEMETHOD(2995_0_init_GE(java.lang.Object(Carre(java.lang.Object(ARRAY(x0[3], x1[3])))), +(x2[3], 1), 0), java.lang.Object(Carre(java.lang.Object(ARRAY(x0[3], x1[3])))))

The following pairs are in P_bound:

2995_1_MAIN_INVOKEMETHOD(2995_0_init_GE(java.lang.Object(Carre(java.lang.Object(ARRAY(x0[0], x1[0])))), x2[0], x3[0]), java.lang.Object(Carre(java.lang.Object(ARRAY(x0[0], x1[0]))))) → COND_2995_1_MAIN_INVOKEMETHOD(&&(&&(&&(&&(>=(x3[0], 0), <=(x3[0], 2)), <(x3[0], 3)), >=(x2[0], 0)), <=(x2[0], 2)), 2995_0_init_GE(java.lang.Object(Carre(java.lang.Object(ARRAY(x0[0], x1[0])))), x2[0], x3[0]), java.lang.Object(Carre(java.lang.Object(ARRAY(x0[0], x1[0])))))
2995_1_MAIN_INVOKEMETHOD(2995_0_init_GE(java.lang.Object(Carre(java.lang.Object(ARRAY(x0[2], x1[2])))), x2[2], x3[2]), java.lang.Object(Carre(java.lang.Object(ARRAY(x0[2], x1[2]))))) → COND_2995_1_MAIN_INVOKEMETHOD1(&&(&&(>=(x3[2], 3), >=(x2[2], 0)), >(3, +(x2[2], 1))), 2995_0_init_GE(java.lang.Object(Carre(java.lang.Object(ARRAY(x0[2], x1[2])))), x2[2], x3[2]), java.lang.Object(Carre(java.lang.Object(ARRAY(x0[2], x1[2])))))

The following pairs are in P_≥:

2995_1_MAIN_INVOKEMETHOD(2995_0_init_GE(java.lang.Object(Carre(java.lang.Object(ARRAY(x0[0], x1[0])))), x2[0], x3[0]), java.lang.Object(Carre(java.lang.Object(ARRAY(x0[0], x1[0]))))) → COND_2995_1_MAIN_INVOKEMETHOD(&&(&&(&&(&&(>=(x3[0], 0), <=(x3[0], 2)), <(x3[0], 3)), >=(x2[0], 0)), <=(x2[0], 2)), 2995_0_init_GE(java.lang.Object(Carre(java.lang.Object(ARRAY(x0[0], x1[0])))), x2[0], x3[0]), java.lang.Object(Carre(java.lang.Object(ARRAY(x0[0], x1[0])))))
COND_2995_1_MAIN_INVOKEMETHOD(TRUE, 2995_0_init_GE(java.lang.Object(Carre(java.lang.Object(ARRAY(x0[1], x1[1])))), x2[1], x3[1]), java.lang.Object(Carre(java.lang.Object(ARRAY(x0[1], x1[1]))))) → 2995_1_MAIN_INVOKEMETHOD(2995_0_init_GE(java.lang.Object(Carre(java.lang.Object(ARRAY(x0[1], x1[1])))), x2[1], +(x3[1], 1)), java.lang.Object(Carre(java.lang.Object(ARRAY(x0[1], x1[1])))))
2995_1_MAIN_INVOKEMETHOD(2995_0_init_GE(java.lang.Object(Carre(java.lang.Object(ARRAY(x0[2], x1[2])))), x2[2], x3[2]), java.lang.Object(Carre(java.lang.Object(ARRAY(x0[2], x1[2]))))) → COND_2995_1_MAIN_INVOKEMETHOD1(&&(&&(>=(x3[2], 3), >=(x2[2], 0)), >(3, +(x2[2], 1))), 2995_0_init_GE(java.lang.Object(Carre(java.lang.Object(ARRAY(x0[2], x1[2])))), x2[2], x3[2]), java.lang.Object(Carre(java.lang.Object(ARRAY(x0[2], x1[2])))))

There are no usable rules.

