Termination of the given JBC could be proven:

(0) Obligation:

JBC Problem based on JBC Program:
Manifest-Version: 1.0 Created-By: 1.6.0_16 (Sun Microsystems Inc.) Main-Class: PastaC1

/**
 * Example taken from "A Term Rewriting Approach to the Automated Termination
 * Analysis of Imperative Programs" (http://www.cs.unm.edu/~spf/papers/2009-02.pdf)
 * and converted to Java.
 */

public class PastaC1 {
    public static void main(String[] args) {
        Random.args = args;
        int x = Random.random();

		while (x >= 0) {
			int y = 1;
			while (x > y) {
				y = 2*y;
			}
			x--;
		}
    }
}


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

  public static int random() {
    String string = args[index];
    index++;
    return string.length();
  }
}

(1) JBCToGraph (SOUND transformation)

Constructed TerminationGraph.

(2) Obligation:

Termination Graph based on JBC Program:
PastaC1.main([Ljava/lang/String;)V: Graph of 124 nodes with 1 SCC.

(3) TerminationGraphToSCCProof (SOUND transformation)

Splitted TerminationGraph to 1 SCCs.

(4) Obligation:

SCC of termination graph based on JBC Program.
SCC contains nodes from the following methods: PastaC1.main([Ljava/lang/String;)V
SCC calls the following helper methods:
Performed SCC analyses: UsedFieldsAnalysis

(5) SCCToIDPv1Proof (SOUND transformation)

Transformed FIGraph SCCs to IDPs. Log:

Generated 24 rules for P and 0 rules for R.

P rules:
224_0_main_LT(EOS(STATIC_224), i32, i32) → 233_0_main_LT(EOS(STATIC_233), i32, i32)
233_0_main_LT(EOS(STATIC_233), i32, i32) → 242_0_main_ConstantStackPush(EOS(STATIC_242), i32) | >=(i32, 0)
242_0_main_ConstantStackPush(EOS(STATIC_242), i32) → 248_0_main_Store(EOS(STATIC_248), i32, 1)
248_0_main_Store(EOS(STATIC_248), i32, matching1) → 253_0_main_Load(EOS(STATIC_253), i32, 1) | =(matching1, 1)
253_0_main_Load(EOS(STATIC_253), i32, matching1) → 611_0_main_Load(EOS(STATIC_611), i32, 1) | =(matching1, 1)
611_0_main_Load(EOS(STATIC_611), i91, i92) → 715_0_main_Load(EOS(STATIC_715), i91, i92)
715_0_main_Load(EOS(STATIC_715), i91, i132) → 806_0_main_Load(EOS(STATIC_806), i91, i132)
806_0_main_Load(EOS(STATIC_806), i91, i169) → 891_0_main_Load(EOS(STATIC_891), i91, i169)
891_0_main_Load(EOS(STATIC_891), i91, i206) → 893_0_main_Load(EOS(STATIC_893), i91, i206, i91)
893_0_main_Load(EOS(STATIC_893), i91, i206, i91) → 894_0_main_LE(EOS(STATIC_894), i91, i206, i91, i206)
894_0_main_LE(EOS(STATIC_894), i91, i206, i91, i206) → 895_0_main_LE(EOS(STATIC_895), i91, i206, i91, i206)
894_0_main_LE(EOS(STATIC_894), i91, i206, i91, i206) → 896_0_main_LE(EOS(STATIC_896), i91, i206, i91, i206)
895_0_main_LE(EOS(STATIC_895), i91, i206, i91, i206) → 898_0_main_Inc(EOS(STATIC_898), i91) | <=(i91, i206)
898_0_main_Inc(EOS(STATIC_898), i91) → 900_0_main_JMP(EOS(STATIC_900), +(i91, -1)) | >=(i91, 0)
900_0_main_JMP(EOS(STATIC_900), i208) → 903_0_main_Load(EOS(STATIC_903), i208)
903_0_main_Load(EOS(STATIC_903), i208) → 213_0_main_Load(EOS(STATIC_213), i208)
213_0_main_Load(EOS(STATIC_213), i28) → 224_0_main_LT(EOS(STATIC_224), i28, i28)
896_0_main_LE(EOS(STATIC_896), i91, i206, i91, i206) → 899_0_main_ConstantStackPush(EOS(STATIC_899), i91, i206) | >(i91, i206)
899_0_main_ConstantStackPush(EOS(STATIC_899), i91, i206) → 902_0_main_Load(EOS(STATIC_902), i91, i206, 2)
902_0_main_Load(EOS(STATIC_902), i91, i206, matching1) → 905_0_main_IntArithmetic(EOS(STATIC_905), i91, 2, i206) | =(matching1, 2)
905_0_main_IntArithmetic(EOS(STATIC_905), i91, matching1, i206) → 1044_0_main_Store(EOS(STATIC_1044), i91, *(2, i206)) | &&(>=(i206, 1), =(matching1, 2))
1044_0_main_Store(EOS(STATIC_1044), i91, i245) → 1046_0_main_JMP(EOS(STATIC_1046), i91, i245)
1046_0_main_JMP(EOS(STATIC_1046), i91, i245) → 1050_0_main_Load(EOS(STATIC_1050), i91, i245)
1050_0_main_Load(EOS(STATIC_1050), i91, i245) → 891_0_main_Load(EOS(STATIC_891), i91, i245)
R rules:

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

P rules:
894_0_main_LE(EOS(STATIC_894), x0, x1, x0, x1) → 894_0_main_LE(EOS(STATIC_894), +(x0, -1), 1, +(x0, -1), 1) | &&(>=(x1, x0), >(+(x0, 1), 1))
894_0_main_LE(EOS(STATIC_894), x0, x1, x0, x1) → 894_0_main_LE(EOS(STATIC_894), x0, *(2, x1), x0, *(2, x1)) | &&(>(+(x1, 1), 1), <(x1, x0))
R rules:

Filtered ground terms:

894_0_main_LE(x1, x2, x3, x4, x5) → 894_0_main_LE(x2, x3, x4, x5)
EOS(x1) → EOS
Cond_894_0_main_LE1(x1, x2, x3, x4, x5, x6) → Cond_894_0_main_LE1(x1, x3, x4, x5, x6)
Cond_894_0_main_LE(x1, x2, x3, x4, x5, x6) → Cond_894_0_main_LE(x1, x3, x4, x5, x6)

