Termination proof of /tmp/tmpCYpgfk/PastaA1.jar

Termination of the given JBC could be proven:

0 JBC

↳1 JBCToGraph (⇒, 350 ms)

↳2 JBCTerminationGraph

↳3 TerminationGraphToSCCProof (⇒, 0 ms)

↳4 JBCTerminationSCC

↳5 SCCToIDPv1Proof (⇒, 70 ms)

↳6 IDP

↳7 IDPNonInfProof (⇒, 210 ms)

↳8 AND

    ↳9 IDP

        ↳10 IDependencyGraphProof (⇔, 0 ms)

        ↳11 IDP

        ↳12 IDPNonInfProof (⇒, 70 ms)

        ↳13 IDP

        ↳14 IDependencyGraphProof (⇔, 0 ms)

        ↳15 TRUE

    ↳16 IDP

        ↳17 IDependencyGraphProof (⇔, 0 ms)

        ↳18 TRUE

(0) Obligation:

JBC Problem based on JBC Program:
Manifest-Version: 1.0 Created-By: 1.6.0_16 (Sun Microsystems Inc.) Main-Class: PastaA1

/**
 * 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 PastaA1 {
    public static void main(String[] args) {
        Random.args = args;
        int x = Random.random();
        while (x > 0) {
            int y = 0;
            while (y < x) {
                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:
PastaA1.main([Ljava/lang/String;)V: Graph of 120 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: PastaA1.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 21 rules for P and 0 rules for R.

P rules:
146_0_main_LE(EOS(STATIC_146), i22, i22) → 149_0_main_LE(EOS(STATIC_149), i22, i22)
149_0_main_LE(EOS(STATIC_149), i22, i22) → 156_0_main_ConstantStackPush(EOS(STATIC_156), i22) | >(i22, 0)
156_0_main_ConstantStackPush(EOS(STATIC_156), i22) → 164_0_main_Store(EOS(STATIC_164), i22, 0)
164_0_main_Store(EOS(STATIC_164), i22, matching1) → 169_0_main_Load(EOS(STATIC_169), i22, 0) | =(matching1, 0)
169_0_main_Load(EOS(STATIC_169), i22, matching1) → 217_0_main_Load(EOS(STATIC_217), i22, 0) | =(matching1, 0)
217_0_main_Load(EOS(STATIC_217), i22, i23) → 317_0_main_Load(EOS(STATIC_317), i22, i23)
317_0_main_Load(EOS(STATIC_317), i22, i33) → 533_0_main_Load(EOS(STATIC_533), i22, i33)
533_0_main_Load(EOS(STATIC_533), i22, i48) → 596_0_main_Load(EOS(STATIC_596), i22, i48)
596_0_main_Load(EOS(STATIC_596), i22, i59) → 635_0_main_Load(EOS(STATIC_635), i22, i59, i59)
635_0_main_Load(EOS(STATIC_635), i22, i59, i59) → 636_0_main_GE(EOS(STATIC_636), i22, i59, i59, i22)
636_0_main_GE(EOS(STATIC_636), i22, i59, i59, i22) → 637_0_main_GE(EOS(STATIC_637), i22, i59, i59, i22)
636_0_main_GE(EOS(STATIC_636), i22, i59, i59, i22) → 638_0_main_GE(EOS(STATIC_638), i22, i59, i59, i22)
637_0_main_GE(EOS(STATIC_637), i22, i59, i59, i22) → 640_0_main_Inc(EOS(STATIC_640), i22) | >=(i59, i22)
640_0_main_Inc(EOS(STATIC_640), i22) → 642_0_main_JMP(EOS(STATIC_642), +(i22, -1)) | >(i22, 0)
642_0_main_JMP(EOS(STATIC_642), i67) → 646_0_main_Load(EOS(STATIC_646), i67)
646_0_main_Load(EOS(STATIC_646), i67) → 143_0_main_Load(EOS(STATIC_143), i67)
143_0_main_Load(EOS(STATIC_143), i18) → 146_0_main_LE(EOS(STATIC_146), i18, i18)
638_0_main_GE(EOS(STATIC_638), i22, i59, i59, i22) → 641_0_main_Inc(EOS(STATIC_641), i22, i59) | <(i59, i22)
641_0_main_Inc(EOS(STATIC_641), i22, i59) → 644_0_main_JMP(EOS(STATIC_644), i22, +(i59, 1)) | >=(i59, 0)
644_0_main_JMP(EOS(STATIC_644), i22, i68) → 648_0_main_Load(EOS(STATIC_648), i22, i68)
648_0_main_Load(EOS(STATIC_648), i22, i68) → 596_0_main_Load(EOS(STATIC_596), i22, i68)
R rules:

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

P rules:
636_0_main_GE(EOS(STATIC_636), x0, x1, x1, x0) → 636_0_main_GE(EOS(STATIC_636), +(x0, -1), 0, 0, +(x0, -1)) | &&(>=(x1, x0), >(x0, 1))
636_0_main_GE(EOS(STATIC_636), x0, x1, x1, x0) → 636_0_main_GE(EOS(STATIC_636), x0, +(x1, 1), +(x1, 1), x0) | &&(>(+(x1, 1), 0), <(x1, x0))
R rules:

Filtered ground terms:

636_0_main_GE(x1, x2, x3, x4, x5) → 636_0_main_GE(x2, x3, x4, x5)
EOS(x1) → EOS
Cond_636_0_main_GE1(x1, x2, x3, x4, x5, x6) → Cond_636_0_main_GE1(x1, x3, x4, x5, x6)
Cond_636_0_main_GE(x1, x2, x3, x4, x5, x6) → Cond_636_0_main_GE(x1, x3, x4, x5, x6)

Filtered duplicate args:

636_0_main_GE(x1, x2, x3, x4) → 636_0_main_GE(x3, x4)
Cond_636_0_main_GE(x1, x2, x3, x4, x5) → Cond_636_0_main_GE(x1, x4, x5)
Cond_636_0_main_GE1(x1, x2, x3, x4, x5) → Cond_636_0_main_GE1(x1, x4, x5)

Filtered unneeded arguments:

Cond_636_0_main_GE(x1, x2, x3) → Cond_636_0_main_GE(x1, x3)

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

P rules:
636_0_main_GE(x1, x0) → 636_0_main_GE(0, +(x0, -1)) | &&(>=(x1, x0), >(x0, 1))
636_0_main_GE(x1, x0) → 636_0_main_GE(+(x1, 1), x0) | &&(>(x1, -1), <(x1, x0))
R rules:

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

P rules:
636_0_MAIN_GE(x1, x0) → COND_636_0_MAIN_GE(&&(>=(x1, x0), >(x0, 1)), x1, x0)
COND_636_0_MAIN_GE(TRUE, x1, x0) → 636_0_MAIN_GE(0, +(x0, -1))
636_0_MAIN_GE(x1, x0) → COND_636_0_MAIN_GE1(&&(>(x1, -1), <(x1, x0)), x1, x0)
COND_636_0_MAIN_GE1(TRUE, x1, x0) → 636_0_MAIN_GE(+(x1, 1), x0)
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): 636_0_MAIN_GE(x1[0], x0[0]) → COND_636_0_MAIN_GE(x1[0] >= x0[0] && x0[0] > 1, x1[0], x0[0])
(1): COND_636_0_MAIN_GE(TRUE, x1[1], x0[1]) → 636_0_MAIN_GE(0, x0[1] + -1)
(2): 636_0_MAIN_GE(x1[2], x0[2]) → COND_636_0_MAIN_GE1(x1[2] > -1 && x1[2] < x0[2], x1[2], x0[2])
(3): COND_636_0_MAIN_GE1(TRUE, x1[3], x0[3]) → 636_0_MAIN_GE(x1[3] + 1, x0[3])

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

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

(1) -> (2), if (0 →^* x1[2]∧x0[1] + -1 →^* x0[2])

(2) -> (3), if (x1[2] > -1 && x1[2] < x0[2] ∧x1[2] →^* x1[3]∧x0[2] →^* x0[3])

(3) -> (0), if (x1[3] + 1 →^* x1[0]∧x0[3] →^* x0[0])

(3) -> (2), if (x1[3] + 1 →^* x1[2]∧x0[3] →^* x0[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.IdpCand1ShapeHeuristic@19aea6e2 Constraint Generator: NonInfConstraintGenerator: PathGenerator: MetricPathGenerator: Max Left Steps: 0 Max Right Steps: 0

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 636_0_MAIN_GE(x1, x0) → COND_636_0_MAIN_GE(&&(>=(x1, x0), >(x0, 1)), x1, x0) the following chains were created:

We consider the chain 636_0_MAIN_GE(x1[0], x0[0]) → COND_636_0_MAIN_GE(&&(>=(x1[0], x0[0]), >(x0[0], 1)), x1[0], x0[0]), COND_636_0_MAIN_GE(TRUE, x1[1], x0[1]) → 636_0_MAIN_GE(0, +(x0[1], -1)) which results in the following constraint:
(1)    (&&(>=(x1[0], x0[0]), >(x0[0], 1))=TRUE∧x1[0]=x1[1]∧x0[0]=x0[1] ⇒ 636_0_MAIN_GE(x1[0], x0[0])≥NonInfC∧636_0_MAIN_GE(x1[0], x0[0])≥COND_636_0_MAIN_GE(&&(>=(x1[0], x0[0]), >(x0[0], 1)), x1[0], x0[0])∧(U^Increasing(COND_636_0_MAIN_GE(&&(>=(x1[0], x0[0]), >(x0[0], 1)), x1[0], x0[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], 1)=TRUE ⇒ 636_0_MAIN_GE(x1[0], x0[0])≥NonInfC∧636_0_MAIN_GE(x1[0], x0[0])≥COND_636_0_MAIN_GE(&&(>=(x1[0], x0[0]), >(x0[0], 1)), x1[0], x0[0])∧(U^Increasing(COND_636_0_MAIN_GE(&&(>=(x1[0], x0[0]), >(x0[0], 1)), x1[0], x0[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] + [-2] ≥ 0 ⇒ (U^Increasing(COND_636_0_MAIN_GE(&&(>=(x1[0], x0[0]), >(x0[0], 1)), x1[0], x0[0])), ≥)∧[(-1)bni_12 + (-1)Bound*bni_12] + [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]x0[0] ≥ 0∧x0[0] + [-2] ≥ 0 ⇒ (U^Increasing(COND_636_0_MAIN_GE(&&(>=(x1[0], x0[0]), >(x0[0], 1)), x1[0], x0[0])), ≥)∧[(-1)bni_12 + (-1)Bound*bni_12] + [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]x0[0] ≥ 0∧x0[0] + [-2] ≥ 0 ⇒ (U^Increasing(COND_636_0_MAIN_GE(&&(>=(x1[0], x0[0]), >(x0[0], 1)), x1[0], x0[0])), ≥)∧[(-1)bni_12 + (-1)Bound*bni_12] + [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] + [-2] ≥ 0 ⇒ (U^Increasing(COND_636_0_MAIN_GE(&&(>=(x1[0], x0[0]), >(x0[0], 1)), x1[0], x0[0])), ≥)∧[(-1)bni_12 + (-1)Bound*bni_12] + [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_636_0_MAIN_GE(&&(>=(x1[0], x0[0]), >(x0[0], 1)), x1[0], x0[0])), ≥)∧[bni_12 + (-1)Bound*bni_12] + [bni_12]x0[0] ≥ 0∧[(-1)bso_13] ≥ 0)

For Pair COND_636_0_MAIN_GE(TRUE, x1, x0) → 636_0_MAIN_GE(0, +(x0, -1)) the following chains were created:

We consider the chain COND_636_0_MAIN_GE(TRUE, x1[1], x0[1]) → 636_0_MAIN_GE(0, +(x0[1], -1)) which results in the following constraint:
(8)    (COND_636_0_MAIN_GE(TRUE, x1[1], x0[1])≥NonInfC∧COND_636_0_MAIN_GE(TRUE, x1[1], x0[1])≥636_0_MAIN_GE(0, +(x0[1], -1))∧(U^Increasing(636_0_MAIN_GE(0, +(x0[1], -1))), ≥))

We simplified constraint (8) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(9)    ((U^Increasing(636_0_MAIN_GE(0, +(x0[1], -1))), ≥)∧[bni_14] = 0∧[1 + (-1)bso_15] ≥ 0)

We simplified constraint (9) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(10)    ((U^Increasing(636_0_MAIN_GE(0, +(x0[1], -1))), ≥)∧[bni_14] = 0∧[1 + (-1)bso_15] ≥ 0)

