(0) Obligation:

JBC Problem based on JBC Program:
Manifest-Version: 1.0 Created-By: 1.6.0_16 (Sun Microsystems Inc.) Main-Class: MinusBuiltIn
`public class MinusBuiltIn{  public static void main(String[] args) {    Random.args = args;    int x = Random.random();    int y = Random.random();    int res = 0;    while (x > y) {      y++;      res++;    }  }}public class Random {  static String[] args;  static int index = 0;  public static int random() {    String string = args[index];    index++;    return string.length();  }}`

(1) JBC2FIG (SOUND transformation)

Constructed FIGraph.

(2) Obligation:

FIGraph based on JBC Program:
Graph of 188 nodes with 1 SCC.

(3) FIGtoITRSProof (SOUND transformation)

Transformed FIGraph to ITRS rules

(4) Obligation:

ITRS 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 TRS R consists of the following rules:
Load920(i14, i61, i62) → Cond_Load920(i14 > i61 && i62 + 1 > 0, i14, i61, i62)
Cond_Load920(TRUE, i14, i61, i62) → Load920(i14, i61 + 1, i62 + 1)
The set Q consists of the following terms:

(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

The ITRS R consists of the following rules:
Load920(i14, i61, i62) → Cond_Load920(i14 > i61 && i62 + 1 > 0, i14, i61, i62)
Cond_Load920(TRUE, i14, i61, i62) → Load920(i14, i61 + 1, i62 + 1)

The integer pair graph contains the following rules and edges:
(0): LOAD920(i14[0], i61[0], i62[0]) → COND_LOAD920(i14[0] > i61[0] && i62[0] + 1 > 0, i14[0], i61[0], i62[0])
(1): COND_LOAD920(TRUE, i14[1], i61[1], i62[1]) → LOAD920(i14[1], i61[1] + 1, i62[1] + 1)

(0) -> (1), if ((i14[0]* i14[1])∧(i14[0] > i61[0] && i62[0] + 1 > 0* TRUE)∧(i62[0]* i62[1])∧(i61[0]* i61[1]))

(1) -> (0), if ((i62[1] + 1* i62[0])∧(i14[1]* i14[0])∧(i61[1] + 1* i61[0]))

The set Q consists of the following terms:

(7) UsableRulesProof (EQUIVALENT transformation)

As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [LPAR04] we can delete all non-usable rules [FROCOS05] from R.

(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): LOAD920(i14[0], i61[0], i62[0]) → COND_LOAD920(i14[0] > i61[0] && i62[0] + 1 > 0, i14[0], i61[0], i62[0])
(1): COND_LOAD920(TRUE, i14[1], i61[1], i62[1]) → LOAD920(i14[1], i61[1] + 1, i62[1] + 1)

(0) -> (1), if ((i14[0]* i14[1])∧(i14[0] > i61[0] && i62[0] + 1 > 0* TRUE)∧(i62[0]* i62[1])∧(i61[0]* i61[1]))

(1) -> (0), if ((i62[1] + 1* i62[0])∧(i14[1]* i14[0])∧(i61[1] + 1* i61[0]))

The set Q consists of the following terms:

(9) 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 LOAD920(i14, i61, i62) → COND_LOAD920(&&(>(i14, i61), >(+(i62, 1), 0)), i14, i61, i62) the following chains were created:
• We consider the chain LOAD920(i14[0], i61[0], i62[0]) → COND_LOAD920(&&(>(i14[0], i61[0]), >(+(i62[0], 1), 0)), i14[0], i61[0], i62[0]), COND_LOAD920(TRUE, i14[1], i61[1], i62[1]) → LOAD920(i14[1], +(i61[1], 1), +(i62[1], 1)) which results in the following constraint:

(1)    (i14[0]=i14[1]&&(>(i14[0], i61[0]), >(+(i62[0], 1), 0))=TRUEi62[0]=i62[1]i61[0]=i61[1]LOAD920(i14[0], i61[0], i62[0])≥NonInfC∧LOAD920(i14[0], i61[0], i62[0])≥COND_LOAD920(&&(>(i14[0], i61[0]), >(+(i62[0], 1), 0)), i14[0], i61[0], i62[0])∧(UIncreasing(COND_LOAD920(&&(>(i14[0], i61[0]), >(+(i62[0], 1), 0)), i14[0], i61[0], i62[0])), ≥))

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

(2)    (>(i14[0], i61[0])=TRUE>(+(i62[0], 1), 0)=TRUELOAD920(i14[0], i61[0], i62[0])≥NonInfC∧LOAD920(i14[0], i61[0], i62[0])≥COND_LOAD920(&&(>(i14[0], i61[0]), >(+(i62[0], 1), 0)), i14[0], i61[0], i62[0])∧(UIncreasing(COND_LOAD920(&&(>(i14[0], i61[0]), >(+(i62[0], 1), 0)), i14[0], i61[0], i62[0])), ≥))

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

(3)    (i14[0] + [-1] + [-1]i61[0] ≥ 0∧i62[0] ≥ 0 ⇒ (UIncreasing(COND_LOAD920(&&(>(i14[0], i61[0]), >(+(i62[0], 1), 0)), i14[0], i61[0], i62[0])), ≥)∧[(2)bni_14 + (-1)Bound*bni_14] + [(-1)bni_14]i61[0] + [bni_14]i14[0] ≥ 0∧[(-1)bso_15] ≥ 0)

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

(4)    (i14[0] + [-1] + [-1]i61[0] ≥ 0∧i62[0] ≥ 0 ⇒ (UIncreasing(COND_LOAD920(&&(>(i14[0], i61[0]), >(+(i62[0], 1), 0)), i14[0], i61[0], i62[0])), ≥)∧[(2)bni_14 + (-1)Bound*bni_14] + [(-1)bni_14]i61[0] + [bni_14]i14[0] ≥ 0∧[(-1)bso_15] ≥ 0)

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

(5)    (i14[0] + [-1] + [-1]i61[0] ≥ 0∧i62[0] ≥ 0 ⇒ (UIncreasing(COND_LOAD920(&&(>(i14[0], i61[0]), >(+(i62[0], 1), 0)), i14[0], i61[0], i62[0])), ≥)∧[(2)bni_14 + (-1)Bound*bni_14] + [(-1)bni_14]i61[0] + [bni_14]i14[0] ≥ 0∧[(-1)bso_15] ≥ 0)

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

(6)    (i14[0] ≥ 0∧i62[0] ≥ 0 ⇒ (UIncreasing(COND_LOAD920(&&(>(i14[0], i61[0]), >(+(i62[0], 1), 0)), i14[0], i61[0], i62[0])), ≥)∧[(3)bni_14 + (-1)Bound*bni_14] + [bni_14]i14[0] ≥ 0∧[(-1)bso_15] ≥ 0)

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

(7)    (i14[0] ≥ 0∧i62[0] ≥ 0∧i61[0] ≥ 0 ⇒ (UIncreasing(COND_LOAD920(&&(>(i14[0], i61[0]), >(+(i62[0], 1), 0)), i14[0], i61[0], i62[0])), ≥)∧[(3)bni_14 + (-1)Bound*bni_14] + [bni_14]i14[0] ≥ 0∧[(-1)bso_15] ≥ 0)

(8)    (i14[0] ≥ 0∧i62[0] ≥ 0∧i61[0] ≥ 0 ⇒ (UIncreasing(COND_LOAD920(&&(>(i14[0], i61[0]), >(+(i62[0], 1), 0)), i14[0], i61[0], i62[0])), ≥)∧[(3)bni_14 + (-1)Bound*bni_14] + [bni_14]i14[0] ≥ 0∧[(-1)bso_15] ≥ 0)

For Pair COND_LOAD920(TRUE, i14, i61, i62) → LOAD920(i14, +(i61, 1), +(i62, 1)) the following chains were created:
• We consider the chain LOAD920(i14[0], i61[0], i62[0]) → COND_LOAD920(&&(>(i14[0], i61[0]), >(+(i62[0], 1), 0)), i14[0], i61[0], i62[0]), COND_LOAD920(TRUE, i14[1], i61[1], i62[1]) → LOAD920(i14[1], +(i61[1], 1), +(i62[1], 1)), LOAD920(i14[0], i61[0], i62[0]) → COND_LOAD920(&&(>(i14[0], i61[0]), >(+(i62[0], 1), 0)), i14[0], i61[0], i62[0]) which results in the following constraint:

(9)    (i14[0]=i14[1]&&(>(i14[0], i61[0]), >(+(i62[0], 1), 0))=TRUEi62[0]=i62[1]i61[0]=i61[1]+(i62[1], 1)=i62[0]1i14[1]=i14[0]1+(i61[1], 1)=i61[0]1COND_LOAD920(TRUE, i14[1], i61[1], i62[1])≥NonInfC∧COND_LOAD920(TRUE, i14[1], i61[1], i62[1])≥LOAD920(i14[1], +(i61[1], 1), +(i62[1], 1))∧(UIncreasing(LOAD920(i14[1], +(i61[1], 1), +(i62[1], 1))), ≥))

We simplified constraint (9) using rules (III), (IV), (IDP_BOOLEAN) which results in the following new constraint:

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

(11)    (i14[0] + [-1] + [-1]i61[0] ≥ 0∧i62[0] ≥ 0 ⇒ (UIncreasing(LOAD920(i14[1], +(i61[1], 1), +(i62[1], 1))), ≥)∧[(2)bni_16 + (-1)Bound*bni_16] + [(-1)bni_16]i61[0] + [bni_16]i14[0] ≥ 0∧[1 + (-1)bso_17] ≥ 0)

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

(12)    (i14[0] + [-1] + [-1]i61[0] ≥ 0∧i62[0] ≥ 0 ⇒ (UIncreasing(LOAD920(i14[1], +(i61[1], 1), +(i62[1], 1))), ≥)∧[(2)bni_16 + (-1)Bound*bni_16] + [(-1)bni_16]i61[0] + [bni_16]i14[0] ≥ 0∧[1 + (-1)bso_17] ≥ 0)

We simplified constraint (12) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:

(13)    (i14[0] + [-1] + [-1]i61[0] ≥ 0∧i62[0] ≥ 0 ⇒ (UIncreasing(LOAD920(i14[1], +(i61[1], 1), +(i62[1], 1))), ≥)∧[(2)bni_16 + (-1)Bound*bni_16] + [(-1)bni_16]i61[0] + [bni_16]i14[0] ≥ 0∧[1 + (-1)bso_17] ≥ 0)

We simplified constraint (13) using rule (IDP_SMT_SPLIT) which results in the following new constraint:

(14)    (i14[0] ≥ 0∧i62[0] ≥ 0 ⇒ (UIncreasing(LOAD920(i14[1], +(i61[1], 1), +(i62[1], 1))), ≥)∧[(3)bni_16 + (-1)Bound*bni_16] + [bni_16]i14[0] ≥ 0∧[1 + (-1)bso_17] ≥ 0)

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

(15)    (i14[0] ≥ 0∧i62[0] ≥ 0∧i61[0] ≥ 0 ⇒ (UIncreasing(LOAD920(i14[1], +(i61[1], 1), +(i62[1], 1))), ≥)∧[(3)bni_16 + (-1)Bound*bni_16] + [bni_16]i14[0] ≥ 0∧[1 + (-1)bso_17] ≥ 0)

(16)    (i14[0] ≥ 0∧i62[0] ≥ 0∧i61[0] ≥ 0 ⇒ (UIncreasing(LOAD920(i14[1], +(i61[1], 1), +(i62[1], 1))), ≥)∧[(3)bni_16 + (-1)Bound*bni_16] + [bni_16]i14[0] ≥ 0∧[1 + (-1)bso_17] ≥ 0)

To summarize, we get the following constraints P for the following pairs.
• LOAD920(i14, i61, i62) → COND_LOAD920(&&(>(i14, i61), >(+(i62, 1), 0)), i14, i61, i62)
• (i14[0] ≥ 0∧i62[0] ≥ 0∧i61[0] ≥ 0 ⇒ (UIncreasing(COND_LOAD920(&&(>(i14[0], i61[0]), >(+(i62[0], 1), 0)), i14[0], i61[0], i62[0])), ≥)∧[(3)bni_14 + (-1)Bound*bni_14] + [bni_14]i14[0] ≥ 0∧[(-1)bso_15] ≥ 0)
• (i14[0] ≥ 0∧i62[0] ≥ 0∧i61[0] ≥ 0 ⇒ (UIncreasing(COND_LOAD920(&&(>(i14[0], i61[0]), >(+(i62[0], 1), 0)), i14[0], i61[0], i62[0])), ≥)∧[(3)bni_14 + (-1)Bound*bni_14] + [bni_14]i14[0] ≥ 0∧[(-1)bso_15] ≥ 0)

• (i14[0] ≥ 0∧i62[0] ≥ 0∧i61[0] ≥ 0 ⇒ (UIncreasing(LOAD920(i14[1], +(i61[1], 1), +(i62[1], 1))), ≥)∧[(3)bni_16 + (-1)Bound*bni_16] + [bni_16]i14[0] ≥ 0∧[1 + (-1)bso_17] ≥ 0)
• (i14[0] ≥ 0∧i62[0] ≥ 0∧i61[0] ≥ 0 ⇒ (UIncreasing(LOAD920(i14[1], +(i61[1], 1), +(i62[1], 1))), ≥)∧[(3)bni_16 + (-1)Bound*bni_16] + [bni_16]i14[0] ≥ 0∧[1 + (-1)bso_17] ≥ 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(LOAD920(x1, x2, x3)) = [2] + [-1]x2 + x1
POL(COND_LOAD920(x1, x2, x3, x4)) = [2] + [-1]x3 + x2 + [-1]x1
POL(&&(x1, x2)) = 0
POL(>(x1, x2)) = [-1]
POL(+(x1, x2)) = x1 + x2
POL(1) = [1]
POL(0) = 0

The following pairs are in P>:

The following pairs are in Pbound:

LOAD920(i14[0], i61[0], i62[0]) → COND_LOAD920(&&(>(i14[0], i61[0]), >(+(i62[0], 1), 0)), i14[0], i61[0], i62[0])

The following pairs are in P:

LOAD920(i14[0], i61[0], i62[0]) → COND_LOAD920(&&(>(i14[0], i61[0]), >(+(i62[0], 1), 0)), i14[0], i61[0], i62[0])

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

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

(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:

Boolean, Integer

R is empty.

The integer pair graph contains the following rules and edges:
(0): LOAD920(i14[0], i61[0], i62[0]) → COND_LOAD920(i14[0] > i61[0] && i62[0] + 1 > 0, i14[0], i61[0], i62[0])

The set Q consists of the following terms:

(12) IDependencyGraphProof (EQUIVALENT transformation)

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

(14) 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 consists of the following terms: