(0) Obligation:

JBC Problem based on JBC Program:
Manifest-Version: 1.0 Created-By: 1.6.0_16 (Sun Microsystems Inc.) Main-Class: PastaC7
/**
* 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 PastaC7 {
public static void main(String[] args) {
Random.args = args;
int i = Random.random();
int j = Random.random();
int k = Random.random();

while (i <= 100 && j <= k) {
int t = i;
i = j;
j = i + 1;
k--;
}
}
}


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:
PastaC7.main([Ljava/lang/String;)V: Graph of 260 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: PastaC7.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 19 rules for P and 0 rules for R.


P rules:
899_0_main_ConstantStackPush(EOS(STATIC_899), i198, i199, i200, i198) → 901_0_main_GT(EOS(STATIC_901), i198, i199, i200, i198, 100)
901_0_main_GT(EOS(STATIC_901), i208, i199, i200, i208, matching1) → 904_0_main_GT(EOS(STATIC_904), i208, i199, i200, i208, 100) | =(matching1, 100)
904_0_main_GT(EOS(STATIC_904), i208, i199, i200, i208, matching1) → 907_0_main_Load(EOS(STATIC_907), i208, i199, i200) | &&(<=(i208, 100), =(matching1, 100))
907_0_main_Load(EOS(STATIC_907), i208, i199, i200) → 910_0_main_Load(EOS(STATIC_910), i208, i199, i200, i199)
910_0_main_Load(EOS(STATIC_910), i208, i199, i200, i199) → 915_0_main_GT(EOS(STATIC_915), i208, i199, i200, i199, i200)
915_0_main_GT(EOS(STATIC_915), i208, i199, i200, i199, i200) → 918_0_main_GT(EOS(STATIC_918), i208, i199, i200, i199, i200)
918_0_main_GT(EOS(STATIC_918), i208, i199, i200, i199, i200) → 925_0_main_Load(EOS(STATIC_925), i208, i199, i200) | <=(i199, i200)
925_0_main_Load(EOS(STATIC_925), i208, i199, i200) → 928_0_main_Store(EOS(STATIC_928), i199, i200, i208)
928_0_main_Store(EOS(STATIC_928), i199, i200, i208) → 931_0_main_Load(EOS(STATIC_931), i199, i200)
931_0_main_Load(EOS(STATIC_931), i199, i200) → 932_0_main_Store(EOS(STATIC_932), i200, i199)
932_0_main_Store(EOS(STATIC_932), i200, i199) → 934_0_main_Load(EOS(STATIC_934), i199, i200)
934_0_main_Load(EOS(STATIC_934), i199, i200) → 936_0_main_ConstantStackPush(EOS(STATIC_936), i199, i200, i199)
936_0_main_ConstantStackPush(EOS(STATIC_936), i199, i200, i199) → 938_0_main_IntArithmetic(EOS(STATIC_938), i199, i200, i199, 1)
938_0_main_IntArithmetic(EOS(STATIC_938), i199, i200, i199, matching1) → 940_0_main_Store(EOS(STATIC_940), i199, i200, +(i199, 1)) | =(matching1, 1)
940_0_main_Store(EOS(STATIC_940), i199, i200, i212) → 942_0_main_Inc(EOS(STATIC_942), i199, i212, i200)
942_0_main_Inc(EOS(STATIC_942), i199, i212, i200) → 944_0_main_JMP(EOS(STATIC_944), i199, i212, +(i200, -1))
944_0_main_JMP(EOS(STATIC_944), i199, i212, i213) → 961_0_main_Load(EOS(STATIC_961), i199, i212, i213)
961_0_main_Load(EOS(STATIC_961), i199, i212, i213) → 896_0_main_Load(EOS(STATIC_896), i199, i212, i213)
896_0_main_Load(EOS(STATIC_896), i198, i199, i200) → 899_0_main_ConstantStackPush(EOS(STATIC_899), i198, i199, i200, i198)
R rules:

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


P rules:
899_0_main_ConstantStackPush(EOS(STATIC_899), x0, x1, x2, x0) → 899_0_main_ConstantStackPush(EOS(STATIC_899), x1, +(x1, 1), +(x2, -1), x1) | &&(>=(x2, x1), <=(x0, 100))
R rules:

Filtered ground terms:



899_0_main_ConstantStackPush(x1, x2, x3, x4, x5) → 899_0_main_ConstantStackPush(x2, x3, x4, x5)
EOS(x1) → EOS
Cond_899_0_main_ConstantStackPush(x1, x2, x3, x4, x5, x6) → Cond_899_0_main_ConstantStackPush(x1, x3, x4, x5, x6)

Filtered duplicate args:



899_0_main_ConstantStackPush(x1, x2, x3, x4) → 899_0_main_ConstantStackPush(x2, x3, x4)
Cond_899_0_main_ConstantStackPush(x1, x2, x3, x4, x5) → Cond_899_0_main_ConstantStackPush(x1, x3, x4, x5)

Filtered unneeded arguments:



Cond_899_0_main_ConstantStackPush(x1, x2, x3, x4) → Cond_899_0_main_ConstantStackPush(x1, x2, x3)

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


P rules:
899_0_main_ConstantStackPush(x1, x2, x0) → 899_0_main_ConstantStackPush(+(x1, 1), +(x2, -1), x1) | &&(>=(x2, x1), <=(x0, 100))
R rules:

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


P rules:
899_0_MAIN_CONSTANTSTACKPUSH(x1, x2, x0) → COND_899_0_MAIN_CONSTANTSTACKPUSH(&&(>=(x2, x1), <=(x0, 100)), x1, x2, x0)
COND_899_0_MAIN_CONSTANTSTACKPUSH(TRUE, x1, x2, x0) → 899_0_MAIN_CONSTANTSTACKPUSH(+(x1, 1), +(x2, -1), 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): 899_0_MAIN_CONSTANTSTACKPUSH(x1[0], x2[0], x0[0]) → COND_899_0_MAIN_CONSTANTSTACKPUSH(x2[0] >= x1[0] && x0[0] <= 100, x1[0], x2[0], x0[0])
(1): COND_899_0_MAIN_CONSTANTSTACKPUSH(TRUE, x1[1], x2[1], x0[1]) → 899_0_MAIN_CONSTANTSTACKPUSH(x1[1] + 1, x2[1] + -1, x1[1])

(0) -> (1), if (x2[0] >= x1[0] && x0[0] <= 100x1[0]* x1[1]x2[0]* x2[1]x0[0]* x0[1])


(1) -> (0), if (x1[1] + 1* x1[0]x2[1] + -1* x2[0]x1[1]* x0[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@1fff293 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 899_0_MAIN_CONSTANTSTACKPUSH(x1, x2, x0) → COND_899_0_MAIN_CONSTANTSTACKPUSH(&&(>=(x2, x1), <=(x0, 100)), x1, x2, x0) the following chains were created:
  • We consider the chain 899_0_MAIN_CONSTANTSTACKPUSH(x1[0], x2[0], x0[0]) → COND_899_0_MAIN_CONSTANTSTACKPUSH(&&(>=(x2[0], x1[0]), <=(x0[0], 100)), x1[0], x2[0], x0[0]), COND_899_0_MAIN_CONSTANTSTACKPUSH(TRUE, x1[1], x2[1], x0[1]) → 899_0_MAIN_CONSTANTSTACKPUSH(+(x1[1], 1), +(x2[1], -1), x1[1]) which results in the following constraint:

    (1)    (&&(>=(x2[0], x1[0]), <=(x0[0], 100))=TRUEx1[0]=x1[1]x2[0]=x2[1]x0[0]=x0[1]899_0_MAIN_CONSTANTSTACKPUSH(x1[0], x2[0], x0[0])≥NonInfC∧899_0_MAIN_CONSTANTSTACKPUSH(x1[0], x2[0], x0[0])≥COND_899_0_MAIN_CONSTANTSTACKPUSH(&&(>=(x2[0], x1[0]), <=(x0[0], 100)), x1[0], x2[0], x0[0])∧(UIncreasing(COND_899_0_MAIN_CONSTANTSTACKPUSH(&&(>=(x2[0], x1[0]), <=(x0[0], 100)), x1[0], x2[0], x0[0])), ≥))



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

    (2)    (>=(x2[0], x1[0])=TRUE<=(x0[0], 100)=TRUE899_0_MAIN_CONSTANTSTACKPUSH(x1[0], x2[0], x0[0])≥NonInfC∧899_0_MAIN_CONSTANTSTACKPUSH(x1[0], x2[0], x0[0])≥COND_899_0_MAIN_CONSTANTSTACKPUSH(&&(>=(x2[0], x1[0]), <=(x0[0], 100)), x1[0], x2[0], x0[0])∧(UIncreasing(COND_899_0_MAIN_CONSTANTSTACKPUSH(&&(>=(x2[0], x1[0]), <=(x0[0], 100)), x1[0], x2[0], x0[0])), ≥))



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

    (3)    (x2[0] + [-1]x1[0] ≥ 0∧[100] + [-1]x0[0] ≥ 0 ⇒ (UIncreasing(COND_899_0_MAIN_CONSTANTSTACKPUSH(&&(>=(x2[0], x1[0]), <=(x0[0], 100)), x1[0], x2[0], x0[0])), ≥)∧[(-1)Bound*bni_15] + [bni_15]x2[0] + [(-1)bni_15]x1[0] ≥ 0∧[1 + (-1)bso_16] ≥ 0)



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

    (4)    (x2[0] + [-1]x1[0] ≥ 0∧[100] + [-1]x0[0] ≥ 0 ⇒ (UIncreasing(COND_899_0_MAIN_CONSTANTSTACKPUSH(&&(>=(x2[0], x1[0]), <=(x0[0], 100)), x1[0], x2[0], x0[0])), ≥)∧[(-1)Bound*bni_15] + [bni_15]x2[0] + [(-1)bni_15]x1[0] ≥ 0∧[1 + (-1)bso_16] ≥ 0)



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

    (5)    (x2[0] + [-1]x1[0] ≥ 0∧[100] + [-1]x0[0] ≥ 0 ⇒ (UIncreasing(COND_899_0_MAIN_CONSTANTSTACKPUSH(&&(>=(x2[0], x1[0]), <=(x0[0], 100)), x1[0], x2[0], x0[0])), ≥)∧[(-1)Bound*bni_15] + [bni_15]x2[0] + [(-1)bni_15]x1[0] ≥ 0∧[1 + (-1)bso_16] ≥ 0)



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

    (6)    (x2[0] ≥ 0∧[100] + [-1]x0[0] ≥ 0 ⇒ (UIncreasing(COND_899_0_MAIN_CONSTANTSTACKPUSH(&&(>=(x2[0], x1[0]), <=(x0[0], 100)), x1[0], x2[0], x0[0])), ≥)∧[(-1)Bound*bni_15] + [bni_15]x2[0] ≥ 0∧[1 + (-1)bso_16] ≥ 0)



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

    (7)    (x2[0] ≥ 0∧[100] + [-1]x0[0] ≥ 0∧x1[0] ≥ 0 ⇒ (UIncreasing(COND_899_0_MAIN_CONSTANTSTACKPUSH(&&(>=(x2[0], x1[0]), <=(x0[0], 100)), x1[0], x2[0], x0[0])), ≥)∧[(-1)Bound*bni_15] + [bni_15]x2[0] ≥ 0∧[1 + (-1)bso_16] ≥ 0)


    (8)    (x2[0] ≥ 0∧[100] + [-1]x0[0] ≥ 0∧x1[0] ≥ 0 ⇒ (UIncreasing(COND_899_0_MAIN_CONSTANTSTACKPUSH(&&(>=(x2[0], x1[0]), <=(x0[0], 100)), x1[0], x2[0], x0[0])), ≥)∧[(-1)Bound*bni_15] + [bni_15]x2[0] ≥ 0∧[1 + (-1)bso_16] ≥ 0)



    We simplified constraint (7) using rule (IDP_SMT_SPLIT) which results in the following new constraints:

    (9)    (x2[0] ≥ 0∧[100] + [-1]x0[0] ≥ 0∧x1[0] ≥ 0∧x0[0] ≥ 0 ⇒ (UIncreasing(COND_899_0_MAIN_CONSTANTSTACKPUSH(&&(>=(x2[0], x1[0]), <=(x0[0], 100)), x1[0], x2[0], x0[0])), ≥)∧[(-1)Bound*bni_15] + [bni_15]x2[0] ≥ 0∧[1 + (-1)bso_16] ≥ 0)


    (10)    (x2[0] ≥ 0∧[100] + x0[0] ≥ 0∧x1[0] ≥ 0∧x0[0] ≥ 0 ⇒ (UIncreasing(COND_899_0_MAIN_CONSTANTSTACKPUSH(&&(>=(x2[0], x1[0]), <=(x0[0], 100)), x1[0], x2[0], x0[0])), ≥)∧[(-1)Bound*bni_15] + [bni_15]x2[0] ≥ 0∧[1 + (-1)bso_16] ≥ 0)



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

    (11)    (x2[0] ≥ 0∧[100] + x0[0] ≥ 0∧x1[0] ≥ 0∧x0[0] ≥ 0 ⇒ (UIncreasing(COND_899_0_MAIN_CONSTANTSTACKPUSH(&&(>=(x2[0], x1[0]), <=(x0[0], 100)), x1[0], x2[0], x0[0])), ≥)∧[(-1)Bound*bni_15] + [bni_15]x2[0] ≥ 0∧[1 + (-1)bso_16] ≥ 0)


    (12)    (x2[0] ≥ 0∧[100] + [-1]x0[0] ≥ 0∧x1[0] ≥ 0∧x0[0] ≥ 0 ⇒ (UIncreasing(COND_899_0_MAIN_CONSTANTSTACKPUSH(&&(>=(x2[0], x1[0]), <=(x0[0], 100)), x1[0], x2[0], x0[0])), ≥)∧[(-1)Bound*bni_15] + [bni_15]x2[0] ≥ 0∧[1 + (-1)bso_16] ≥ 0)







For Pair COND_899_0_MAIN_CONSTANTSTACKPUSH(TRUE, x1, x2, x0) → 899_0_MAIN_CONSTANTSTACKPUSH(+(x1, 1), +(x2, -1), x1) the following chains were created:
  • We consider the chain 899_0_MAIN_CONSTANTSTACKPUSH(x1[0], x2[0], x0[0]) → COND_899_0_MAIN_CONSTANTSTACKPUSH(&&(>=(x2[0], x1[0]), <=(x0[0], 100)), x1[0], x2[0], x0[0]), COND_899_0_MAIN_CONSTANTSTACKPUSH(TRUE, x1[1], x2[1], x0[1]) → 899_0_MAIN_CONSTANTSTACKPUSH(+(x1[1], 1), +(x2[1], -1), x1[1]), 899_0_MAIN_CONSTANTSTACKPUSH(x1[0], x2[0], x0[0]) → COND_899_0_MAIN_CONSTANTSTACKPUSH(&&(>=(x2[0], x1[0]), <=(x0[0], 100)), x1[0], x2[0], x0[0]) which results in the following constraint:

    (13)    (&&(>=(x2[0], x1[0]), <=(x0[0], 100))=TRUEx1[0]=x1[1]x2[0]=x2[1]x0[0]=x0[1]+(x1[1], 1)=x1[0]1+(x2[1], -1)=x2[0]1x1[1]=x0[0]1COND_899_0_MAIN_CONSTANTSTACKPUSH(TRUE, x1[1], x2[1], x0[1])≥NonInfC∧COND_899_0_MAIN_CONSTANTSTACKPUSH(TRUE, x1[1], x2[1], x0[1])≥899_0_MAIN_CONSTANTSTACKPUSH(+(x1[1], 1), +(x2[1], -1), x1[1])∧(UIncreasing(899_0_MAIN_CONSTANTSTACKPUSH(+(x1[1], 1), +(x2[1], -1), x1[1])), ≥))



    We simplified constraint (13) using rules (III), (IV), (IDP_BOOLEAN) which results in the following new constraint:

    (14)    (>=(x2[0], x1[0])=TRUE<=(x0[0], 100)=TRUECOND_899_0_MAIN_CONSTANTSTACKPUSH(TRUE, x1[0], x2[0], x0[0])≥NonInfC∧COND_899_0_MAIN_CONSTANTSTACKPUSH(TRUE, x1[0], x2[0], x0[0])≥899_0_MAIN_CONSTANTSTACKPUSH(+(x1[0], 1), +(x2[0], -1), x1[0])∧(UIncreasing(899_0_MAIN_CONSTANTSTACKPUSH(+(x1[1], 1), +(x2[1], -1), x1[1])), ≥))



    We simplified constraint (14) using rule (POLY_CONSTRAINTS) which results in the following new constraint:

    (15)    (x2[0] + [-1]x1[0] ≥ 0∧[100] + [-1]x0[0] ≥ 0 ⇒ (UIncreasing(899_0_MAIN_CONSTANTSTACKPUSH(+(x1[1], 1), +(x2[1], -1), x1[1])), ≥)∧[(-1)bni_17 + (-1)Bound*bni_17] + [bni_17]x2[0] + [(-1)bni_17]x1[0] ≥ 0∧[1 + (-1)bso_18] ≥ 0)



    We simplified constraint (15) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:

    (16)    (x2[0] + [-1]x1[0] ≥ 0∧[100] + [-1]x0[0] ≥ 0 ⇒ (UIncreasing(899_0_MAIN_CONSTANTSTACKPUSH(+(x1[1], 1), +(x2[1], -1), x1[1])), ≥)∧[(-1)bni_17 + (-1)Bound*bni_17] + [bni_17]x2[0] + [(-1)bni_17]x1[0] ≥ 0∧[1 + (-1)bso_18] ≥ 0)



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

    (17)    (x2[0] + [-1]x1[0] ≥ 0∧[100] + [-1]x0[0] ≥ 0 ⇒ (UIncreasing(899_0_MAIN_CONSTANTSTACKPUSH(+(x1[1], 1), +(x2[1], -1), x1[1])), ≥)∧[(-1)bni_17 + (-1)Bound*bni_17] + [bni_17]x2[0] + [(-1)bni_17]x1[0] ≥ 0∧[1 + (-1)bso_18] ≥ 0)



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

    (18)    (x2[0] ≥ 0∧[100] + [-1]x0[0] ≥ 0 ⇒ (UIncreasing(899_0_MAIN_CONSTANTSTACKPUSH(+(x1[1], 1), +(x2[1], -1), x1[1])), ≥)∧[(-1)bni_17 + (-1)Bound*bni_17] + [bni_17]x2[0] ≥ 0∧[1 + (-1)bso_18] ≥ 0)



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

    (19)    (x2[0] ≥ 0∧[100] + [-1]x0[0] ≥ 0∧x1[0] ≥ 0 ⇒ (UIncreasing(899_0_MAIN_CONSTANTSTACKPUSH(+(x1[1], 1), +(x2[1], -1), x1[1])), ≥)∧[(-1)bni_17 + (-1)Bound*bni_17] + [bni_17]x2[0] ≥ 0∧[1 + (-1)bso_18] ≥ 0)


    (20)    (x2[0] ≥ 0∧[100] + [-1]x0[0] ≥ 0∧x1[0] ≥ 0 ⇒ (UIncreasing(899_0_MAIN_CONSTANTSTACKPUSH(+(x1[1], 1), +(x2[1], -1), x1[1])), ≥)∧[(-1)bni_17 + (-1)Bound*bni_17] + [bni_17]x2[0] ≥ 0∧[1 + (-1)bso_18] ≥ 0)



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

    (21)    (x2[0] ≥ 0∧[100] + [-1]x0[0] ≥ 0∧x1[0] ≥ 0∧x0[0] ≥ 0 ⇒ (UIncreasing(899_0_MAIN_CONSTANTSTACKPUSH(+(x1[1], 1), +(x2[1], -1), x1[1])), ≥)∧[(-1)bni_17 + (-1)Bound*bni_17] + [bni_17]x2[0] ≥ 0∧[1 + (-1)bso_18] ≥ 0)


    (22)    (x2[0] ≥ 0∧[100] + x0[0] ≥ 0∧x1[0] ≥ 0∧x0[0] ≥ 0 ⇒ (UIncreasing(899_0_MAIN_CONSTANTSTACKPUSH(+(x1[1], 1), +(x2[1], -1), x1[1])), ≥)∧[(-1)bni_17 + (-1)Bound*bni_17] + [bni_17]x2[0] ≥ 0∧[1 + (-1)bso_18] ≥ 0)



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

    (23)    (x2[0] ≥ 0∧[100] + x0[0] ≥ 0∧x1[0] ≥ 0∧x0[0] ≥ 0 ⇒ (UIncreasing(899_0_MAIN_CONSTANTSTACKPUSH(+(x1[1], 1), +(x2[1], -1), x1[1])), ≥)∧[(-1)bni_17 + (-1)Bound*bni_17] + [bni_17]x2[0] ≥ 0∧[1 + (-1)bso_18] ≥ 0)


    (24)    (x2[0] ≥ 0∧[100] + [-1]x0[0] ≥ 0∧x1[0] ≥ 0∧x0[0] ≥ 0 ⇒ (UIncreasing(899_0_MAIN_CONSTANTSTACKPUSH(+(x1[1], 1), +(x2[1], -1), x1[1])), ≥)∧[(-1)bni_17 + (-1)Bound*bni_17] + [bni_17]x2[0] ≥ 0∧[1 + (-1)bso_18] ≥ 0)







To summarize, we get the following constraints P for the following pairs.
  • 899_0_MAIN_CONSTANTSTACKPUSH(x1, x2, x0) → COND_899_0_MAIN_CONSTANTSTACKPUSH(&&(>=(x2, x1), <=(x0, 100)), x1, x2, x0)
    • (x2[0] ≥ 0∧[100] + [-1]x0[0] ≥ 0∧x1[0] ≥ 0∧x0[0] ≥ 0 ⇒ (UIncreasing(COND_899_0_MAIN_CONSTANTSTACKPUSH(&&(>=(x2[0], x1[0]), <=(x0[0], 100)), x1[0], x2[0], x0[0])), ≥)∧[(-1)Bound*bni_15] + [bni_15]x2[0] ≥ 0∧[1 + (-1)bso_16] ≥ 0)
    • (x2[0] ≥ 0∧[100] + x0[0] ≥ 0∧x1[0] ≥ 0∧x0[0] ≥ 0 ⇒ (UIncreasing(COND_899_0_MAIN_CONSTANTSTACKPUSH(&&(>=(x2[0], x1[0]), <=(x0[0], 100)), x1[0], x2[0], x0[0])), ≥)∧[(-1)Bound*bni_15] + [bni_15]x2[0] ≥ 0∧[1 + (-1)bso_16] ≥ 0)
    • (x2[0] ≥ 0∧[100] + x0[0] ≥ 0∧x1[0] ≥ 0∧x0[0] ≥ 0 ⇒ (UIncreasing(COND_899_0_MAIN_CONSTANTSTACKPUSH(&&(>=(x2[0], x1[0]), <=(x0[0], 100)), x1[0], x2[0], x0[0])), ≥)∧[(-1)Bound*bni_15] + [bni_15]x2[0] ≥ 0∧[1 + (-1)bso_16] ≥ 0)
    • (x2[0] ≥ 0∧[100] + [-1]x0[0] ≥ 0∧x1[0] ≥ 0∧x0[0] ≥ 0 ⇒ (UIncreasing(COND_899_0_MAIN_CONSTANTSTACKPUSH(&&(>=(x2[0], x1[0]), <=(x0[0], 100)), x1[0], x2[0], x0[0])), ≥)∧[(-1)Bound*bni_15] + [bni_15]x2[0] ≥ 0∧[1 + (-1)bso_16] ≥ 0)

  • COND_899_0_MAIN_CONSTANTSTACKPUSH(TRUE, x1, x2, x0) → 899_0_MAIN_CONSTANTSTACKPUSH(+(x1, 1), +(x2, -1), x1)
    • (x2[0] ≥ 0∧[100] + [-1]x0[0] ≥ 0∧x1[0] ≥ 0∧x0[0] ≥ 0 ⇒ (UIncreasing(899_0_MAIN_CONSTANTSTACKPUSH(+(x1[1], 1), +(x2[1], -1), x1[1])), ≥)∧[(-1)bni_17 + (-1)Bound*bni_17] + [bni_17]x2[0] ≥ 0∧[1 + (-1)bso_18] ≥ 0)
    • (x2[0] ≥ 0∧[100] + x0[0] ≥ 0∧x1[0] ≥ 0∧x0[0] ≥ 0 ⇒ (UIncreasing(899_0_MAIN_CONSTANTSTACKPUSH(+(x1[1], 1), +(x2[1], -1), x1[1])), ≥)∧[(-1)bni_17 + (-1)Bound*bni_17] + [bni_17]x2[0] ≥ 0∧[1 + (-1)bso_18] ≥ 0)
    • (x2[0] ≥ 0∧[100] + x0[0] ≥ 0∧x1[0] ≥ 0∧x0[0] ≥ 0 ⇒ (UIncreasing(899_0_MAIN_CONSTANTSTACKPUSH(+(x1[1], 1), +(x2[1], -1), x1[1])), ≥)∧[(-1)bni_17 + (-1)Bound*bni_17] + [bni_17]x2[0] ≥ 0∧[1 + (-1)bso_18] ≥ 0)
    • (x2[0] ≥ 0∧[100] + [-1]x0[0] ≥ 0∧x1[0] ≥ 0∧x0[0] ≥ 0 ⇒ (UIncreasing(899_0_MAIN_CONSTANTSTACKPUSH(+(x1[1], 1), +(x2[1], -1), x1[1])), ≥)∧[(-1)bni_17 + (-1)Bound*bni_17] + [bni_17]x2[0] ≥ 0∧[1 + (-1)bso_18] ≥ 0)




The constraints for P> respective Pbound 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 Pbound.
Using the following integer polynomial ordering the resulting constraints can be solved
Polynomial interpretation over integers[POLO]:

POL(TRUE) = 0   
POL(FALSE) = 0   
POL(899_0_MAIN_CONSTANTSTACKPUSH(x1, x2, x3)) = x2 + [-1]x1   
POL(COND_899_0_MAIN_CONSTANTSTACKPUSH(x1, x2, x3, x4)) = [-1] + x3 + [-1]x2 + [-1]x1   
POL(&&(x1, x2)) = 0   
POL(>=(x1, x2)) = [-1]   
POL(<=(x1, x2)) = [-1]   
POL(100) = [100]   
POL(+(x1, x2)) = x1 + x2   
POL(1) = [1]   
POL(-1) = [-1]   

The following pairs are in P>:

899_0_MAIN_CONSTANTSTACKPUSH(x1[0], x2[0], x0[0]) → COND_899_0_MAIN_CONSTANTSTACKPUSH(&&(>=(x2[0], x1[0]), <=(x0[0], 100)), x1[0], x2[0], x0[0])
COND_899_0_MAIN_CONSTANTSTACKPUSH(TRUE, x1[1], x2[1], x0[1]) → 899_0_MAIN_CONSTANTSTACKPUSH(+(x1[1], 1), +(x2[1], -1), x1[1])

The following pairs are in Pbound:

899_0_MAIN_CONSTANTSTACKPUSH(x1[0], x2[0], x0[0]) → COND_899_0_MAIN_CONSTANTSTACKPUSH(&&(>=(x2[0], x1[0]), <=(x0[0], 100)), x1[0], x2[0], x0[0])
COND_899_0_MAIN_CONSTANTSTACKPUSH(TRUE, x1[1], x2[1], x0[1]) → 899_0_MAIN_CONSTANTSTACKPUSH(+(x1[1], 1), +(x2[1], -1), x1[1])

The following pairs are in P:
none

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

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

(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:
none


R is empty.

The integer pair graph is empty.

The set Q is empty.

(9) IDependencyGraphProof (EQUIVALENT transformation)

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

(10) TRUE