We simplified constraint (10) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(11)    ((U^Increasing(636_0_MAIN_GE(0, +(x0[1], -1))), ≥)∧[bni_14] = 0∧[1 + (-1)bso_15] ≥ 0)

We simplified constraint (11) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint:
(12)    ((U^Increasing(636_0_MAIN_GE(0, +(x0[1], -1))), ≥)∧[bni_14] = 0∧0 = 0∧0 = 0∧[1 + (-1)bso_15] ≥ 0)

For Pair 636_0_MAIN_GE(x1, x0) → COND_636_0_MAIN_GE1(&&(>(x1, -1), <(x1, x0)), x1, x0) the following chains were created:

We consider the chain 636_0_MAIN_GE(x1[2], x0[2]) → COND_636_0_MAIN_GE1(&&(>(x1[2], -1), <(x1[2], x0[2])), x1[2], x0[2]), COND_636_0_MAIN_GE1(TRUE, x1[3], x0[3]) → 636_0_MAIN_GE(+(x1[3], 1), x0[3]) which results in the following constraint:
(13)    (&&(>(x1[2], -1), <(x1[2], x0[2]))=TRUE∧x1[2]=x1[3]∧x0[2]=x0[3] ⇒ 636_0_MAIN_GE(x1[2], x0[2])≥NonInfC∧636_0_MAIN_GE(x1[2], x0[2])≥COND_636_0_MAIN_GE1(&&(>(x1[2], -1), <(x1[2], x0[2])), x1[2], x0[2])∧(U^Increasing(COND_636_0_MAIN_GE1(&&(>(x1[2], -1), <(x1[2], x0[2])), x1[2], x0[2])), ≥))

We simplified constraint (13) using rules (IV), (IDP_BOOLEAN) which results in the following new constraint:
(14)    (>(x1[2], -1)=TRUE∧<(x1[2], x0[2])=TRUE ⇒ 636_0_MAIN_GE(x1[2], x0[2])≥NonInfC∧636_0_MAIN_GE(x1[2], x0[2])≥COND_636_0_MAIN_GE1(&&(>(x1[2], -1), <(x1[2], x0[2])), x1[2], x0[2])∧(U^Increasing(COND_636_0_MAIN_GE1(&&(>(x1[2], -1), <(x1[2], x0[2])), x1[2], x0[2])), ≥))

We simplified constraint (14) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(15)    (x1[2] ≥ 0∧x0[2] + [-1] + [-1]x1[2] ≥ 0 ⇒ (U^Increasing(COND_636_0_MAIN_GE1(&&(>(x1[2], -1), <(x1[2], x0[2])), x1[2], x0[2])), ≥)∧[(-1)bni_16 + (-1)Bound*bni_16] + [bni_16]x0[2] ≥ 0∧[(-1)bso_17] ≥ 0)

We simplified constraint (15) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(16)    (x1[2] ≥ 0∧x0[2] + [-1] + [-1]x1[2] ≥ 0 ⇒ (U^Increasing(COND_636_0_MAIN_GE1(&&(>(x1[2], -1), <(x1[2], x0[2])), x1[2], x0[2])), ≥)∧[(-1)bni_16 + (-1)Bound*bni_16] + [bni_16]x0[2] ≥ 0∧[(-1)bso_17] ≥ 0)

We simplified constraint (16) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(17)    (x1[2] ≥ 0∧x0[2] + [-1] + [-1]x1[2] ≥ 0 ⇒ (U^Increasing(COND_636_0_MAIN_GE1(&&(>(x1[2], -1), <(x1[2], x0[2])), x1[2], x0[2])), ≥)∧[(-1)bni_16 + (-1)Bound*bni_16] + [bni_16]x0[2] ≥ 0∧[(-1)bso_17] ≥ 0)

We simplified constraint (17) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(18)    (x1[2] ≥ 0∧x0[2] ≥ 0 ⇒ (U^Increasing(COND_636_0_MAIN_GE1(&&(>(x1[2], -1), <(x1[2], x0[2])), x1[2], x0[2])), ≥)∧[(-1)Bound*bni_16] + [bni_16]x1[2] + [bni_16]x0[2] ≥ 0∧[(-1)bso_17] ≥ 0)

