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_22 (Sun Microsystems Inc.) Main-Class: TerminatorRec01

public class TerminatorRec01 {
	static int z = 0;

	public static void main(String[] args) {
		z = args.length;
		f(z);
	}

	public static void f(int x) {
		int y = 0;
		if (x > 0) {
			y = 2;
			while (y > 0) {
				z = z - 1;
				f(x - y);
				y = y - 1;
			}
		}
	}
}

(1) JBC2FIG (SOUND transformation)

Constructed FIGraph.

(2) Obligation:

FIGraph based on JBC Program:
TerminatorRec01.main([Ljava/lang/String;)V: Graph of 18 nodes with 0 SCCs.

TerminatorRec01.f(I)V: Graph of 37 nodes with 1 SCC.

(3) FIGtoITRSProof (SOUND transformation)

Transformed FIGraph SCCs to IDPs. Logs:

---

Log for SCC 0:

Generated 33 rules for P and 4 rules for R.

Combined rules. Obtained 3 rules for P and 0 rules for R.

Filtered ground terms:

223_0_f_Store(x1, x2) → 223_0_f_Store(x2)
Cond_370_1_f_InvokeMethod1(x1, x2, x3, x4, x5) → Cond_370_1_f_InvokeMethod1(x1, x3, x4, x5)
341_0_f_Return(x1) → 341_0_f_Return
Cond_370_1_f_InvokeMethod(x1, x2, x3, x4, x5) → Cond_370_1_f_InvokeMethod(x1, x3, x4, x5)
233_0_f_Return(x1) → 233_0_f_Return
Cond_223_0_f_Store(x1, x2, x3) → Cond_223_0_f_Store(x1, x3)

Filtered unneeded arguments:

370_1_f_InvokeMethod(x1, x2, x3, x4) → 370_1_f_InvokeMethod(x1, x2, x3)
Cond_370_1_f_InvokeMethod(x1, x2, x3, x4) → Cond_370_1_f_InvokeMethod(x1, x2, x3)
Cond_370_1_f_InvokeMethod1(x1, x2, x3, x4) → Cond_370_1_f_InvokeMethod1(x1, x2, x3)

Combined rules. Obtained 3 rules for P and 0 rules for R.

Finished conversion. Obtained 3 rules for P and 0 rules for R. System has predefined symbols.

(4) 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): 223_0_F_STORE(x0[0]) → COND_223_0_F_STORE(x0[0] > 0 && 2 > 0, x0[0])
(1): COND_223_0_F_STORE(TRUE, x0[1]) → 370_1_F_INVOKEMETHOD(223_0_f_Store(x0[1] - 2), x0[1], 2)
(2): COND_223_0_F_STORE(TRUE, x0[2]) → 223_0_F_STORE(x0[2] - 2)
(3): 370_1_F_INVOKEMETHOD(233_0_f_Return, x0[3], x1[3]) → COND_370_1_F_INVOKEMETHOD(x1[3] > 0 && x0[3] > 0 && 0 < x1[3] - 1, 233_0_f_Return, x0[3], x1[3])
(4): COND_370_1_F_INVOKEMETHOD(TRUE, 233_0_f_Return, x0[4], x1[4]) → 370_1_F_INVOKEMETHOD(223_0_f_Store(x0[4] - x1[4] - 1), x0[4], x1[4] - 1)
(5): COND_370_1_F_INVOKEMETHOD(TRUE, 233_0_f_Return, x0[5], x1[5]) → 223_0_F_STORE(x0[5] - x1[5] - 1)
(6): 370_1_F_INVOKEMETHOD(341_0_f_Return, x0[6], x1[6]) → COND_370_1_F_INVOKEMETHOD1(x1[6] > 0 && x0[6] > 0 && 0 < x1[6] - 1, 341_0_f_Return, x0[6], x1[6])
(7): COND_370_1_F_INVOKEMETHOD1(TRUE, 341_0_f_Return, x0[7], x1[7]) → 370_1_F_INVOKEMETHOD(223_0_f_Store(x0[7] - x1[7] - 1), x0[7], x1[7] - 1)
(8): COND_370_1_F_INVOKEMETHOD1(TRUE, 341_0_f_Return, x0[8], x1[8]) → 223_0_F_STORE(x0[8] - x1[8] - 1)

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

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

(1) -> (3), if ((223_0_f_Store(x0[1] - 2) →^* 233_0_f_Return)∧(x0[1] →^* x0[3])∧(2 →^* x1[3]))

(1) -> (6), if ((223_0_f_Store(x0[1] - 2) →^* 341_0_f_Return)∧(x0[1] →^* x0[6])∧(2 →^* x1[6]))

(2) -> (0), if ((x0[2] - 2 →^* x0[0]))

(3) -> (4), if ((x1[3] > 0 && x0[3] > 0 && 0 < x1[3] - 1 →^* TRUE)∧(x0[3] →^* x0[4])∧(x1[3] →^* x1[4]))

(3) -> (5), if ((x1[3] > 0 && x0[3] > 0 && 0 < x1[3] - 1 →^* TRUE)∧(x0[3] →^* x0[5])∧(x1[3] →^* x1[5]))

(4) -> (3), if ((223_0_f_Store(x0[4] - x1[4] - 1) →^* 233_0_f_Return)∧(x0[4] →^* x0[3])∧(x1[4] - 1 →^* x1[3]))

(4) -> (6), if ((223_0_f_Store(x0[4] - x1[4] - 1) →^* 341_0_f_Return)∧(x0[4] →^* x0[6])∧(x1[4] - 1 →^* x1[6]))

(5) -> (0), if ((x0[5] - x1[5] - 1 →^* x0[0]))

(6) -> (7), if ((x1[6] > 0 && x0[6] > 0 && 0 < x1[6] - 1 →^* TRUE)∧(x0[6] →^* x0[7])∧(x1[6] →^* x1[7]))

(6) -> (8), if ((x1[6] > 0 && x0[6] > 0 && 0 < x1[6] - 1 →^* TRUE)∧(x0[6] →^* x0[8])∧(x1[6] →^* x1[8]))

(7) -> (3), if ((223_0_f_Store(x0[7] - x1[7] - 1) →^* 233_0_f_Return)∧(x0[7] →^* x0[3])∧(x1[7] - 1 →^* x1[3]))

(7) -> (6), if ((223_0_f_Store(x0[7] - x1[7] - 1) →^* 341_0_f_Return)∧(x0[7] →^* x0[6])∧(x1[7] - 1 →^* x1[6]))

(8) -> (0), if ((x0[8] - x1[8] - 1 →^* x0[0]))

The set Q is empty.

(5) 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 223_0_F_STORE(x0) → COND_223_0_F_STORE(&&(>(x0, 0), >(2, 0)), x0) the following chains were created:

• We consider the chain 223_0_F_STORE(x0[0]) → COND_223_0_F_STORE(&&(>(x0[0], 0), >(2, 0)), x0[0]), COND_223_0_F_STORE(TRUE, x0[1]) → 370_1_F_INVOKEMETHOD(223_0_f_Store(-(x0[1], 2)), x0[1], 2) which results in the following constraint:
  

  (1)    (&&(>(x0[0], 0), >(2, 0))=TRUE∧x0[0]=x0[1] ⇒ 223_0_F_STORE(x0[0])≥NonInfC∧223_0_F_STORE(x0[0])≥COND_223_0_F_STORE(&&(>(x0[0], 0), >(2, 0)), x0[0])∧(U^Increasing(COND_223_0_F_STORE(&&(>(x0[0], 0), >(2, 0)), x0[0])), ≥))

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

  (2)    (&&(>(x0[0], 0), TRUE)=TRUE ⇒ 223_0_F_STORE(x0[0])≥NonInfC∧223_0_F_STORE(x0[0])≥COND_223_0_F_STORE(&&(>(x0[0], 0), >(2, 0)), x0[0])∧(U^Increasing(COND_223_0_F_STORE(&&(>(x0[0], 0), >(2, 0)), x0[0])), ≥))

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

  (3)    (0 ≥ 0 ⇒ (U^Increasing(COND_223_0_F_STORE(&&(>(x0[0], 0), >(2, 0)), x0[0])), ≥)∧[(-1)bni_24 + (-1)Bound*bni_24] + [bni_24]x0[0] ≥ 0∧[(-1)bso_25] ≥ 0)

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

  (4)    (0 ≥ 0 ⇒ (U^Increasing(COND_223_0_F_STORE(&&(>(x0[0], 0), >(2, 0)), x0[0])), ≥)∧[(-1)bni_24 + (-1)Bound*bni_24] + [bni_24]x0[0] ≥ 0∧[(-1)bso_25] ≥ 0)

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

  (5)    (0 ≥ 0 ⇒ (U^Increasing(COND_223_0_F_STORE(&&(>(x0[0], 0), >(2, 0)), x0[0])), ≥)∧[(-1)bni_24 + (-1)Bound*bni_24] + [bni_24]x0[0] ≥ 0∧[(-1)bso_25] ≥ 0)

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

  (6)    (0 ≥ 0 ⇒ (U^Increasing(COND_223_0_F_STORE(&&(>(x0[0], 0), >(2, 0)), x0[0])), ≥)∧[bni_24] ≥ 0∧[(-1)bni_24 + (-1)Bound*bni_24] ≥ 0∧0 ≥ 0∧[(-1)bso_25] ≥ 0)

  
  
  
• We consider the chain 223_0_F_STORE(x0[0]) → COND_223_0_F_STORE(&&(>(x0[0], 0), >(2, 0)), x0[0]), COND_223_0_F_STORE(TRUE, x0[2]) → 223_0_F_STORE(-(x0[2], 2)) which results in the following constraint:
  

  (7)    (&&(>(x0[0], 0), >(2, 0))=TRUE∧x0[0]=x0[2] ⇒ 223_0_F_STORE(x0[0])≥NonInfC∧223_0_F_STORE(x0[0])≥COND_223_0_F_STORE(&&(>(x0[0], 0), >(2, 0)), x0[0])∧(U^Increasing(COND_223_0_F_STORE(&&(>(x0[0], 0), >(2, 0)), x0[0])), ≥))

  
  
  We simplified constraint (7) using rules (IV), (IDP_CONSTANT_FOLD) which results in the following new constraint:
  

  (8)    (&&(>(x0[0], 0), TRUE)=TRUE ⇒ 223_0_F_STORE(x0[0])≥NonInfC∧223_0_F_STORE(x0[0])≥COND_223_0_F_STORE(&&(>(x0[0], 0), >(2, 0)), x0[0])∧(U^Increasing(COND_223_0_F_STORE(&&(>(x0[0], 0), >(2, 0)), x0[0])), ≥))

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

  (9)    (0 ≥ 0 ⇒ (U^Increasing(COND_223_0_F_STORE(&&(>(x0[0], 0), >(2, 0)), x0[0])), ≥)∧[(-1)bni_24 + (-1)Bound*bni_24] + [bni_24]x0[0] ≥ 0∧[(-1)bso_25] ≥ 0)

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

  (10)    (0 ≥ 0 ⇒ (U^Increasing(COND_223_0_F_STORE(&&(>(x0[0], 0), >(2, 0)), x0[0])), ≥)∧[(-1)bni_24 + (-1)Bound*bni_24] + [bni_24]x0[0] ≥ 0∧[(-1)bso_25] ≥ 0)

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

  (11)    (0 ≥ 0 ⇒ (U^Increasing(COND_223_0_F_STORE(&&(>(x0[0], 0), >(2, 0)), x0[0])), ≥)∧[(-1)bni_24 + (-1)Bound*bni_24] + [bni_24]x0[0] ≥ 0∧[(-1)bso_25] ≥ 0)

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

  (12)    (0 ≥ 0 ⇒ (U^Increasing(COND_223_0_F_STORE(&&(>(x0[0], 0), >(2, 0)), x0[0])), ≥)∧[bni_24] ≥ 0∧[(-1)bni_24 + (-1)Bound*bni_24] ≥ 0∧0 ≥ 0∧[(-1)bso_25] ≥ 0)

  
  
  

For Pair COND_223_0_F_STORE(TRUE, x0) → 370_1_F_INVOKEMETHOD(223_0_f_Store(-(x0, 2)), x0, 2) the following chains were created:

• We consider the chain 223_0_F_STORE(x0[0]) → COND_223_0_F_STORE(&&(>(x0[0], 0), >(2, 0)), x0[0]), COND_223_0_F_STORE(TRUE, x0[1]) → 370_1_F_INVOKEMETHOD(223_0_f_Store(-(x0[1], 2)), x0[1], 2), 370_1_F_INVOKEMETHOD(233_0_f_Return, x0[3], x1[3]) → COND_370_1_F_INVOKEMETHOD(&&(&&(>(x1[3], 0), >(x0[3], 0)), <(0, -(x1[3], 1))), 233_0_f_Return, x0[3], x1[3]) which results in the following constraint:
  

  (13)    (&&(>(x0[0], 0), >(2, 0))=TRUE∧x0[0]=x0[1]∧223_0_f_Store(-(x0[1], 2))=233_0_f_Return∧x0[1]=x0[3]∧2=x1[3] ⇒ COND_223_0_F_STORE(TRUE, x0[1])≥NonInfC∧COND_223_0_F_STORE(TRUE, x0[1])≥370_1_F_INVOKEMETHOD(223_0_f_Store(-(x0[1], 2)), x0[1], 2)∧(U^Increasing(370_1_F_INVOKEMETHOD(223_0_f_Store(-(x0[1], 2)), x0[1], 2)), ≥))

  
  
  We solved constraint (13) using rules (I), (II), (IDP_CONSTANT_FOLD).
• We consider the chain 223_0_F_STORE(x0[0]) → COND_223_0_F_STORE(&&(>(x0[0], 0), >(2, 0)), x0[0]), COND_223_0_F_STORE(TRUE, x0[1]) → 370_1_F_INVOKEMETHOD(223_0_f_Store(-(x0[1], 2)), x0[1], 2), 370_1_F_INVOKEMETHOD(341_0_f_Return, x0[6], x1[6]) → COND_370_1_F_INVOKEMETHOD1(&&(&&(>(x1[6], 0), >(x0[6], 0)), <(0, -(x1[6], 1))), 341_0_f_Return, x0[6], x1[6]) which results in the following constraint:
  

  (14)    (&&(>(x0[0], 0), >(2, 0))=TRUE∧x0[0]=x0[1]∧223_0_f_Store(-(x0[1], 2))=341_0_f_Return∧x0[1]=x0[6]∧2=x1[6] ⇒ COND_223_0_F_STORE(TRUE, x0[1])≥NonInfC∧COND_223_0_F_STORE(TRUE, x0[1])≥370_1_F_INVOKEMETHOD(223_0_f_Store(-(x0[1], 2)), x0[1], 2)∧(U^Increasing(370_1_F_INVOKEMETHOD(223_0_f_Store(-(x0[1], 2)), x0[1], 2)), ≥))

  
  
  We solved constraint (14) using rules (I), (II), (IDP_CONSTANT_FOLD).

For Pair COND_223_0_F_STORE(TRUE, x0) → 223_0_F_STORE(-(x0, 2)) the following chains were created:

• We consider the chain 223_0_F_STORE(x0[0]) → COND_223_0_F_STORE(&&(>(x0[0], 0), >(2, 0)), x0[0]), COND_223_0_F_STORE(TRUE, x0[2]) → 223_0_F_STORE(-(x0[2], 2)), 223_0_F_STORE(x0[0]) → COND_223_0_F_STORE(&&(>(x0[0], 0), >(2, 0)), x0[0]) which results in the following constraint:
  

  (15)    (&&(>(x0[0], 0), >(2, 0))=TRUE∧x0[0]=x0[2]∧-(x0[2], 2)=x0[0]1 ⇒ COND_223_0_F_STORE(TRUE, x0[2])≥NonInfC∧COND_223_0_F_STORE(TRUE, x0[2])≥223_0_F_STORE(-(x0[2], 2))∧(U^Increasing(223_0_F_STORE(-(x0[2], 2))), ≥))

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

  (16)    (&&(>(x0[0], 0), TRUE)=TRUE ⇒ COND_223_0_F_STORE(TRUE, x0[0])≥NonInfC∧COND_223_0_F_STORE(TRUE, x0[0])≥223_0_F_STORE(-(x0[0], 2))∧(U^Increasing(223_0_F_STORE(-(x0[2], 2))), ≥))

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

  (17)    (0 ≥ 0 ⇒ (U^Increasing(223_0_F_STORE(-(x0[2], 2))), ≥)∧[(-1)bni_26 + (-1)Bound*bni_26] + [bni_26]x0[0] ≥ 0∧[(-1)bso_27] + x0[0] ≥ 0)

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

  (18)    (0 ≥ 0 ⇒ (U^Increasing(223_0_F_STORE(-(x0[2], 2))), ≥)∧[(-1)bni_26 + (-1)Bound*bni_26] + [bni_26]x0[0] ≥ 0∧[(-1)bso_27] + x0[0] ≥ 0)

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

  (19)    (0 ≥ 0 ⇒ (U^Increasing(223_0_F_STORE(-(x0[2], 2))), ≥)∧[(-1)bni_26 + (-1)Bound*bni_26] + [bni_26]x0[0] ≥ 0∧[(-1)bso_27] + x0[0] ≥ 0)

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

  (20)    (0 ≥ 0 ⇒ (U^Increasing(223_0_F_STORE(-(x0[2], 2))), ≥)∧[bni_26] ≥ 0∧[(-1)bni_26 + (-1)Bound*bni_26] ≥ 0∧[1] ≥ 0∧[(-1)bso_27] ≥ 0)

  
  
  

For Pair 370_1_F_INVOKEMETHOD(233_0_f_Return, x0, x1) → COND_370_1_F_INVOKEMETHOD(&&(&&(>(x1, 0), >(x0, 0)), <(0, -(x1, 1))), 233_0_f_Return, x0, x1) the following chains were created:

• We consider the chain 370_1_F_INVOKEMETHOD(233_0_f_Return, x0[3], x1[3]) → COND_370_1_F_INVOKEMETHOD(&&(&&(>(x1[3], 0), >(x0[3], 0)), <(0, -(x1[3], 1))), 233_0_f_Return, x0[3], x1[3]), COND_370_1_F_INVOKEMETHOD(TRUE, 233_0_f_Return, x0[4], x1[4]) → 370_1_F_INVOKEMETHOD(223_0_f_Store(-(x0[4], -(x1[4], 1))), x0[4], -(x1[4], 1)) which results in the following constraint:
  

  (21)    (&&(&&(>(x1[3], 0), >(x0[3], 0)), <(0, -(x1[3], 1)))=TRUE∧x0[3]=x0[4]∧x1[3]=x1[4] ⇒ 370_1_F_INVOKEMETHOD(233_0_f_Return, x0[3], x1[3])≥NonInfC∧370_1_F_INVOKEMETHOD(233_0_f_Return, x0[3], x1[3])≥COND_370_1_F_INVOKEMETHOD(&&(&&(>(x1[3], 0), >(x0[3], 0)), <(0, -(x1[3], 1))), 233_0_f_Return, x0[3], x1[3])∧(U^Increasing(COND_370_1_F_INVOKEMETHOD(&&(&&(>(x1[3], 0), >(x0[3], 0)), <(0, -(x1[3], 1))), 233_0_f_Return, x0[3], x1[3])), ≥))

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

  (22)    (&&(&&(>(x1[3], 0), >(x0[3], 0)), <(0, -(x1[3], 1)))=TRUE ⇒ 370_1_F_INVOKEMETHOD(233_0_f_Return, x0[3], x1[3])≥NonInfC∧370_1_F_INVOKEMETHOD(233_0_f_Return, x0[3], x1[3])≥COND_370_1_F_INVOKEMETHOD(&&(&&(>(x1[3], 0), >(x0[3], 0)), <(0, -(x1[3], 1))), 233_0_f_Return, x0[3], x1[3])∧(U^Increasing(COND_370_1_F_INVOKEMETHOD(&&(&&(>(x1[3], 0), >(x0[3], 0)), <(0, -(x1[3], 1))), 233_0_f_Return, x0[3], x1[3])), ≥))

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

  (23)    (0 ≥ 0 ⇒ (U^Increasing(COND_370_1_F_INVOKEMETHOD(&&(&&(>(x1[3], 0), >(x0[3], 0)), <(0, -(x1[3], 1))), 233_0_f_Return, x0[3], x1[3])), ≥)∧[(-1)Bound*bni_28] + [bni_28]x1[3] + [(2)bni_28]x0[3] ≥ 0∧[1 + (-1)bso_29] + x1[3] + x0[3] ≥ 0)

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

  (24)    (0 ≥ 0 ⇒ (U^Increasing(COND_370_1_F_INVOKEMETHOD(&&(&&(>(x1[3], 0), >(x0[3], 0)), <(0, -(x1[3], 1))), 233_0_f_Return, x0[3], x1[3])), ≥)∧[(-1)Bound*bni_28] + [bni_28]x1[3] + [(2)bni_28]x0[3] ≥ 0∧[1 + (-1)bso_29] + x1[3] + x0[3] ≥ 0)

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

  (25)    (0 ≥ 0 ⇒ (U^Increasing(COND_370_1_F_INVOKEMETHOD(&&(&&(>(x1[3], 0), >(x0[3], 0)), <(0, -(x1[3], 1))), 233_0_f_Return, x0[3], x1[3])), ≥)∧[(-1)Bound*bni_28] + [bni_28]x1[3] + [(2)bni_28]x0[3] ≥ 0∧[1 + (-1)bso_29] + x1[3] + x0[3] ≥ 0)

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

  (26)    (0 ≥ 0 ⇒ (U^Increasing(COND_370_1_F_INVOKEMETHOD(&&(&&(>(x1[3], 0), >(x0[3], 0)), <(0, -(x1[3], 1))), 233_0_f_Return, x0[3], x1[3])), ≥)∧[bni_28] ≥ 0∧[(2)bni_28] ≥ 0∧[(-1)Bound*bni_28] ≥ 0∧[1] ≥ 0∧[1] ≥ 0∧[1 + (-1)bso_29] ≥ 0)

  
  
  
• We consider the chain 370_1_F_INVOKEMETHOD(233_0_f_Return, x0[3], x1[3]) → COND_370_1_F_INVOKEMETHOD(&&(&&(>(x1[3], 0), >(x0[3], 0)), <(0, -(x1[3], 1))), 233_0_f_Return, x0[3], x1[3]), COND_370_1_F_INVOKEMETHOD(TRUE, 233_0_f_Return, x0[5], x1[5]) → 223_0_F_STORE(-(x0[5], -(x1[5], 1))) which results in the following constraint:
  

  (27)    (&&(&&(>(x1[3], 0), >(x0[3], 0)), <(0, -(x1[3], 1)))=TRUE∧x0[3]=x0[5]∧x1[3]=x1[5] ⇒ 370_1_F_INVOKEMETHOD(233_0_f_Return, x0[3], x1[3])≥NonInfC∧370_1_F_INVOKEMETHOD(233_0_f_Return, x0[3], x1[3])≥COND_370_1_F_INVOKEMETHOD(&&(&&(>(x1[3], 0), >(x0[3], 0)), <(0, -(x1[3], 1))), 233_0_f_Return, x0[3], x1[3])∧(U^Increasing(COND_370_1_F_INVOKEMETHOD(&&(&&(>(x1[3], 0), >(x0[3], 0)), <(0, -(x1[3], 1))), 233_0_f_Return, x0[3], x1[3])), ≥))

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

  (28)    (&&(&&(>(x1[3], 0), >(x0[3], 0)), <(0, -(x1[3], 1)))=TRUE ⇒ 370_1_F_INVOKEMETHOD(233_0_f_Return, x0[3], x1[3])≥NonInfC∧370_1_F_INVOKEMETHOD(233_0_f_Return, x0[3], x1[3])≥COND_370_1_F_INVOKEMETHOD(&&(&&(>(x1[3], 0), >(x0[3], 0)), <(0, -(x1[3], 1))), 233_0_f_Return, x0[3], x1[3])∧(U^Increasing(COND_370_1_F_INVOKEMETHOD(&&(&&(>(x1[3], 0), >(x0[3], 0)), <(0, -(x1[3], 1))), 233_0_f_Return, x0[3], x1[3])), ≥))

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

  (29)    (0 ≥ 0 ⇒ (U^Increasing(COND_370_1_F_INVOKEMETHOD(&&(&&(>(x1[3], 0), >(x0[3], 0)), <(0, -(x1[3], 1))), 233_0_f_Return, x0[3], x1[3])), ≥)∧[(-1)Bound*bni_28] + [bni_28]x1[3] + [(2)bni_28]x0[3] ≥ 0∧[1 + (-1)bso_29] + x1[3] + x0[3] ≥ 0)

  
  
  We simplified constraint (29) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
  

  (30)    (0 ≥ 0 ⇒ (U^Increasing(COND_370_1_F_INVOKEMETHOD(&&(&&(>(x1[3], 0), >(x0[3], 0)), <(0, -(x1[3], 1))), 233_0_f_Return, x0[3], x1[3])), ≥)∧[(-1)Bound*bni_28] + [bni_28]x1[3] + [(2)bni_28]x0[3] ≥ 0∧[1 + (-1)bso_29] + x1[3] + x0[3] ≥ 0)

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

  (31)    (0 ≥ 0 ⇒ (U^Increasing(COND_370_1_F_INVOKEMETHOD(&&(&&(>(x1[3], 0), >(x0[3], 0)), <(0, -(x1[3], 1))), 233_0_f_Return, x0[3], x1[3])), ≥)∧[(-1)Bound*bni_28] + [bni_28]x1[3] + [(2)bni_28]x0[3] ≥ 0∧[1 + (-1)bso_29] + x1[3] + x0[3] ≥ 0)

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

  (32)    (0 ≥ 0 ⇒ (U^Increasing(COND_370_1_F_INVOKEMETHOD(&&(&&(>(x1[3], 0), >(x0[3], 0)), <(0, -(x1[3], 1))), 233_0_f_Return, x0[3], x1[3])), ≥)∧[bni_28] ≥ 0∧[(2)bni_28] ≥ 0∧[(-1)Bound*bni_28] ≥ 0∧[1] ≥ 0∧[1] ≥ 0∧[1 + (-1)bso_29] ≥ 0)

  
  
  

For Pair COND_370_1_F_INVOKEMETHOD(TRUE, 233_0_f_Return, x0, x1) → 370_1_F_INVOKEMETHOD(223_0_f_Store(-(x0, -(x1, 1))), x0, -(x1, 1)) the following chains were created:

• We consider the chain 370_1_F_INVOKEMETHOD(233_0_f_Return, x0[3], x1[3]) → COND_370_1_F_INVOKEMETHOD(&&(&&(>(x1[3], 0), >(x0[3], 0)), <(0, -(x1[3], 1))), 233_0_f_Return, x0[3], x1[3]), COND_370_1_F_INVOKEMETHOD(TRUE, 233_0_f_Return, x0[4], x1[4]) → 370_1_F_INVOKEMETHOD(223_0_f_Store(-(x0[4], -(x1[4], 1))), x0[4], -(x1[4], 1)), 370_1_F_INVOKEMETHOD(233_0_f_Return, x0[3], x1[3]) → COND_370_1_F_INVOKEMETHOD(&&(&&(>(x1[3], 0), >(x0[3], 0)), <(0, -(x1[3], 1))), 233_0_f_Return, x0[3], x1[3]) which results in the following constraint:
  

  (33)    (&&(&&(>(x1[3], 0), >(x0[3], 0)), <(0, -(x1[3], 1)))=TRUE∧x0[3]=x0[4]∧x1[3]=x1[4]∧223_0_f_Store(-(x0[4], -(x1[4], 1)))=233_0_f_Return∧x0[4]=x0[3]1∧-(x1[4], 1)=x1[3]1 ⇒ COND_370_1_F_INVOKEMETHOD(TRUE, 233_0_f_Return, x0[4], x1[4])≥NonInfC∧COND_370_1_F_INVOKEMETHOD(TRUE, 233_0_f_Return, x0[4], x1[4])≥370_1_F_INVOKEMETHOD(223_0_f_Store(-(x0[4], -(x1[4], 1))), x0[4], -(x1[4], 1))∧(U^Increasing(370_1_F_INVOKEMETHOD(223_0_f_Store(-(x0[4], -(x1[4], 1))), x0[4], -(x1[4], 1))), ≥))

  
  
  We solved constraint (33) using rules (I), (II).
• We consider the chain 370_1_F_INVOKEMETHOD(233_0_f_Return, x0[3], x1[3]) → COND_370_1_F_INVOKEMETHOD(&&(&&(>(x1[3], 0), >(x0[3], 0)), <(0, -(x1[3], 1))), 233_0_f_Return, x0[3], x1[3]), COND_370_1_F_INVOKEMETHOD(TRUE, 233_0_f_Return, x0[4], x1[4]) → 370_1_F_INVOKEMETHOD(223_0_f_Store(-(x0[4], -(x1[4], 1))), x0[4], -(x1[4], 1)), 370_1_F_INVOKEMETHOD(341_0_f_Return, x0[6], x1[6]) → COND_370_1_F_INVOKEMETHOD1(&&(&&(>(x1[6], 0), >(x0[6], 0)), <(0, -(x1[6], 1))), 341_0_f_Return, x0[6], x1[6]) which results in the following constraint:
  

  (34)    (&&(&&(>(x1[3], 0), >(x0[3], 0)), <(0, -(x1[3], 1)))=TRUE∧x0[3]=x0[4]∧x1[3]=x1[4]∧223_0_f_Store(-(x0[4], -(x1[4], 1)))=341_0_f_Return∧x0[4]=x0[6]∧-(x1[4], 1)=x1[6] ⇒ COND_370_1_F_INVOKEMETHOD(TRUE, 233_0_f_Return, x0[4], x1[4])≥NonInfC∧COND_370_1_F_INVOKEMETHOD(TRUE, 233_0_f_Return, x0[4], x1[4])≥370_1_F_INVOKEMETHOD(223_0_f_Store(-(x0[4], -(x1[4], 1))), x0[4], -(x1[4], 1))∧(U^Increasing(370_1_F_INVOKEMETHOD(223_0_f_Store(-(x0[4], -(x1[4], 1))), x0[4], -(x1[4], 1))), ≥))

  
  
  We solved constraint (34) using rules (I), (II).

For Pair COND_370_1_F_INVOKEMETHOD(TRUE, 233_0_f_Return, x0, x1) → 223_0_F_STORE(-(x0, -(x1, 1))) the following chains were created:

• We consider the chain 370_1_F_INVOKEMETHOD(233_0_f_Return, x0[3], x1[3]) → COND_370_1_F_INVOKEMETHOD(&&(&&(>(x1[3], 0), >(x0[3], 0)), <(0, -(x1[3], 1))), 233_0_f_Return, x0[3], x1[3]), COND_370_1_F_INVOKEMETHOD(TRUE, 233_0_f_Return, x0[5], x1[5]) → 223_0_F_STORE(-(x0[5], -(x1[5], 1))), 223_0_F_STORE(x0[0]) → COND_223_0_F_STORE(&&(>(x0[0], 0), >(2, 0)), x0[0]) which results in the following constraint:
  

  (35)    (&&(&&(>(x1[3], 0), >(x0[3], 0)), <(0, -(x1[3], 1)))=TRUE∧x0[3]=x0[5]∧x1[3]=x1[5]∧-(x0[5], -(x1[5], 1))=x0[0] ⇒ COND_370_1_F_INVOKEMETHOD(TRUE, 233_0_f_Return, x0[5], x1[5])≥NonInfC∧COND_370_1_F_INVOKEMETHOD(TRUE, 233_0_f_Return, x0[5], x1[5])≥223_0_F_STORE(-(x0[5], -(x1[5], 1)))∧(U^Increasing(223_0_F_STORE(-(x0[5], -(x1[5], 1)))), ≥))

  
  
  We simplified constraint (35) using rules (III), (IV) which results in the following new constraint:
  

  (36)    (&&(&&(>(x1[3], 0), >(x0[3], 0)), <(0, -(x1[3], 1)))=TRUE ⇒ COND_370_1_F_INVOKEMETHOD(TRUE, 233_0_f_Return, x0[3], x1[3])≥NonInfC∧COND_370_1_F_INVOKEMETHOD(TRUE, 233_0_f_Return, x0[3], x1[3])≥223_0_F_STORE(-(x0[3], -(x1[3], 1)))∧(U^Increasing(223_0_F_STORE(-(x0[5], -(x1[5], 1)))), ≥))

  
  
  We simplified constraint (36) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
  

  (37)    (0 ≥ 0 ⇒ (U^Increasing(223_0_F_STORE(-(x0[5], -(x1[5], 1)))), ≥)∧[(-1)bni_30 + (-1)Bound*bni_30] + [bni_30]x0[3] ≥ 0∧[(-1)bso_31] + x0[3] ≥ 0)

  
  
  We simplified constraint (37) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
  

  (38)    (0 ≥ 0 ⇒ (U^Increasing(223_0_F_STORE(-(x0[5], -(x1[5], 1)))), ≥)∧[(-1)bni_30 + (-1)Bound*bni_30] + [bni_30]x0[3] ≥ 0∧[(-1)bso_31] + x0[3] ≥ 0)

  
  
  We simplified constraint (38) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
  

  (39)    (0 ≥ 0 ⇒ (U^Increasing(223_0_F_STORE(-(x0[5], -(x1[5], 1)))), ≥)∧[(-1)bni_30 + (-1)Bound*bni_30] + [bni_30]x0[3] ≥ 0∧[(-1)bso_31] + x0[3] ≥ 0)

  
  
  We simplified constraint (39) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint:
  

  (40)    (0 ≥ 0 ⇒ (U^Increasing(223_0_F_STORE(-(x0[5], -(x1[5], 1)))), ≥)∧0 ≥ 0∧[bni_30] ≥ 0∧[(-1)bni_30 + (-1)Bound*bni_30] ≥ 0∧0 ≥ 0∧[1] ≥ 0∧[(-1)bso_31] ≥ 0)

  
  
  

For Pair 370_1_F_INVOKEMETHOD(341_0_f_Return, x0, x1) → COND_370_1_F_INVOKEMETHOD1(&&(&&(>(x1, 0), >(x0, 0)), <(0, -(x1, 1))), 341_0_f_Return, x0, x1) the following chains were created:

• We consider the chain 370_1_F_INVOKEMETHOD(341_0_f_Return, x0[6], x1[6]) → COND_370_1_F_INVOKEMETHOD1(&&(&&(>(x1[6], 0), >(x0[6], 0)), <(0, -(x1[6], 1))), 341_0_f_Return, x0[6], x1[6]), COND_370_1_F_INVOKEMETHOD1(TRUE, 341_0_f_Return, x0[7], x1[7]) → 370_1_F_INVOKEMETHOD(223_0_f_Store(-(x0[7], -(x1[7], 1))), x0[7], -(x1[7], 1)) which results in the following constraint:
  

  (41)    (&&(&&(>(x1[6], 0), >(x0[6], 0)), <(0, -(x1[6], 1)))=TRUE∧x0[6]=x0[7]∧x1[6]=x1[7] ⇒ 370_1_F_INVOKEMETHOD(341_0_f_Return, x0[6], x1[6])≥NonInfC∧370_1_F_INVOKEMETHOD(341_0_f_Return, x0[6], x1[6])≥COND_370_1_F_INVOKEMETHOD1(&&(&&(>(x1[6], 0), >(x0[6], 0)), <(0, -(x1[6], 1))), 341_0_f_Return, x0[6], x1[6])∧(U^Increasing(COND_370_1_F_INVOKEMETHOD1(&&(&&(>(x1[6], 0), >(x0[6], 0)), <(0, -(x1[6], 1))), 341_0_f_Return, x0[6], x1[6])), ≥))

