(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:
227_0_main_LT(EOS(STATIC_227), i32, i32) → 237_0_main_LT(EOS(STATIC_237), i32, i32)
237_0_main_LT(EOS(STATIC_237), i32, i32) → 246_0_main_ConstantStackPush(EOS(STATIC_246), i32) | >=(i32, 0)
246_0_main_ConstantStackPush(EOS(STATIC_246), i32) → 252_0_main_Store(EOS(STATIC_252), i32, 1)
252_0_main_Store(EOS(STATIC_252), i32, matching1) → 257_0_main_Load(EOS(STATIC_257), i32, 1) | =(matching1, 1)
257_0_main_Load(EOS(STATIC_257), i32, matching1) → 307_0_main_Load(EOS(STATIC_307), i32, 1) | =(matching1, 1)
307_0_main_Load(EOS(STATIC_307), i42, i43) → 348_0_main_Load(EOS(STATIC_348), i42, i43)
348_0_main_Load(EOS(STATIC_348), i42, i53) → 385_0_main_Load(EOS(STATIC_385), i42, i53)
385_0_main_Load(EOS(STATIC_385), i42, i60) → 422_0_main_Load(EOS(STATIC_422), i42, i60)
422_0_main_Load(EOS(STATIC_422), i42, i66) → 426_0_main_Load(EOS(STATIC_426), i42, i66, i42)
426_0_main_Load(EOS(STATIC_426), i42, i66, i42) → 428_0_main_LE(EOS(STATIC_428), i42, i66, i42, i66)
428_0_main_LE(EOS(STATIC_428), i42, i66, i42, i66) → 430_0_main_LE(EOS(STATIC_430), i42, i66, i42, i66)
428_0_main_LE(EOS(STATIC_428), i42, i66, i42, i66) → 431_0_main_LE(EOS(STATIC_431), i42, i66, i42, i66)
430_0_main_LE(EOS(STATIC_430), i42, i66, i42, i66) → 432_0_main_Inc(EOS(STATIC_432), i42) | <=(i42, i66)
432_0_main_Inc(EOS(STATIC_432), i42) → 436_0_main_JMP(EOS(STATIC_436), +(i42, -1)) | >=(i42, 0)
436_0_main_JMP(EOS(STATIC_436), i68) → 441_0_main_Load(EOS(STATIC_441), i68)
441_0_main_Load(EOS(STATIC_441), i68) → 218_0_main_Load(EOS(STATIC_218), i68)
218_0_main_Load(EOS(STATIC_218), i28) → 227_0_main_LT(EOS(STATIC_227), i28, i28)
431_0_main_LE(EOS(STATIC_431), i42, i66, i42, i66) → 434_0_main_ConstantStackPush(EOS(STATIC_434), i42, i66) | >(i42, i66)
434_0_main_ConstantStackPush(EOS(STATIC_434), i42, i66) → 438_0_main_Load(EOS(STATIC_438), i42, i66, 2)
438_0_main_Load(EOS(STATIC_438), i42, i66, matching1) → 443_0_main_IntArithmetic(EOS(STATIC_443), i42, 2, i66) | =(matching1, 2)
443_0_main_IntArithmetic(EOS(STATIC_443), i42, matching1, i66) → 445_0_main_Store(EOS(STATIC_445), i42, *(2, i66)) | &&(>=(i66, 1), =(matching1, 2))
445_0_main_Store(EOS(STATIC_445), i42, i70) → 447_0_main_JMP(EOS(STATIC_447), i42, i70)
447_0_main_JMP(EOS(STATIC_447), i42, i70) → 450_0_main_Load(EOS(STATIC_450), i42, i70)
450_0_main_Load(EOS(STATIC_450), i42, i70) → 422_0_main_Load(EOS(STATIC_422), i42, i70)
R rules:

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


P rules:
428_0_main_LE(EOS(STATIC_428), x0, x1, x0, x1) → 428_0_main_LE(EOS(STATIC_428), +(x0, -1), 1, +(x0, -1), 1) | &&(>=(x1, x0), >(+(x0, 1), 1))
428_0_main_LE(EOS(STATIC_428), x0, x1, x0, x1) → 428_0_main_LE(EOS(STATIC_428), x0, *(2, x1), x0, *(2, x1)) | &&(>(+(x1, 1), 1), <(x1, x0))
R rules:

Filtered ground terms:



428_0_main_LE(x1, x2, x3, x4, x5) → 428_0_main_LE(x2, x3, x4, x5)
EOS(x1) → EOS
Cond_428_0_main_LE1(x1, x2, x3, x4, x5, x6) → Cond_428_0_main_LE1(x1, x3, x4, x5, x6)
Cond_428_0_main_LE(x1, x2, x3, x4, x5, x6) → Cond_428_0_main_LE(x1, x3, x4, x5, x6)

Filtered duplicate args:



428_0_main_LE(x1, x2, x3, x4) → 428_0_main_LE(x3, x4)
Cond_428_0_main_LE(x1, x2, x3, x4, x5) → Cond_428_0_main_LE(x1, x4, x5)
Cond_428_0_main_LE1(x1, x2, x3, x4, x5) → Cond_428_0_main_LE1(x1, x4, x5)

Filtered unneeded arguments:



Cond_428_0_main_LE(x1, x2, x3) → Cond_428_0_main_LE(x1, x2)

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


P rules:
428_0_main_LE(x0, x1) → 428_0_main_LE(+(x0, -1), 1) | &&(>=(x1, x0), >(x0, 0))
428_0_main_LE(x0, x1) → 428_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:
428_0_MAIN_LE(x0, x1) → COND_428_0_MAIN_LE(&&(>=(x1, x0), >(x0, 0)), x0, x1)
COND_428_0_MAIN_LE(TRUE, x0, x1) → 428_0_MAIN_LE(+(x0, -1), 1)
428_0_MAIN_LE(x0, x1) → COND_428_0_MAIN_LE1(&&(>(x1, 0), <(x1, x0)), x0, x1)
COND_428_0_MAIN_LE1(TRUE, x0, x1) → 428_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): 428_0_MAIN_LE(x0[0], x1[0]) → COND_428_0_MAIN_LE(x1[0] >= x0[0] && x0[0] > 0, x0[0], x1[0])
(1): COND_428_0_MAIN_LE(TRUE, x0[1], x1[1]) → 428_0_MAIN_LE(x0[1] + -1, 1)
(2): 428_0_MAIN_LE(x0[2], x1[2]) → COND_428_0_MAIN_LE1(x1[2] > 0 && x1[2] < x0[2], x0[2], x1[2])
(3): COND_428_0_MAIN_LE1(TRUE, x0[3], x1[3]) → 428_0_MAIN_LE(x0[3], 2 * x1[3])

(0) -> (1), if (x1[0] >= x0[0] && x0[0] > 0x0[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@ea6e48f 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 428_0_MAIN_LE(x0, x1) → COND_428_0_MAIN_LE(&&(>=(x1, x0), >(x0, 0)), x0, x1) the following chains were created:
  • We consider the chain 428_0_MAIN_LE(x0[0], x1[0]) → COND_428_0_MAIN_LE(&&(>=(x1[0], x0[0]), >(x0[0], 0)), x0[0], x1[0]), COND_428_0_MAIN_LE(TRUE, x0[1], x1[1]) → 428_0_MAIN_LE(+(x0[1], -1), 1) which results in the following constraint:

    (1)    (&&(>=(x1[0], x0[0]), >(x0[0], 0))=TRUEx0[0]=x0[1]x1[0]=x1[1]428_0_MAIN_LE(x0[0], x1[0])≥NonInfC∧428_0_MAIN_LE(x0[0], x1[0])≥COND_428_0_MAIN_LE(&&(>=(x1[0], x0[0]), >(x0[0], 0)), x0[0], x1[0])∧(UIncreasing(COND_428_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)=TRUE428_0_MAIN_LE(x0[0], x1[0])≥NonInfC∧428_0_MAIN_LE(x0[0], x1[0])≥COND_428_0_MAIN_LE(&&(>=(x1[0], x0[0]), >(x0[0], 0)), x0[0], x1[0])∧(UIncreasing(COND_428_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 ⇒ (UIncreasing(COND_428_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 ⇒ (UIncreasing(COND_428_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 ⇒ (UIncreasing(COND_428_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 ⇒ (UIncreasing(COND_428_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 ⇒ (UIncreasing(COND_428_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_428_0_MAIN_LE(TRUE, x0, x1) → 428_0_MAIN_LE(+(x0, -1), 1) the following chains were created:
  • We consider the chain 428_0_MAIN_LE(x0[0], x1[0]) → COND_428_0_MAIN_LE(&&(>=(x1[0], x0[0]), >(x0[0], 0)), x0[0], x1[0]), COND_428_0_MAIN_LE(TRUE, x0[1], x1[1]) → 428_0_MAIN_LE(+(x0[1], -1), 1), 428_0_MAIN_LE(x0[0], x1[0]) → COND_428_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))=TRUEx0[0]=x0[1]x1[0]=x1[1]+(x0[1], -1)=x0[0]11=x1[0]1COND_428_0_MAIN_LE(TRUE, x0[1], x1[1])≥NonInfC∧COND_428_0_MAIN_LE(TRUE, x0[1], x1[1])≥428_0_MAIN_LE(+(x0[1], -1), 1)∧(UIncreasing(428_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)=TRUECOND_428_0_MAIN_LE(TRUE, x0[0], x1[0])≥NonInfC∧COND_428_0_MAIN_LE(TRUE, x0[0], x1[0])≥428_0_MAIN_LE(+(x0[0], -1), 1)∧(UIncreasing(428_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 ⇒ (UIncreasing(428_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 ⇒ (UIncreasing(428_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 ⇒ (UIncreasing(428_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 ⇒ (UIncreasing(428_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 ⇒ (UIncreasing(428_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 428_0_MAIN_LE(x0[0], x1[0]) → COND_428_0_MAIN_LE(&&(>=(x1[0], x0[0]), >(x0[0], 0)), x0[0], x1[0]), COND_428_0_MAIN_LE(TRUE, x0[1], x1[1]) → 428_0_MAIN_LE(+(x0[1], -1), 1), 428_0_MAIN_LE(x0[2], x1[2]) → COND_428_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))=TRUEx0[0]=x0[1]x1[0]=x1[1]+(x0[1], -1)=x0[2]1=x1[2]COND_428_0_MAIN_LE(TRUE, x0[1], x1[1])≥NonInfC∧COND_428_0_MAIN_LE(TRUE, x0[1], x1[1])≥428_0_MAIN_LE(+(x0[1], -1), 1)∧(UIncreasing(428_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)=TRUECOND_428_0_MAIN_LE(TRUE, x0[0], x1[0])≥NonInfC∧COND_428_0_MAIN_LE(TRUE, x0[0], x1[0])≥428_0_MAIN_LE(+(x0[0], -1), 1)∧(UIncreasing(428_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 ⇒ (UIncreasing(428_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 ⇒ (UIncreasing(428_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 ⇒ (UIncreasing(428_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 ⇒ (UIncreasing(428_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 ⇒ (UIncreasing(428_0_MAIN_LE(+(x0[1], -1), 1)), ≥)∧[(-1)Bound*bni_20] + [bni_20]x0[0] ≥ 0∧[1 + (-1)bso_21] ≥ 0)







For Pair 428_0_MAIN_LE(x0, x1) → COND_428_0_MAIN_LE1(&&(>(x1, 0), <(x1, x0)), x0, x1) the following chains were created:
  • We consider the chain 428_0_MAIN_LE(x0[2], x1[2]) → COND_428_0_MAIN_LE1(&&(>(x1[2], 0), <(x1[2], x0[2])), x0[2], x1[2]), COND_428_0_MAIN_LE1(TRUE, x0[3], x1[3]) → 428_0_MAIN_LE(x0[3], *(2, x1[3])) which results in the following constraint:

    (22)    (&&(>(x1[2], 0), <(x1[2], x0[2]))=TRUEx0[2]=x0[3]x1[2]=x1[3]428_0_MAIN_LE(x0[2], x1[2])≥NonInfC∧428_0_MAIN_LE(x0[2], x1[2])≥COND_428_0_MAIN_LE1(&&(>(x1[2], 0), <(x1[2], x0[2])), x0[2], x1[2])∧(UIncreasing(COND_428_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])=TRUE428_0_MAIN_LE(x0[2], x1[2])≥NonInfC∧428_0_MAIN_LE(x0[2], x1[2])≥COND_428_0_MAIN_LE1(&&(>(x1[2], 0), <(x1[2], x0[2])), x0[2], x1[2])∧(UIncreasing(COND_428_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 ⇒ (UIncreasing(COND_428_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 ⇒ (UIncreasing(COND_428_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 ⇒ (UIncreasing(COND_428_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 ⇒ (UIncreasing(COND_428_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 ⇒ (UIncreasing(COND_428_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_428_0_MAIN_LE1(TRUE, x0, x1) → 428_0_MAIN_LE(x0, *(2, x1)) the following chains were created:
  • We consider the chain 428_0_MAIN_LE(x0[2], x1[2]) → COND_428_0_MAIN_LE1(&&(>(x1[2], 0), <(x1[2], x0[2])), x0[2], x1[2]), COND_428_0_MAIN_LE1(TRUE, x0[3], x1[3]) → 428_0_MAIN_LE(x0[3], *(2, x1[3])), 428_0_MAIN_LE(x0[0], x1[0]) → COND_428_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]))=TRUEx0[2]=x0[3]x1[2]=x1[3]x0[3]=x0[0]*(2, x1[3])=x1[0]COND_428_0_MAIN_LE1(TRUE, x0[3], x1[3])≥NonInfC∧COND_428_0_MAIN_LE1(TRUE, x0[3], x1[3])≥428_0_MAIN_LE(x0[3], *(2, x1[3]))∧(UIncreasing(428_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])=TRUECOND_428_0_MAIN_LE1(TRUE, x0[2], x1[2])≥NonInfC∧COND_428_0_MAIN_LE1(TRUE, x0[2], x1[2])≥428_0_MAIN_LE(x0[2], *(2, x1[2]))∧(UIncreasing(428_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 ⇒ (UIncreasing(428_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 ⇒ (UIncreasing(428_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 ⇒ (UIncreasing(428_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 ⇒ (UIncreasing(428_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 ⇒ (UIncreasing(428_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 428_0_MAIN_LE(x0[2], x1[2]) → COND_428_0_MAIN_LE1(&&(>(x1[2], 0), <(x1[2], x0[2])), x0[2], x1[2]), COND_428_0_MAIN_LE1(TRUE, x0[3], x1[3]) → 428_0_MAIN_LE(x0[3], *(2, x1[3])), 428_0_MAIN_LE(x0[2], x1[2]) → COND_428_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]))=TRUEx0[2]=x0[3]x1[2]=x1[3]x0[3]=x0[2]1*(2, x1[3])=x1[2]1COND_428_0_MAIN_LE1(TRUE, x0[3], x1[3])≥NonInfC∧COND_428_0_MAIN_LE1(TRUE, x0[3], x1[3])≥428_0_MAIN_LE(x0[3], *(2, x1[3]))∧(UIncreasing(428_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])=TRUECOND_428_0_MAIN_LE1(TRUE, x0[2], x1[2])≥NonInfC∧COND_428_0_MAIN_LE1(TRUE, x0[2], x1[2])≥428_0_MAIN_LE(x0[2], *(2, x1[2]))∧(UIncreasing(428_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 ⇒ (UIncreasing(428_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 ⇒ (UIncreasing(428_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 ⇒ (UIncreasing(428_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 ⇒ (UIncreasing(428_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 ⇒ (UIncreasing(428_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.
  • 428_0_MAIN_LE(x0, x1) → COND_428_0_MAIN_LE(&&(>=(x1, x0), >(x0, 0)), x0, x1)
    • (x1[0] ≥ 0∧x0[0] ≥ 0 ⇒ (UIncreasing(COND_428_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_428_0_MAIN_LE(TRUE, x0, x1) → 428_0_MAIN_LE(+(x0, -1), 1)
    • (x1[0] ≥ 0∧x0[0] ≥ 0 ⇒ (UIncreasing(428_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 ⇒ (UIncreasing(428_0_MAIN_LE(+(x0[1], -1), 1)), ≥)∧[(-1)Bound*bni_20] + [bni_20]x0[0] ≥ 0∧[1 + (-1)bso_21] ≥ 0)

  • 428_0_MAIN_LE(x0, x1) → COND_428_0_MAIN_LE1(&&(>(x1, 0), <(x1, x0)), x0, x1)
    • (x1[2] ≥ 0∧x0[2] ≥ 0 ⇒ (UIncreasing(COND_428_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_428_0_MAIN_LE1(TRUE, x0, x1) → 428_0_MAIN_LE(x0, *(2, x1))
    • (x1[2] ≥ 0∧x0[2] ≥ 0 ⇒ (UIncreasing(428_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 ⇒ (UIncreasing(428_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 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) = [1]   
POL(428_0_MAIN_LE(x1, x2)) = [-1] + x1   
POL(COND_428_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_428_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_428_0_MAIN_LE(TRUE, x0[1], x1[1]) → 428_0_MAIN_LE(+(x0[1], -1), 1)

The following pairs are in Pbound:

428_0_MAIN_LE(x0[0], x1[0]) → COND_428_0_MAIN_LE(&&(>=(x1[0], x0[0]), >(x0[0], 0)), x0[0], x1[0])
COND_428_0_MAIN_LE(TRUE, x0[1], x1[1]) → 428_0_MAIN_LE(+(x0[1], -1), 1)
428_0_MAIN_LE(x0[2], x1[2]) → COND_428_0_MAIN_LE1(&&(>(x1[2], 0), <(x1[2], x0[2])), x0[2], x1[2])
COND_428_0_MAIN_LE1(TRUE, x0[3], x1[3]) → 428_0_MAIN_LE(x0[3], *(2, x1[3]))

The following pairs are in P:

428_0_MAIN_LE(x0[0], x1[0]) → COND_428_0_MAIN_LE(&&(>=(x1[0], x0[0]), >(x0[0], 0)), x0[0], x1[0])
428_0_MAIN_LE(x0[2], x1[2]) → COND_428_0_MAIN_LE1(&&(>(x1[2], 0), <(x1[2], x0[2])), x0[2], x1[2])
COND_428_0_MAIN_LE1(TRUE, x0[3], x1[3]) → 428_0_MAIN_LE(x0[3], *(2, x1[3]))

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

FALSE1&&(TRUE, FALSE)1
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): 428_0_MAIN_LE(x0[0], x1[0]) → COND_428_0_MAIN_LE(x1[0] >= x0[0] && x0[0] > 0, x0[0], x1[0])
(2): 428_0_MAIN_LE(x0[2], x1[2]) → COND_428_0_MAIN_LE1(x1[2] > 0 && x1[2] < x0[2], x0[2], x1[2])
(3): COND_428_0_MAIN_LE1(TRUE, x0[3], x1[3]) → 428_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_428_0_MAIN_LE1(TRUE, x0[3], x1[3]) → 428_0_MAIN_LE(x0[3], 2 * x1[3])
(2): 428_0_MAIN_LE(x0[2], x1[2]) → COND_428_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@ea6e48f 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_428_0_MAIN_LE1(TRUE, x0[3], x1[3]) → 428_0_MAIN_LE(x0[3], *(2, x1[3])) the following chains were created:
  • We consider the chain 428_0_MAIN_LE(x0[2], x1[2]) → COND_428_0_MAIN_LE1(&&(>(x1[2], 0), <(x1[2], x0[2])), x0[2], x1[2]), COND_428_0_MAIN_LE1(TRUE, x0[3], x1[3]) → 428_0_MAIN_LE(x0[3], *(2, x1[3])), 428_0_MAIN_LE(x0[2], x1[2]) → COND_428_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]))=TRUEx0[2]=x0[3]x1[2]=x1[3]x0[3]=x0[2]1*(2, x1[3])=x1[2]1COND_428_0_MAIN_LE1(TRUE, x0[3], x1[3])≥NonInfC∧COND_428_0_MAIN_LE1(TRUE, x0[3], x1[3])≥428_0_MAIN_LE(x0[3], *(2, x1[3]))∧(UIncreasing(428_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])=TRUECOND_428_0_MAIN_LE1(TRUE, x0[2], x1[2])≥NonInfC∧COND_428_0_MAIN_LE1(TRUE, x0[2], x1[2])≥428_0_MAIN_LE(x0[2], *(2, x1[2]))∧(UIncreasing(428_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 ⇒ (UIncreasing(428_0_MAIN_LE(x0[3], *(2, x1[3]))), ≥)∧[(2)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 ⇒ (UIncreasing(428_0_MAIN_LE(x0[3], *(2, x1[3]))), ≥)∧[(2)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 ⇒ (UIncreasing(428_0_MAIN_LE(x0[3], *(2, x1[3]))), ≥)∧[(2)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 ⇒ (UIncreasing(428_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 + (-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 ⇒ (UIncreasing(428_0_MAIN_LE(x0[3], *(2, x1[3]))), ≥)∧[(3)bni_13 + (-1)Bound*bni_13] + [bni_13]x0[2] ≥ 0∧[1 + (-1)bso_14] + x1[2] ≥ 0)







For Pair 428_0_MAIN_LE(x0[2], x1[2]) → COND_428_0_MAIN_LE1(&&(>(x1[2], 0), <(x1[2], x0[2])), x0[2], x1[2]) the following chains were created:
  • We consider the chain 428_0_MAIN_LE(x0[2], x1[2]) → COND_428_0_MAIN_LE1(&&(>(x1[2], 0), <(x1[2], x0[2])), x0[2], x1[2]), COND_428_0_MAIN_LE1(TRUE, x0[3], x1[3]) → 428_0_MAIN_LE(x0[3], *(2, x1[3])) which results in the following constraint:

    (8)    (&&(>(x1[2], 0), <(x1[2], x0[2]))=TRUEx0[2]=x0[3]x1[2]=x1[3]428_0_MAIN_LE(x0[2], x1[2])≥NonInfC∧428_0_MAIN_LE(x0[2], x1[2])≥COND_428_0_MAIN_LE1(&&(>(x1[2], 0), <(x1[2], x0[2])), x0[2], x1[2])∧(UIncreasing(COND_428_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])=TRUE428_0_MAIN_LE(x0[2], x1[2])≥NonInfC∧428_0_MAIN_LE(x0[2], x1[2])≥COND_428_0_MAIN_LE1(&&(>(x1[2], 0), <(x1[2], x0[2])), x0[2], x1[2])∧(UIncreasing(COND_428_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 ⇒ (UIncreasing(COND_428_0_MAIN_LE1(&&(>(x1[2], 0), <(x1[2], x0[2])), x0[2], x1[2])), ≥)∧[(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 ⇒ (UIncreasing(COND_428_0_MAIN_LE1(&&(>(x1[2], 0), <(x1[2], x0[2])), x0[2], x1[2])), ≥)∧[(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 ⇒ (UIncreasing(COND_428_0_MAIN_LE1(&&(>(x1[2], 0), <(x1[2], x0[2])), x0[2], x1[2])), ≥)∧[(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 ⇒ (UIncreasing(COND_428_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 (13) using rule (IDP_SMT_SPLIT) which results in the following new constraint:

    (14)    (x1[2] ≥ 0∧x0[2] ≥ 0 ⇒ (UIncreasing(COND_428_0_MAIN_LE1(&&(>(x1[2], 0), <(x1[2], x0[2])), x0[2], x1[2])), ≥)∧[(3)bni_15 + (-1)Bound*bni_15] + [bni_15]x0[2] ≥ 0∧[(-1)bso_16] ≥ 0)







To summarize, we get the following constraints P for the following pairs.
  • COND_428_0_MAIN_LE1(TRUE, x0[3], x1[3]) → 428_0_MAIN_LE(x0[3], *(2, x1[3]))
    • (x1[2] ≥ 0∧x0[2] ≥ 0 ⇒ (UIncreasing(428_0_MAIN_LE(x0[3], *(2, x1[3]))), ≥)∧[(3)bni_13 + (-1)Bound*bni_13] + [bni_13]x0[2] ≥ 0∧[1 + (-1)bso_14] + x1[2] ≥ 0)

  • 428_0_MAIN_LE(x0[2], x1[2]) → COND_428_0_MAIN_LE1(&&(>(x1[2], 0), <(x1[2], x0[2])), x0[2], x1[2])
    • (x1[2] ≥ 0∧x0[2] ≥ 0 ⇒ (UIncreasing(COND_428_0_MAIN_LE1(&&(>(x1[2], 0), <(x1[2], x0[2])), x0[2], x1[2])), ≥)∧[(3)bni_15 + (-1)Bound*bni_15] + [bni_15]x0[2] ≥ 0∧[(-1)bso_16] ≥ 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(COND_428_0_MAIN_LE1(x1, x2, x3)) = [2] + [-1]x3 + x2 + [-1]x1   
POL(428_0_MAIN_LE(x1, x2)) = [2] + [-1]x2 + x1   
POL(*(x1, x2)) = x1·x2   
POL(2) = [2]   
POL(&&(x1, x2)) = 0   
POL(>(x1, x2)) = [-1]   
POL(0) = 0   
POL(<(x1, x2)) = [-1]   

The following pairs are in P>:

COND_428_0_MAIN_LE1(TRUE, x0[3], x1[3]) → 428_0_MAIN_LE(x0[3], *(2, x1[3]))

The following pairs are in Pbound:

COND_428_0_MAIN_LE1(TRUE, x0[3], x1[3]) → 428_0_MAIN_LE(x0[3], *(2, x1[3]))
428_0_MAIN_LE(x0[2], x1[2]) → COND_428_0_MAIN_LE1(&&(>(x1[2], 0), <(x1[2], x0[2])), x0[2], x1[2])

The following pairs are in P:

428_0_MAIN_LE(x0[2], x1[2]) → COND_428_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)1TRUE1
&&(TRUE, FALSE)1FALSE1
&&(FALSE, TRUE)1FALSE1
&&(FALSE, FALSE)1FALSE1

(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): 428_0_MAIN_LE(x0[2], x1[2]) → COND_428_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