For Pair COND_636_0_MAIN_GE1(TRUE, x1, x0) → 636_0_MAIN_GE(+(x1, 1), x0) the following chains were created:

We consider the chain COND_636_0_MAIN_GE1(TRUE, x1[3], x0[3]) → 636_0_MAIN_GE(+(x1[3], 1), x0[3]) which results in the following constraint:
(19)    (COND_636_0_MAIN_GE1(TRUE, x1[3], x0[3])≥NonInfC∧COND_636_0_MAIN_GE1(TRUE, x1[3], x0[3])≥636_0_MAIN_GE(+(x1[3], 1), x0[3])∧(U^Increasing(636_0_MAIN_GE(+(x1[3], 1), x0[3])), ≥))

We simplified constraint (19) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(20)    ((U^Increasing(636_0_MAIN_GE(+(x1[3], 1), x0[3])), ≥)∧[bni_18] = 0∧[(-1)bso_19] ≥ 0)

We simplified constraint (20) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(21)    ((U^Increasing(636_0_MAIN_GE(+(x1[3], 1), x0[3])), ≥)∧[bni_18] = 0∧[(-1)bso_19] ≥ 0)

We simplified constraint (21) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(22)    ((U^Increasing(636_0_MAIN_GE(+(x1[3], 1), x0[3])), ≥)∧[bni_18] = 0∧[(-1)bso_19] ≥ 0)

We simplified constraint (22) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint:
(23)    ((U^Increasing(636_0_MAIN_GE(+(x1[3], 1), x0[3])), ≥)∧[bni_18] = 0∧0 = 0∧0 = 0∧[(-1)bso_19] ≥ 0)

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

636_0_MAIN_GE(x1, x0) → COND_636_0_MAIN_GE(&&(>=(x1, x0), >(x0, 1)), x1, x0)
- (x1[0] ≥ 0∧x0[0] ≥ 0 ⇒ (U^Increasing(COND_636_0_MAIN_GE(&&(>=(x1[0], x0[0]), >(x0[0], 1)), x1[0], x0[0])), ≥)∧[bni_12 + (-1)Bound*bni_12] + [bni_12]x0[0] ≥ 0∧[(-1)bso_13] ≥ 0)
COND_636_0_MAIN_GE(TRUE, x1, x0) → 636_0_MAIN_GE(0, +(x0, -1))
- ((U^Increasing(636_0_MAIN_GE(0, +(x0[1], -1))), ≥)∧[bni_14] = 0∧0 = 0∧0 = 0∧[1 + (-1)bso_15] ≥ 0)
636_0_MAIN_GE(x1, x0) → COND_636_0_MAIN_GE1(&&(>(x1, -1), <(x1, x0)), x1, x0)
- (x1[2] ≥ 0∧x0[2] ≥ 0 ⇒ (U^Increasing(COND_636_0_MAIN_GE1(&&(>(x1[2], -1), <(x1[2], x0[2])), x1[2], x0[2])), ≥)∧[(-1)Bound*bni_16] + [bni_16]x1[2] + [bni_16]x0[2] ≥ 0∧[(-1)bso_17] ≥ 0)
COND_636_0_MAIN_GE1(TRUE, x1, x0) → 636_0_MAIN_GE(+(x1, 1), x0)
- ((U^Increasing(636_0_MAIN_GE(+(x1[3], 1), x0[3])), ≥)∧[bni_18] = 0∧0 = 0∧0 = 0∧[(-1)bso_19] ≥ 0)

The constraints for P_> respective P_bound are constructed from P_≥ where we just replace every occurence of "t ≥ s" in P_≥ by "t > s" respective "t ≥ c". Here c stands for the fresh constant used for P_bound.
Using the following integer polynomial ordering the resulting constraints can be solved
Polynomial interpretation over integers[POLO]:

POL(TRUE) = 0
POL(FALSE) = 0
POL(636_0_MAIN_GE(x₁, x₂)) = [-1] + x₂
POL(COND_636_0_MAIN_GE(x₁, x₂, x₃)) = [-1] + x₃
POL(&&(x₁, x₂)) = [-1]
POL(>=(x₁, x₂)) = [-1]
POL(>(x₁, x₂)) = [-1]
POL(1) = [1]
POL(0) = 0
POL(+(x₁, x₂)) = x₁ + x₂
POL(-1) = [-1]
POL(COND_636_0_MAIN_GE1(x₁, x₂, x₃)) = [-1] + x₃
POL(<(x₁, x₂)) = [-1]