(28) Complex Obligation (AND)

(29) 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): 2995_1_MAIN_INVOKEMETHOD(2995_0_init_GE(java.lang.Object(Carre(java.lang.Object(ARRAY(x0[0], x1[0])))), x2[0], x3[0]), java.lang.Object(Carre(java.lang.Object(ARRAY(x0[0], x1[0]))))) → COND_2995_1_MAIN_INVOKEMETHOD(x3[0] >= 0 && x3[0] <= 2 && x3[0] < 3 && x2[0] >= 0 && x2[0] <= 2, 2995_0_init_GE(java.lang.Object(Carre(java.lang.Object(ARRAY(x0[0], x1[0])))), x2[0], x3[0]), java.lang.Object(Carre(java.lang.Object(ARRAY(x0[0], x1[0])))))
(1): COND_2995_1_MAIN_INVOKEMETHOD(TRUE, 2995_0_init_GE(java.lang.Object(Carre(java.lang.Object(ARRAY(x0[1], x1[1])))), x2[1], x3[1]), java.lang.Object(Carre(java.lang.Object(ARRAY(x0[1], x1[1]))))) → 2995_1_MAIN_INVOKEMETHOD(2995_0_init_GE(java.lang.Object(Carre(java.lang.Object(ARRAY(x0[1], x1[1])))), x2[1], x3[1] + 1), java.lang.Object(Carre(java.lang.Object(ARRAY(x0[1], x1[1])))))
(2): 2995_1_MAIN_INVOKEMETHOD(2995_0_init_GE(java.lang.Object(Carre(java.lang.Object(ARRAY(x0[2], x1[2])))), x2[2], x3[2]), java.lang.Object(Carre(java.lang.Object(ARRAY(x0[2], x1[2]))))) → COND_2995_1_MAIN_INVOKEMETHOD1(x3[2] >= 3 && x2[2] >= 0 && 3 > x2[2] + 1, 2995_0_init_GE(java.lang.Object(Carre(java.lang.Object(ARRAY(x0[2], x1[2])))), x2[2], x3[2]), java.lang.Object(Carre(java.lang.Object(ARRAY(x0[2], x1[2])))))

(1) -> (0), if ((2995_0_init_GE(java.lang.Object(Carre(java.lang.Object(ARRAY(x0[1], x1[1])))), x2[1], x3[1] + 1) →^* 2995_0_init_GE(java.lang.Object(Carre(java.lang.Object(ARRAY(x0[0], x1[0])))), x2[0], x3[0]))∧(java.lang.Object(Carre(java.lang.Object(ARRAY(x0[1], x1[1])))) →^* java.lang.Object(Carre(java.lang.Object(ARRAY(x0[0], x1[0]))))))

(0) -> (1), if ((x3[0] >= 0 && x3[0] <= 2 && x3[0] < 3 && x2[0] >= 0 && x2[0] <= 2 →^* TRUE)∧(2995_0_init_GE(java.lang.Object(Carre(java.lang.Object(ARRAY(x0[0], x1[0])))), x2[0], x3[0]) →^* 2995_0_init_GE(java.lang.Object(Carre(java.lang.Object(ARRAY(x0[1], x1[1])))), x2[1], x3[1]))∧(java.lang.Object(Carre(java.lang.Object(ARRAY(x0[0], x1[0])))) →^* java.lang.Object(Carre(java.lang.Object(ARRAY(x0[1], x1[1]))))))

(1) -> (2), if ((2995_0_init_GE(java.lang.Object(Carre(java.lang.Object(ARRAY(x0[1], x1[1])))), x2[1], x3[1] + 1) →^* 2995_0_init_GE(java.lang.Object(Carre(java.lang.Object(ARRAY(x0[2], x1[2])))), x2[2], x3[2]))∧(java.lang.Object(Carre(java.lang.Object(ARRAY(x0[1], x1[1])))) →^* java.lang.Object(Carre(java.lang.Object(ARRAY(x0[2], x1[2]))))))

The set Q is empty.

(30) IDependencyGraphProof (EQUIVALENT transformation)

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

(31) Obligation:

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

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

The following domains are used:

Integer, Boolean

R is empty.

