(0) Obligation:

JBC Problem based on JBC Program:
Manifest-Version: 1.0 Created-By: 1.6.0_20 (Apple Inc.) Main-Class: Test11
public class Test11 {
public static void main(String[] args) {
Random.args = args;

int x = args.length * 100, y = args.length * 200 / 13;

while (x + y > 0) {
if (Random.random() * Random.random() > 9)
x--;
else
y--;
}
}
}

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

public static int random() {
if (index >= args.length)
return 0;

String string = args[index];
index++;
return string.length();
}
}


(1) JBC2FIG (SOUND transformation)

Constructed FIGraph.

(2) Obligation:

FIGraph based on JBC Program:
Test11.main([Ljava/lang/String;)V: Graph of 331 nodes with 1 SCC.


(3) FIGtoITRSProof (SOUND transformation)

Transformed FIGraph SCCs to IDPs. Logs:


Log for SCC 0:

Generated 108 rules for P and 190 rules for R.


Combined rules. Obtained 17 rules for P and 0 rules for R.


Filtered ground terms:


5887_0_main_Load(x1, x2, x3, x4) → 5887_0_main_Load(x2, x3, x4)
5954_0_random_LT(x1, x2, x3) → 5954_0_random_LT(x2, x3)
6091_0_random_IntArithmetic(x1, x2, x3, x4) → 6091_0_random_IntArithmetic(x2, x3)
6012_0_random_ArrayAccess(x1, x2, x3) → 6012_0_random_ArrayAccess(x2, x3)
5952_0_random_LT(x1, x2, x3) → 5952_0_random_LT(x2, x3)
5905_0_random_LT(x1, x2, x3) → 5905_0_random_LT(x2, x3)
6286_0_random_LT(x1, x2, x3) → 6286_0_random_LT(x2, x3)
6591_0_random_IntArithmetic(x1, x2, x3, x4) → 6591_0_random_IntArithmetic(x2, x3)
6442_0_random_ArrayAccess(x1, x2, x3) → 6442_0_random_ArrayAccess(x2, x3)
6283_0_random_LT(x1, x2, x3) → 6283_0_random_LT(x2, x3)
5957_0_random_IntArithmetic(x1, x2, x3, x4) → 5957_0_random_IntArithmetic(x2, x3)
5924_0_random_ArrayAccess(x1, x2, x3) → 5924_0_random_ArrayAccess(x2, x3)
5904_0_random_LT(x1, x2, x3) → 5904_0_random_LT(x2, x3)
Cond_5887_0_main_Load1(x1, x2, x3, x4, x5) → Cond_5887_0_main_Load1(x1, x3, x4, x5)
Cond_5887_0_main_Load(x1, x2, x3, x4, x5) → Cond_5887_0_main_Load(x1, x3, x4, x5)

Filtered duplicate args:


5887_0_main_Load(x1, x2, x3) → 5887_0_main_Load(x2, x3)
Cond_5887_0_main_Load1(x1, x2, x3, x4) → Cond_5887_0_main_Load1(x1, x3, x4)
Cond_5887_0_main_Load(x1, x2, x3, x4) → Cond_5887_0_main_Load(x1, x3, x4)

Filtered unneeded arguments:


Cond_6591_1_main_InvokeMethod(x1, x2, x3, x4, x5) → Cond_6591_1_main_InvokeMethod(x1, x2, x3, x4)
Cond_6591_1_main_InvokeMethod1(x1, x2, x3, x4, x5) → Cond_6591_1_main_InvokeMethod1(x1, x2, x3, x4)
6286_1_main_InvokeMethod(x1, x2, x3, x4) → 6286_1_main_InvokeMethod(x1, x2, x3)
Cond_6286_1_main_InvokeMethod(x1, x2, x3, x4, x5) → Cond_6286_1_main_InvokeMethod(x1, x2, x3, x4)

Filtered all free variables:


5904_1_main_InvokeMethod(x1, x2, x3) → 5904_1_main_InvokeMethod(x2, x3)
5905_1_main_InvokeMethod(x1, x2, x3) → 5905_1_main_InvokeMethod(x2, x3)
Cond_5904_1_main_InvokeMethod(x1, x2, x3, x4) → Cond_5904_1_main_InvokeMethod(x1, x3, x4)
5924_1_main_InvokeMethod(x1, x2, x3) → 5924_1_main_InvokeMethod(x2, x3)
Cond_5924_1_main_InvokeMethod(x1, x2, x3, x4) → Cond_5924_1_main_InvokeMethod(x1, x3, x4)
5957_1_main_InvokeMethod(x1, x2, x3) → 5957_1_main_InvokeMethod(x2, x3)
Cond_5957_1_main_InvokeMethod(x1, x2, x3, x4) → Cond_5957_1_main_InvokeMethod(x1, x3, x4)
6283_1_main_InvokeMethod(x1, x2, x3, x4) → 6283_1_main_InvokeMethod(x2, x3, x4)
Cond_5957_1_main_InvokeMethod1(x1, x2, x3, x4) → Cond_5957_1_main_InvokeMethod1(x1, x3, x4)
6286_1_main_InvokeMethod(x1, x2, x3) → 6286_1_main_InvokeMethod(x2, x3)
Cond_6283_1_main_InvokeMethod(x1, x2, x3, x4, x5) → Cond_6283_1_main_InvokeMethod(x1, x3, x4, x5)
6442_1_main_InvokeMethod(x1, x2, x3, x4) → 6442_1_main_InvokeMethod(x2, x3, x4)
Cond_6442_1_main_InvokeMethod(x1, x2, x3, x4, x5) → Cond_6442_1_main_InvokeMethod(x1, x3, x4, x5)
6591_1_main_InvokeMethod(x1, x2, x3, x4) → 6591_1_main_InvokeMethod(x2, x3, x4)
Cond_6591_1_main_InvokeMethod(x1, x2, x3, x4) → Cond_6591_1_main_InvokeMethod(x1, x3, x4)
Cond_6591_1_main_InvokeMethod1(x1, x2, x3, x4) → Cond_6591_1_main_InvokeMethod1(x1, x3, x4)
Cond_6286_1_main_InvokeMethod(x1, x2, x3, x4) → Cond_6286_1_main_InvokeMethod(x1, x3, x4)
Cond_5905_1_main_InvokeMethod(x1, x2, x3, x4) → Cond_5905_1_main_InvokeMethod(x1, x3, x4)
5952_1_main_InvokeMethod(x1, x2, x3) → 5952_1_main_InvokeMethod(x2, x3)
Cond_5905_1_main_InvokeMethod1(x1, x2, x3, x4) → Cond_5905_1_main_InvokeMethod1(x1, x3, x4)
5954_1_main_InvokeMethod(x1, x2, x3) → 5954_1_main_InvokeMethod(x2, x3)
Cond_5952_1_main_InvokeMethod(x1, x2, x3, x4) → Cond_5952_1_main_InvokeMethod(x1, x3, x4)
6012_1_main_InvokeMethod(x1, x2, x3) → 6012_1_main_InvokeMethod(x2, x3)
Cond_6012_1_main_InvokeMethod(x1, x2, x3, x4) → Cond_6012_1_main_InvokeMethod(x1, x3, x4)
6091_1_main_InvokeMethod(x1, x2, x3) → 6091_1_main_InvokeMethod(x2, x3)
Cond_6091_1_main_InvokeMethod(x1, x2, x3, x4) → Cond_6091_1_main_InvokeMethod(x1, x3, x4)
Cond_5954_1_main_InvokeMethod(x1, x2, x3, x4) → Cond_5954_1_main_InvokeMethod(x1, x3, x4)

Combined rules. Obtained 6 rules for P and 0 rules for R.


Finished conversion. Obtained 6 rules for P and 0 rules for R. System has predefined symbols.


(4) 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:
(0): 6091_1_MAIN_INVOKEMETHOD(x4[0], x5[0]) → 5887_0_MAIN_LOAD(x5[0] + -1, x4[0])
(1): 5887_0_MAIN_LOAD(x0[1], x1[1]) → COND_5887_0_MAIN_LOAD(0 < x1[1] + x0[1], x0[1], x1[1])
(2): COND_5887_0_MAIN_LOAD(TRUE, x0[2], x1[2]) → 6091_1_MAIN_INVOKEMETHOD(x1[2], x0[2])
(3): 5887_0_MAIN_LOAD(x0[3], x1[3]) → COND_5887_0_MAIN_LOAD1(0 < x1[3] + x0[3], x0[3], x1[3])
(4): COND_5887_0_MAIN_LOAD1(TRUE, x0[4], x1[4]) → 5887_0_MAIN_LOAD(x0[4] + -1, x1[4])
(5): 5887_0_MAIN_LOAD(x0[5], x1[5]) → COND_5887_0_MAIN_LOAD2(0 < x1[5] + x0[5], x0[5], x1[5])
(6): COND_5887_0_MAIN_LOAD2(TRUE, x0[6], x1[6]) → 5887_0_MAIN_LOAD(x0[6], x1[6] + -1)

(0) -> (1), if ((x5[0] + -1* x0[1])∧(x4[0]* x1[1]))


(0) -> (3), if ((x5[0] + -1* x0[3])∧(x4[0]* x1[3]))


(0) -> (5), if ((x5[0] + -1* x0[5])∧(x4[0]* x1[5]))


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


(2) -> (0), if ((x1[2]* x4[0])∧(x0[2]* x5[0]))


(3) -> (4), if ((0 < x1[3] + x0[3]* TRUE)∧(x0[3]* x0[4])∧(x1[3]* x1[4]))


(4) -> (1), if ((x0[4] + -1* x0[1])∧(x1[4]* x1[1]))


(4) -> (3), if ((x0[4] + -1* x0[3])∧(x1[4]* x1[3]))


(4) -> (5), if ((x0[4] + -1* x0[5])∧(x1[4]* x1[5]))


(5) -> (6), if ((0 < x1[5] + x0[5]* TRUE)∧(x0[5]* x0[6])∧(x1[5]* x1[6]))


(6) -> (1), if ((x0[6]* x0[1])∧(x1[6] + -1* x1[1]))


(6) -> (3), if ((x0[6]* x0[3])∧(x1[6] + -1* x1[3]))


(6) -> (5), if ((x0[6]* x0[5])∧(x1[6] + -1* x1[5]))



The set Q is empty.