Filtered duplicate args:

894_0_main_LE(x1, x2, x3, x4) → 894_0_main_LE(x3, x4)
Cond_894_0_main_LE(x1, x2, x3, x4, x5) → Cond_894_0_main_LE(x1, x4, x5)
Cond_894_0_main_LE1(x1, x2, x3, x4, x5) → Cond_894_0_main_LE1(x1, x4, x5)

Filtered unneeded arguments:

Cond_894_0_main_LE(x1, x2, x3) → Cond_894_0_main_LE(x1, x2)

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

P rules:
894_0_main_LE(x0, x1) → 894_0_main_LE(+(x0, -1), 1) | &&(>=(x1, x0), >(x0, 0))
894_0_main_LE(x0, x1) → 894_0_main_LE(x0, *(2, x1)) | &&(>(x1, 0), <(x1, x0))
R rules:

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

P rules:
894_0_MAIN_LE(x0, x1) → COND_894_0_MAIN_LE(&&(>=(x1, x0), >(x0, 0)), x0, x1)
COND_894_0_MAIN_LE(TRUE, x0, x1) → 894_0_MAIN_LE(+(x0, -1), 1)
894_0_MAIN_LE(x0, x1) → COND_894_0_MAIN_LE1(&&(>(x1, 0), <(x1, x0)), x0, x1)
COND_894_0_MAIN_LE1(TRUE, x0, x1) → 894_0_MAIN_LE(x0, *(2, x1))
R rules:

(6) Obligation:

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

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

The following domains are used:

Boolean, Integer

R is empty.

The integer pair graph contains the following rules and edges:
(0): 894_0_MAIN_LE(x0[0], x1[0]) → COND_894_0_MAIN_LE(x1[0] >= x0[0] && x0[0] > 0, x0[0], x1[0])
(1): COND_894_0_MAIN_LE(TRUE, x0[1], x1[1]) → 894_0_MAIN_LE(x0[1] + -1, 1)
(2): 894_0_MAIN_LE(x0[2], x1[2]) → COND_894_0_MAIN_LE1(x1[2] > 0 && x1[2] < x0[2], x0[2], x1[2])
(3): COND_894_0_MAIN_LE1(TRUE, x0[3], x1[3]) → 894_0_MAIN_LE(x0[3], 2 * x1[3])

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

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

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

(2) -> (3), if (x1[2] > 0 && x1[2] < x0[2] ∧x0[2] →^* x0[3]∧x1[2] →^* x1[3])

(3) -> (0), if (x0[3] →^* x0[0]∧2 * x1[3] →^* x1[0])

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

The set Q is empty.

(7) IDPNonInfProof (SOUND transformation)

Used the following options for this NonInfProof:
IDPGPoloSolver: Range: [(-1,2)] IsNat: false Interpretation Shape Heuristic: aprove.DPFramework.IDPProblem.Processors.nonInf.poly.IdpDefaultShapeHeuristic@d6206b5 Constraint Generator: NonInfConstraintGenerator: PathGenerator: MetricPathGenerator: Max Left Steps: 1 Max Right Steps: 1

The constraints were generated the following way:
The DP Problem is simplified using the Induction Calculus [NONINF] with the following steps:
Note that final constraints are written in bold face.


For Pair 894_0_MAIN_LE(x0, x1) → COND_894_0_MAIN_LE(&&(>=(x1, x0), >(x0, 0)), x0, x1) the following chains were created:

• We consider the chain 894_0_MAIN_LE(x0[0], x1[0]) → COND_894_0_MAIN_LE(&&(>=(x1[0], x0[0]), >(x0[0], 0)), x0[0], x1[0]), COND_894_0_MAIN_LE(TRUE, x0[1], x1[1]) → 894_0_MAIN_LE(+(x0[1], -1), 1) which results in the following constraint:
  

  (1)    (&&(>=(x1[0], x0[0]), >(x0[0], 0))=TRUE∧x0[0]=x0[1]∧x1[0]=x1[1] ⇒ 894_0_MAIN_LE(x0[0], x1[0])≥NonInfC∧894_0_MAIN_LE(x0[0], x1[0])≥COND_894_0_MAIN_LE(&&(>=(x1[0], x0[0]), >(x0[0], 0)), x0[0], x1[0])∧(U^Increasing(COND_894_0_MAIN_LE(&&(>=(x1[0], x0[0]), >(x0[0], 0)), x0[0], x1[0])), ≥))

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

  (2)    (>=(x1[0], x0[0])=TRUE∧>(x0[0], 0)=TRUE ⇒ 894_0_MAIN_LE(x0[0], x1[0])≥NonInfC∧894_0_MAIN_LE(x0[0], x1[0])≥COND_894_0_MAIN_LE(&&(>=(x1[0], x0[0]), >(x0[0], 0)), x0[0], x1[0])∧(U^Increasing(COND_894_0_MAIN_LE(&&(>=(x1[0], x0[0]), >(x0[0], 0)), x0[0], x1[0])), ≥))

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

  (3)    (x1[0] + [-1]x0[0] ≥ 0∧x0[0] + [-1] ≥ 0 ⇒ (U^Increasing(COND_894_0_MAIN_LE(&&(>=(x1[0], x0[0]), >(x0[0], 0)), x0[0], x1[0])), ≥)∧[(-1)bni_18 + (-1)Bound*bni_18] + [bni_18]x0[0] ≥ 0∧[(-1)bso_19] ≥ 0)

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

  (4)    (x1[0] + [-1]x0[0] ≥ 0∧x0[0] + [-1] ≥ 0 ⇒ (U^Increasing(COND_894_0_MAIN_LE(&&(>=(x1[0], x0[0]), >(x0[0], 0)), x0[0], x1[0])), ≥)∧[(-1)bni_18 + (-1)Bound*bni_18] + [bni_18]x0[0] ≥ 0∧[(-1)bso_19] ≥ 0)

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

  (5)    (x1[0] + [-1]x0[0] ≥ 0∧x0[0] + [-1] ≥ 0 ⇒ (U^Increasing(COND_894_0_MAIN_LE(&&(>=(x1[0], x0[0]), >(x0[0], 0)), x0[0], x1[0])), ≥)∧[(-1)bni_18 + (-1)Bound*bni_18] + [bni_18]x0[0] ≥ 0∧[(-1)bso_19] ≥ 0)

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

  (6)    (x1[0] ≥ 0∧x0[0] + [-1] ≥ 0 ⇒ (U^Increasing(COND_894_0_MAIN_LE(&&(>=(x1[0], x0[0]), >(x0[0], 0)), x0[0], x1[0])), ≥)∧[(-1)bni_18 + (-1)Bound*bni_18] + [bni_18]x0[0] ≥ 0∧[(-1)bso_19] ≥ 0)

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

  (7)    (x1[0] ≥ 0∧x0[0] ≥ 0 ⇒ (U^Increasing(COND_894_0_MAIN_LE(&&(>=(x1[0], x0[0]), >(x0[0], 0)), x0[0], x1[0])), ≥)∧[(-1)Bound*bni_18] + [bni_18]x0[0] ≥ 0∧[(-1)bso_19] ≥ 0)

  
  
  

For Pair COND_894_0_MAIN_LE(TRUE, x0, x1) → 894_0_MAIN_LE(+(x0, -1), 1) the following chains were created:

• We consider the chain 894_0_MAIN_LE(x0[0], x1[0]) → COND_894_0_MAIN_LE(&&(>=(x1[0], x0[0]), >(x0[0], 0)), x0[0], x1[0]), COND_894_0_MAIN_LE(TRUE, x0[1], x1[1]) → 894_0_MAIN_LE(+(x0[1], -1), 1), 894_0_MAIN_LE(x0[0], x1[0]) → COND_894_0_MAIN_LE(&&(>=(x1[0], x0[0]), >(x0[0], 0)), x0[0], x1[0]) which results in the following constraint:
  

  (8)    (&&(>=(x1[0], x0[0]), >(x0[0], 0))=TRUE∧x0[0]=x0[1]∧x1[0]=x1[1]∧+(x0[1], -1)=x0[0]1∧1=x1[0]1 ⇒ COND_894_0_MAIN_LE(TRUE, x0[1], x1[1])≥NonInfC∧COND_894_0_MAIN_LE(TRUE, x0[1], x1[1])≥894_0_MAIN_LE(+(x0[1], -1), 1)∧(U^Increasing(894_0_MAIN_LE(+(x0[1], -1), 1)), ≥))

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

  (9)    (>=(x1[0], x0[0])=TRUE∧>(x0[0], 0)=TRUE ⇒ COND_894_0_MAIN_LE(TRUE, x0[0], x1[0])≥NonInfC∧COND_894_0_MAIN_LE(TRUE, x0[0], x1[0])≥894_0_MAIN_LE(+(x0[0], -1), 1)∧(U^Increasing(894_0_MAIN_LE(+(x0[1], -1), 1)), ≥))

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

  (10)    (x1[0] + [-1]x0[0] ≥ 0∧x0[0] + [-1] ≥ 0 ⇒ (U^Increasing(894_0_MAIN_LE(+(x0[1], -1), 1)), ≥)∧[(-1)bni_20 + (-1)Bound*bni_20] + [bni_20]x0[0] ≥ 0∧[1 + (-1)bso_21] ≥ 0)

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

  (11)    (x1[0] + [-1]x0[0] ≥ 0∧x0[0] + [-1] ≥ 0 ⇒ (U^Increasing(894_0_MAIN_LE(+(x0[1], -1), 1)), ≥)∧[(-1)bni_20 + (-1)Bound*bni_20] + [bni_20]x0[0] ≥ 0∧[1 + (-1)bso_21] ≥ 0)

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

  (12)    (x1[0] + [-1]x0[0] ≥ 0∧x0[0] + [-1] ≥ 0 ⇒ (U^Increasing(894_0_MAIN_LE(+(x0[1], -1), 1)), ≥)∧[(-1)bni_20 + (-1)Bound*bni_20] + [bni_20]x0[0] ≥ 0∧[1 + (-1)bso_21] ≥ 0)

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

  (13)    (x1[0] ≥ 0∧x0[0] + [-1] ≥ 0 ⇒ (U^Increasing(894_0_MAIN_LE(+(x0[1], -1), 1)), ≥)∧[(-1)bni_20 + (-1)Bound*bni_20] + [bni_20]x0[0] ≥ 0∧[1 + (-1)bso_21] ≥ 0)

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

  (14)    (x1[0] ≥ 0∧x0[0] ≥ 0 ⇒ (U^Increasing(894_0_MAIN_LE(+(x0[1], -1), 1)), ≥)∧[(-1)Bound*bni_20] + [bni_20]x0[0] ≥ 0∧[1 + (-1)bso_21] ≥ 0)

  
  
  
• We consider the chain 894_0_MAIN_LE(x0[0], x1[0]) → COND_894_0_MAIN_LE(&&(>=(x1[0], x0[0]), >(x0[0], 0)), x0[0], x1[0]), COND_894_0_MAIN_LE(TRUE, x0[1], x1[1]) → 894_0_MAIN_LE(+(x0[1], -1), 1), 894_0_MAIN_LE(x0[2], x1[2]) → COND_894_0_MAIN_LE1(&&(>(x1[2], 0), <(x1[2], x0[2])), x0[2], x1[2]) which results in the following constraint:
  

  (15)    (&&(>=(x1[0], x0[0]), >(x0[0], 0))=TRUE∧x0[0]=x0[1]∧x1[0]=x1[1]∧+(x0[1], -1)=x0[2]∧1=x1[2] ⇒ COND_894_0_MAIN_LE(TRUE, x0[1], x1[1])≥NonInfC∧COND_894_0_MAIN_LE(TRUE, x0[1], x1[1])≥894_0_MAIN_LE(+(x0[1], -1), 1)∧(U^Increasing(894_0_MAIN_LE(+(x0[1], -1), 1)), ≥))

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

  (16)    (>=(x1[0], x0[0])=TRUE∧>(x0[0], 0)=TRUE ⇒ COND_894_0_MAIN_LE(TRUE, x0[0], x1[0])≥NonInfC∧COND_894_0_MAIN_LE(TRUE, x0[0], x1[0])≥894_0_MAIN_LE(+(x0[0], -1), 1)∧(U^Increasing(894_0_MAIN_LE(+(x0[1], -1), 1)), ≥))

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

  (17)    (x1[0] + [-1]x0[0] ≥ 0∧x0[0] + [-1] ≥ 0 ⇒ (U^Increasing(894_0_MAIN_LE(+(x0[1], -1), 1)), ≥)∧[(-1)bni_20 + (-1)Bound*bni_20] + [bni_20]x0[0] ≥ 0∧[1 + (-1)bso_21] ≥ 0)

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

  (18)    (x1[0] + [-1]x0[0] ≥ 0∧x0[0] + [-1] ≥ 0 ⇒ (U^Increasing(894_0_MAIN_LE(+(x0[1], -1), 1)), ≥)∧[(-1)bni_20 + (-1)Bound*bni_20] + [bni_20]x0[0] ≥ 0∧[1 + (-1)bso_21] ≥ 0)

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

  (19)    (x1[0] + [-1]x0[0] ≥ 0∧x0[0] + [-1] ≥ 0 ⇒ (U^Increasing(894_0_MAIN_LE(+(x0[1], -1), 1)), ≥)∧[(-1)bni_20 + (-1)Bound*bni_20] + [bni_20]x0[0] ≥ 0∧[1 + (-1)bso_21] ≥ 0)

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

  (20)    (x1[0] ≥ 0∧x0[0] + [-1] ≥ 0 ⇒ (U^Increasing(894_0_MAIN_LE(+(x0[1], -1), 1)), ≥)∧[(-1)bni_20 + (-1)Bound*bni_20] + [bni_20]x0[0] ≥ 0∧[1 + (-1)bso_21] ≥ 0)

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

  (21)    (x1[0] ≥ 0∧x0[0] ≥ 0 ⇒ (U^Increasing(894_0_MAIN_LE(+(x0[1], -1), 1)), ≥)∧[(-1)Bound*bni_20] + [bni_20]x0[0] ≥ 0∧[1 + (-1)bso_21] ≥ 0)

  
  
  

For Pair 894_0_MAIN_LE(x0, x1) → COND_894_0_MAIN_LE1(&&(>(x1, 0), <(x1, x0)), x0, x1) the following chains were created:

• We consider the chain 894_0_MAIN_LE(x0[2], x1[2]) → COND_894_0_MAIN_LE1(&&(>(x1[2], 0), <(x1[2], x0[2])), x0[2], x1[2]), COND_894_0_MAIN_LE1(TRUE, x0[3], x1[3]) → 894_0_MAIN_LE(x0[3], *(2, x1[3])) which results in the following constraint:
  

  (22)    (&&(>(x1[2], 0), <(x1[2], x0[2]))=TRUE∧x0[2]=x0[3]∧x1[2]=x1[3] ⇒ 894_0_MAIN_LE(x0[2], x1[2])≥NonInfC∧894_0_MAIN_LE(x0[2], x1[2])≥COND_894_0_MAIN_LE1(&&(>(x1[2], 0), <(x1[2], x0[2])), x0[2], x1[2])∧(U^Increasing(COND_894_0_MAIN_LE1(&&(>(x1[2], 0), <(x1[2], x0[2])), x0[2], x1[2])), ≥))

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

  (23)    (>(x1[2], 0)=TRUE∧<(x1[2], x0[2])=TRUE ⇒ 894_0_MAIN_LE(x0[2], x1[2])≥NonInfC∧894_0_MAIN_LE(x0[2], x1[2])≥COND_894_0_MAIN_LE1(&&(>(x1[2], 0), <(x1[2], x0[2])), x0[2], x1[2])∧(U^Increasing(COND_894_0_MAIN_LE1(&&(>(x1[2], 0), <(x1[2], x0[2])), x0[2], x1[2])), ≥))

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

  (24)    (x1[2] + [-1] ≥ 0∧x0[2] + [-1] + [-1]x1[2] ≥ 0 ⇒ (U^Increasing(COND_894_0_MAIN_LE1(&&(>(x1[2], 0), <(x1[2], x0[2])), x0[2], x1[2])), ≥)∧[(-1)bni_22 + (-1)Bound*bni_22] + [bni_22]x0[2] ≥ 0∧[(-1)bso_23] ≥ 0)

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

  (25)    (x1[2] + [-1] ≥ 0∧x0[2] + [-1] + [-1]x1[2] ≥ 0 ⇒ (U^Increasing(COND_894_0_MAIN_LE1(&&(>(x1[2], 0), <(x1[2], x0[2])), x0[2], x1[2])), ≥)∧[(-1)bni_22 + (-1)Bound*bni_22] + [bni_22]x0[2] ≥ 0∧[(-1)bso_23] ≥ 0)

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

  (26)    (x1[2] + [-1] ≥ 0∧x0[2] + [-1] + [-1]x1[2] ≥ 0 ⇒ (U^Increasing(COND_894_0_MAIN_LE1(&&(>(x1[2], 0), <(x1[2], x0[2])), x0[2], x1[2])), ≥)∧[(-1)bni_22 + (-1)Bound*bni_22] + [bni_22]x0[2] ≥ 0∧[(-1)bso_23] ≥ 0)

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

  (27)    (x1[2] ≥ 0∧x0[2] + [-2] + [-1]x1[2] ≥ 0 ⇒ (U^Increasing(COND_894_0_MAIN_LE1(&&(>(x1[2], 0), <(x1[2], x0[2])), x0[2], x1[2])), ≥)∧[(-1)bni_22 + (-1)Bound*bni_22] + [bni_22]x0[2] ≥ 0∧[(-1)bso_23] ≥ 0)

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

  (28)    (x1[2] ≥ 0∧x0[2] ≥ 0 ⇒ (U^Increasing(COND_894_0_MAIN_LE1(&&(>(x1[2], 0), <(x1[2], x0[2])), x0[2], x1[2])), ≥)∧[bni_22 + (-1)Bound*bni_22] + [bni_22]x1[2] + [bni_22]x0[2] ≥ 0∧[(-1)bso_23] ≥ 0)

  
  
  

For Pair COND_894_0_MAIN_LE1(TRUE, x0, x1) → 894_0_MAIN_LE(x0, *(2, x1)) the following chains were created:

• We consider the chain 894_0_MAIN_LE(x0[2], x1[2]) → COND_894_0_MAIN_LE1(&&(>(x1[2], 0), <(x1[2], x0[2])), x0[2], x1[2]), COND_894_0_MAIN_LE1(TRUE, x0[3], x1[3]) → 894_0_MAIN_LE(x0[3], *(2, x1[3])), 894_0_MAIN_LE(x0[0], x1[0]) → COND_894_0_MAIN_LE(&&(>=(x1[0], x0[0]), >(x0[0], 0)), x0[0], x1[0]) which results in the following constraint:
  

  (29)    (&&(>(x1[2], 0), <(x1[2], x0[2]))=TRUE∧x0[2]=x0[3]∧x1[2]=x1[3]∧x0[3]=x0[0]∧*(2, x1[3])=x1[0] ⇒ COND_894_0_MAIN_LE1(TRUE, x0[3], x1[3])≥NonInfC∧COND_894_0_MAIN_LE1(TRUE, x0[3], x1[3])≥894_0_MAIN_LE(x0[3], *(2, x1[3]))∧(U^Increasing(894_0_MAIN_LE(x0[3], *(2, x1[3]))), ≥))

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

  (30)    (>(x1[2], 0)=TRUE∧<(x1[2], x0[2])=TRUE ⇒ COND_894_0_MAIN_LE1(TRUE, x0[2], x1[2])≥NonInfC∧COND_894_0_MAIN_LE1(TRUE, x0[2], x1[2])≥894_0_MAIN_LE(x0[2], *(2, x1[2]))∧(U^Increasing(894_0_MAIN_LE(x0[3], *(2, x1[3]))), ≥))

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

  (31)    (x1[2] + [-1] ≥ 0∧x0[2] + [-1] + [-1]x1[2] ≥ 0 ⇒ (U^Increasing(894_0_MAIN_LE(x0[3], *(2, x1[3]))), ≥)∧[(-1)bni_24 + (-1)Bound*bni_24] + [bni_24]x0[2] ≥ 0∧[(-1)bso_25] ≥ 0)

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

  (32)    (x1[2] + [-1] ≥ 0∧x0[2] + [-1] + [-1]x1[2] ≥ 0 ⇒ (U^Increasing(894_0_MAIN_LE(x0[3], *(2, x1[3]))), ≥)∧[(-1)bni_24 + (-1)Bound*bni_24] + [bni_24]x0[2] ≥ 0∧[(-1)bso_25] ≥ 0)

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

  (33)    (x1[2] + [-1] ≥ 0∧x0[2] + [-1] + [-1]x1[2] ≥ 0 ⇒ (U^Increasing(894_0_MAIN_LE(x0[3], *(2, x1[3]))), ≥)∧[(-1)bni_24 + (-1)Bound*bni_24] + [bni_24]x0[2] ≥ 0∧[(-1)bso_25] ≥ 0)

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

  (34)    (x1[2] ≥ 0∧x0[2] + [-2] + [-1]x1[2] ≥ 0 ⇒ (U^Increasing(894_0_MAIN_LE(x0[3], *(2, x1[3]))), ≥)∧[(-1)bni_24 + (-1)Bound*bni_24] + [bni_24]x0[2] ≥ 0∧[(-1)bso_25] ≥ 0)

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

  (35)    (x1[2] ≥ 0∧x0[2] ≥ 0 ⇒ (U^Increasing(894_0_MAIN_LE(x0[3], *(2, x1[3]))), ≥)∧[bni_24 + (-1)Bound*bni_24] + [bni_24]x1[2] + [bni_24]x0[2] ≥ 0∧[(-1)bso_25] ≥ 0)

  
  
  
• We consider the chain 894_0_MAIN_LE(x0[2], x1[2]) → COND_894_0_MAIN_LE1(&&(>(x1[2], 0), <(x1[2], x0[2])), x0[2], x1[2]), COND_894_0_MAIN_LE1(TRUE, x0[3], x1[3]) → 894_0_MAIN_LE(x0[3], *(2, x1[3])), 894_0_MAIN_LE(x0[2], x1[2]) → COND_894_0_MAIN_LE1(&&(>(x1[2], 0), <(x1[2], x0[2])), x0[2], x1[2]) which results in the following constraint:
  

  (36)    (&&(>(x1[2], 0), <(x1[2], x0[2]))=TRUE∧x0[2]=x0[3]∧x1[2]=x1[3]∧x0[3]=x0[2]1∧*(2, x1[3])=x1[2]1 ⇒ COND_894_0_MAIN_LE1(TRUE, x0[3], x1[3])≥NonInfC∧COND_894_0_MAIN_LE1(TRUE, x0[3], x1[3])≥894_0_MAIN_LE(x0[3], *(2, x1[3]))∧(U^Increasing(894_0_MAIN_LE(x0[3], *(2, x1[3]))), ≥))

  
  
  We simplified constraint (36) using rules (III), (IV), (IDP_BOOLEAN) which results in the following new constraint:
  

  (37)    (>(x1[2], 0)=TRUE∧<(x1[2], x0[2])=TRUE ⇒ COND_894_0_MAIN_LE1(TRUE, x0[2], x1[2])≥NonInfC∧COND_894_0_MAIN_LE1(TRUE, x0[2], x1[2])≥894_0_MAIN_LE(x0[2], *(2, x1[2]))∧(U^Increasing(894_0_MAIN_LE(x0[3], *(2, x1[3]))), ≥))

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

  (38)    (x1[2] + [-1] ≥ 0∧x0[2] + [-1] + [-1]x1[2] ≥ 0 ⇒ (U^Increasing(894_0_MAIN_LE(x0[3], *(2, x1[3]))), ≥)∧[(-1)bni_24 + (-1)Bound*bni_24] + [bni_24]x0[2] ≥ 0∧[(-1)bso_25] ≥ 0)

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

  (39)    (x1[2] + [-1] ≥ 0∧x0[2] + [-1] + [-1]x1[2] ≥ 0 ⇒ (U^Increasing(894_0_MAIN_LE(x0[3], *(2, x1[3]))), ≥)∧[(-1)bni_24 + (-1)Bound*bni_24] + [bni_24]x0[2] ≥ 0∧[(-1)bso_25] ≥ 0)

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

  (40)    (x1[2] + [-1] ≥ 0∧x0[2] + [-1] + [-1]x1[2] ≥ 0 ⇒ (U^Increasing(894_0_MAIN_LE(x0[3], *(2, x1[3]))), ≥)∧[(-1)bni_24 + (-1)Bound*bni_24] + [bni_24]x0[2] ≥ 0∧[(-1)bso_25] ≥ 0)

  
  
  We simplified constraint (40) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
  