The integer pair graph contains the following rules and edges:
(1): COND_2995_1_MAIN_INVOKEMETHOD(TRUE, 2995_0_init_GE(java.lang.Object(Carre(java.lang.Object(ARRAY(x0[1], x1[1])))), x2[1], x3[1]), java.lang.Object(Carre(java.lang.Object(ARRAY(x0[1], x1[1]))))) → 2995_1_MAIN_INVOKEMETHOD(2995_0_init_GE(java.lang.Object(Carre(java.lang.Object(ARRAY(x0[1], x1[1])))), x2[1], x3[1] + 1), java.lang.Object(Carre(java.lang.Object(ARRAY(x0[1], x1[1])))))
(0): 2995_1_MAIN_INVOKEMETHOD(2995_0_init_GE(java.lang.Object(Carre(java.lang.Object(ARRAY(x0[0], x1[0])))), x2[0], x3[0]), java.lang.Object(Carre(java.lang.Object(ARRAY(x0[0], x1[0]))))) → COND_2995_1_MAIN_INVOKEMETHOD(x3[0] >= 0 && x3[0] <= 2 && x3[0] < 3 && x2[0] >= 0 && x2[0] <= 2, 2995_0_init_GE(java.lang.Object(Carre(java.lang.Object(ARRAY(x0[0], x1[0])))), x2[0], x3[0]), java.lang.Object(Carre(java.lang.Object(ARRAY(x0[0], x1[0])))))

(1) -> (0), if ((2995_0_init_GE(java.lang.Object(Carre(java.lang.Object(ARRAY(x0[1], x1[1])))), x2[1], x3[1] + 1) →^* 2995_0_init_GE(java.lang.Object(Carre(java.lang.Object(ARRAY(x0[0], x1[0])))), x2[0], x3[0]))∧(java.lang.Object(Carre(java.lang.Object(ARRAY(x0[1], x1[1])))) →^* java.lang.Object(Carre(java.lang.Object(ARRAY(x0[0], x1[0]))))))

(0) -> (1), if ((x3[0] >= 0 && x3[0] <= 2 && x3[0] < 3 && x2[0] >= 0 && x2[0] <= 2 →^* TRUE)∧(2995_0_init_GE(java.lang.Object(Carre(java.lang.Object(ARRAY(x0[0], x1[0])))), x2[0], x3[0]) →^* 2995_0_init_GE(java.lang.Object(Carre(java.lang.Object(ARRAY(x0[1], x1[1])))), x2[1], x3[1]))∧(java.lang.Object(Carre(java.lang.Object(ARRAY(x0[0], x1[0])))) →^* java.lang.Object(Carre(java.lang.Object(ARRAY(x0[1], x1[1]))))))

The set Q is empty.

(32) IDPNonInfProof (SOUND transformation)

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_2995_1_MAIN_INVOKEMETHOD(TRUE, 2995_0_init_GE(java.lang.Object(Carre(java.lang.Object(ARRAY(x0[1], x1[1])))), x2[1], x3[1]), java.lang.Object(Carre(java.lang.Object(ARRAY(x0[1], x1[1]))))) → 2995_1_MAIN_INVOKEMETHOD(2995_0_init_GE(java.lang.Object(Carre(java.lang.Object(ARRAY(x0[1], x1[1])))), x2[1], +(x3[1], 1)), java.lang.Object(Carre(java.lang.Object(ARRAY(x0[1], x1[1]))))) the following chains were created:

• We consider the chain COND_2995_1_MAIN_INVOKEMETHOD(TRUE, 2995_0_init_GE(java.lang.Object(Carre(java.lang.Object(ARRAY(x0[1], x1[1])))), x2[1], x3[1]), java.lang.Object(Carre(java.lang.Object(ARRAY(x0[1], x1[1]))))) → 2995_1_MAIN_INVOKEMETHOD(2995_0_init_GE(java.lang.Object(Carre(java.lang.Object(ARRAY(x0[1], x1[1])))), x2[1], +(x3[1], 1)), java.lang.Object(Carre(java.lang.Object(ARRAY(x0[1], x1[1]))))) which results in the following constraint:
  

  (1)    (COND_2995_1_MAIN_INVOKEMETHOD(TRUE, 2995_0_init_GE(java.lang.Object(Carre(java.lang.Object(ARRAY(x0[1], x1[1])))), x2[1], x3[1]), java.lang.Object(Carre(java.lang.Object(ARRAY(x0[1], x1[1])))))≥NonInfC∧COND_2995_1_MAIN_INVOKEMETHOD(TRUE, 2995_0_init_GE(java.lang.Object(Carre(java.lang.Object(ARRAY(x0[1], x1[1])))), x2[1], x3[1]), java.lang.Object(Carre(java.lang.Object(ARRAY(x0[1], x1[1])))))≥2995_1_MAIN_INVOKEMETHOD(2995_0_init_GE(java.lang.Object(Carre(java.lang.Object(ARRAY(x0[1], x1[1])))), x2[1], +(x3[1], 1)), java.lang.Object(Carre(java.lang.Object(ARRAY(x0[1], x1[1])))))∧(U^Increasing(2995_1_MAIN_INVOKEMETHOD(2995_0_init_GE(java.lang.Object(Carre(java.lang.Object(ARRAY(x0[1], x1[1])))), x2[1], +(x3[1], 1)), java.lang.Object(Carre(java.lang.Object(ARRAY(x0[1], x1[1])))))), ≥))

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

  (2)    ((U^Increasing(2995_1_MAIN_INVOKEMETHOD(2995_0_init_GE(java.lang.Object(Carre(java.lang.Object(ARRAY(x0[1], x1[1])))), x2[1], +(x3[1], 1)), java.lang.Object(Carre(java.lang.Object(ARRAY(x0[1], x1[1])))))), ≥)∧[(-1)bso_14] ≥ 0)

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

  (3)    ((U^Increasing(2995_1_MAIN_INVOKEMETHOD(2995_0_init_GE(java.lang.Object(Carre(java.lang.Object(ARRAY(x0[1], x1[1])))), x2[1], +(x3[1], 1)), java.lang.Object(Carre(java.lang.Object(ARRAY(x0[1], x1[1])))))), ≥)∧[(-1)bso_14] ≥ 0)

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

  (4)    ((U^Increasing(2995_1_MAIN_INVOKEMETHOD(2995_0_init_GE(java.lang.Object(Carre(java.lang.Object(ARRAY(x0[1], x1[1])))), x2[1], +(x3[1], 1)), java.lang.Object(Carre(java.lang.Object(ARRAY(x0[1], x1[1])))))), ≥)∧[(-1)bso_14] ≥ 0)

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

  (5)    ((U^Increasing(2995_1_MAIN_INVOKEMETHOD(2995_0_init_GE(java.lang.Object(Carre(java.lang.Object(ARRAY(x0[1], x1[1])))), x2[1], +(x3[1], 1)), java.lang.Object(Carre(java.lang.Object(ARRAY(x0[1], x1[1])))))), ≥)∧0 = 0∧0 = 0∧0 = 0∧0 = 0∧[(-1)bso_14] ≥ 0)

  
  
  

For Pair 2995_1_MAIN_INVOKEMETHOD(2995_0_init_GE(java.lang.Object(Carre(java.lang.Object(ARRAY(x0[0], x1[0])))), x2[0], x3[0]), java.lang.Object(Carre(java.lang.Object(ARRAY(x0[0], x1[0]))))) → COND_2995_1_MAIN_INVOKEMETHOD(&&(&&(&&(&&(>=(x3[0], 0), <=(x3[0], 2)), <(x3[0], 3)), >=(x2[0], 0)), <=(x2[0], 2)), 2995_0_init_GE(java.lang.Object(Carre(java.lang.Object(ARRAY(x0[0], x1[0])))), x2[0], x3[0]), java.lang.Object(Carre(java.lang.Object(ARRAY(x0[0], x1[0]))))) the following chains were created:

• We consider the chain 2995_1_MAIN_INVOKEMETHOD(2995_0_init_GE(java.lang.Object(Carre(java.lang.Object(ARRAY(x0[0], x1[0])))), x2[0], x3[0]), java.lang.Object(Carre(java.lang.Object(ARRAY(x0[0], x1[0]))))) → COND_2995_1_MAIN_INVOKEMETHOD(&&(&&(&&(&&(>=(x3[0], 0), <=(x3[0], 2)), <(x3[0], 3)), >=(x2[0], 0)), <=(x2[0], 2)), 2995_0_init_GE(java.lang.Object(Carre(java.lang.Object(ARRAY(x0[0], x1[0])))), x2[0], x3[0]), java.lang.Object(Carre(java.lang.Object(ARRAY(x0[0], x1[0]))))), COND_2995_1_MAIN_INVOKEMETHOD(TRUE, 2995_0_init_GE(java.lang.Object(Carre(java.lang.Object(ARRAY(x0[1], x1[1])))), x2[1], x3[1]), java.lang.Object(Carre(java.lang.Object(ARRAY(x0[1], x1[1]))))) → 2995_1_MAIN_INVOKEMETHOD(2995_0_init_GE(java.lang.Object(Carre(java.lang.Object(ARRAY(x0[1], x1[1])))), x2[1], +(x3[1], 1)), java.lang.Object(Carre(java.lang.Object(ARRAY(x0[1], x1[1]))))) which results in the following constraint:
  