(5) IDPNonInfProof (SOUND transformation)

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 6091_1_MAIN_INVOKEMETHOD(x4, x5) → 5887_0_MAIN_LOAD(+(x5, -1), x4) the following chains were created:
  • We consider the chain 6091_1_MAIN_INVOKEMETHOD(x4[0], x5[0]) → 5887_0_MAIN_LOAD(+(x5[0], -1), x4[0]) which results in the following constraint:

    (1)    (6091_1_MAIN_INVOKEMETHOD(x4[0], x5[0])≥NonInfC∧6091_1_MAIN_INVOKEMETHOD(x4[0], x5[0])≥5887_0_MAIN_LOAD(+(x5[0], -1), x4[0])∧(UIncreasing(5887_0_MAIN_LOAD(+(x5[0], -1), x4[0])), ≥))



    We simplified constraint (1) using rule (POLY_CONSTRAINTS) which results in the following new constraint:

    (2)    ((UIncreasing(5887_0_MAIN_LOAD(+(x5[0], -1), x4[0])), ≥)∧[(-1)bso_12] ≥ 0)



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

    (3)    ((UIncreasing(5887_0_MAIN_LOAD(+(x5[0], -1), x4[0])), ≥)∧[(-1)bso_12] ≥ 0)



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

    (4)    ((UIncreasing(5887_0_MAIN_LOAD(+(x5[0], -1), x4[0])), ≥)∧[(-1)bso_12] ≥ 0)



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

    (5)    ((UIncreasing(5887_0_MAIN_LOAD(+(x5[0], -1), x4[0])), ≥)∧0 = 0∧0 = 0∧[(-1)bso_12] ≥ 0)







For Pair 5887_0_MAIN_LOAD(x0, x1) → COND_5887_0_MAIN_LOAD(<(0, +(x1, x0)), x0, x1) the following chains were created:
  • We consider the chain 5887_0_MAIN_LOAD(x0[1], x1[1]) → COND_5887_0_MAIN_LOAD(<(0, +(x1[1], x0[1])), x0[1], x1[1]), COND_5887_0_MAIN_LOAD(TRUE, x0[2], x1[2]) → 6091_1_MAIN_INVOKEMETHOD(x1[2], x0[2]) which results in the following constraint:

    (6)    (<(0, +(x1[1], x0[1]))=TRUEx0[1]=x0[2]x1[1]=x1[2]5887_0_MAIN_LOAD(x0[1], x1[1])≥NonInfC∧5887_0_MAIN_LOAD(x0[1], x1[1])≥COND_5887_0_MAIN_LOAD(<(0, +(x1[1], x0[1])), x0[1], x1[1])∧(UIncreasing(COND_5887_0_MAIN_LOAD(<(0, +(x1[1], x0[1])), x0[1], x1[1])), ≥))



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

    (7)    (<(0, +(x1[1], x0[1]))=TRUE5887_0_MAIN_LOAD(x0[1], x1[1])≥NonInfC∧5887_0_MAIN_LOAD(x0[1], x1[1])≥COND_5887_0_MAIN_LOAD(<(0, +(x1[1], x0[1])), x0[1], x1[1])∧(UIncreasing(COND_5887_0_MAIN_LOAD(<(0, +(x1[1], x0[1])), x0[1], x1[1])), ≥))



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

    (8)    (x1[1] + [-1] + x0[1] ≥ 0 ⇒ (UIncreasing(COND_5887_0_MAIN_LOAD(<(0, +(x1[1], x0[1])), x0[1], x1[1])), ≥)∧[bni_13 + (-1)Bound*bni_13] + [bni_13]x0[1] + [bni_13]x1[1] ≥ 0∧[(-1)bso_14] ≥ 0)



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

    (9)    (x1[1] + [-1] + x0[1] ≥ 0 ⇒ (UIncreasing(COND_5887_0_MAIN_LOAD(<(0, +(x1[1], x0[1])), x0[1], x1[1])), ≥)∧[bni_13 + (-1)Bound*bni_13] + [bni_13]x0[1] + [bni_13]x1[1] ≥ 0∧[(-1)bso_14] ≥ 0)



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

    (10)    (x1[1] + [-1] + x0[1] ≥ 0 ⇒ (UIncreasing(COND_5887_0_MAIN_LOAD(<(0, +(x1[1], x0[1])), x0[1], x1[1])), ≥)∧[bni_13 + (-1)Bound*bni_13] + [bni_13]x0[1] + [bni_13]x1[1] ≥ 0∧[(-1)bso_14] ≥ 0)



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

    (11)    (x1[1] ≥ 0 ⇒ (UIncreasing(COND_5887_0_MAIN_LOAD(<(0, +(x1[1], x0[1])), x0[1], x1[1])), ≥)∧[(2)bni_13 + (-1)Bound*bni_13] + [bni_13]x1[1] ≥ 0∧[(-1)bso_14] ≥ 0)



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

    (12)    (x1[1] ≥ 0∧x0[1] ≥ 0 ⇒ (UIncreasing(COND_5887_0_MAIN_LOAD(<(0, +(x1[1], x0[1])), x0[1], x1[1])), ≥)∧[(2)bni_13 + (-1)Bound*bni_13] + [bni_13]x1[1] ≥ 0∧[(-1)bso_14] ≥ 0)


    (13)    (x1[1] ≥ 0∧x0[1] ≥ 0 ⇒ (UIncreasing(COND_5887_0_MAIN_LOAD(<(0, +(x1[1], x0[1])), x0[1], x1[1])), ≥)∧[(2)bni_13 + (-1)Bound*bni_13] + [bni_13]x1[1] ≥ 0∧[(-1)bso_14] ≥ 0)







For Pair COND_5887_0_MAIN_LOAD(TRUE, x0, x1) → 6091_1_MAIN_INVOKEMETHOD(x1, x0) the following chains were created:
  • We consider the chain COND_5887_0_MAIN_LOAD(TRUE, x0[2], x1[2]) → 6091_1_MAIN_INVOKEMETHOD(x1[2], x0[2]), 6091_1_MAIN_INVOKEMETHOD(x4[0], x5[0]) → 5887_0_MAIN_LOAD(+(x5[0], -1), x4[0]) which results in the following constraint:

    (14)    (x1[2]=x4[0]x0[2]=x5[0]COND_5887_0_MAIN_LOAD(TRUE, x0[2], x1[2])≥NonInfC∧COND_5887_0_MAIN_LOAD(TRUE, x0[2], x1[2])≥6091_1_MAIN_INVOKEMETHOD(x1[2], x0[2])∧(UIncreasing(6091_1_MAIN_INVOKEMETHOD(x1[2], x0[2])), ≥))



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

    (15)    (COND_5887_0_MAIN_LOAD(TRUE, x0[2], x1[2])≥NonInfC∧COND_5887_0_MAIN_LOAD(TRUE, x0[2], x1[2])≥6091_1_MAIN_INVOKEMETHOD(x1[2], x0[2])∧(UIncreasing(6091_1_MAIN_INVOKEMETHOD(x1[2], x0[2])), ≥))



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

    (16)    ((UIncreasing(6091_1_MAIN_INVOKEMETHOD(x1[2], x0[2])), ≥)∧[1 + (-1)bso_16] ≥ 0)



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

    (17)    ((UIncreasing(6091_1_MAIN_INVOKEMETHOD(x1[2], x0[2])), ≥)∧[1 + (-1)bso_16] ≥ 0)



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

    (18)    ((UIncreasing(6091_1_MAIN_INVOKEMETHOD(x1[2], x0[2])), ≥)∧[1 + (-1)bso_16] ≥ 0)



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

    (19)    ((UIncreasing(6091_1_MAIN_INVOKEMETHOD(x1[2], x0[2])), ≥)∧0 = 0∧0 = 0∧[1 + (-1)bso_16] ≥ 0)







For Pair 5887_0_MAIN_LOAD(x0, x1) → COND_5887_0_MAIN_LOAD1(<(0, +(x1, x0)), x0, x1) the following chains were created:
  • We consider the chain 5887_0_MAIN_LOAD(x0[3], x1[3]) → COND_5887_0_MAIN_LOAD1(<(0, +(x1[3], x0[3])), x0[3], x1[3]), COND_5887_0_MAIN_LOAD1(TRUE, x0[4], x1[4]) → 5887_0_MAIN_LOAD(+(x0[4], -1), x1[4]) which results in the following constraint:

    (20)    (<(0, +(x1[3], x0[3]))=TRUEx0[3]=x0[4]x1[3]=x1[4]5887_0_MAIN_LOAD(x0[3], x1[3])≥NonInfC∧5887_0_MAIN_LOAD(x0[3], x1[3])≥COND_5887_0_MAIN_LOAD1(<(0, +(x1[3], x0[3])), x0[3], x1[3])∧(UIncreasing(COND_5887_0_MAIN_LOAD1(<(0, +(x1[3], x0[3])), x0[3], x1[3])), ≥))



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

    (21)    (<(0, +(x1[3], x0[3]))=TRUE5887_0_MAIN_LOAD(x0[3], x1[3])≥NonInfC∧5887_0_MAIN_LOAD(x0[3], x1[3])≥COND_5887_0_MAIN_LOAD1(<(0, +(x1[3], x0[3])), x0[3], x1[3])∧(UIncreasing(COND_5887_0_MAIN_LOAD1(<(0, +(x1[3], x0[3])), x0[3], x1[3])), ≥))



    We simplified constraint (21) using rule (POLY_CONSTRAINTS) which results in the following new constraint:

    (22)    (x1[3] + [-1] + x0[3] ≥ 0 ⇒ (UIncreasing(COND_5887_0_MAIN_LOAD1(<(0, +(x1[3], x0[3])), x0[3], x1[3])), ≥)∧[bni_17 + (-1)Bound*bni_17] + [bni_17]x0[3] + [bni_17]x1[3] ≥ 0∧[(-1)bso_18] ≥ 0)



    We simplified constraint (22) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:

    (23)    (x1[3] + [-1] + x0[3] ≥ 0 ⇒ (UIncreasing(COND_5887_0_MAIN_LOAD1(<(0, +(x1[3], x0[3])), x0[3], x1[3])), ≥)∧[bni_17 + (-1)Bound*bni_17] + [bni_17]x0[3] + [bni_17]x1[3] ≥ 0∧[(-1)bso_18] ≥ 0)



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

    (24)    (x1[3] + [-1] + x0[3] ≥ 0 ⇒ (UIncreasing(COND_5887_0_MAIN_LOAD1(<(0, +(x1[3], x0[3])), x0[3], x1[3])), ≥)∧[bni_17 + (-1)Bound*bni_17] + [bni_17]x0[3] + [bni_17]x1[3] ≥ 0∧[(-1)bso_18] ≥ 0)



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

    (25)    (x1[3] ≥ 0 ⇒ (UIncreasing(COND_5887_0_MAIN_LOAD1(<(0, +(x1[3], x0[3])), x0[3], x1[3])), ≥)∧[(2)bni_17 + (-1)Bound*bni_17] + [bni_17]x1[3] ≥ 0∧[(-1)bso_18] ≥ 0)



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

    (26)    (x1[3] ≥ 0∧x0[3] ≥ 0 ⇒ (UIncreasing(COND_5887_0_MAIN_LOAD1(<(0, +(x1[3], x0[3])), x0[3], x1[3])), ≥)∧[(2)bni_17 + (-1)Bound*bni_17] + [bni_17]x1[3] ≥ 0∧[(-1)bso_18] ≥ 0)


    (27)    (x1[3] ≥ 0∧x0[3] ≥ 0 ⇒ (UIncreasing(COND_5887_0_MAIN_LOAD1(<(0, +(x1[3], x0[3])), x0[3], x1[3])), ≥)∧[(2)bni_17 + (-1)Bound*bni_17] + [bni_17]x1[3] ≥ 0∧[(-1)bso_18] ≥ 0)







For Pair COND_5887_0_MAIN_LOAD1(TRUE, x0, x1) → 5887_0_MAIN_LOAD(+(x0, -1), x1) the following chains were created:
  • We consider the chain COND_5887_0_MAIN_LOAD1(TRUE, x0[4], x1[4]) → 5887_0_MAIN_LOAD(+(x0[4], -1), x1[4]) which results in the following constraint:

    (28)    (COND_5887_0_MAIN_LOAD1(TRUE, x0[4], x1[4])≥NonInfC∧COND_5887_0_MAIN_LOAD1(TRUE, x0[4], x1[4])≥5887_0_MAIN_LOAD(+(x0[4], -1), x1[4])∧(UIncreasing(5887_0_MAIN_LOAD(+(x0[4], -1), x1[4])), ≥))



    We simplified constraint (28) using rule (POLY_CONSTRAINTS) which results in the following new constraint:

    (29)    ((UIncreasing(5887_0_MAIN_LOAD(+(x0[4], -1), x1[4])), ≥)∧[1 + (-1)bso_20] ≥ 0)



    We simplified constraint (29) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:

    (30)    ((UIncreasing(5887_0_MAIN_LOAD(+(x0[4], -1), x1[4])), ≥)∧[1 + (-1)bso_20] ≥ 0)



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

    (31)    ((UIncreasing(5887_0_MAIN_LOAD(+(x0[4], -1), x1[4])), ≥)∧[1 + (-1)bso_20] ≥ 0)



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

    (32)    ((UIncreasing(5887_0_MAIN_LOAD(+(x0[4], -1), x1[4])), ≥)∧0 = 0∧0 = 0∧[1 + (-1)bso_20] ≥ 0)







For Pair 5887_0_MAIN_LOAD(x0, x1) → COND_5887_0_MAIN_LOAD2(<(0, +(x1, x0)), x0, x1) the following chains were created:
  • We consider the chain 5887_0_MAIN_LOAD(x0[5], x1[5]) → COND_5887_0_MAIN_LOAD2(<(0, +(x1[5], x0[5])), x0[5], x1[5]), COND_5887_0_MAIN_LOAD2(TRUE, x0[6], x1[6]) → 5887_0_MAIN_LOAD(x0[6], +(x1[6], -1)) which results in the following constraint:

    (33)    (<(0, +(x1[5], x0[5]))=TRUEx0[5]=x0[6]x1[5]=x1[6]5887_0_MAIN_LOAD(x0[5], x1[5])≥NonInfC∧5887_0_MAIN_LOAD(x0[5], x1[5])≥COND_5887_0_MAIN_LOAD2(<(0, +(x1[5], x0[5])), x0[5], x1[5])∧(UIncreasing(COND_5887_0_MAIN_LOAD2(<(0, +(x1[5], x0[5])), x0[5], x1[5])), ≥))



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

    (34)    (<(0, +(x1[5], x0[5]))=TRUE5887_0_MAIN_LOAD(x0[5], x1[5])≥NonInfC∧5887_0_MAIN_LOAD(x0[5], x1[5])≥COND_5887_0_MAIN_LOAD2(<(0, +(x1[5], x0[5])), x0[5], x1[5])∧(UIncreasing(COND_5887_0_MAIN_LOAD2(<(0, +(x1[5], x0[5])), x0[5], x1[5])), ≥))



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

    (35)    (x1[5] + [-1] + x0[5] ≥ 0 ⇒ (UIncreasing(COND_5887_0_MAIN_LOAD2(<(0, +(x1[5], x0[5])), x0[5], x1[5])), ≥)∧[bni_21 + (-1)Bound*bni_21] + [bni_21]x0[5] + [bni_21]x1[5] ≥ 0∧[(-1)bso_22] ≥ 0)



    We simplified constraint (35) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:

    (36)    (x1[5] + [-1] + x0[5] ≥ 0 ⇒ (UIncreasing(COND_5887_0_MAIN_LOAD2(<(0, +(x1[5], x0[5])), x0[5], x1[5])), ≥)∧[bni_21 + (-1)Bound*bni_21] + [bni_21]x0[5] + [bni_21]x1[5] ≥ 0∧[(-1)bso_22] ≥ 0)



    We simplified constraint (36) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:

    (37)    (x1[5] + [-1] + x0[5] ≥ 0 ⇒ (UIncreasing(COND_5887_0_MAIN_LOAD2(<(0, +(x1[5], x0[5])), x0[5], x1[5])), ≥)∧[bni_21 + (-1)Bound*bni_21] + [bni_21]x0[5] + [bni_21]x1[5] ≥ 0∧[(-1)bso_22] ≥ 0)



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

    (38)    (x1[5] ≥ 0 ⇒ (UIncreasing(COND_5887_0_MAIN_LOAD2(<(0, +(x1[5], x0[5])), x0[5], x1[5])), ≥)∧[(2)bni_21 + (-1)Bound*bni_21] + [bni_21]x1[5] ≥ 0∧[(-1)bso_22] ≥ 0)



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

    (39)    (x1[5] ≥ 0∧x0[5] ≥ 0 ⇒ (UIncreasing(COND_5887_0_MAIN_LOAD2(<(0, +(x1[5], x0[5])), x0[5], x1[5])), ≥)∧[(2)bni_21 + (-1)Bound*bni_21] + [bni_21]x1[5] ≥ 0∧[(-1)bso_22] ≥ 0)


    (40)    (x1[5] ≥ 0∧x0[5] ≥ 0 ⇒ (UIncreasing(COND_5887_0_MAIN_LOAD2(<(0, +(x1[5], x0[5])), x0[5], x1[5])), ≥)∧[(2)bni_21 + (-1)Bound*bni_21] + [bni_21]x1[5] ≥ 0∧[(-1)bso_22] ≥ 0)







For Pair COND_5887_0_MAIN_LOAD2(TRUE, x0, x1) → 5887_0_MAIN_LOAD(x0, +(x1, -1)) the following chains were created:
  • We consider the chain COND_5887_0_MAIN_LOAD2(TRUE, x0[6], x1[6]) → 5887_0_MAIN_LOAD(x0[6], +(x1[6], -1)) which results in the following constraint:

    (41)    (COND_5887_0_MAIN_LOAD2(TRUE, x0[6], x1[6])≥NonInfC∧COND_5887_0_MAIN_LOAD2(TRUE, x0[6], x1[6])≥5887_0_MAIN_LOAD(x0[6], +(x1[6], -1))∧(UIncreasing(5887_0_MAIN_LOAD(x0[6], +(x1[6], -1))), ≥))



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

    (42)    ((UIncreasing(5887_0_MAIN_LOAD(x0[6], +(x1[6], -1))), ≥)∧[1 + (-1)bso_24] ≥ 0)