The following pairs are in P_>:

COND_636_0_MAIN_GE(TRUE, x1[1], x0[1]) → 636_0_MAIN_GE(0, +(x0[1], -1))

The following pairs are in P_bound:

636_0_MAIN_GE(x1[0], x0[0]) → COND_636_0_MAIN_GE(&&(>=(x1[0], x0[0]), >(x0[0], 1)), x1[0], x0[0])
636_0_MAIN_GE(x1[2], x0[2]) → COND_636_0_MAIN_GE1(&&(>(x1[2], -1), <(x1[2], x0[2])), x1[2], x0[2])

The following pairs are in P_≥:

636_0_MAIN_GE(x1[0], x0[0]) → COND_636_0_MAIN_GE(&&(>=(x1[0], x0[0]), >(x0[0], 1)), x1[0], x0[0])
636_0_MAIN_GE(x1[2], x0[2]) → COND_636_0_MAIN_GE1(&&(>(x1[2], -1), <(x1[2], x0[2])), x1[2], x0[2])
COND_636_0_MAIN_GE1(TRUE, x1[3], x0[3]) → 636_0_MAIN_GE(+(x1[3], 1), x0[3])

There are no usable rules.

(8) Complex Obligation (AND)

(9) 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): 636_0_MAIN_GE(x1[0], x0[0]) → COND_636_0_MAIN_GE(x1[0] >= x0[0] && x0[0] > 1, x1[0], x0[0])
(2): 636_0_MAIN_GE(x1[2], x0[2]) → COND_636_0_MAIN_GE1(x1[2] > -1 && x1[2] < x0[2], x1[2], x0[2])
(3): COND_636_0_MAIN_GE1(TRUE, x1[3], x0[3]) → 636_0_MAIN_GE(x1[3] + 1, x0[3])

(3) -> (0), if (x1[3] + 1 →^* x1[0]∧x0[3] →^* x0[0])

(3) -> (2), if (x1[3] + 1 →^* x1[2]∧x0[3] →^* x0[2])

(2) -> (3), if (x1[2] > -1 && x1[2] < x0[2] ∧x1[2] →^* x1[3]∧x0[2] →^* x0[3])

The set Q is empty.

(10) IDependencyGraphProof (EQUIVALENT transformation)

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

(11) 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_636_0_MAIN_GE1(TRUE, x1[3], x0[3]) → 636_0_MAIN_GE(x1[3] + 1, x0[3])
(2): 636_0_MAIN_GE(x1[2], x0[2]) → COND_636_0_MAIN_GE1(x1[2] > -1 && x1[2] < x0[2], x1[2], x0[2])

(3) -> (2), if (x1[3] + 1 →^* x1[2]∧x0[3] →^* x0[2])

(2) -> (3), if (x1[2] > -1 && x1[2] < x0[2] ∧x1[2] →^* x1[3]∧x0[2] →^* x0[3])

The set Q is empty.

(12) 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.IdpCand1ShapeHeuristic@19aea6e2 Constraint Generator: NonInfConstraintGenerator: PathGenerator: MetricPathGenerator: Max Left Steps: 0 Max Right Steps: 0

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_636_0_MAIN_GE1(TRUE, x1[3], x0[3]) → 636_0_MAIN_GE(+(x1[3], 1), x0[3]) the following chains were created:

We consider the chain COND_636_0_MAIN_GE1(TRUE, x1[3], x0[3]) → 636_0_MAIN_GE(+(x1[3], 1), x0[3]) which results in the following constraint:
(1)    (COND_636_0_MAIN_GE1(TRUE, x1[3], x0[3])≥NonInfC∧COND_636_0_MAIN_GE1(TRUE, x1[3], x0[3])≥636_0_MAIN_GE(+(x1[3], 1), x0[3])∧(U^Increasing(636_0_MAIN_GE(+(x1[3], 1), x0[3])), ≥))

We simplified constraint (1) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(2)    ((U^Increasing(636_0_MAIN_GE(+(x1[3], 1), x0[3])), ≥)∧[bni_10] = 0∧[(-1)bso_11] ≥ 0)