  (41)    (x1[2] ≥ 0∧x0[2] + [-2] + [-1]x1[2] ≥ 0 ⇒ (U^Increasing(894_0_MAIN_LE(x0[3], *(2, x1[3]))), ≥)∧[(-1)bni_24 + (-1)Bound*bni_24] + [bni_24]x0[2] ≥ 0∧[(-1)bso_25] ≥ 0)

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

  (42)    (x1[2] ≥ 0∧x0[2] ≥ 0 ⇒ (U^Increasing(894_0_MAIN_LE(x0[3], *(2, x1[3]))), ≥)∧[bni_24 + (-1)Bound*bni_24] + [bni_24]x1[2] + [bni_24]x0[2] ≥ 0∧[(-1)bso_25] ≥ 0)

  
  
  

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

• 894_0_MAIN_LE(x0, x1) → COND_894_0_MAIN_LE(&&(>=(x1, x0), >(x0, 0)), x0, x1)
  
  • (x1[0] ≥ 0∧x0[0] ≥ 0 ⇒ (U^Increasing(COND_894_0_MAIN_LE(&&(>=(x1[0], x0[0]), >(x0[0], 0)), x0[0], x1[0])), ≥)∧[(-1)Bound*bni_18] + [bni_18]x0[0] ≥ 0∧[(-1)bso_19] ≥ 0)
    
  
  
• COND_894_0_MAIN_LE(TRUE, x0, x1) → 894_0_MAIN_LE(+(x0, -1), 1)
  
  • (x1[0] ≥ 0∧x0[0] ≥ 0 ⇒ (U^Increasing(894_0_MAIN_LE(+(x0[1], -1), 1)), ≥)∧[(-1)Bound*bni_20] + [bni_20]x0[0] ≥ 0∧[1 + (-1)bso_21] ≥ 0)
    
  • (x1[0] ≥ 0∧x0[0] ≥ 0 ⇒ (U^Increasing(894_0_MAIN_LE(+(x0[1], -1), 1)), ≥)∧[(-1)Bound*bni_20] + [bni_20]x0[0] ≥ 0∧[1 + (-1)bso_21] ≥ 0)
    
  
  
• 894_0_MAIN_LE(x0, x1) → COND_894_0_MAIN_LE1(&&(>(x1, 0), <(x1, x0)), x0, x1)
  
  • (x1[2] ≥ 0∧x0[2] ≥ 0 ⇒ (U^Increasing(COND_894_0_MAIN_LE1(&&(>(x1[2], 0), <(x1[2], x0[2])), x0[2], x1[2])), ≥)∧[bni_22 + (-1)Bound*bni_22] + [bni_22]x1[2] + [bni_22]x0[2] ≥ 0∧[(-1)bso_23] ≥ 0)
    
  
  
• COND_894_0_MAIN_LE1(TRUE, x0, x1) → 894_0_MAIN_LE(x0, *(2, x1))
  
  • (x1[2] ≥ 0∧x0[2] ≥ 0 ⇒ (U^Increasing(894_0_MAIN_LE(x0[3], *(2, x1[3]))), ≥)∧[bni_24 + (-1)Bound*bni_24] + [bni_24]x1[2] + [bni_24]x0[2] ≥ 0∧[(-1)bso_25] ≥ 0)
    
  • (x1[2] ≥ 0∧x0[2] ≥ 0 ⇒ (U^Increasing(894_0_MAIN_LE(x0[3], *(2, x1[3]))), ≥)∧[bni_24 + (-1)Bound*bni_24] + [bni_24]x1[2] + [bni_24]x0[2] ≥ 0∧[(-1)bso_25] ≥ 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) = [3]   
POL(894_0_MAIN_LE(x₁, x₂)) = [-1] + x₁   
POL(COND_894_0_MAIN_LE(x₁, x₂, x₃)) = [-1] + x₂   
POL(&&(x₁, x₂)) = [-1]   
POL(>=(x₁, x₂)) = [-1]   
POL(>(x₁, x₂)) = [-1]   
POL(0) = 0   
POL(+(x₁, x₂)) = x₁ + x₂   
POL(-1) = [-1]   
POL(1) = [1]   
POL(COND_894_0_MAIN_LE1(x₁, x₂, x₃)) = [-1] + x₂   
POL(<(x₁, x₂)) = [-1]   
POL(*(x₁, x₂)) = x₁·x₂   
POL(2) = [2]   

The following pairs are in P_>:

COND_894_0_MAIN_LE(TRUE, x0[1], x1[1]) → 894_0_MAIN_LE(+(x0[1], -1), 1)

The following pairs are in P_bound:

894_0_MAIN_LE(x0[0], x1[0]) → COND_894_0_MAIN_LE(&&(>=(x1[0], x0[0]), >(x0[0], 0)), x0[0], x1[0])
COND_894_0_MAIN_LE(TRUE, x0[1], x1[1]) → 894_0_MAIN_LE(+(x0[1], -1), 1)
894_0_MAIN_LE(x0[2], x1[2]) → COND_894_0_MAIN_LE1(&&(>(x1[2], 0), <(x1[2], x0[2])), x0[2], x1[2])
COND_894_0_MAIN_LE1(TRUE, x0[3], x1[3]) → 894_0_MAIN_LE(x0[3], *(2, x1[3]))

The following pairs are in P_≥:

894_0_MAIN_LE(x0[0], x1[0]) → COND_894_0_MAIN_LE(&&(>=(x1[0], x0[0]), >(x0[0], 0)), x0[0], x1[0])
894_0_MAIN_LE(x0[2], x1[2]) → COND_894_0_MAIN_LE1(&&(>(x1[2], 0), <(x1[2], x0[2])), x0[2], x1[2])
COND_894_0_MAIN_LE1(TRUE, x0[3], x1[3]) → 894_0_MAIN_LE(x0[3], *(2, x1[3]))

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

FALSE¹ → &&(FALSE, TRUE)¹

(8) Obligation:

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

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

The following domains are used:

Boolean, Integer

R is empty.

The integer pair graph contains the following rules and edges:
(0): 894_0_MAIN_LE(x0[0], x1[0]) → COND_894_0_MAIN_LE(x1[0] >= x0[0] && x0[0] > 0, x0[0], x1[0])
(2): 894_0_MAIN_LE(x0[2], x1[2]) → COND_894_0_MAIN_LE1(x1[2] > 0 && x1[2] < x0[2], x0[2], x1[2])
(3): COND_894_0_MAIN_LE1(TRUE, x0[3], x1[3]) → 894_0_MAIN_LE(x0[3], 2 * x1[3])

(3) -> (0), if (x0[3] →^* x0[0]∧2 * x1[3] →^* x1[0])

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

(2) -> (3), if (x1[2] > 0 && x1[2] < x0[2] ∧x0[2] →^* x0[3]∧x1[2] →^* x1[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_894_0_MAIN_LE1(TRUE, x0[3], x1[3]) → 894_0_MAIN_LE(x0[3], 2 * x1[3])
(2): 894_0_MAIN_LE(x0[2], x1[2]) → COND_894_0_MAIN_LE1(x1[2] > 0 && x1[2] < x0[2], x0[2], x1[2])

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

(2) -> (3), if (x1[2] > 0 && x1[2] < x0[2] ∧x0[2] →^* x0[3]∧x1[2] →^* x1[3])

The set Q is empty.

(11) IDPNonInfProof (SOUND transformation)

Used the following options for this NonInfProof:
IDPGPoloSolver: Range: [(-1,2)] IsNat: false Interpretation Shape Heuristic: aprove.DPFramework.IDPProblem.Processors.nonInf.poly.IdpDefaultShapeHeuristic@d6206b5 Constraint Generator: NonInfConstraintGenerator: PathGenerator: MetricPathGenerator: Max Left Steps: 1 Max Right Steps: 1

The constraints were generated the following way:
The DP Problem is simplified using the Induction Calculus [NONINF] with the following steps:
Note that final constraints are written in bold face.


For Pair COND_894_0_MAIN_LE1(TRUE, x0[3], x1[3]) → 894_0_MAIN_LE(x0[3], *(2, x1[3])) the following chains were created:

• We consider the chain 894_0_MAIN_LE(x0[2], x1[2]) → COND_894_0_MAIN_LE1(&&(>(x1[2], 0), <(x1[2], x0[2])), x0[2], x1[2]), COND_894_0_MAIN_LE1(TRUE, x0[3], x1[3]) → 894_0_MAIN_LE(x0[3], *(2, x1[3])), 894_0_MAIN_LE(x0[2], x1[2]) → COND_894_0_MAIN_LE1(&&(>(x1[2], 0), <(x1[2], x0[2])), x0[2], x1[2]) which results in the following constraint:
  

  (1)    (&&(>(x1[2], 0), <(x1[2], x0[2]))=TRUE∧x0[2]=x0[3]∧x1[2]=x1[3]∧x0[3]=x0[2]1∧*(2, x1[3])=x1[2]1 ⇒ COND_894_0_MAIN_LE1(TRUE, x0[3], x1[3])≥NonInfC∧COND_894_0_MAIN_LE1(TRUE, x0[3], x1[3])≥894_0_MAIN_LE(x0[3], *(2, x1[3]))∧(U^Increasing(894_0_MAIN_LE(x0[3], *(2, x1[3]))), ≥))

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

  (2)    (>(x1[2], 0)=TRUE∧<(x1[2], x0[2])=TRUE ⇒ COND_894_0_MAIN_LE1(TRUE, x0[2], x1[2])≥NonInfC∧COND_894_0_MAIN_LE1(TRUE, x0[2], x1[2])≥894_0_MAIN_LE(x0[2], *(2, x1[2]))∧(U^Increasing(894_0_MAIN_LE(x0[3], *(2, x1[3]))), ≥))

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

  (3)    (x1[2] + [-1] ≥ 0∧x0[2] + [-1] + [-1]x1[2] ≥ 0 ⇒ (U^Increasing(894_0_MAIN_LE(x0[3], *(2, x1[3]))), ≥)∧[bni_13 + (-1)Bound*bni_13] + [(-1)bni_13]x1[2] + [bni_13]x0[2] ≥ 0∧[(-1)bso_14] + x1[2] ≥ 0)

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

  (4)    (x1[2] + [-1] ≥ 0∧x0[2] + [-1] + [-1]x1[2] ≥ 0 ⇒ (U^Increasing(894_0_MAIN_LE(x0[3], *(2, x1[3]))), ≥)∧[bni_13 + (-1)Bound*bni_13] + [(-1)bni_13]x1[2] + [bni_13]x0[2] ≥ 0∧[(-1)bso_14] + x1[2] ≥ 0)

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

  (5)    (x1[2] + [-1] ≥ 0∧x0[2] + [-1] + [-1]x1[2] ≥ 0 ⇒ (U^Increasing(894_0_MAIN_LE(x0[3], *(2, x1[3]))), ≥)∧[bni_13 + (-1)Bound*bni_13] + [(-1)bni_13]x1[2] + [bni_13]x0[2] ≥ 0∧[(-1)bso_14] + x1[2] ≥ 0)

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

  (6)    (x1[2] ≥ 0∧x0[2] + [-2] + [-1]x1[2] ≥ 0 ⇒ (U^Increasing(894_0_MAIN_LE(x0[3], *(2, x1[3]))), ≥)∧[(-1)Bound*bni_13] + [(-1)bni_13]x1[2] + [bni_13]x0[2] ≥ 0∧[1 + (-1)bso_14] + x1[2] ≥ 0)

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

  (7)    (x1[2] ≥ 0∧x0[2] ≥ 0 ⇒ (U^Increasing(894_0_MAIN_LE(x0[3], *(2, x1[3]))), ≥)∧[(-1)Bound*bni_13 + (2)bni_13] + [bni_13]x0[2] ≥ 0∧[1 + (-1)bso_14] + x1[2] ≥ 0)

  
  
  

For Pair 894_0_MAIN_LE(x0[2], x1[2]) → COND_894_0_MAIN_LE1(&&(>(x1[2], 0), <(x1[2], x0[2])), x0[2], x1[2]) the following chains were created:

• We consider the chain 894_0_MAIN_LE(x0[2], x1[2]) → COND_894_0_MAIN_LE1(&&(>(x1[2], 0), <(x1[2], x0[2])), x0[2], x1[2]), COND_894_0_MAIN_LE1(TRUE, x0[3], x1[3]) → 894_0_MAIN_LE(x0[3], *(2, x1[3])) which results in the following constraint:
  