  
  
  We simplified constraint (41) using rule (IV) which results in the following new constraint:
  

  (42)    (&&(&&(>(x1[6], 0), >(x0[6], 0)), <(0, -(x1[6], 1)))=TRUE ⇒ 370_1_F_INVOKEMETHOD(341_0_f_Return, x0[6], x1[6])≥NonInfC∧370_1_F_INVOKEMETHOD(341_0_f_Return, x0[6], x1[6])≥COND_370_1_F_INVOKEMETHOD1(&&(&&(>(x1[6], 0), >(x0[6], 0)), <(0, -(x1[6], 1))), 341_0_f_Return, x0[6], x1[6])∧(U^Increasing(COND_370_1_F_INVOKEMETHOD1(&&(&&(>(x1[6], 0), >(x0[6], 0)), <(0, -(x1[6], 1))), 341_0_f_Return, x0[6], x1[6])), ≥))

  
  
  We simplified constraint (42) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
  

  (43)    (0 ≥ 0 ⇒ (U^Increasing(COND_370_1_F_INVOKEMETHOD1(&&(&&(>(x1[6], 0), >(x0[6], 0)), <(0, -(x1[6], 1))), 341_0_f_Return, x0[6], x1[6])), ≥)∧[(-1)Bound*bni_32] + [bni_32]x1[6] + [(2)bni_32]x0[6] ≥ 0∧[1 + (-1)bso_33] + x1[6] + [2]x0[6] ≥ 0)

  
  
  We simplified constraint (43) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
  

  (44)    (0 ≥ 0 ⇒ (U^Increasing(COND_370_1_F_INVOKEMETHOD1(&&(&&(>(x1[6], 0), >(x0[6], 0)), <(0, -(x1[6], 1))), 341_0_f_Return, x0[6], x1[6])), ≥)∧[(-1)Bound*bni_32] + [bni_32]x1[6] + [(2)bni_32]x0[6] ≥ 0∧[1 + (-1)bso_33] + x1[6] + [2]x0[6] ≥ 0)

  
  
  We simplified constraint (44) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
  

  (45)    (0 ≥ 0 ⇒ (U^Increasing(COND_370_1_F_INVOKEMETHOD1(&&(&&(>(x1[6], 0), >(x0[6], 0)), <(0, -(x1[6], 1))), 341_0_f_Return, x0[6], x1[6])), ≥)∧[(-1)Bound*bni_32] + [bni_32]x1[6] + [(2)bni_32]x0[6] ≥ 0∧[1 + (-1)bso_33] + x1[6] + [2]x0[6] ≥ 0)

  
  
  We simplified constraint (45) using rules (IDP_UNRESTRICTED_VARS), (IDP_POLY_GCD) which results in the following new constraint:
  

  (46)    (0 ≥ 0 ⇒ (U^Increasing(COND_370_1_F_INVOKEMETHOD1(&&(&&(>(x1[6], 0), >(x0[6], 0)), <(0, -(x1[6], 1))), 341_0_f_Return, x0[6], x1[6])), ≥)∧[bni_32] ≥ 0∧[(2)bni_32] ≥ 0∧[(-1)Bound*bni_32] ≥ 0∧[1] ≥ 0∧[1 + (-1)bso_33] ≥ 0∧[1] ≥ 0)

  
  
  
• We consider the chain 370_1_F_INVOKEMETHOD(341_0_f_Return, x0[6], x1[6]) → COND_370_1_F_INVOKEMETHOD1(&&(&&(>(x1[6], 0), >(x0[6], 0)), <(0, -(x1[6], 1))), 341_0_f_Return, x0[6], x1[6]), COND_370_1_F_INVOKEMETHOD1(TRUE, 341_0_f_Return, x0[8], x1[8]) → 223_0_F_STORE(-(x0[8], -(x1[8], 1))) which results in the following constraint:
  

  (47)    (&&(&&(>(x1[6], 0), >(x0[6], 0)), <(0, -(x1[6], 1)))=TRUE∧x0[6]=x0[8]∧x1[6]=x1[8] ⇒ 370_1_F_INVOKEMETHOD(341_0_f_Return, x0[6], x1[6])≥NonInfC∧370_1_F_INVOKEMETHOD(341_0_f_Return, x0[6], x1[6])≥COND_370_1_F_INVOKEMETHOD1(&&(&&(>(x1[6], 0), >(x0[6], 0)), <(0, -(x1[6], 1))), 341_0_f_Return, x0[6], x1[6])∧(U^Increasing(COND_370_1_F_INVOKEMETHOD1(&&(&&(>(x1[6], 0), >(x0[6], 0)), <(0, -(x1[6], 1))), 341_0_f_Return, x0[6], x1[6])), ≥))

  
  
  We simplified constraint (47) using rule (IV) which results in the following new constraint:
  

  (48)    (&&(&&(>(x1[6], 0), >(x0[6], 0)), <(0, -(x1[6], 1)))=TRUE ⇒ 370_1_F_INVOKEMETHOD(341_0_f_Return, x0[6], x1[6])≥NonInfC∧370_1_F_INVOKEMETHOD(341_0_f_Return, x0[6], x1[6])≥COND_370_1_F_INVOKEMETHOD1(&&(&&(>(x1[6], 0), >(x0[6], 0)), <(0, -(x1[6], 1))), 341_0_f_Return, x0[6], x1[6])∧(U^Increasing(COND_370_1_F_INVOKEMETHOD1(&&(&&(>(x1[6], 0), >(x0[6], 0)), <(0, -(x1[6], 1))), 341_0_f_Return, x0[6], x1[6])), ≥))

  
  
  We simplified constraint (48) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
  

  (49)    (0 ≥ 0 ⇒ (U^Increasing(COND_370_1_F_INVOKEMETHOD1(&&(&&(>(x1[6], 0), >(x0[6], 0)), <(0, -(x1[6], 1))), 341_0_f_Return, x0[6], x1[6])), ≥)∧[(-1)Bound*bni_32] + [bni_32]x1[6] + [(2)bni_32]x0[6] ≥ 0∧[1 + (-1)bso_33] + x1[6] + [2]x0[6] ≥ 0)

  
  
  We simplified constraint (49) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
  

  (50)    (0 ≥ 0 ⇒ (U^Increasing(COND_370_1_F_INVOKEMETHOD1(&&(&&(>(x1[6], 0), >(x0[6], 0)), <(0, -(x1[6], 1))), 341_0_f_Return, x0[6], x1[6])), ≥)∧[(-1)Bound*bni_32] + [bni_32]x1[6] + [(2)bni_32]x0[6] ≥ 0∧[1 + (-1)bso_33] + x1[6] + [2]x0[6] ≥ 0)

  
  
  We simplified constraint (50) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
  

  (51)    (0 ≥ 0 ⇒ (U^Increasing(COND_370_1_F_INVOKEMETHOD1(&&(&&(>(x1[6], 0), >(x0[6], 0)), <(0, -(x1[6], 1))), 341_0_f_Return, x0[6], x1[6])), ≥)∧[(-1)Bound*bni_32] + [bni_32]x1[6] + [(2)bni_32]x0[6] ≥ 0∧[1 + (-1)bso_33] + x1[6] + [2]x0[6] ≥ 0)

  
  
  We simplified constraint (51) using rules (IDP_UNRESTRICTED_VARS), (IDP_POLY_GCD) which results in the following new constraint:
  

  (52)    (0 ≥ 0 ⇒ (U^Increasing(COND_370_1_F_INVOKEMETHOD1(&&(&&(>(x1[6], 0), >(x0[6], 0)), <(0, -(x1[6], 1))), 341_0_f_Return, x0[6], x1[6])), ≥)∧[bni_32] ≥ 0∧[(2)bni_32] ≥ 0∧[(-1)Bound*bni_32] ≥ 0∧[1] ≥ 0∧[1 + (-1)bso_33] ≥ 0∧[1] ≥ 0)

  
  
  

For Pair COND_370_1_F_INVOKEMETHOD1(TRUE, 341_0_f_Return, x0, x1) → 370_1_F_INVOKEMETHOD(223_0_f_Store(-(x0, -(x1, 1))), x0, -(x1, 1)) the following chains were created:

• We consider the chain 370_1_F_INVOKEMETHOD(341_0_f_Return, x0[6], x1[6]) → COND_370_1_F_INVOKEMETHOD1(&&(&&(>(x1[6], 0), >(x0[6], 0)), <(0, -(x1[6], 1))), 341_0_f_Return, x0[6], x1[6]), COND_370_1_F_INVOKEMETHOD1(TRUE, 341_0_f_Return, x0[7], x1[7]) → 370_1_F_INVOKEMETHOD(223_0_f_Store(-(x0[7], -(x1[7], 1))), x0[7], -(x1[7], 1)), 370_1_F_INVOKEMETHOD(233_0_f_Return, x0[3], x1[3]) → COND_370_1_F_INVOKEMETHOD(&&(&&(>(x1[3], 0), >(x0[3], 0)), <(0, -(x1[3], 1))), 233_0_f_Return, x0[3], x1[3]) which results in the following constraint:
  