We simplified constraint (2) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(3)    ((U^Increasing(636_0_MAIN_GE(+(x1[3], 1), x0[3])), ≥)∧[bni_10] = 0∧[(-1)bso_11] ≥ 0)

We simplified constraint (3) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(4)    ((U^Increasing(636_0_MAIN_GE(+(x1[3], 1), x0[3])), ≥)∧[bni_10] = 0∧[(-1)bso_11] ≥ 0)

We simplified constraint (4) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint:
(5)    ((U^Increasing(636_0_MAIN_GE(+(x1[3], 1), x0[3])), ≥)∧[bni_10] = 0∧0 = 0∧0 = 0∧[(-1)bso_11] ≥ 0)

For Pair 636_0_MAIN_GE(x1[2], x0[2]) → COND_636_0_MAIN_GE1(&&(>(x1[2], -1), <(x1[2], x0[2])), x1[2], x0[2]) the following chains were created:

We consider the chain 636_0_MAIN_GE(x1[2], x0[2]) → COND_636_0_MAIN_GE1(&&(>(x1[2], -1), <(x1[2], x0[2])), x1[2], x0[2]), COND_636_0_MAIN_GE1(TRUE, x1[3], x0[3]) → 636_0_MAIN_GE(+(x1[3], 1), x0[3]) which results in the following constraint:
(6)    (&&(>(x1[2], -1), <(x1[2], x0[2]))=TRUE∧x1[2]=x1[3]∧x0[2]=x0[3] ⇒ 636_0_MAIN_GE(x1[2], x0[2])≥NonInfC∧636_0_MAIN_GE(x1[2], x0[2])≥COND_636_0_MAIN_GE1(&&(>(x1[2], -1), <(x1[2], x0[2])), x1[2], x0[2])∧(U^Increasing(COND_636_0_MAIN_GE1(&&(>(x1[2], -1), <(x1[2], x0[2])), x1[2], x0[2])), ≥))

We simplified constraint (6) using rules (IV), (IDP_BOOLEAN) which results in the following new constraint:
(7)    (>(x1[2], -1)=TRUE∧<(x1[2], x0[2])=TRUE ⇒ 636_0_MAIN_GE(x1[2], x0[2])≥NonInfC∧636_0_MAIN_GE(x1[2], x0[2])≥COND_636_0_MAIN_GE1(&&(>(x1[2], -1), <(x1[2], x0[2])), x1[2], x0[2])∧(U^Increasing(COND_636_0_MAIN_GE1(&&(>(x1[2], -1), <(x1[2], x0[2])), x1[2], x0[2])), ≥))

We simplified constraint (7) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(8)    (x1[2] ≥ 0∧x0[2] + [-1] + [-1]x1[2] ≥ 0 ⇒ (U^Increasing(COND_636_0_MAIN_GE1(&&(>(x1[2], -1), <(x1[2], x0[2])), x1[2], x0[2])), ≥)∧[(-1)Bound*bni_12] + [(-1)bni_12]x1[2] + [(2)bni_12]x0[2] ≥ 0∧[1 + (-1)bso_13] ≥ 0)

We simplified constraint (8) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(9)    (x1[2] ≥ 0∧x0[2] + [-1] + [-1]x1[2] ≥ 0 ⇒ (U^Increasing(COND_636_0_MAIN_GE1(&&(>(x1[2], -1), <(x1[2], x0[2])), x1[2], x0[2])), ≥)∧[(-1)Bound*bni_12] + [(-1)bni_12]x1[2] + [(2)bni_12]x0[2] ≥ 0∧[1 + (-1)bso_13] ≥ 0)

We simplified constraint (9) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(10)    (x1[2] ≥ 0∧x0[2] + [-1] + [-1]x1[2] ≥ 0 ⇒ (U^Increasing(COND_636_0_MAIN_GE1(&&(>(x1[2], -1), <(x1[2], x0[2])), x1[2], x0[2])), ≥)∧[(-1)Bound*bni_12] + [(-1)bni_12]x1[2] + [(2)bni_12]x0[2] ≥ 0∧[1 + (-1)bso_13] ≥ 0)

