Termination proof of /tmp/tmpStqvFR/costa09-example

Termination of the given JBC could be proven:

0 JBC

↳1 JBC2FIG (⇒)

↳2 JBCTerminationGraph

↳3 FIGtoITRSProof (⇒)

↳4 AND

    ↳5 IDP

        ↳6 IDPNonInfProof (⇒)

        ↳7 IDP

        ↳8 IDependencyGraphProof (⇔)

        ↳9 TRUE

    ↳10 IDP

        ↳11 IDPNonInfProof (⇒)

        ↳12 AND

            ↳13 IDP

                ↳14 IDependencyGraphProof (⇔)

                ↳15 TRUE

            ↳16 IDP

                ↳17 IDependencyGraphProof (⇔)

                ↳18 TRUE

    ↳19 IDP

        ↳20 IDPNonInfProof (⇒)

        ↳21 AND

            ↳22 IDP

                ↳23 IDependencyGraphProof (⇔)

                ↳24 TRUE

            ↳25 IDP

                ↳26 IDependencyGraphProof (⇔)

                ↳27 TRUE

(0) Obligation:

JBC Problem based on JBC Program:
Manifest-Version: 1.0 Created-By: 1.6.0_16 (Sun Microsystems Inc.) Main-Class: example_1/Test

package example_1;

public class A {

	int incr(int i) {
		return i=i+1;
	}
}


package example_1;

public class B extends A {
	int incr(int i) {
		return i = i+2;
	}
}



package example_1;

public class C extends B {
	int incr(int i) {
		return i=i+3;
	}
}


package example_1;

public class Test {

	public int add(int n,A o){
		int res=0;
		int i=0;
		while (i<=n){
			res=res+i;
			i=o.incr(i);
		}    
		return res;
	}

	public static void main(String[] args) {
		int test = 1000;
		Test testClass = new Test();
		A a = new A();
		int result1 = testClass.add(test,a);
		a = new B();
		int result2 = testClass.add(test,a);
		a = new C(); 
		int result3 = testClass.add(test,a);     
		// System.out.println("Result: "+result1 + result2 + result3);
	}
}

(1) JBC2FIG (SOUND transformation)

Constructed FIGraph.

(2) Obligation:

FIGraph based on JBC Program:
example_1.Test.main([Ljava/lang/String;)V: Graph of 153 nodes with 3 SCCs.

(3) FIGtoITRSProof (SOUND transformation)

Transformed FIGraph SCCs to IDPs. Logs:

Log for SCC 0:

Generated 20 rules for P and 5 rules for R.

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

Filtered ground terms:

1080_1_main_InvokeMethod(x1, x2) → 1080_1_main_InvokeMethod(x1)
1080_0_add_Load(x1, x2, x3, x4) → 1080_0_add_Load(x3, x4)
Cond_1080_1_main_InvokeMethod(x1, x2, x3) → Cond_1080_1_main_InvokeMethod(x1, x2)

Filtered duplicate args:

1080_0_add_Load(x1, x2) → 1080_0_add_Load(x2)

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

Finished conversion. Obtained 1 rules for P and 0 rules for R. System has predefined symbols.

Log for SCC 1:

Generated 20 rules for P and 56 rules for R.

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

Filtered ground terms:

700_1_main_InvokeMethod(x1, x2, x3) → 700_1_main_InvokeMethod(x1)
700_0_add_Load(x1, x2, x3, x4) → 700_0_add_Load(x3, x4)
Cond_700_1_main_InvokeMethod(x1, x2, x3, x4) → Cond_700_1_main_InvokeMethod(x1, x2)

Filtered duplicate args:

700_0_add_Load(x1, x2) → 700_0_add_Load(x2)

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

Finished conversion. Obtained 1 rules for P and 0 rules for R. System has predefined symbols.

Log for SCC 2:

Generated 20 rules for P and 104 rules for R.