  (53)    (&&(&&(>(x1[6], 0), >(x0[6], 0)), <(0, -(x1[6], 1)))=TRUE∧x0[6]=x0[7]∧x1[6]=x1[7]∧223_0_f_Store(-(x0[7], -(x1[7], 1)))=233_0_f_Return∧x0[7]=x0[3]∧-(x1[7], 1)=x1[3] ⇒ COND_370_1_F_INVOKEMETHOD1(TRUE, 341_0_f_Return, x0[7], x1[7])≥NonInfC∧COND_370_1_F_INVOKEMETHOD1(TRUE, 341_0_f_Return, x0[7], x1[7])≥370_1_F_INVOKEMETHOD(223_0_f_Store(-(x0[7], -(x1[7], 1))), x0[7], -(x1[7], 1))∧(U^Increasing(370_1_F_INVOKEMETHOD(223_0_f_Store(-(x0[7], -(x1[7], 1))), x0[7], -(x1[7], 1))), ≥))

  
  
  We solved constraint (53) using rules (I), (II).
• We consider the chain 370_1_F_INVOKEMETHOD(341_0_f_Return, x0[6], x1[6]) → COND_370_1_F_INVOKEMETHOD1(&&(&&(>(x1[6], 0), >(x0[6], 0)), <(0, -(x1[6], 1))), 341_0_f_Return, x0[6], x1[6]), COND_370_1_F_INVOKEMETHOD1(TRUE, 341_0_f_Return, x0[7], x1[7]) → 370_1_F_INVOKEMETHOD(223_0_f_Store(-(x0[7], -(x1[7], 1))), x0[7], -(x1[7], 1)), 370_1_F_INVOKEMETHOD(341_0_f_Return, x0[6], x1[6]) → COND_370_1_F_INVOKEMETHOD1(&&(&&(>(x1[6], 0), >(x0[6], 0)), <(0, -(x1[6], 1))), 341_0_f_Return, x0[6], x1[6]) which results in the following constraint:
  

  (54)    (&&(&&(>(x1[6], 0), >(x0[6], 0)), <(0, -(x1[6], 1)))=TRUE∧x0[6]=x0[7]∧x1[6]=x1[7]∧223_0_f_Store(-(x0[7], -(x1[7], 1)))=341_0_f_Return∧x0[7]=x0[6]1∧-(x1[7], 1)=x1[6]1 ⇒ COND_370_1_F_INVOKEMETHOD1(TRUE, 341_0_f_Return, x0[7], x1[7])≥NonInfC∧COND_370_1_F_INVOKEMETHOD1(TRUE, 341_0_f_Return, x0[7], x1[7])≥370_1_F_INVOKEMETHOD(223_0_f_Store(-(x0[7], -(x1[7], 1))), x0[7], -(x1[7], 1))∧(U^Increasing(370_1_F_INVOKEMETHOD(223_0_f_Store(-(x0[7], -(x1[7], 1))), x0[7], -(x1[7], 1))), ≥))

  
  
  We solved constraint (54) using rules (I), (II).

For Pair COND_370_1_F_INVOKEMETHOD1(TRUE, 341_0_f_Return, x0, x1) → 223_0_F_STORE(-(x0, -(x1, 1))) the following chains were created:

• We consider the chain 370_1_F_INVOKEMETHOD(341_0_f_Return, x0[6], x1[6]) → COND_370_1_F_INVOKEMETHOD1(&&(&&(>(x1[6], 0), >(x0[6], 0)), <(0, -(x1[6], 1))), 341_0_f_Return, x0[6], x1[6]), COND_370_1_F_INVOKEMETHOD1(TRUE, 341_0_f_Return, x0[8], x1[8]) → 223_0_F_STORE(-(x0[8], -(x1[8], 1))), 223_0_F_STORE(x0[0]) → COND_223_0_F_STORE(&&(>(x0[0], 0), >(2, 0)), x0[0]) which results in the following constraint:
  

  (55)    (&&(&&(>(x1[6], 0), >(x0[6], 0)), <(0, -(x1[6], 1)))=TRUE∧x0[6]=x0[8]∧x1[6]=x1[8]∧-(x0[8], -(x1[8], 1))=x0[0] ⇒ COND_370_1_F_INVOKEMETHOD1(TRUE, 341_0_f_Return, x0[8], x1[8])≥NonInfC∧COND_370_1_F_INVOKEMETHOD1(TRUE, 341_0_f_Return, x0[8], x1[8])≥223_0_F_STORE(-(x0[8], -(x1[8], 1)))∧(U^Increasing(223_0_F_STORE(-(x0[8], -(x1[8], 1)))), ≥))

  
  
  We simplified constraint (55) using rules (III), (IV) which results in the following new constraint:
  

  (56)    (&&(&&(>(x1[6], 0), >(x0[6], 0)), <(0, -(x1[6], 1)))=TRUE ⇒ COND_370_1_F_INVOKEMETHOD1(TRUE, 341_0_f_Return, x0[6], x1[6])≥NonInfC∧COND_370_1_F_INVOKEMETHOD1(TRUE, 341_0_f_Return, x0[6], x1[6])≥223_0_F_STORE(-(x0[6], -(x1[6], 1)))∧(U^Increasing(223_0_F_STORE(-(x0[8], -(x1[8], 1)))), ≥))

  
  
  We simplified constraint (56) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
  

  (57)    (0 ≥ 0 ⇒ (U^Increasing(223_0_F_STORE(-(x0[8], -(x1[8], 1)))), ≥)∧[(-1)bni_34 + (-1)Bound*bni_34] ≥ 0∧[(-1)bso_35] ≥ 0)

  
  
  We simplified constraint (57) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
  

  (58)    (0 ≥ 0 ⇒ (U^Increasing(223_0_F_STORE(-(x0[8], -(x1[8], 1)))), ≥)∧[(-1)bni_34 + (-1)Bound*bni_34] ≥ 0∧[(-1)bso_35] ≥ 0)

  
  
  We simplified constraint (58) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
  

  (59)    (0 ≥ 0 ⇒ (U^Increasing(223_0_F_STORE(-(x0[8], -(x1[8], 1)))), ≥)∧[(-1)bni_34 + (-1)Bound*bni_34] ≥ 0∧[(-1)bso_35] ≥ 0)

  
  
  We simplified constraint (59) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint:
  

  (60)    (0 ≥ 0 ⇒ (U^Increasing(223_0_F_STORE(-(x0[8], -(x1[8], 1)))), ≥)∧0 ≥ 0∧0 ≥ 0∧[(-1)bni_34 + (-1)Bound*bni_34] ≥ 0∧0 ≥ 0∧0 ≥ 0∧[(-1)bso_35] ≥ 0)

  
  
  

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

• 223_0_F_STORE(x0) → COND_223_0_F_STORE(&&(>(x0, 0), >(2, 0)), x0)
  
  • (0 ≥ 0 ⇒ (U^Increasing(COND_223_0_F_STORE(&&(>(x0[0], 0), >(2, 0)), x0[0])), ≥)∧[bni_24] ≥ 0∧[(-1)bni_24 + (-1)Bound*bni_24] ≥ 0∧0 ≥ 0∧[(-1)bso_25] ≥ 0)
    
  • (0 ≥ 0 ⇒ (U^Increasing(COND_223_0_F_STORE(&&(>(x0[0], 0), >(2, 0)), x0[0])), ≥)∧[bni_24] ≥ 0∧[(-1)bni_24 + (-1)Bound*bni_24] ≥ 0∧0 ≥ 0∧[(-1)bso_25] ≥ 0)
    
  
  
• COND_223_0_F_STORE(TRUE, x0) → 370_1_F_INVOKEMETHOD(223_0_f_Store(-(x0, 2)), x0, 2)
  
  
• COND_223_0_F_STORE(TRUE, x0) → 223_0_F_STORE(-(x0, 2))
  
  • (0 ≥ 0 ⇒ (U^Increasing(223_0_F_STORE(-(x0[2], 2))), ≥)∧[bni_26] ≥ 0∧[(-1)bni_26 + (-1)Bound*bni_26] ≥ 0∧[1] ≥ 0∧[(-1)bso_27] ≥ 0)
    
  
  
• 370_1_F_INVOKEMETHOD(233_0_f_Return, x0, x1) → COND_370_1_F_INVOKEMETHOD(&&(&&(>(x1, 0), >(x0, 0)), <(0, -(x1, 1))), 233_0_f_Return, x0, x1)
  
  • (0 ≥ 0 ⇒ (U^Increasing(COND_370_1_F_INVOKEMETHOD(&&(&&(>(x1[3], 0), >(x0[3], 0)), <(0, -(x1[3], 1))), 233_0_f_Return, x0[3], x1[3])), ≥)∧[bni_28] ≥ 0∧[(2)bni_28] ≥ 0∧[(-1)Bound*bni_28] ≥ 0∧[1] ≥ 0∧[1] ≥ 0∧[1 + (-1)bso_29] ≥ 0)
    