  (6)    (&&(&&(&&(&&(>=(x3[0], 0), <=(x3[0], 2)), <(x3[0], 3)), >=(x2[0], 0)), <=(x2[0], 2))=TRUE∧2995_0_init_GE(java.lang.Object(Carre(java.lang.Object(ARRAY(x0[0], x1[0])))), x2[0], x3[0])=2995_0_init_GE(java.lang.Object(Carre(java.lang.Object(ARRAY(x0[1], x1[1])))), x2[1], x3[1])∧java.lang.Object(Carre(java.lang.Object(ARRAY(x0[0], x1[0]))))=java.lang.Object(Carre(java.lang.Object(ARRAY(x0[1], x1[1])))) ⇒ 2995_1_MAIN_INVOKEMETHOD(2995_0_init_GE(java.lang.Object(Carre(java.lang.Object(ARRAY(x0[0], x1[0])))), x2[0], x3[0]), java.lang.Object(Carre(java.lang.Object(ARRAY(x0[0], x1[0])))))≥NonInfC∧2995_1_MAIN_INVOKEMETHOD(2995_0_init_GE(java.lang.Object(Carre(java.lang.Object(ARRAY(x0[0], x1[0])))), x2[0], x3[0]), java.lang.Object(Carre(java.lang.Object(ARRAY(x0[0], x1[0])))))≥COND_2995_1_MAIN_INVOKEMETHOD(&&(&&(&&(&&(>=(x3[0], 0), <=(x3[0], 2)), <(x3[0], 3)), >=(x2[0], 0)), <=(x2[0], 2)), 2995_0_init_GE(java.lang.Object(Carre(java.lang.Object(ARRAY(x0[0], x1[0])))), x2[0], x3[0]), java.lang.Object(Carre(java.lang.Object(ARRAY(x0[0], x1[0])))))∧(U^Increasing(COND_2995_1_MAIN_INVOKEMETHOD(&&(&&(&&(&&(>=(x3[0], 0), <=(x3[0], 2)), <(x3[0], 3)), >=(x2[0], 0)), <=(x2[0], 2)), 2995_0_init_GE(java.lang.Object(Carre(java.lang.Object(ARRAY(x0[0], x1[0])))), x2[0], x3[0]), java.lang.Object(Carre(java.lang.Object(ARRAY(x0[0], x1[0])))))), ≥))

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

  (7)    (<=(x2[0], 2)=TRUE∧>=(x2[0], 0)=TRUE∧<(x3[0], 3)=TRUE∧>=(x3[0], 0)=TRUE∧<=(x3[0], 2)=TRUE ⇒ 2995_1_MAIN_INVOKEMETHOD(2995_0_init_GE(java.lang.Object(Carre(java.lang.Object(ARRAY(x0[0], x1[0])))), x2[0], x3[0]), java.lang.Object(Carre(java.lang.Object(ARRAY(x0[0], x1[0])))))≥NonInfC∧2995_1_MAIN_INVOKEMETHOD(2995_0_init_GE(java.lang.Object(Carre(java.lang.Object(ARRAY(x0[0], x1[0])))), x2[0], x3[0]), java.lang.Object(Carre(java.lang.Object(ARRAY(x0[0], x1[0])))))≥COND_2995_1_MAIN_INVOKEMETHOD(&&(&&(&&(&&(>=(x3[0], 0), <=(x3[0], 2)), <(x3[0], 3)), >=(x2[0], 0)), <=(x2[0], 2)), 2995_0_init_GE(java.lang.Object(Carre(java.lang.Object(ARRAY(x0[0], x1[0])))), x2[0], x3[0]), java.lang.Object(Carre(java.lang.Object(ARRAY(x0[0], x1[0])))))∧(U^Increasing(COND_2995_1_MAIN_INVOKEMETHOD(&&(&&(&&(&&(>=(x3[0], 0), <=(x3[0], 2)), <(x3[0], 3)), >=(x2[0], 0)), <=(x2[0], 2)), 2995_0_init_GE(java.lang.Object(Carre(java.lang.Object(ARRAY(x0[0], x1[0])))), x2[0], x3[0]), java.lang.Object(Carre(java.lang.Object(ARRAY(x0[0], x1[0])))))), ≥))

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

  (8)    ([2] + [-1]x2[0] ≥ 0∧x2[0] ≥ 0∧[2] + [-1]x3[0] ≥ 0∧x3[0] ≥ 0∧[2] + [-1]x3[0] ≥ 0 ⇒ (U^Increasing(COND_2995_1_MAIN_INVOKEMETHOD(&&(&&(&&(&&(>=(x3[0], 0), <=(x3[0], 2)), <(x3[0], 3)), >=(x2[0], 0)), <=(x2[0], 2)), 2995_0_init_GE(java.lang.Object(Carre(java.lang.Object(ARRAY(x0[0], x1[0])))), x2[0], x3[0]), java.lang.Object(Carre(java.lang.Object(ARRAY(x0[0], x1[0])))))), ≥)∧[bni_15 + (-1)Bound*bni_15] + [(-1)bni_15]x3[0] ≥ 0∧[1 + (-1)bso_16] ≥ 0)

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

  (9)    ([2] + [-1]x2[0] ≥ 0∧x2[0] ≥ 0∧[2] + [-1]x3[0] ≥ 0∧x3[0] ≥ 0∧[2] + [-1]x3[0] ≥ 0 ⇒ (U^Increasing(COND_2995_1_MAIN_INVOKEMETHOD(&&(&&(&&(&&(>=(x3[0], 0), <=(x3[0], 2)), <(x3[0], 3)), >=(x2[0], 0)), <=(x2[0], 2)), 2995_0_init_GE(java.lang.Object(Carre(java.lang.Object(ARRAY(x0[0], x1[0])))), x2[0], x3[0]), java.lang.Object(Carre(java.lang.Object(ARRAY(x0[0], x1[0])))))), ≥)∧[bni_15 + (-1)Bound*bni_15] + [(-1)bni_15]x3[0] ≥ 0∧[1 + (-1)bso_16] ≥ 0)

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

  (10)    ([2] + [-1]x2[0] ≥ 0∧x2[0] ≥ 0∧[2] + [-1]x3[0] ≥ 0∧x3[0] ≥ 0∧[2] + [-1]x3[0] ≥ 0 ⇒ (U^Increasing(COND_2995_1_MAIN_INVOKEMETHOD(&&(&&(&&(&&(>=(x3[0], 0), <=(x3[0], 2)), <(x3[0], 3)), >=(x2[0], 0)), <=(x2[0], 2)), 2995_0_init_GE(java.lang.Object(Carre(java.lang.Object(ARRAY(x0[0], x1[0])))), x2[0], x3[0]), java.lang.Object(Carre(java.lang.Object(ARRAY(x0[0], x1[0])))))), ≥)∧[bni_15 + (-1)Bound*bni_15] + [(-1)bni_15]x3[0] ≥ 0∧[1 + (-1)bso_16] ≥ 0)

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

  (11)    ([2] + [-1]x2[0] ≥ 0∧x2[0] ≥ 0∧[2] + [-1]x3[0] ≥ 0∧x3[0] ≥ 0∧[2] + [-1]x3[0] ≥ 0 ⇒ (U^Increasing(COND_2995_1_MAIN_INVOKEMETHOD(&&(&&(&&(&&(>=(x3[0], 0), <=(x3[0], 2)), <(x3[0], 3)), >=(x2[0], 0)), <=(x2[0], 2)), 2995_0_init_GE(java.lang.Object(Carre(java.lang.Object(ARRAY(x0[0], x1[0])))), x2[0], x3[0]), java.lang.Object(Carre(java.lang.Object(ARRAY(x0[0], x1[0])))))), ≥)∧0 = 0∧0 = 0∧[bni_15 + (-1)Bound*bni_15] + [(-1)bni_15]x3[0] ≥ 0∧0 = 0∧0 = 0∧[1 + (-1)bso_16] ≥ 0)

  
  
  

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

• COND_2995_1_MAIN_INVOKEMETHOD(TRUE, 2995_0_init_GE(java.lang.Object(Carre(java.lang.Object(ARRAY(x0[1], x1[1])))), x2[1], x3[1]), java.lang.Object(Carre(java.lang.Object(ARRAY(x0[1], x1[1]))))) → 2995_1_MAIN_INVOKEMETHOD(2995_0_init_GE(java.lang.Object(Carre(java.lang.Object(ARRAY(x0[1], x1[1])))), x2[1], +(x3[1], 1)), java.lang.Object(Carre(java.lang.Object(ARRAY(x0[1], x1[1])))))
  
  • ((U^Increasing(2995_1_MAIN_INVOKEMETHOD(2995_0_init_GE(java.lang.Object(Carre(java.lang.Object(ARRAY(x0[1], x1[1])))), x2[1], +(x3[1], 1)), java.lang.Object(Carre(java.lang.Object(ARRAY(x0[1], x1[1])))))), ≥)∧0 = 0∧0 = 0∧0 = 0∧0 = 0∧[(-1)bso_14] ≥ 0)
    
  
  
• 2995_1_MAIN_INVOKEMETHOD(2995_0_init_GE(java.lang.Object(Carre(java.lang.Object(ARRAY(x0[0], x1[0])))), x2[0], x3[0]), java.lang.Object(Carre(java.lang.Object(ARRAY(x0[0], x1[0]))))) → COND_2995_1_MAIN_INVOKEMETHOD(&&(&&(&&(&&(>=(x3[0], 0), <=(x3[0], 2)), <(x3[0], 3)), >=(x2[0], 0)), <=(x2[0], 2)), 2995_0_init_GE(java.lang.Object(Carre(java.lang.Object(ARRAY(x0[0], x1[0])))), x2[0], x3[0]), java.lang.Object(Carre(java.lang.Object(ARRAY(x0[0], x1[0])))))
  
  • ([2] + [-1]x2[0] ≥ 0∧x2[0] ≥ 0∧[2] + [-1]x3[0] ≥ 0∧x3[0] ≥ 0∧[2] + [-1]x3[0] ≥ 0 ⇒ (U^Increasing(COND_2995_1_MAIN_INVOKEMETHOD(&&(&&(&&(&&(>=(x3[0], 0), <=(x3[0], 2)), <(x3[0], 3)), >=(x2[0], 0)), <=(x2[0], 2)), 2995_0_init_GE(java.lang.Object(Carre(java.lang.Object(ARRAY(x0[0], x1[0])))), x2[0], x3[0]), java.lang.Object(Carre(java.lang.Object(ARRAY(x0[0], x1[0])))))), ≥)∧0 = 0∧0 = 0∧[bni_15 + (-1)Bound*bni_15] + [(-1)bni_15]x3[0] ≥ 0∧0 = 0∧0 = 0∧[1 + (-1)bso_16] ≥ 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_2995_1_MAIN_INVOKEMETHOD(x₁, x₂, x₃)) = [-1] + [-1]x₃ + [-1]x₂   
POL(2995_0_init_GE(x₁, x₂, x₃)) = [-1] + x₃ + [-1]x₁   
POL(java.lang.Object(x₁)) = x₁   
POL(Carre(x₁)) = x₁   
POL(ARRAY(x₁, x₂)) = [1] + [-1]x₂ + [-1]x₁   
POL(2995_1_MAIN_INVOKEMETHOD(x₁, x₂)) = [-1]x₁ + [-1]x₂   
POL(+(x₁, x₂)) = x₁ + x₂   
POL(1) = [1]   
POL(&&(x₁, x₂)) = [-1]   
POL(>=(x₁, x₂)) = [-1]   
POL(0) = 0   
POL(<=(x₁, x₂)) = [-1]   
POL(2) = [2]   
POL(<(x₁, x₂)) = [-1]   
POL(3) = [3]   

The following pairs are in P_>:

2995_1_MAIN_INVOKEMETHOD(2995_0_init_GE(java.lang.Object(Carre(java.lang.Object(ARRAY(x0[0], x1[0])))), x2[0], x3[0]), java.lang.Object(Carre(java.lang.Object(ARRAY(x0[0], x1[0]))))) → COND_2995_1_MAIN_INVOKEMETHOD(&&(&&(&&(&&(>=(x3[0], 0), <=(x3[0], 2)), <(x3[0], 3)), >=(x2[0], 0)), <=(x2[0], 2)), 2995_0_init_GE(java.lang.Object(Carre(java.lang.Object(ARRAY(x0[0], x1[0])))), x2[0], x3[0]), java.lang.Object(Carre(java.lang.Object(ARRAY(x0[0], x1[0])))))

The following pairs are in P_bound:

2995_1_MAIN_INVOKEMETHOD(2995_0_init_GE(java.lang.Object(Carre(java.lang.Object(ARRAY(x0[0], x1[0])))), x2[0], x3[0]), java.lang.Object(Carre(java.lang.Object(ARRAY(x0[0], x1[0]))))) → COND_2995_1_MAIN_INVOKEMETHOD(&&(&&(&&(&&(>=(x3[0], 0), <=(x3[0], 2)), <(x3[0], 3)), >=(x2[0], 0)), <=(x2[0], 2)), 2995_0_init_GE(java.lang.Object(Carre(java.lang.Object(ARRAY(x0[0], x1[0])))), x2[0], x3[0]), java.lang.Object(Carre(java.lang.Object(ARRAY(x0[0], x1[0])))))

The following pairs are in P_≥:

COND_2995_1_MAIN_INVOKEMETHOD(TRUE, 2995_0_init_GE(java.lang.Object(Carre(java.lang.Object(ARRAY(x0[1], x1[1])))), x2[1], x3[1]), java.lang.Object(Carre(java.lang.Object(ARRAY(x0[1], x1[1]))))) → 2995_1_MAIN_INVOKEMETHOD(2995_0_init_GE(java.lang.Object(Carre(java.lang.Object(ARRAY(x0[1], x1[1])))), x2[1], +(x3[1], 1)), java.lang.Object(Carre(java.lang.Object(ARRAY(x0[1], x1[1])))))

There are no usable rules.

(33) 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_2995_1_MAIN_INVOKEMETHOD(TRUE, 2995_0_init_GE(java.lang.Object(Carre(java.lang.Object(ARRAY(x0[1], x1[1])))), x2[1], x3[1]), java.lang.Object(Carre(java.lang.Object(ARRAY(x0[1], x1[1]))))) → 2995_1_MAIN_INVOKEMETHOD(2995_0_init_GE(java.lang.Object(Carre(java.lang.Object(ARRAY(x0[1], x1[1])))), x2[1], x3[1] + 1), java.lang.Object(Carre(java.lang.Object(ARRAY(x0[1], x1[1])))))


The set Q is empty.

(34) IDependencyGraphProof (EQUIVALENT transformation)

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

(35) TRUE

(36) 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_2995_1_MAIN_INVOKEMETHOD(TRUE, 2995_0_init_GE(java.lang.Object(Carre(java.lang.Object(ARRAY(x0[1], x1[1])))), x2[1], x3[1]), java.lang.Object(Carre(java.lang.Object(ARRAY(x0[1], x1[1]))))) → 2995_1_MAIN_INVOKEMETHOD(2995_0_init_GE(java.lang.Object(Carre(java.lang.Object(ARRAY(x0[1], x1[1])))), x2[1], x3[1] + 1), java.lang.Object(Carre(java.lang.Object(ARRAY(x0[1], x1[1])))))
(3): COND_2995_1_MAIN_INVOKEMETHOD1(TRUE, 2995_0_init_GE(java.lang.Object(Carre(java.lang.Object(ARRAY(x0[3], x1[3])))), x2[3], x3[3]), java.lang.Object(Carre(java.lang.Object(ARRAY(x0[3], x1[3]))))) → 2995_1_MAIN_INVOKEMETHOD(2995_0_init_GE(java.lang.Object(Carre(java.lang.Object(ARRAY(x0[3], x1[3])))), x2[3] + 1, 0), java.lang.Object(Carre(java.lang.Object(ARRAY(x0[3], x1[3])))))


The set Q is empty.

(37) IDependencyGraphProof (EQUIVALENT transformation)

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

(38) TRUE