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

Filtered ground terms:

237_1_main_InvokeMethod(x1, x2, x3) → 237_1_main_InvokeMethod(x1)
237_0_add_Load(x1, x2, x3, x4) → 237_0_add_Load(x3, x4)
Cond_237_1_main_InvokeMethod(x1, x2, x3, x4) → Cond_237_1_main_InvokeMethod(x1, x2)

Filtered duplicate args:

237_0_add_Load(x1, x2) → 237_0_add_Load(x2)

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

Finished conversion. Obtained 1 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): 1080_1_MAIN_INVOKEMETHOD(1080_0_add_Load(x1[0])) → COND_1080_1_MAIN_INVOKEMETHOD(x1[0] >= 0 && x1[0] <= 1000, 1080_0_add_Load(x1[0]))
(1): COND_1080_1_MAIN_INVOKEMETHOD(TRUE, 1080_0_add_Load(x1[1])) → 1080_1_MAIN_INVOKEMETHOD(1080_0_add_Load(x1[1] + 3))

(0) -> (1), if ((x1[0] >= 0 && x1[0] <= 1000 →^* TRUE)∧(1080_0_add_Load(x1[0]) →^* 1080_0_add_Load(x1[1])))

(1) -> (0), if ((1080_0_add_Load(x1[1] + 3) →^* 1080_0_add_Load(x1[0])))

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 1080_1_MAIN_INVOKEMETHOD(1080_0_add_Load(x1)) → COND_1080_1_MAIN_INVOKEMETHOD(&&(>=(x1, 0), <=(x1, 1000)), 1080_0_add_Load(x1)) the following chains were created:

We consider the chain 1080_1_MAIN_INVOKEMETHOD(1080_0_add_Load(x1[0])) → COND_1080_1_MAIN_INVOKEMETHOD(&&(>=(x1[0], 0), <=(x1[0], 1000)), 1080_0_add_Load(x1[0])), COND_1080_1_MAIN_INVOKEMETHOD(TRUE, 1080_0_add_Load(x1[1])) → 1080_1_MAIN_INVOKEMETHOD(1080_0_add_Load(+(x1[1], 3))) which results in the following constraint:
(1)    (&&(>=(x1[0], 0), <=(x1[0], 1000))=TRUE∧1080_0_add_Load(x1[0])=1080_0_add_Load(x1[1]) ⇒ 1080_1_MAIN_INVOKEMETHOD(1080_0_add_Load(x1[0]))≥NonInfC∧1080_1_MAIN_INVOKEMETHOD(1080_0_add_Load(x1[0]))≥COND_1080_1_MAIN_INVOKEMETHOD(&&(>=(x1[0], 0), <=(x1[0], 1000)), 1080_0_add_Load(x1[0]))∧(U^Increasing(COND_1080_1_MAIN_INVOKEMETHOD(&&(>=(x1[0], 0), <=(x1[0], 1000)), 1080_0_add_Load(x1[0]))), ≥))

We simplified constraint (1) using rules (I), (II), (IV), (IDP_BOOLEAN) which results in the following new constraint:
(2)    (>=(x1[0], 0)=TRUE∧<=(x1[0], 1000)=TRUE ⇒ 1080_1_MAIN_INVOKEMETHOD(1080_0_add_Load(x1[0]))≥NonInfC∧1080_1_MAIN_INVOKEMETHOD(1080_0_add_Load(x1[0]))≥COND_1080_1_MAIN_INVOKEMETHOD(&&(>=(x1[0], 0), <=(x1[0], 1000)), 1080_0_add_Load(x1[0]))∧(U^Increasing(COND_1080_1_MAIN_INVOKEMETHOD(&&(>=(x1[0], 0), <=(x1[0], 1000)), 1080_0_add_Load(x1[0]))), ≥))

We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(3)    (x1[0] ≥ 0∧[1000] + [-1]x1[0] ≥ 0 ⇒ (U^Increasing(COND_1080_1_MAIN_INVOKEMETHOD(&&(>=(x1[0], 0), <=(x1[0], 1000)), 1080_0_add_Load(x1[0]))), ≥)∧[bni_10 + (-1)Bound*bni_10] + [(-1)bni_10]x1[0] ≥ 0∧[2 + (-1)bso_11] ≥ 0)

We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(4)    (x1[0] ≥ 0∧[1000] + [-1]x1[0] ≥ 0 ⇒ (U^Increasing(COND_1080_1_MAIN_INVOKEMETHOD(&&(>=(x1[0], 0), <=(x1[0], 1000)), 1080_0_add_Load(x1[0]))), ≥)∧[bni_10 + (-1)Bound*bni_10] + [(-1)bni_10]x1[0] ≥ 0∧[2 + (-1)bso_11] ≥ 0)

We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(5)    (x1[0] ≥ 0∧[1000] + [-1]x1[0] ≥ 0 ⇒ (U^Increasing(COND_1080_1_MAIN_INVOKEMETHOD(&&(>=(x1[0], 0), <=(x1[0], 1000)), 1080_0_add_Load(x1[0]))), ≥)∧[bni_10 + (-1)Bound*bni_10] + [(-1)bni_10]x1[0] ≥ 0∧[2 + (-1)bso_11] ≥ 0)

For Pair COND_1080_1_MAIN_INVOKEMETHOD(TRUE, 1080_0_add_Load(x1)) → 1080_1_MAIN_INVOKEMETHOD(1080_0_add_Load(+(x1, 3))) the following chains were created:

We consider the chain COND_1080_1_MAIN_INVOKEMETHOD(TRUE, 1080_0_add_Load(x1[1])) → 1080_1_MAIN_INVOKEMETHOD(1080_0_add_Load(+(x1[1], 3))) which results in the following constraint:
(6)    (COND_1080_1_MAIN_INVOKEMETHOD(TRUE, 1080_0_add_Load(x1[1]))≥NonInfC∧COND_1080_1_MAIN_INVOKEMETHOD(TRUE, 1080_0_add_Load(x1[1]))≥1080_1_MAIN_INVOKEMETHOD(1080_0_add_Load(+(x1[1], 3)))∧(U^Increasing(1080_1_MAIN_INVOKEMETHOD(1080_0_add_Load(+(x1[1], 3)))), ≥))

We simplified constraint (6) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(7)    ((U^Increasing(1080_1_MAIN_INVOKEMETHOD(1080_0_add_Load(+(x1[1], 3)))), ≥)∧[1 + (-1)bso_13] ≥ 0)

We simplified constraint (7) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(8)    ((U^Increasing(1080_1_MAIN_INVOKEMETHOD(1080_0_add_Load(+(x1[1], 3)))), ≥)∧[1 + (-1)bso_13] ≥ 0)

We simplified constraint (8) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(9)    ((U^Increasing(1080_1_MAIN_INVOKEMETHOD(1080_0_add_Load(+(x1[1], 3)))), ≥)∧[1 + (-1)bso_13] ≥ 0)

We simplified constraint (9) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint:
(10)    ((U^Increasing(1080_1_MAIN_INVOKEMETHOD(1080_0_add_Load(+(x1[1], 3)))), ≥)∧0 = 0∧[1 + (-1)bso_13] ≥ 0)

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

1080_1_MAIN_INVOKEMETHOD(1080_0_add_Load(x1)) → COND_1080_1_MAIN_INVOKEMETHOD(&&(>=(x1, 0), <=(x1, 1000)), 1080_0_add_Load(x1))
- (x1[0] ≥ 0∧[1000] + [-1]x1[0] ≥ 0 ⇒ (U^Increasing(COND_1080_1_MAIN_INVOKEMETHOD(&&(>=(x1[0], 0), <=(x1[0], 1000)), 1080_0_add_Load(x1[0]))), ≥)∧[bni_10 + (-1)Bound*bni_10] + [(-1)bni_10]x1[0] ≥ 0∧[2 + (-1)bso_11] ≥ 0)
COND_1080_1_MAIN_INVOKEMETHOD(TRUE, 1080_0_add_Load(x1)) → 1080_1_MAIN_INVOKEMETHOD(1080_0_add_Load(+(x1, 3)))
- ((U^Increasing(1080_1_MAIN_INVOKEMETHOD(1080_0_add_Load(+(x1[1], 3)))), ≥)∧0 = 0∧[1 + (-1)bso_13] ≥ 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(1080_1_MAIN_INVOKEMETHOD(x₁)) = [1] + [-1]x₁
POL(1080_0_add_Load(x₁)) = x₁
POL(COND_1080_1_MAIN_INVOKEMETHOD(x₁, x₂)) = [-1] + [-1]x₂
POL(&&(x₁, x₂)) = [-1]
POL(>=(x₁, x₂)) = [-1]
POL(0) = 0
POL(<=(x₁, x₂)) = [-1]
POL(1000) = [1000]
POL(+(x₁, x₂)) = x₁ + x₂
POL(3) = [3]

The following pairs are in P_>:

1080_1_MAIN_INVOKEMETHOD(1080_0_add_Load(x1[0])) → COND_1080_1_MAIN_INVOKEMETHOD(&&(>=(x1[0], 0), <=(x1[0], 1000)), 1080_0_add_Load(x1[0]))
COND_1080_1_MAIN_INVOKEMETHOD(TRUE, 1080_0_add_Load(x1[1])) → 1080_1_MAIN_INVOKEMETHOD(1080_0_add_Load(+(x1[1], 3)))

The following pairs are in P_bound:

1080_1_MAIN_INVOKEMETHOD(1080_0_add_Load(x1[0])) → COND_1080_1_MAIN_INVOKEMETHOD(&&(>=(x1[0], 0), <=(x1[0], 1000)), 1080_0_add_Load(x1[0]))

The following pairs are in P_≥:
none

There are no usable rules.

(7) Obligation:

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

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

The following domains are used:

Integer

R is empty.

The integer pair graph contains the following rules and edges:
(1): COND_1080_1_MAIN_INVOKEMETHOD(TRUE, 1080_0_add_Load(x1[1])) → 1080_1_MAIN_INVOKEMETHOD(1080_0_add_Load(x1[1] + 3))

The set Q is empty.

(8) IDependencyGraphProof (EQUIVALENT transformation)

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

(9) TRUE

(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:

Boolean, Integer

R is empty.

The integer pair graph contains the following rules and edges:
(0): 700_1_MAIN_INVOKEMETHOD(700_0_add_Load(x1[0])) → COND_700_1_MAIN_INVOKEMETHOD(x1[0] >= 0 && x1[0] <= 1000, 700_0_add_Load(x1[0]))
(1): COND_700_1_MAIN_INVOKEMETHOD(TRUE, 700_0_add_Load(x1[1])) → 700_1_MAIN_INVOKEMETHOD(700_0_add_Load(x1[1] + 2))

(0) -> (1), if ((x1[0] >= 0 && x1[0] <= 1000 →^* TRUE)∧(700_0_add_Load(x1[0]) →^* 700_0_add_Load(x1[1])))

(1) -> (0), if ((700_0_add_Load(x1[1] + 2) →^* 700_0_add_Load(x1[0])))

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 700_1_MAIN_INVOKEMETHOD(700_0_add_Load(x1)) → COND_700_1_MAIN_INVOKEMETHOD(&&(>=(x1, 0), <=(x1, 1000)), 700_0_add_Load(x1)) the following chains were created:

We consider the chain 700_1_MAIN_INVOKEMETHOD(700_0_add_Load(x1[0])) → COND_700_1_MAIN_INVOKEMETHOD(&&(>=(x1[0], 0), <=(x1[0], 1000)), 700_0_add_Load(x1[0])), COND_700_1_MAIN_INVOKEMETHOD(TRUE, 700_0_add_Load(x1[1])) → 700_1_MAIN_INVOKEMETHOD(700_0_add_Load(+(x1[1], 2))) which results in the following constraint:
(1)    (&&(>=(x1[0], 0), <=(x1[0], 1000))=TRUE∧700_0_add_Load(x1[0])=700_0_add_Load(x1[1]) ⇒ 700_1_MAIN_INVOKEMETHOD(700_0_add_Load(x1[0]))≥NonInfC∧700_1_MAIN_INVOKEMETHOD(700_0_add_Load(x1[0]))≥COND_700_1_MAIN_INVOKEMETHOD(&&(>=(x1[0], 0), <=(x1[0], 1000)), 700_0_add_Load(x1[0]))∧(U^Increasing(COND_700_1_MAIN_INVOKEMETHOD(&&(>=(x1[0], 0), <=(x1[0], 1000)), 700_0_add_Load(x1[0]))), ≥))

We simplified constraint (1) using rules (I), (II), (IV), (IDP_BOOLEAN) which results in the following new constraint:
(2)    (>=(x1[0], 0)=TRUE∧<=(x1[0], 1000)=TRUE ⇒ 700_1_MAIN_INVOKEMETHOD(700_0_add_Load(x1[0]))≥NonInfC∧700_1_MAIN_INVOKEMETHOD(700_0_add_Load(x1[0]))≥COND_700_1_MAIN_INVOKEMETHOD(&&(>=(x1[0], 0), <=(x1[0], 1000)), 700_0_add_Load(x1[0]))∧(U^Increasing(COND_700_1_MAIN_INVOKEMETHOD(&&(>=(x1[0], 0), <=(x1[0], 1000)), 700_0_add_Load(x1[0]))), ≥))

We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(3)    (x1[0] ≥ 0∧[1000] + [-1]x1[0] ≥ 0 ⇒ (U^Increasing(COND_700_1_MAIN_INVOKEMETHOD(&&(>=(x1[0], 0), <=(x1[0], 1000)), 700_0_add_Load(x1[0]))), ≥)∧[(-1)bni_10 + (-1)Bound*bni_10] + [(-1)bni_10]x1[0] ≥ 0∧[(-1)bso_11] ≥ 0)

We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(4)    (x1[0] ≥ 0∧[1000] + [-1]x1[0] ≥ 0 ⇒ (U^Increasing(COND_700_1_MAIN_INVOKEMETHOD(&&(>=(x1[0], 0), <=(x1[0], 1000)), 700_0_add_Load(x1[0]))), ≥)∧[(-1)bni_10 + (-1)Bound*bni_10] + [(-1)bni_10]x1[0] ≥ 0∧[(-1)bso_11] ≥ 0)

We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(5)    (x1[0] ≥ 0∧[1000] + [-1]x1[0] ≥ 0 ⇒ (U^Increasing(COND_700_1_MAIN_INVOKEMETHOD(&&(>=(x1[0], 0), <=(x1[0], 1000)), 700_0_add_Load(x1[0]))), ≥)∧[(-1)bni_10 + (-1)Bound*bni_10] + [(-1)bni_10]x1[0] ≥ 0∧[(-1)bso_11] ≥ 0)

For Pair COND_700_1_MAIN_INVOKEMETHOD(TRUE, 700_0_add_Load(x1)) → 700_1_MAIN_INVOKEMETHOD(700_0_add_Load(+(x1, 2))) the following chains were created:

We consider the chain COND_700_1_MAIN_INVOKEMETHOD(TRUE, 700_0_add_Load(x1[1])) → 700_1_MAIN_INVOKEMETHOD(700_0_add_Load(+(x1[1], 2))) which results in the following constraint:
(6)    (COND_700_1_MAIN_INVOKEMETHOD(TRUE, 700_0_add_Load(x1[1]))≥NonInfC∧COND_700_1_MAIN_INVOKEMETHOD(TRUE, 700_0_add_Load(x1[1]))≥700_1_MAIN_INVOKEMETHOD(700_0_add_Load(+(x1[1], 2)))∧(U^Increasing(700_1_MAIN_INVOKEMETHOD(700_0_add_Load(+(x1[1], 2)))), ≥))

We simplified constraint (6) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(7)    ((U^Increasing(700_1_MAIN_INVOKEMETHOD(700_0_add_Load(+(x1[1], 2)))), ≥)∧[2 + (-1)bso_13] ≥ 0)

We simplified constraint (7) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(8)    ((U^Increasing(700_1_MAIN_INVOKEMETHOD(700_0_add_Load(+(x1[1], 2)))), ≥)∧[2 + (-1)bso_13] ≥ 0)

We simplified constraint (8) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(9)    ((U^Increasing(700_1_MAIN_INVOKEMETHOD(700_0_add_Load(+(x1[1], 2)))), ≥)∧[2 + (-1)bso_13] ≥ 0)

We simplified constraint (9) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint:
(10)    ((U^Increasing(700_1_MAIN_INVOKEMETHOD(700_0_add_Load(+(x1[1], 2)))), ≥)∧0 = 0∧[2 + (-1)bso_13] ≥ 0)

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

700_1_MAIN_INVOKEMETHOD(700_0_add_Load(x1)) → COND_700_1_MAIN_INVOKEMETHOD(&&(>=(x1, 0), <=(x1, 1000)), 700_0_add_Load(x1))
- (x1[0] ≥ 0∧[1000] + [-1]x1[0] ≥ 0 ⇒ (U^Increasing(COND_700_1_MAIN_INVOKEMETHOD(&&(>=(x1[0], 0), <=(x1[0], 1000)), 700_0_add_Load(x1[0]))), ≥)∧[(-1)bni_10 + (-1)Bound*bni_10] + [(-1)bni_10]x1[0] ≥ 0∧[(-1)bso_11] ≥ 0)
COND_700_1_MAIN_INVOKEMETHOD(TRUE, 700_0_add_Load(x1)) → 700_1_MAIN_INVOKEMETHOD(700_0_add_Load(+(x1, 2)))
- ((U^Increasing(700_1_MAIN_INVOKEMETHOD(700_0_add_Load(+(x1[1], 2)))), ≥)∧0 = 0∧[2 + (-1)bso_13] ≥ 0)