  • (0 ≥ 0 ⇒ (U^Increasing(COND_370_1_F_INVOKEMETHOD(&&(&&(>(x1[3], 0), >(x0[3], 0)), <(0, -(x1[3], 1))), 233_0_f_Return, x0[3], x1[3])), ≥)∧[bni_28] ≥ 0∧[(2)bni_28] ≥ 0∧[(-1)Bound*bni_28] ≥ 0∧[1] ≥ 0∧[1] ≥ 0∧[1 + (-1)bso_29] ≥ 0)
    
  
  
• COND_370_1_F_INVOKEMETHOD(TRUE, 233_0_f_Return, x0, x1) → 370_1_F_INVOKEMETHOD(223_0_f_Store(-(x0, -(x1, 1))), x0, -(x1, 1))
  
  
• COND_370_1_F_INVOKEMETHOD(TRUE, 233_0_f_Return, x0, x1) → 223_0_F_STORE(-(x0, -(x1, 1)))
  
  • (0 ≥ 0 ⇒ (U^Increasing(223_0_F_STORE(-(x0[5], -(x1[5], 1)))), ≥)∧0 ≥ 0∧[bni_30] ≥ 0∧[(-1)bni_30 + (-1)Bound*bni_30] ≥ 0∧0 ≥ 0∧[1] ≥ 0∧[(-1)bso_31] ≥ 0)
    
  
  
• 370_1_F_INVOKEMETHOD(341_0_f_Return, x0, x1) → COND_370_1_F_INVOKEMETHOD1(&&(&&(>(x1, 0), >(x0, 0)), <(0, -(x1, 1))), 341_0_f_Return, x0, x1)
  
  • (0 ≥ 0 ⇒ (U^Increasing(COND_370_1_F_INVOKEMETHOD1(&&(&&(>(x1[6], 0), >(x0[6], 0)), <(0, -(x1[6], 1))), 341_0_f_Return, x0[6], x1[6])), ≥)∧[bni_32] ≥ 0∧[(2)bni_32] ≥ 0∧[(-1)Bound*bni_32] ≥ 0∧[1] ≥ 0∧[1 + (-1)bso_33] ≥ 0∧[1] ≥ 0)
    
  • (0 ≥ 0 ⇒ (U^Increasing(COND_370_1_F_INVOKEMETHOD1(&&(&&(>(x1[6], 0), >(x0[6], 0)), <(0, -(x1[6], 1))), 341_0_f_Return, x0[6], x1[6])), ≥)∧[bni_32] ≥ 0∧[(2)bni_32] ≥ 0∧[(-1)Bound*bni_32] ≥ 0∧[1] ≥ 0∧[1 + (-1)bso_33] ≥ 0∧[1] ≥ 0)
    
  
  
• COND_370_1_F_INVOKEMETHOD1(TRUE, 341_0_f_Return, x0, x1) → 370_1_F_INVOKEMETHOD(223_0_f_Store(-(x0, -(x1, 1))), x0, -(x1, 1))
  
  
• COND_370_1_F_INVOKEMETHOD1(TRUE, 341_0_f_Return, x0, x1) → 223_0_F_STORE(-(x0, -(x1, 1)))
  
  • (0 ≥ 0 ⇒ (U^Increasing(223_0_F_STORE(-(x0[8], -(x1[8], 1)))), ≥)∧0 ≥ 0∧0 ≥ 0∧[(-1)bni_34 + (-1)Bound*bni_34] ≥ 0∧0 ≥ 0∧0 ≥ 0∧[(-1)bso_35] ≥ 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 with natural coefficients for non-tuple symbols [NONINF][POLO]:

POL(TRUE) = 0   
POL(FALSE) = 0   
POL(223_0_F_STORE(x₁)) = [-1] + x₁   
POL(COND_223_0_F_STORE(x₁, x₂)) = [-1] + x₂ + [-1]x₁   
POL(&&(x₁, x₂)) = 0   
POL(>(x₁, x₂)) = 0   
POL(0) = 0   
POL(2) = 0   
POL(370_1_F_INVOKEMETHOD(x₁, x₂, x₃)) = x₃ + [2]x₂ + [-1]x₁   
POL(223_0_f_Store(x₁)) = 0   
POL(-(x₁, x₂)) = 0   
POL(233_0_f_Return) = 0   
POL(COND_370_1_F_INVOKEMETHOD(x₁, x₂, x₃, x₄)) = [-1] + x₃ + [-1]x₂ + [2]x₁   
POL(<(x₁, x₂)) = 0   
POL(1) = 0   
POL(341_0_f_Return) = 0   
POL(COND_370_1_F_INVOKEMETHOD1(x₁, x₂, x₃, x₄)) = [-1] + [-1]x₂ + [-1]x₁   

The following pairs are in P_>:

COND_223_0_F_STORE(TRUE, x0[1]) → 370_1_F_INVOKEMETHOD(223_0_f_Store(-(x0[1], 2)), x0[1], 2)
370_1_F_INVOKEMETHOD(233_0_f_Return, x0[3], x1[3]) → COND_370_1_F_INVOKEMETHOD(&&(&&(>(x1[3], 0), >(x0[3], 0)), <(0, -(x1[3], 1))), 233_0_f_Return, x0[3], x1[3])
COND_370_1_F_INVOKEMETHOD(TRUE, 233_0_f_Return, x0[4], x1[4]) → 370_1_F_INVOKEMETHOD(223_0_f_Store(-(x0[4], -(x1[4], 1))), x0[4], -(x1[4], 1))
370_1_F_INVOKEMETHOD(341_0_f_Return, x0[6], x1[6]) → COND_370_1_F_INVOKEMETHOD1(&&(&&(>(x1[6], 0), >(x0[6], 0)), <(0, -(x1[6], 1))), 341_0_f_Return, x0[6], x1[6])
COND_370_1_F_INVOKEMETHOD1(TRUE, 341_0_f_Return, x0[7], x1[7]) → 370_1_F_INVOKEMETHOD(223_0_f_Store(-(x0[7], -(x1[7], 1))), x0[7], -(x1[7], 1))

The following pairs are in P_bound:

223_0_F_STORE(x0[0]) → COND_223_0_F_STORE(&&(>(x0[0], 0), >(2, 0)), x0[0])
COND_223_0_F_STORE(TRUE, x0[1]) → 370_1_F_INVOKEMETHOD(223_0_f_Store(-(x0[1], 2)), x0[1], 2)
COND_223_0_F_STORE(TRUE, x0[2]) → 223_0_F_STORE(-(x0[2], 2))
370_1_F_INVOKEMETHOD(233_0_f_Return, x0[3], x1[3]) → COND_370_1_F_INVOKEMETHOD(&&(&&(>(x1[3], 0), >(x0[3], 0)), <(0, -(x1[3], 1))), 233_0_f_Return, x0[3], x1[3])
COND_370_1_F_INVOKEMETHOD(TRUE, 233_0_f_Return, x0[4], x1[4]) → 370_1_F_INVOKEMETHOD(223_0_f_Store(-(x0[4], -(x1[4], 1))), x0[4], -(x1[4], 1))
COND_370_1_F_INVOKEMETHOD(TRUE, 233_0_f_Return, x0[5], x1[5]) → 223_0_F_STORE(-(x0[5], -(x1[5], 1)))
370_1_F_INVOKEMETHOD(341_0_f_Return, x0[6], x1[6]) → COND_370_1_F_INVOKEMETHOD1(&&(&&(>(x1[6], 0), >(x0[6], 0)), <(0, -(x1[6], 1))), 341_0_f_Return, x0[6], x1[6])
COND_370_1_F_INVOKEMETHOD1(TRUE, 341_0_f_Return, x0[7], x1[7]) → 370_1_F_INVOKEMETHOD(223_0_f_Store(-(x0[7], -(x1[7], 1))), x0[7], -(x1[7], 1))
COND_370_1_F_INVOKEMETHOD1(TRUE, 341_0_f_Return, x0[8], x1[8]) → 223_0_F_STORE(-(x0[8], -(x1[8], 1)))

The following pairs are in P_≥:

223_0_F_STORE(x0[0]) → COND_223_0_F_STORE(&&(>(x0[0], 0), >(2, 0)), x0[0])
COND_223_0_F_STORE(TRUE, x0[2]) → 223_0_F_STORE(-(x0[2], 2))
COND_370_1_F_INVOKEMETHOD(TRUE, 233_0_f_Return, x0[5], x1[5]) → 223_0_F_STORE(-(x0[5], -(x1[5], 1)))
COND_370_1_F_INVOKEMETHOD1(TRUE, 341_0_f_Return, x0[8], x1[8]) → 223_0_F_STORE(-(x0[8], -(x1[8], 1)))

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

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

(6) Obligation:

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

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

The following domains are used:

Boolean, Integer

R is empty.

The integer pair graph contains the following rules and edges:
(0): 223_0_F_STORE(x0[0]) → COND_223_0_F_STORE(x0[0] > 0 && 2 > 0, x0[0])
(2): COND_223_0_F_STORE(TRUE, x0[2]) → 223_0_F_STORE(x0[2] - 2)
(5): COND_370_1_F_INVOKEMETHOD(TRUE, 233_0_f_Return, x0[5], x1[5]) → 223_0_F_STORE(x0[5] - x1[5] - 1)
(8): COND_370_1_F_INVOKEMETHOD1(TRUE, 341_0_f_Return, x0[8], x1[8]) → 223_0_F_STORE(x0[8] - x1[8] - 1)

(2) -> (0), if ((x0[2] - 2 →^* x0[0]))

(5) -> (0), if ((x0[5] - x1[5] - 1 →^* x0[0]))

(8) -> (0), if ((x0[8] - x1[8] - 1 →^* x0[0]))

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

The set Q is empty.

(7) IDependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 2 less nodes.

(8) Obligation:

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

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

The following domains are used:

Integer, Boolean

R is empty.

The integer pair graph contains the following rules and edges:
(2): COND_223_0_F_STORE(TRUE, x0[2]) → 223_0_F_STORE(x0[2] - 2)
(0): 223_0_F_STORE(x0[0]) → COND_223_0_F_STORE(x0[0] > 0 && 2 > 0, x0[0])

(2) -> (0), if ((x0[2] - 2 →^* x0[0]))

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

The set Q is empty.

(9) 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_223_0_F_STORE(TRUE, x0[2]) → 223_0_F_STORE(-(x0[2], 2)) the following chains were created:

• We consider the chain 223_0_F_STORE(x0[0]) → COND_223_0_F_STORE(&&(>(x0[0], 0), >(2, 0)), x0[0]), COND_223_0_F_STORE(TRUE, x0[2]) → 223_0_F_STORE(-(x0[2], 2)), 223_0_F_STORE(x0[0]) → COND_223_0_F_STORE(&&(>(x0[0], 0), >(2, 0)), x0[0]) which results in the following constraint:
  

  (1)    (&&(>(x0[0], 0), >(2, 0))=TRUE∧x0[0]=x0[2]∧-(x0[2], 2)=x0[0]1 ⇒ COND_223_0_F_STORE(TRUE, x0[2])≥NonInfC∧COND_223_0_F_STORE(TRUE, x0[2])≥223_0_F_STORE(-(x0[2], 2))∧(U^Increasing(223_0_F_STORE(-(x0[2], 2))), ≥))

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

  (2)    (>(x0[0], 0)=TRUE ⇒ COND_223_0_F_STORE(TRUE, x0[0])≥NonInfC∧COND_223_0_F_STORE(TRUE, x0[0])≥223_0_F_STORE(-(x0[0], 2))∧(U^Increasing(223_0_F_STORE(-(x0[2], 2))), ≥))

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

  (3)    (x0[0] + [-1] ≥ 0 ⇒ (U^Increasing(223_0_F_STORE(-(x0[2], 2))), ≥)∧[(-1)bni_10 + (-1)Bound*bni_10] + [bni_10]x0[0] ≥ 0∧[2 + (-1)bso_11] ≥ 0)

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

  (4)    (x0[0] + [-1] ≥ 0 ⇒ (U^Increasing(223_0_F_STORE(-(x0[2], 2))), ≥)∧[(-1)bni_10 + (-1)Bound*bni_10] + [bni_10]x0[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)    (x0[0] + [-1] ≥ 0 ⇒ (U^Increasing(223_0_F_STORE(-(x0[2], 2))), ≥)∧[(-1)bni_10 + (-1)Bound*bni_10] + [bni_10]x0[0] ≥ 0∧[2 + (-1)bso_11] ≥ 0)

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

  (6)    (x0[0] ≥ 0 ⇒ (U^Increasing(223_0_F_STORE(-(x0[2], 2))), ≥)∧[(-1)Bound*bni_10] + [bni_10]x0[0] ≥ 0∧[2 + (-1)bso_11] ≥ 0)

  
  
  

For Pair 223_0_F_STORE(x0[0]) → COND_223_0_F_STORE(&&(>(x0[0], 0), >(2, 0)), x0[0]) the following chains were created:

• We consider the chain 223_0_F_STORE(x0[0]) → COND_223_0_F_STORE(&&(>(x0[0], 0), >(2, 0)), x0[0]), COND_223_0_F_STORE(TRUE, x0[2]) → 223_0_F_STORE(-(x0[2], 2)) which results in the following constraint:
  

  (7)    (&&(>(x0[0], 0), >(2, 0))=TRUE∧x0[0]=x0[2] ⇒ 223_0_F_STORE(x0[0])≥NonInfC∧223_0_F_STORE(x0[0])≥COND_223_0_F_STORE(&&(>(x0[0], 0), >(2, 0)), x0[0])∧(U^Increasing(COND_223_0_F_STORE(&&(>(x0[0], 0), >(2, 0)), x0[0])), ≥))

  
  
  We simplified constraint (7) using rules (IV), (IDP_CONSTANT_FOLD), (DELETE_TRIVIAL_REDUCESTO), (IDP_BOOLEAN) which results in the following new constraint:
  

  (8)    (>(x0[0], 0)=TRUE ⇒ 223_0_F_STORE(x0[0])≥NonInfC∧223_0_F_STORE(x0[0])≥COND_223_0_F_STORE(&&(>(x0[0], 0), >(2, 0)), x0[0])∧(U^Increasing(COND_223_0_F_STORE(&&(>(x0[0], 0), >(2, 0)), x0[0])), ≥))

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

  (9)    (x0[0] + [-1] ≥ 0 ⇒ (U^Increasing(COND_223_0_F_STORE(&&(>(x0[0], 0), >(2, 0)), x0[0])), ≥)∧[(-1)bni_12 + (-1)Bound*bni_12] + [bni_12]x0[0] ≥ 0∧[(-1)bso_13] ≥ 0)

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

  (10)    (x0[0] + [-1] ≥ 0 ⇒ (U^Increasing(COND_223_0_F_STORE(&&(>(x0[0], 0), >(2, 0)), x0[0])), ≥)∧[(-1)bni_12 + (-1)Bound*bni_12] + [bni_12]x0[0] ≥ 0∧[(-1)bso_13] ≥ 0)

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

  (11)    (x0[0] + [-1] ≥ 0 ⇒ (U^Increasing(COND_223_0_F_STORE(&&(>(x0[0], 0), >(2, 0)), x0[0])), ≥)∧[(-1)bni_12 + (-1)Bound*bni_12] + [bni_12]x0[0] ≥ 0∧[(-1)bso_13] ≥ 0)

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

  (12)    (x0[0] ≥ 0 ⇒ (U^Increasing(COND_223_0_F_STORE(&&(>(x0[0], 0), >(2, 0)), x0[0])), ≥)∧[(-1)Bound*bni_12] + [bni_12]x0[0] ≥ 0∧[(-1)bso_13] ≥ 0)

  
  
  

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

• COND_223_0_F_STORE(TRUE, x0[2]) → 223_0_F_STORE(-(x0[2], 2))
  
  • (x0[0] ≥ 0 ⇒ (U^Increasing(223_0_F_STORE(-(x0[2], 2))), ≥)∧[(-1)Bound*bni_10] + [bni_10]x0[0] ≥ 0∧[2 + (-1)bso_11] ≥ 0)
    
  
  
• 223_0_F_STORE(x0[0]) → COND_223_0_F_STORE(&&(>(x0[0], 0), >(2, 0)), x0[0])
  
  • (x0[0] ≥ 0 ⇒ (U^Increasing(COND_223_0_F_STORE(&&(>(x0[0], 0), >(2, 0)), x0[0])), ≥)∧[(-1)Bound*bni_12] + [bni_12]x0[0] ≥ 0∧[(-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) = [1]   
POL(COND_223_0_F_STORE(x₁, x₂)) = [-1] + x₂ + [-1]x₁   
POL(223_0_F_STORE(x₁)) = [-1] + x₁   
POL(-(x₁, x₂)) = x₁ + [-1]x₂   
POL(2) = [2]   
POL(&&(x₁, x₂)) = 0   
POL(>(x₁, x₂)) = [-1]   
POL(0) = 0   

The following pairs are in P_>:

COND_223_0_F_STORE(TRUE, x0[2]) → 223_0_F_STORE(-(x0[2], 2))

The following pairs are in P_bound:

COND_223_0_F_STORE(TRUE, x0[2]) → 223_0_F_STORE(-(x0[2], 2))
223_0_F_STORE(x0[0]) → COND_223_0_F_STORE(&&(>(x0[0], 0), >(2, 0)), x0[0])

The following pairs are in P_≥:

223_0_F_STORE(x0[0]) → COND_223_0_F_STORE(&&(>(x0[0], 0), >(2, 0)), x0[0])

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

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

(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): 223_0_F_STORE(x0[0]) → COND_223_0_F_STORE(x0[0] > 0 && 2 > 0, x0[0])


The set Q is empty.

(11) IDependencyGraphProof (EQUIVALENT transformation)

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

(12) TRUE