Termination proof of /tmp/tmpfmo0jQ/AG313.jar

Termination of the given JBC could be proven:

0 JBC

↳1 JBCToGraph (⇒, 260 ms)

↳2 JBCTerminationGraph

↳3 TerminationGraphToSCCProof (⇒, 0 ms)

↳4 JBCTerminationSCC

↳5 SCCToIDPv1Proof (⇒, 40 ms)

↳6 IDP

↳7 IDPNonInfProof (⇒, 180 ms)

↳8 IDP

↳9 IDependencyGraphProof (⇔, 0 ms)

↳10 TRUE

(0) Obligation:

JBC Problem based on JBC Program:
Manifest-Version: 1.0 Created-By: 1.6.0_20 (Sun Microsystems Inc.) Main-Class: AG313

public class AG313 {
  public static void main(String[] args) {
    int x, y;
    x = args[0].length();
    y = args[1].length() + 1;
    quot(x,y);

  }


  public static int quot(int x, int y) {
    int i = 0;
    if(x==0) return 0;
    while (x > 0 && y > 0) {
      i += 1;
      x = (x - 1)- (y - 1);

    }
    return i;
  }
}

(1) JBCToGraph (SOUND transformation)

Constructed TerminationGraph.

(2) Obligation:

Termination Graph based on JBC Program:
AG313.main([Ljava/lang/String;)V: Graph of 180 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: AG313.main([Ljava/lang/String;)V
SCC calls the following helper methods:
Performed SCC analyses: UsedFieldsAnalysis

(5) SCCToIDPv1Proof (SOUND transformation)

Transformed FIGraph SCCs to IDPs. Log:

Generated 16 rules for P and 0 rules for R.

P rules:
588_0_quot_LE(EOS(STATIC_588), i40, i142, i40, i142) → 591_0_quot_LE(EOS(STATIC_591), i40, i142, i40, i142)
591_0_quot_LE(EOS(STATIC_591), i40, i142, i40, i142) → 595_0_quot_Load(EOS(STATIC_595), i40, i142, i40) | >(i142, 0)
595_0_quot_Load(EOS(STATIC_595), i40, i142, i40) → 600_0_quot_LE(EOS(STATIC_600), i40, i142, i40, i40)
600_0_quot_LE(EOS(STATIC_600), i40, i142, i40, i40) → 606_0_quot_Inc(EOS(STATIC_606), i40, i142, i40) | >(i40, 0)
606_0_quot_Inc(EOS(STATIC_606), i40, i142, i40) → 610_0_quot_Load(EOS(STATIC_610), i40, i142, i40)
610_0_quot_Load(EOS(STATIC_610), i40, i142, i40) → 613_0_quot_ConstantStackPush(EOS(STATIC_613), i40, i40, i142)
613_0_quot_ConstantStackPush(EOS(STATIC_613), i40, i40, i142) → 615_0_quot_IntArithmetic(EOS(STATIC_615), i40, i40, i142, 1)
615_0_quot_IntArithmetic(EOS(STATIC_615), i40, i40, i142, matching1) → 618_0_quot_Load(EOS(STATIC_618), i40, i40, -(i142, 1)) | &&(>(i142, 0), =(matching1, 1))
618_0_quot_Load(EOS(STATIC_618), i40, i40, i149) → 620_0_quot_ConstantStackPush(EOS(STATIC_620), i40, i40, i149, i40)
620_0_quot_ConstantStackPush(EOS(STATIC_620), i40, i40, i149, i40) → 622_0_quot_IntArithmetic(EOS(STATIC_622), i40, i40, i149, i40, 1)
622_0_quot_IntArithmetic(EOS(STATIC_622), i40, i40, i149, i40, matching1) → 624_0_quot_IntArithmetic(EOS(STATIC_624), i40, i40, i149, -(i40, 1)) | &&(>(i40, 0), =(matching1, 1))
624_0_quot_IntArithmetic(EOS(STATIC_624), i40, i40, i149, i150) → 626_0_quot_Store(EOS(STATIC_626), i40, i40, -(i149, i150)) | &&(>=(i149, 0), >=(i150, 0))
626_0_quot_Store(EOS(STATIC_626), i40, i40, i152) → 629_0_quot_JMP(EOS(STATIC_629), i40, i152, i40)
629_0_quot_JMP(EOS(STATIC_629), i40, i152, i40) → 645_0_quot_Load(EOS(STATIC_645), i40, i152, i40)
645_0_quot_Load(EOS(STATIC_645), i40, i152, i40) → 584_0_quot_Load(EOS(STATIC_584), i40, i152, i40)
584_0_quot_Load(EOS(STATIC_584), i40, i133, i40) → 588_0_quot_LE(EOS(STATIC_588), i40, i133, i40, i133)
R rules:

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

P rules:
588_0_quot_LE(EOS(STATIC_588), x0, x1, x0, x1) → 588_0_quot_LE(EOS(STATIC_588), x0, -(-(x1, 1), -(x0, 1)), x0, -(-(x1, 1), -(x0, 1))) | &&(>(+(x1, 1), 1), >(+(x0, 1), 1))
R rules:

Filtered ground terms:

588_0_quot_LE(x1, x2, x3, x4, x5) → 588_0_quot_LE(x2, x3, x4, x5)
EOS(x1) → EOS
Cond_588_0_quot_LE(x1, x2, x3, x4, x5, x6) → Cond_588_0_quot_LE(x1, x3, x4, x5, x6)

Filtered duplicate args:

588_0_quot_LE(x1, x2, x3, x4) → 588_0_quot_LE(x3, x4)
Cond_588_0_quot_LE(x1, x2, x3, x4, x5) → Cond_588_0_quot_LE(x1, x4, x5)

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

P rules:
588_0_quot_LE(x0, x1) → 588_0_quot_LE(x0, -(-(x1, 1), -(x0, 1))) | &&(>(x1, 0), >(x0, 0))
R rules:

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

P rules:
588_0_QUOT_LE(x0, x1) → COND_588_0_QUOT_LE(&&(>(x1, 0), >(x0, 0)), x0, x1)
COND_588_0_QUOT_LE(TRUE, x0, x1) → 588_0_QUOT_LE(x0, -(-(x1, 1), -(x0, 1)))
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): 588_0_QUOT_LE(x0[0], x1[0]) → COND_588_0_QUOT_LE(x1[0] > 0 && x0[0] > 0, x0[0], x1[0])
(1): COND_588_0_QUOT_LE(TRUE, x0[1], x1[1]) → 588_0_QUOT_LE(x0[1], x1[1] - 1 - x0[1] - 1)

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

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

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@4e93b16c 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 588_0_QUOT_LE(x0, x1) → COND_588_0_QUOT_LE(&&(>(x1, 0), >(x0, 0)), x0, x1) the following chains were created:

We consider the chain 588_0_QUOT_LE(x0[0], x1[0]) → COND_588_0_QUOT_LE(&&(>(x1[0], 0), >(x0[0], 0)), x0[0], x1[0]), COND_588_0_QUOT_LE(TRUE, x0[1], x1[1]) → 588_0_QUOT_LE(x0[1], -(-(x1[1], 1), -(x0[1], 1))) which results in the following constraint:
(1)    (&&(>(x1[0], 0), >(x0[0], 0))=TRUE∧x0[0]=x0[1]∧x1[0]=x1[1] ⇒ 588_0_QUOT_LE(x0[0], x1[0])≥NonInfC∧588_0_QUOT_LE(x0[0], x1[0])≥COND_588_0_QUOT_LE(&&(>(x1[0], 0), >(x0[0], 0)), x0[0], x1[0])∧(U^Increasing(COND_588_0_QUOT_LE(&&(>(x1[0], 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], 0)=TRUE∧>(x0[0], 0)=TRUE ⇒ 588_0_QUOT_LE(x0[0], x1[0])≥NonInfC∧588_0_QUOT_LE(x0[0], x1[0])≥COND_588_0_QUOT_LE(&&(>(x1[0], 0), >(x0[0], 0)), x0[0], x1[0])∧(U^Increasing(COND_588_0_QUOT_LE(&&(>(x1[0], 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] ≥ 0∧x0[0] + [-1] ≥ 0 ⇒ (U^Increasing(COND_588_0_QUOT_LE(&&(>(x1[0], 0), >(x0[0], 0)), x0[0], x1[0])), ≥)∧[(-1)bni_12 + (-1)Bound*bni_12] + [bni_12]x1[0] + [bni_12]x0[0] ≥ 0∧[(-1)bso_13] ≥ 0)