We simplified constraint (10) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(11)    (x1[2] ≥ 0∧x0[2] ≥ 0 ⇒ (U^Increasing(COND_636_0_MAIN_GE1(&&(>(x1[2], -1), <(x1[2], x0[2])), x1[2], x0[2])), ≥)∧[(-1)Bound*bni_12 + (2)bni_12] + [bni_12]x1[2] + [(2)bni_12]x0[2] ≥ 0∧[1 + (-1)bso_13] ≥ 0)

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

COND_636_0_MAIN_GE1(TRUE, x1[3], x0[3]) → 636_0_MAIN_GE(+(x1[3], 1), x0[3])
- ((U^Increasing(636_0_MAIN_GE(+(x1[3], 1), x0[3])), ≥)∧[bni_10] = 0∧0 = 0∧0 = 0∧[(-1)bso_11] ≥ 0)
636_0_MAIN_GE(x1[2], x0[2]) → COND_636_0_MAIN_GE1(&&(>(x1[2], -1), <(x1[2], x0[2])), x1[2], x0[2])
- (x1[2] ≥ 0∧x0[2] ≥ 0 ⇒ (U^Increasing(COND_636_0_MAIN_GE1(&&(>(x1[2], -1), <(x1[2], x0[2])), x1[2], x0[2])), ≥)∧[(-1)Bound*bni_12 + (2)bni_12] + [bni_12]x1[2] + [(2)bni_12]x0[2] ≥ 0∧[1 + (-1)bso_13] ≥ 0)

POL(TRUE) = 0
POL(FALSE) = 0
POL(COND_636_0_MAIN_GE1(x₁, x₂, x₃)) = [-1] + [2]x₃ + [-1]x₂
POL(636_0_MAIN_GE(x₁, x₂)) = [-1]x₁ + [2]x₂
POL(+(x₁, x₂)) = x₁ + x₂
POL(1) = [1]
POL(&&(x₁, x₂)) = [-1]
POL(>(x₁, x₂)) = [-1]
POL(-1) = [-1]
POL(<(x₁, x₂)) = [-1]

The following pairs are in P_>:

636_0_MAIN_GE(x1[2], x0[2]) → COND_636_0_MAIN_GE1(&&(>(x1[2], -1), <(x1[2], x0[2])), x1[2], x0[2])

The following pairs are in P_bound:

636_0_MAIN_GE(x1[2], x0[2]) → COND_636_0_MAIN_GE1(&&(>(x1[2], -1), <(x1[2], x0[2])), x1[2], x0[2])

The following pairs are in P_≥:

COND_636_0_MAIN_GE1(TRUE, x1[3], x0[3]) → 636_0_MAIN_GE(+(x1[3], 1), x0[3])

There are no usable rules.

(13) Obligation:

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

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

The following domains are used:

Integer

R is empty.

The integer pair graph contains the following rules and edges:
(3): COND_636_0_MAIN_GE1(TRUE, x1[3], x0[3]) → 636_0_MAIN_GE(x1[3] + 1, x0[3])

The set Q is empty.

(14) IDependencyGraphProof (EQUIVALENT transformation)

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

(15) TRUE

(16) Obligation:

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

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

The following domains are used:

Integer

R is empty.

The integer pair graph contains the following rules and edges:
(1): COND_636_0_MAIN_GE(TRUE, x1[1], x0[1]) → 636_0_MAIN_GE(0, x0[1] + -1)
(3): COND_636_0_MAIN_GE1(TRUE, x1[3], x0[3]) → 636_0_MAIN_GE(x1[3] + 1, x0[3])

The set Q is empty.

(17) IDependencyGraphProof (EQUIVALENT transformation)

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

(0) Obligation:

(1) JBCToGraph (SOUND transformation)

(2) Obligation:

(3) TerminationGraphToSCCProof (SOUND transformation)

(4) Obligation:

(5) SCCToIDPv1Proof (SOUND transformation)

(6) Obligation:

(7) IDPNonInfProof (SOUND transformation)

(8) Complex Obligation (AND)

(9) Obligation:

(10) IDependencyGraphProof (EQUIVALENT transformation)

(11) Obligation:

(12) IDPNonInfProof (SOUND transformation)

(13) Obligation:

(14) IDependencyGraphProof (EQUIVALENT transformation)

(15) TRUE

(16) Obligation:

(17) IDependencyGraphProof (EQUIVALENT transformation)

(18) TRUE