  (8)    (&&(>(x1[2], 0), <(x1[2], x0[2]))=TRUE∧x0[2]=x0[3]∧x1[2]=x1[3] ⇒ 894_0_MAIN_LE(x0[2], x1[2])≥NonInfC∧894_0_MAIN_LE(x0[2], x1[2])≥COND_894_0_MAIN_LE1(&&(>(x1[2], 0), <(x1[2], x0[2])), x0[2], x1[2])∧(U^Increasing(COND_894_0_MAIN_LE1(&&(>(x1[2], 0), <(x1[2], x0[2])), x0[2], x1[2])), ≥))

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

  (9)    (>(x1[2], 0)=TRUE∧<(x1[2], x0[2])=TRUE ⇒ 894_0_MAIN_LE(x0[2], x1[2])≥NonInfC∧894_0_MAIN_LE(x0[2], x1[2])≥COND_894_0_MAIN_LE1(&&(>(x1[2], 0), <(x1[2], x0[2])), x0[2], x1[2])∧(U^Increasing(COND_894_0_MAIN_LE1(&&(>(x1[2], 0), <(x1[2], x0[2])), x0[2], x1[2])), ≥))

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

  (10)    (x1[2] + [-1] ≥ 0∧x0[2] + [-1] + [-1]x1[2] ≥ 0 ⇒ (U^Increasing(COND_894_0_MAIN_LE1(&&(>(x1[2], 0), <(x1[2], x0[2])), x0[2], x1[2])), ≥)∧[bni_15 + (-1)Bound*bni_15] + [(-1)bni_15]x1[2] + [bni_15]x0[2] ≥ 0∧[(-1)bso_16] ≥ 0)

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

  (11)    (x1[2] + [-1] ≥ 0∧x0[2] + [-1] + [-1]x1[2] ≥ 0 ⇒ (U^Increasing(COND_894_0_MAIN_LE1(&&(>(x1[2], 0), <(x1[2], x0[2])), x0[2], x1[2])), ≥)∧[bni_15 + (-1)Bound*bni_15] + [(-1)bni_15]x1[2] + [bni_15]x0[2] ≥ 0∧[(-1)bso_16] ≥ 0)

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

  (12)    (x1[2] + [-1] ≥ 0∧x0[2] + [-1] + [-1]x1[2] ≥ 0 ⇒ (U^Increasing(COND_894_0_MAIN_LE1(&&(>(x1[2], 0), <(x1[2], x0[2])), x0[2], x1[2])), ≥)∧[bni_15 + (-1)Bound*bni_15] + [(-1)bni_15]x1[2] + [bni_15]x0[2] ≥ 0∧[(-1)bso_16] ≥ 0)

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

  (13)    (x1[2] ≥ 0∧x0[2] + [-2] + [-1]x1[2] ≥ 0 ⇒ (U^Increasing(COND_894_0_MAIN_LE1(&&(>(x1[2], 0), <(x1[2], x0[2])), x0[2], x1[2])), ≥)∧[(-1)Bound*bni_15] + [(-1)bni_15]x1[2] + [bni_15]x0[2] ≥ 0∧[(-1)bso_16] ≥ 0)

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

  (14)    (x1[2] ≥ 0∧x0[2] ≥ 0 ⇒ (U^Increasing(COND_894_0_MAIN_LE1(&&(>(x1[2], 0), <(x1[2], x0[2])), x0[2], x1[2])), ≥)∧[(-1)Bound*bni_15 + (2)bni_15] + [bni_15]x0[2] ≥ 0∧[(-1)bso_16] ≥ 0)

  
  
  

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

• COND_894_0_MAIN_LE1(TRUE, x0[3], x1[3]) → 894_0_MAIN_LE(x0[3], *(2, x1[3]))
  
  • (x1[2] ≥ 0∧x0[2] ≥ 0 ⇒ (U^Increasing(894_0_MAIN_LE(x0[3], *(2, x1[3]))), ≥)∧[(-1)Bound*bni_13 + (2)bni_13] + [bni_13]x0[2] ≥ 0∧[1 + (-1)bso_14] + x1[2] ≥ 0)
    
  
  
• 894_0_MAIN_LE(x0[2], x1[2]) → COND_894_0_MAIN_LE1(&&(>(x1[2], 0), <(x1[2], x0[2])), x0[2], x1[2])
  
  • (x1[2] ≥ 0∧x0[2] ≥ 0 ⇒ (U^Increasing(COND_894_0_MAIN_LE1(&&(>(x1[2], 0), <(x1[2], x0[2])), x0[2], x1[2])), ≥)∧[(-1)Bound*bni_15 + (2)bni_15] + [bni_15]x0[2] ≥ 0∧[(-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) = [1]   
POL(FALSE) = [1]   
POL(COND_894_0_MAIN_LE1(x₁, x₂, x₃)) = [2] + [-1]x₃ + x₂ + [-1]x₁   
POL(894_0_MAIN_LE(x₁, x₂)) = [1] + [-1]x₂ + x₁   
POL(*(x₁, x₂)) = x₁·x₂   
POL(2) = [2]   
POL(&&(x₁, x₂)) = [1]   
POL(>(x₁, x₂)) = [-1]   
POL(0) = 0   
POL(<(x₁, x₂)) = [-1]   

The following pairs are in P_>:

COND_894_0_MAIN_LE1(TRUE, x0[3], x1[3]) → 894_0_MAIN_LE(x0[3], *(2, x1[3]))

The following pairs are in P_bound:

COND_894_0_MAIN_LE1(TRUE, x0[3], x1[3]) → 894_0_MAIN_LE(x0[3], *(2, x1[3]))
894_0_MAIN_LE(x0[2], x1[2]) → COND_894_0_MAIN_LE1(&&(>(x1[2], 0), <(x1[2], x0[2])), x0[2], x1[2])

The following pairs are in P_≥:

894_0_MAIN_LE(x0[2], x1[2]) → COND_894_0_MAIN_LE1(&&(>(x1[2], 0), <(x1[2], x0[2])), x0[2], x1[2])

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

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

(12) Obligation:

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

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

The following domains are used:

Boolean, Integer

R is empty.

The integer pair graph contains the following rules and edges:
(2): 894_0_MAIN_LE(x0[2], x1[2]) → COND_894_0_MAIN_LE1(x1[2] > 0 && x1[2] < x0[2], x0[2], x1[2])


The set Q is empty.

(13) IDependencyGraphProof (EQUIVALENT transformation)

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

(14) TRUE