    We simplified constraint (42) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:

    (43)    ((UIncreasing(5887_0_MAIN_LOAD(x0[6], +(x1[6], -1))), ≥)∧[1 + (-1)bso_24] ≥ 0)



    We simplified constraint (43) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:

    (44)    ((UIncreasing(5887_0_MAIN_LOAD(x0[6], +(x1[6], -1))), ≥)∧[1 + (-1)bso_24] ≥ 0)



    We simplified constraint (44) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint:

    (45)    ((UIncreasing(5887_0_MAIN_LOAD(x0[6], +(x1[6], -1))), ≥)∧0 = 0∧0 = 0∧[1 + (-1)bso_24] ≥ 0)







To summarize, we get the following constraints P for the following pairs.
  • 6091_1_MAIN_INVOKEMETHOD(x4, x5) → 5887_0_MAIN_LOAD(+(x5, -1), x4)
    • ((UIncreasing(5887_0_MAIN_LOAD(+(x5[0], -1), x4[0])), ≥)∧0 = 0∧0 = 0∧[(-1)bso_12] ≥ 0)

  • 5887_0_MAIN_LOAD(x0, x1) → COND_5887_0_MAIN_LOAD(<(0, +(x1, x0)), x0, x1)
    • (x1[1] ≥ 0∧x0[1] ≥ 0 ⇒ (UIncreasing(COND_5887_0_MAIN_LOAD(<(0, +(x1[1], x0[1])), x0[1], x1[1])), ≥)∧[(2)bni_13 + (-1)Bound*bni_13] + [bni_13]x1[1] ≥ 0∧[(-1)bso_14] ≥ 0)
    • (x1[1] ≥ 0∧x0[1] ≥ 0 ⇒ (UIncreasing(COND_5887_0_MAIN_LOAD(<(0, +(x1[1], x0[1])), x0[1], x1[1])), ≥)∧[(2)bni_13 + (-1)Bound*bni_13] + [bni_13]x1[1] ≥ 0∧[(-1)bso_14] ≥ 0)

  • COND_5887_0_MAIN_LOAD(TRUE, x0, x1) → 6091_1_MAIN_INVOKEMETHOD(x1, x0)
    • ((UIncreasing(6091_1_MAIN_INVOKEMETHOD(x1[2], x0[2])), ≥)∧0 = 0∧0 = 0∧[1 + (-1)bso_16] ≥ 0)

  • 5887_0_MAIN_LOAD(x0, x1) → COND_5887_0_MAIN_LOAD1(<(0, +(x1, x0)), x0, x1)
    • (x1[3] ≥ 0∧x0[3] ≥ 0 ⇒ (UIncreasing(COND_5887_0_MAIN_LOAD1(<(0, +(x1[3], x0[3])), x0[3], x1[3])), ≥)∧[(2)bni_17 + (-1)Bound*bni_17] + [bni_17]x1[3] ≥ 0∧[(-1)bso_18] ≥ 0)
    • (x1[3] ≥ 0∧x0[3] ≥ 0 ⇒ (UIncreasing(COND_5887_0_MAIN_LOAD1(<(0, +(x1[3], x0[3])), x0[3], x1[3])), ≥)∧[(2)bni_17 + (-1)Bound*bni_17] + [bni_17]x1[3] ≥ 0∧[(-1)bso_18] ≥ 0)

  • COND_5887_0_MAIN_LOAD1(TRUE, x0, x1) → 5887_0_MAIN_LOAD(+(x0, -1), x1)
    • ((UIncreasing(5887_0_MAIN_LOAD(+(x0[4], -1), x1[4])), ≥)∧0 = 0∧0 = 0∧[1 + (-1)bso_20] ≥ 0)

  • 5887_0_MAIN_LOAD(x0, x1) → COND_5887_0_MAIN_LOAD2(<(0, +(x1, x0)), x0, x1)
    • (x1[5] ≥ 0∧x0[5] ≥ 0 ⇒ (UIncreasing(COND_5887_0_MAIN_LOAD2(<(0, +(x1[5], x0[5])), x0[5], x1[5])), ≥)∧[(2)bni_21 + (-1)Bound*bni_21] + [bni_21]x1[5] ≥ 0∧[(-1)bso_22] ≥ 0)
    • (x1[5] ≥ 0∧x0[5] ≥ 0 ⇒ (UIncreasing(COND_5887_0_MAIN_LOAD2(<(0, +(x1[5], x0[5])), x0[5], x1[5])), ≥)∧[(2)bni_21 + (-1)Bound*bni_21] + [bni_21]x1[5] ≥ 0∧[(-1)bso_22] ≥ 0)

  • COND_5887_0_MAIN_LOAD2(TRUE, x0, x1) → 5887_0_MAIN_LOAD(x0, +(x1, -1))
    • ((UIncreasing(5887_0_MAIN_LOAD(x0[6], +(x1[6], -1))), ≥)∧0 = 0∧0 = 0∧[1 + (-1)bso_24] ≥ 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(6091_1_MAIN_INVOKEMETHOD(x1, x2)) = x1 + x2   
POL(5887_0_MAIN_LOAD(x1, x2)) = [1] + x1 + x2   
POL(+(x1, x2)) = x1 + x2   
POL(-1) = [-1]   
POL(COND_5887_0_MAIN_LOAD(x1, x2, x3)) = [1] + x3 + x2   
POL(<(x1, x2)) = [-1]   
POL(0) = 0   
POL(COND_5887_0_MAIN_LOAD1(x1, x2, x3)) = [1] + x3 + x2   
POL(COND_5887_0_MAIN_LOAD2(x1, x2, x3)) = [1] + x3 + x2   

The following pairs are in P>:

COND_5887_0_MAIN_LOAD(TRUE, x0[2], x1[2]) → 6091_1_MAIN_INVOKEMETHOD(x1[2], x0[2])
COND_5887_0_MAIN_LOAD1(TRUE, x0[4], x1[4]) → 5887_0_MAIN_LOAD(+(x0[4], -1), x1[4])
COND_5887_0_MAIN_LOAD2(TRUE, x0[6], x1[6]) → 5887_0_MAIN_LOAD(x0[6], +(x1[6], -1))

The following pairs are in Pbound:

5887_0_MAIN_LOAD(x0[1], x1[1]) → COND_5887_0_MAIN_LOAD(<(0, +(x1[1], x0[1])), x0[1], x1[1])
5887_0_MAIN_LOAD(x0[3], x1[3]) → COND_5887_0_MAIN_LOAD1(<(0, +(x1[3], x0[3])), x0[3], x1[3])
5887_0_MAIN_LOAD(x0[5], x1[5]) → COND_5887_0_MAIN_LOAD2(<(0, +(x1[5], x0[5])), x0[5], x1[5])

The following pairs are in P:

6091_1_MAIN_INVOKEMETHOD(x4[0], x5[0]) → 5887_0_MAIN_LOAD(+(x5[0], -1), x4[0])
5887_0_MAIN_LOAD(x0[1], x1[1]) → COND_5887_0_MAIN_LOAD(<(0, +(x1[1], x0[1])), x0[1], x1[1])
5887_0_MAIN_LOAD(x0[3], x1[3]) → COND_5887_0_MAIN_LOAD1(<(0, +(x1[3], x0[3])), x0[3], x1[3])
5887_0_MAIN_LOAD(x0[5], x1[5]) → COND_5887_0_MAIN_LOAD2(<(0, +(x1[5], x0[5])), x0[5], x1[5])

There are no usable rules.

(6) Complex Obligation (AND)

(7) 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:
(0): 6091_1_MAIN_INVOKEMETHOD(x4[0], x5[0]) → 5887_0_MAIN_LOAD(x5[0] + -1, x4[0])
(1): 5887_0_MAIN_LOAD(x0[1], x1[1]) → COND_5887_0_MAIN_LOAD(0 < x1[1] + x0[1], x0[1], x1[1])
(3): 5887_0_MAIN_LOAD(x0[3], x1[3]) → COND_5887_0_MAIN_LOAD1(0 < x1[3] + x0[3], x0[3], x1[3])
(5): 5887_0_MAIN_LOAD(x0[5], x1[5]) → COND_5887_0_MAIN_LOAD2(0 < x1[5] + x0[5], x0[5], x1[5])

(0) -> (1), if ((x5[0] + -1* x0[1])∧(x4[0]* x1[1]))


(0) -> (3), if ((x5[0] + -1* x0[3])∧(x4[0]* x1[3]))


(0) -> (5), if ((x5[0] + -1* x0[5])∧(x4[0]* x1[5]))



The set Q is empty.

(8) IDependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 4 less nodes.

(9) TRUE

(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


R is empty.

The integer pair graph contains the following rules and edges:
(0): 6091_1_MAIN_INVOKEMETHOD(x4[0], x5[0]) → 5887_0_MAIN_LOAD(x5[0] + -1, x4[0])
(2): COND_5887_0_MAIN_LOAD(TRUE, x0[2], x1[2]) → 6091_1_MAIN_INVOKEMETHOD(x1[2], x0[2])
(4): COND_5887_0_MAIN_LOAD1(TRUE, x0[4], x1[4]) → 5887_0_MAIN_LOAD(x0[4] + -1, x1[4])
(6): COND_5887_0_MAIN_LOAD2(TRUE, x0[6], x1[6]) → 5887_0_MAIN_LOAD(x0[6], x1[6] + -1)

(2) -> (0), if ((x1[2]* x4[0])∧(x0[2]* x5[0]))



The set Q is empty.

(11) IDependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 4 less nodes.

(12) TRUE