POL(TRUE) = 0
POL(FALSE) = 0
POL(700_1_MAIN_INVOKEMETHOD(x₁)) = [-1] + [-1]x₁
POL(700_0_add_Load(x₁)) = x₁
POL(COND_700_1_MAIN_INVOKEMETHOD(x₁, x₂)) = [-1] + [-1]x₂
POL(&&(x₁, x₂)) = [-1]
POL(>=(x₁, x₂)) = [-1]
POL(0) = 0
POL(<=(x₁, x₂)) = [-1]
POL(1000) = [1000]
POL(+(x₁, x₂)) = x₁ + x₂
POL(2) = [2]

The following pairs are in P_>:

COND_700_1_MAIN_INVOKEMETHOD(TRUE, 700_0_add_Load(x1[1])) → 700_1_MAIN_INVOKEMETHOD(700_0_add_Load(+(x1[1], 2)))

The following pairs are in P_bound:

700_1_MAIN_INVOKEMETHOD(700_0_add_Load(x1[0])) → COND_700_1_MAIN_INVOKEMETHOD(&&(>=(x1[0], 0), <=(x1[0], 1000)), 700_0_add_Load(x1[0]))

The following pairs are in P_≥:

700_1_MAIN_INVOKEMETHOD(700_0_add_Load(x1[0])) → COND_700_1_MAIN_INVOKEMETHOD(&&(>=(x1[0], 0), <=(x1[0], 1000)), 700_0_add_Load(x1[0]))

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

(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:
(1): COND_700_1_MAIN_INVOKEMETHOD(TRUE, 700_0_add_Load(x1[1])) → 700_1_MAIN_INVOKEMETHOD(700_0_add_Load(x1[1] + 2))

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:

Boolean, Integer

R is empty.

The integer pair graph contains the following rules and edges:
(0): 237_1_MAIN_INVOKEMETHOD(237_0_add_Load(x1[0])) → COND_237_1_MAIN_INVOKEMETHOD(x1[0] >= 0 && x1[0] <= 1000, 237_0_add_Load(x1[0]))
(1): COND_237_1_MAIN_INVOKEMETHOD(TRUE, 237_0_add_Load(x1[1])) → 237_1_MAIN_INVOKEMETHOD(237_0_add_Load(x1[1] + 1))

(0) -> (1), if ((x1[0] >= 0 && x1[0] <= 1000 →^* TRUE)∧(237_0_add_Load(x1[0]) →^* 237_0_add_Load(x1[1])))

(1) -> (0), if ((237_0_add_Load(x1[1] + 1) →^* 237_0_add_Load(x1[0])))

The set Q is empty.

(20) 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 237_1_MAIN_INVOKEMETHOD(237_0_add_Load(x1)) → COND_237_1_MAIN_INVOKEMETHOD(&&(>=(x1, 0), <=(x1, 1000)), 237_0_add_Load(x1)) the following chains were created:

We consider the chain 237_1_MAIN_INVOKEMETHOD(237_0_add_Load(x1[0])) → COND_237_1_MAIN_INVOKEMETHOD(&&(>=(x1[0], 0), <=(x1[0], 1000)), 237_0_add_Load(x1[0])), COND_237_1_MAIN_INVOKEMETHOD(TRUE, 237_0_add_Load(x1[1])) → 237_1_MAIN_INVOKEMETHOD(237_0_add_Load(+(x1[1], 1))) which results in the following constraint:
(1)    (&&(>=(x1[0], 0), <=(x1[0], 1000))=TRUE∧237_0_add_Load(x1[0])=237_0_add_Load(x1[1]) ⇒ 237_1_MAIN_INVOKEMETHOD(237_0_add_Load(x1[0]))≥NonInfC∧237_1_MAIN_INVOKEMETHOD(237_0_add_Load(x1[0]))≥COND_237_1_MAIN_INVOKEMETHOD(&&(>=(x1[0], 0), <=(x1[0], 1000)), 237_0_add_Load(x1[0]))∧(U^Increasing(COND_237_1_MAIN_INVOKEMETHOD(&&(>=(x1[0], 0), <=(x1[0], 1000)), 237_0_add_Load(x1[0]))), ≥))

We simplified constraint (1) using rules (I), (II), (IV), (IDP_BOOLEAN) which results in the following new constraint:
(2)    (>=(x1[0], 0)=TRUE∧<=(x1[0], 1000)=TRUE ⇒ 237_1_MAIN_INVOKEMETHOD(237_0_add_Load(x1[0]))≥NonInfC∧237_1_MAIN_INVOKEMETHOD(237_0_add_Load(x1[0]))≥COND_237_1_MAIN_INVOKEMETHOD(&&(>=(x1[0], 0), <=(x1[0], 1000)), 237_0_add_Load(x1[0]))∧(U^Increasing(COND_237_1_MAIN_INVOKEMETHOD(&&(>=(x1[0], 0), <=(x1[0], 1000)), 237_0_add_Load(x1[0]))), ≥))

We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(3)    (x1[0] ≥ 0∧[1000] + [-1]x1[0] ≥ 0 ⇒ (U^Increasing(COND_237_1_MAIN_INVOKEMETHOD(&&(>=(x1[0], 0), <=(x1[0], 1000)), 237_0_add_Load(x1[0]))), ≥)∧[(-1)bni_10 + (-1)Bound*bni_10] + [(-1)bni_10]x1[0] ≥ 0∧[(-1)bso_11] ≥ 0)

We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(4)    (x1[0] ≥ 0∧[1000] + [-1]x1[0] ≥ 0 ⇒ (U^Increasing(COND_237_1_MAIN_INVOKEMETHOD(&&(>=(x1[0], 0), <=(x1[0], 1000)), 237_0_add_Load(x1[0]))), ≥)∧[(-1)bni_10 + (-1)Bound*bni_10] + [(-1)bni_10]x1[0] ≥ 0∧[(-1)bso_11] ≥ 0)

We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(5)    (x1[0] ≥ 0∧[1000] + [-1]x1[0] ≥ 0 ⇒ (U^Increasing(COND_237_1_MAIN_INVOKEMETHOD(&&(>=(x1[0], 0), <=(x1[0], 1000)), 237_0_add_Load(x1[0]))), ≥)∧[(-1)bni_10 + (-1)Bound*bni_10] + [(-1)bni_10]x1[0] ≥ 0∧[(-1)bso_11] ≥ 0)

For Pair COND_237_1_MAIN_INVOKEMETHOD(TRUE, 237_0_add_Load(x1)) → 237_1_MAIN_INVOKEMETHOD(237_0_add_Load(+(x1, 1))) the following chains were created:

We consider the chain COND_237_1_MAIN_INVOKEMETHOD(TRUE, 237_0_add_Load(x1[1])) → 237_1_MAIN_INVOKEMETHOD(237_0_add_Load(+(x1[1], 1))) which results in the following constraint:
(6)    (COND_237_1_MAIN_INVOKEMETHOD(TRUE, 237_0_add_Load(x1[1]))≥NonInfC∧COND_237_1_MAIN_INVOKEMETHOD(TRUE, 237_0_add_Load(x1[1]))≥237_1_MAIN_INVOKEMETHOD(237_0_add_Load(+(x1[1], 1)))∧(U^Increasing(237_1_MAIN_INVOKEMETHOD(237_0_add_Load(+(x1[1], 1)))), ≥))

We simplified constraint (6) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(7)    ((U^Increasing(237_1_MAIN_INVOKEMETHOD(237_0_add_Load(+(x1[1], 1)))), ≥)∧[1 + (-1)bso_13] ≥ 0)

We simplified constraint (7) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(8)    ((U^Increasing(237_1_MAIN_INVOKEMETHOD(237_0_add_Load(+(x1[1], 1)))), ≥)∧[1 + (-1)bso_13] ≥ 0)

We simplified constraint (8) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(9)    ((U^Increasing(237_1_MAIN_INVOKEMETHOD(237_0_add_Load(+(x1[1], 1)))), ≥)∧[1 + (-1)bso_13] ≥ 0)

We simplified constraint (9) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint:
(10)    ((U^Increasing(237_1_MAIN_INVOKEMETHOD(237_0_add_Load(+(x1[1], 1)))), ≥)∧0 = 0∧[1 + (-1)bso_13] ≥ 0)

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

237_1_MAIN_INVOKEMETHOD(237_0_add_Load(x1)) → COND_237_1_MAIN_INVOKEMETHOD(&&(>=(x1, 0), <=(x1, 1000)), 237_0_add_Load(x1))
- (x1[0] ≥ 0∧[1000] + [-1]x1[0] ≥ 0 ⇒ (U^Increasing(COND_237_1_MAIN_INVOKEMETHOD(&&(>=(x1[0], 0), <=(x1[0], 1000)), 237_0_add_Load(x1[0]))), ≥)∧[(-1)bni_10 + (-1)Bound*bni_10] + [(-1)bni_10]x1[0] ≥ 0∧[(-1)bso_11] ≥ 0)
COND_237_1_MAIN_INVOKEMETHOD(TRUE, 237_0_add_Load(x1)) → 237_1_MAIN_INVOKEMETHOD(237_0_add_Load(+(x1, 1)))
- ((U^Increasing(237_1_MAIN_INVOKEMETHOD(237_0_add_Load(+(x1[1], 1)))), ≥)∧0 = 0∧[1 + (-1)bso_13] ≥ 0)

POL(TRUE) = 0
POL(FALSE) = 0
POL(237_1_MAIN_INVOKEMETHOD(x₁)) = [-1] + [-1]x₁
POL(237_0_add_Load(x₁)) = x₁
POL(COND_237_1_MAIN_INVOKEMETHOD(x₁, x₂)) = [-1] + [-1]x₂
POL(&&(x₁, x₂)) = [-1]
POL(>=(x₁, x₂)) = [-1]
POL(0) = 0
POL(<=(x₁, x₂)) = [-1]
POL(1000) = [1000]
POL(+(x₁, x₂)) = x₁ + x₂
POL(1) = [1]

The following pairs are in P_>:

COND_237_1_MAIN_INVOKEMETHOD(TRUE, 237_0_add_Load(x1[1])) → 237_1_MAIN_INVOKEMETHOD(237_0_add_Load(+(x1[1], 1)))

The following pairs are in P_bound:

237_1_MAIN_INVOKEMETHOD(237_0_add_Load(x1[0])) → COND_237_1_MAIN_INVOKEMETHOD(&&(>=(x1[0], 0), <=(x1[0], 1000)), 237_0_add_Load(x1[0]))

The following pairs are in P_≥:

237_1_MAIN_INVOKEMETHOD(237_0_add_Load(x1[0])) → COND_237_1_MAIN_INVOKEMETHOD(&&(>=(x1[0], 0), <=(x1[0], 1000)), 237_0_add_Load(x1[0]))

There are no usable rules.

(21) Complex Obligation (AND)

(22) Obligation:

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

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

The following domains are used:

Boolean, Integer

(23) IDependencyGraphProof (EQUIVALENT transformation)

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

(24) TRUE

(25) 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_237_1_MAIN_INVOKEMETHOD(TRUE, 237_0_add_Load(x1[1])) → 237_1_MAIN_INVOKEMETHOD(237_0_add_Load(x1[1] + 1))

The set Q is empty.

(26) IDependencyGraphProof (EQUIVALENT transformation)

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

(0) Obligation:

(1) JBC2FIG (SOUND transformation)

(2) Obligation:

(3) FIGtoITRSProof (SOUND transformation)

(4) Complex Obligation (AND)

(5) Obligation:

(6) IDPNonInfProof (SOUND transformation)

(7) Obligation:

(8) IDependencyGraphProof (EQUIVALENT transformation)

(9) TRUE

(10) Obligation:

(11) IDPNonInfProof (SOUND transformation)

(12) Complex Obligation (AND)

(13) Obligation:

(14) IDependencyGraphProof (EQUIVALENT transformation)

(15) TRUE

(16) Obligation:

(17) IDependencyGraphProof (EQUIVALENT transformation)

(18) TRUE

(19) Obligation:

(20) IDPNonInfProof (SOUND transformation)

(21) Complex Obligation (AND)

(22) Obligation:

(23) IDependencyGraphProof (EQUIVALENT transformation)

(24) TRUE

(25) Obligation:

(26) IDependencyGraphProof (EQUIVALENT transformation)

(27) TRUE