We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(4)    (x1[0] + [-1] ≥ 0∧x0[0] + [-1] ≥ 0 ⇒ (U^Increasing(COND_588_0_QUOT_LE(&&(>(x1[0], 0), >(x0[0], 0)), x0[0], x1[0])), ≥)∧[(-1)bni_12 + (-1)Bound*bni_12] + [bni_12]x1[0] + [bni_12]x0[0] ≥ 0∧[(-1)bso_13] ≥ 0)

We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(5)    (x1[0] + [-1] ≥ 0∧x0[0] + [-1] ≥ 0 ⇒ (U^Increasing(COND_588_0_QUOT_LE(&&(>(x1[0], 0), >(x0[0], 0)), x0[0], x1[0])), ≥)∧[(-1)bni_12 + (-1)Bound*bni_12] + [bni_12]x1[0] + [bni_12]x0[0] ≥ 0∧[(-1)bso_13] ≥ 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_588_0_QUOT_LE(&&(>(x1[0], 0), >(x0[0], 0)), x0[0], x1[0])), ≥)∧[(-1)Bound*bni_12] + [bni_12]x1[0] + [bni_12]x0[0] ≥ 0∧[(-1)bso_13] ≥ 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_588_0_QUOT_LE(&&(>(x1[0], 0), >(x0[0], 0)), x0[0], x1[0])), ≥)∧[(-1)Bound*bni_12 + bni_12] + [bni_12]x1[0] + [bni_12]x0[0] ≥ 0∧[(-1)bso_13] ≥ 0)

For Pair COND_588_0_QUOT_LE(TRUE, x0, x1) → 588_0_QUOT_LE(x0, -(-(x1, 1), -(x0, 1))) the following chains were created:

We consider the chain 588_0_QUOT_LE(x0[0], x1[0]) → COND_588_0_QUOT_LE(&&(>(x1[0], 0), >(x0[0], 0)), x0[0], x1[0]), COND_588_0_QUOT_LE(TRUE, x0[1], x1[1]) → 588_0_QUOT_LE(x0[1], -(-(x1[1], 1), -(x0[1], 1))), 588_0_QUOT_LE(x0[0], x1[0]) → COND_588_0_QUOT_LE(&&(>(x1[0], 0), >(x0[0], 0)), x0[0], x1[0]) which results in the following constraint:
(8)    (&&(>(x1[0], 0), >(x0[0], 0))=TRUE∧x0[0]=x0[1]∧x1[0]=x1[1]∧x0[1]=x0[0]1∧-(-(x1[1], 1), -(x0[1], 1))=x1[0]1 ⇒ COND_588_0_QUOT_LE(TRUE, x0[1], x1[1])≥NonInfC∧COND_588_0_QUOT_LE(TRUE, x0[1], x1[1])≥588_0_QUOT_LE(x0[1], -(-(x1[1], 1), -(x0[1], 1)))∧(U^Increasing(588_0_QUOT_LE(x0[1], -(-(x1[1], 1), -(x0[1], 1)))), ≥))

We simplified constraint (8) using rules (III), (IV), (IDP_BOOLEAN) which results in the following new constraint:
(9)    (>(x1[0], 0)=TRUE∧>(x0[0], 0)=TRUE ⇒ COND_588_0_QUOT_LE(TRUE, x0[0], x1[0])≥NonInfC∧COND_588_0_QUOT_LE(TRUE, x0[0], x1[0])≥588_0_QUOT_LE(x0[0], -(-(x1[0], 1), -(x0[0], 1)))∧(U^Increasing(588_0_QUOT_LE(x0[1], -(-(x1[1], 1), -(x0[1], 1)))), ≥))

We simplified constraint (9) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(10)    (x1[0] + [-1] ≥ 0∧x0[0] + [-1] ≥ 0 ⇒ (U^Increasing(588_0_QUOT_LE(x0[1], -(-(x1[1], 1), -(x0[1], 1)))), ≥)∧[(-1)bni_14 + (-1)Bound*bni_14] + [bni_14]x1[0] + [bni_14]x0[0] ≥ 0∧[(-1)bso_15] + x0[0] ≥ 0)

We simplified constraint (10) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(11)    (x1[0] + [-1] ≥ 0∧x0[0] + [-1] ≥ 0 ⇒ (U^Increasing(588_0_QUOT_LE(x0[1], -(-(x1[1], 1), -(x0[1], 1)))), ≥)∧[(-1)bni_14 + (-1)Bound*bni_14] + [bni_14]x1[0] + [bni_14]x0[0] ≥ 0∧[(-1)bso_15] + x0[0] ≥ 0)

We simplified constraint (11) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(12)    (x1[0] + [-1] ≥ 0∧x0[0] + [-1] ≥ 0 ⇒ (U^Increasing(588_0_QUOT_LE(x0[1], -(-(x1[1], 1), -(x0[1], 1)))), ≥)∧[(-1)bni_14 + (-1)Bound*bni_14] + [bni_14]x1[0] + [bni_14]x0[0] ≥ 0∧[(-1)bso_15] + x0[0] ≥ 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(588_0_QUOT_LE(x0[1], -(-(x1[1], 1), -(x0[1], 1)))), ≥)∧[(-1)Bound*bni_14] + [bni_14]x1[0] + [bni_14]x0[0] ≥ 0∧[(-1)bso_15] + x0[0] ≥ 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(588_0_QUOT_LE(x0[1], -(-(x1[1], 1), -(x0[1], 1)))), ≥)∧[(-1)Bound*bni_14 + bni_14] + [bni_14]x1[0] + [bni_14]x0[0] ≥ 0∧[1 + (-1)bso_15] + x0[0] ≥ 0)

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

588_0_QUOT_LE(x0, x1) → COND_588_0_QUOT_LE(&&(>(x1, 0), >(x0, 0)), x0, x1)
- (x1[0] ≥ 0∧x0[0] ≥ 0 ⇒ (U^Increasing(COND_588_0_QUOT_LE(&&(>(x1[0], 0), >(x0[0], 0)), x0[0], x1[0])), ≥)∧[(-1)Bound*bni_12 + bni_12] + [bni_12]x1[0] + [bni_12]x0[0] ≥ 0∧[(-1)bso_13] ≥ 0)
COND_588_0_QUOT_LE(TRUE, x0, x1) → 588_0_QUOT_LE(x0, -(-(x1, 1), -(x0, 1)))
- (x1[0] ≥ 0∧x0[0] ≥ 0 ⇒ (U^Increasing(588_0_QUOT_LE(x0[1], -(-(x1[1], 1), -(x0[1], 1)))), ≥)∧[(-1)Bound*bni_14 + bni_14] + [bni_14]x1[0] + [bni_14]x0[0] ≥ 0∧[1 + (-1)bso_15] + x0[0] ≥ 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(588_0_QUOT_LE(x₁, x₂)) = [-1] + x₂ + x₁
POL(COND_588_0_QUOT_LE(x₁, x₂, x₃)) = [-1] + x₃ + x₂
POL(&&(x₁, x₂)) = [-1]
POL(>(x₁, x₂)) = [-1]
POL(0) = 0
POL(-(x₁, x₂)) = x₁ + [-1]x₂
POL(1) = [1]

The following pairs are in P_>:

COND_588_0_QUOT_LE(TRUE, x0[1], x1[1]) → 588_0_QUOT_LE(x0[1], -(-(x1[1], 1), -(x0[1], 1)))

The following pairs are in P_bound:

588_0_QUOT_LE(x0[0], x1[0]) → COND_588_0_QUOT_LE(&&(>(x1[0], 0), >(x0[0], 0)), x0[0], x1[0])
COND_588_0_QUOT_LE(TRUE, x0[1], x1[1]) → 588_0_QUOT_LE(x0[1], -(-(x1[1], 1), -(x0[1], 1)))

The following pairs are in P_≥:

588_0_QUOT_LE(x0[0], x1[0]) → COND_588_0_QUOT_LE(&&(>(x1[0], 0), >(x0[0], 0)), x0[0], x1[0])

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

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

(8) Obligation: