### (0) Obligation:

JBC Problem based on JBC Program:
Manifest-Version: 1.0 Created-By: 1.6.0_16 (Sun Microsystems Inc.) Main-Class: PastaA9
`/** * 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 PastaA9 {    public static void main(String[] args) {        Random.args = args;        int x = Random.random();        int y = Random.random();        int z = Random.random();        if (y > 0) {            while (x >= z) {                z += y;            }        }    }}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 253 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:
Load1620(i14, i94, i87) → Cond_Load1620(i14 >= i87 && i94 > 0, i14, i94, i87)
Cond_Load1620(TRUE, i14, i94, i87) → Load1620(i14, i94, i87 + i94)
The set Q consists of the following terms:
Cond_Load1620(TRUE, x0, x1, x2)

### (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:
Load1620(i14, i94, i87) → Cond_Load1620(i14 >= i87 && i94 > 0, i14, i94, i87)
Cond_Load1620(TRUE, i14, i94, i87) → Load1620(i14, i94, i87 + i94)

The integer pair graph contains the following rules and edges:
(0): LOAD1620(i14[0], i94[0], i87[0]) → COND_LOAD1620(i14[0] >= i87[0] && i94[0] > 0, i14[0], i94[0], i87[0])
(1): COND_LOAD1620(TRUE, i14[1], i94[1], i87[1]) → LOAD1620(i14[1], i94[1], i87[1] + i94[1])

(0) -> (1), if ((i14[0]* i14[1])∧(i14[0] >= i87[0] && i94[0] > 0* TRUE)∧(i87[0]* i87[1])∧(i94[0]* i94[1]))

(1) -> (0), if ((i87[1] + i94[1]* i87[0])∧(i94[1]* i94[0])∧(i14[1]* i14[0]))

The set Q consists of the following terms:
Cond_Load1620(TRUE, x0, x1, x2)

### (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): LOAD1620(i14[0], i94[0], i87[0]) → COND_LOAD1620(i14[0] >= i87[0] && i94[0] > 0, i14[0], i94[0], i87[0])
(1): COND_LOAD1620(TRUE, i14[1], i94[1], i87[1]) → LOAD1620(i14[1], i94[1], i87[1] + i94[1])

(0) -> (1), if ((i14[0]* i14[1])∧(i14[0] >= i87[0] && i94[0] > 0* TRUE)∧(i87[0]* i87[1])∧(i94[0]* i94[1]))

(1) -> (0), if ((i87[1] + i94[1]* i87[0])∧(i94[1]* i94[0])∧(i14[1]* i14[0]))

The set Q consists of the following terms:
Cond_Load1620(TRUE, x0, x1, x2)

### (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 LOAD1620(i14, i94, i87) → COND_LOAD1620(&&(>=(i14, i87), >(i94, 0)), i14, i94, i87) the following chains were created:
• We consider the chain LOAD1620(i14[0], i94[0], i87[0]) → COND_LOAD1620(&&(>=(i14[0], i87[0]), >(i94[0], 0)), i14[0], i94[0], i87[0]), COND_LOAD1620(TRUE, i14[1], i94[1], i87[1]) → LOAD1620(i14[1], i94[1], +(i87[1], i94[1])) which results in the following constraint:

(1)    (i14[0]=i14[1]&&(>=(i14[0], i87[0]), >(i94[0], 0))=TRUEi87[0]=i87[1]i94[0]=i94[1]LOAD1620(i14[0], i94[0], i87[0])≥NonInfC∧LOAD1620(i14[0], i94[0], i87[0])≥COND_LOAD1620(&&(>=(i14[0], i87[0]), >(i94[0], 0)), i14[0], i94[0], i87[0])∧(UIncreasing(COND_LOAD1620(&&(>=(i14[0], i87[0]), >(i94[0], 0)), i14[0], i94[0], i87[0])), ≥))

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

(2)    (>=(i14[0], i87[0])=TRUE>(i94[0], 0)=TRUELOAD1620(i14[0], i94[0], i87[0])≥NonInfC∧LOAD1620(i14[0], i94[0], i87[0])≥COND_LOAD1620(&&(>=(i14[0], i87[0]), >(i94[0], 0)), i14[0], i94[0], i87[0])∧(UIncreasing(COND_LOAD1620(&&(>=(i14[0], i87[0]), >(i94[0], 0)), i14[0], i94[0], i87[0])), ≥))

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

(3)    (i14[0] + [-1]i87[0] ≥ 0∧i94[0] + [-1] ≥ 0 ⇒ (UIncreasing(COND_LOAD1620(&&(>=(i14[0], i87[0]), >(i94[0], 0)), i14[0], i94[0], i87[0])), ≥)∧[(-1)bni_15 + (-1)Bound*bni_15] + [(-1)bni_15]i87[0] + [bni_15]i14[0] ≥ 0∧[(-1)bso_16] + i94[0] ≥ 0)

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

(4)    (i14[0] + [-1]i87[0] ≥ 0∧i94[0] + [-1] ≥ 0 ⇒ (UIncreasing(COND_LOAD1620(&&(>=(i14[0], i87[0]), >(i94[0], 0)), i14[0], i94[0], i87[0])), ≥)∧[(-1)bni_15 + (-1)Bound*bni_15] + [(-1)bni_15]i87[0] + [bni_15]i14[0] ≥ 0∧[(-1)bso_16] + i94[0] ≥ 0)

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

(5)    (i14[0] + [-1]i87[0] ≥ 0∧i94[0] + [-1] ≥ 0 ⇒ (UIncreasing(COND_LOAD1620(&&(>=(i14[0], i87[0]), >(i94[0], 0)), i14[0], i94[0], i87[0])), ≥)∧[(-1)bni_15 + (-1)Bound*bni_15] + [(-1)bni_15]i87[0] + [bni_15]i14[0] ≥ 0∧[(-1)bso_16] + i94[0] ≥ 0)

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

(6)    (i14[0] ≥ 0∧i94[0] + [-1] ≥ 0 ⇒ (UIncreasing(COND_LOAD1620(&&(>=(i14[0], i87[0]), >(i94[0], 0)), i14[0], i94[0], i87[0])), ≥)∧[(-1)bni_15 + (-1)Bound*bni_15] + [bni_15]i14[0] ≥ 0∧[(-1)bso_16] + i94[0] ≥ 0)

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

(7)    (i14[0] ≥ 0∧i94[0] ≥ 0 ⇒ (UIncreasing(COND_LOAD1620(&&(>=(i14[0], i87[0]), >(i94[0], 0)), i14[0], i94[0], i87[0])), ≥)∧[(-1)bni_15 + (-1)Bound*bni_15] + [bni_15]i14[0] ≥ 0∧[1 + (-1)bso_16] + i94[0] ≥ 0)

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

(8)    (i14[0] ≥ 0∧i94[0] ≥ 0∧i87[0] ≥ 0 ⇒ (UIncreasing(COND_LOAD1620(&&(>=(i14[0], i87[0]), >(i94[0], 0)), i14[0], i94[0], i87[0])), ≥)∧[(-1)bni_15 + (-1)Bound*bni_15] + [bni_15]i14[0] ≥ 0∧[1 + (-1)bso_16] + i94[0] ≥ 0)

(9)    (i14[0] ≥ 0∧i94[0] ≥ 0∧i87[0] ≥ 0 ⇒ (UIncreasing(COND_LOAD1620(&&(>=(i14[0], i87[0]), >(i94[0], 0)), i14[0], i94[0], i87[0])), ≥)∧[(-1)bni_15 + (-1)Bound*bni_15] + [bni_15]i14[0] ≥ 0∧[1 + (-1)bso_16] + i94[0] ≥ 0)

For Pair COND_LOAD1620(TRUE, i14, i94, i87) → LOAD1620(i14, i94, +(i87, i94)) the following chains were created:
• We consider the chain LOAD1620(i14[0], i94[0], i87[0]) → COND_LOAD1620(&&(>=(i14[0], i87[0]), >(i94[0], 0)), i14[0], i94[0], i87[0]), COND_LOAD1620(TRUE, i14[1], i94[1], i87[1]) → LOAD1620(i14[1], i94[1], +(i87[1], i94[1])), LOAD1620(i14[0], i94[0], i87[0]) → COND_LOAD1620(&&(>=(i14[0], i87[0]), >(i94[0], 0)), i14[0], i94[0], i87[0]) which results in the following constraint:

(10)    (i14[0]=i14[1]&&(>=(i14[0], i87[0]), >(i94[0], 0))=TRUEi87[0]=i87[1]i94[0]=i94[1]+(i87[1], i94[1])=i87[0]1i94[1]=i94[0]1i14[1]=i14[0]1COND_LOAD1620(TRUE, i14[1], i94[1], i87[1])≥NonInfC∧COND_LOAD1620(TRUE, i14[1], i94[1], i87[1])≥LOAD1620(i14[1], i94[1], +(i87[1], i94[1]))∧(UIncreasing(LOAD1620(i14[1], i94[1], +(i87[1], i94[1]))), ≥))

We simplified constraint (10) using rules (III), (IV), (IDP_BOOLEAN) which results in the following new constraint:

(11)    (>=(i14[0], i87[0])=TRUE>(i94[0], 0)=TRUECOND_LOAD1620(TRUE, i14[0], i94[0], i87[0])≥NonInfC∧COND_LOAD1620(TRUE, i14[0], i94[0], i87[0])≥LOAD1620(i14[0], i94[0], +(i87[0], i94[0]))∧(UIncreasing(LOAD1620(i14[1], i94[1], +(i87[1], i94[1]))), ≥))

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

(12)    (i14[0] + [-1]i87[0] ≥ 0∧i94[0] + [-1] ≥ 0 ⇒ (UIncreasing(LOAD1620(i14[1], i94[1], +(i87[1], i94[1]))), ≥)∧[(-1)bni_17 + (-1)Bound*bni_17] + [(-1)bni_17]i87[0] + [(-1)bni_17]i94[0] + [bni_17]i14[0] ≥ 0∧[(-1)bso_18] ≥ 0)

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

(13)    (i14[0] + [-1]i87[0] ≥ 0∧i94[0] + [-1] ≥ 0 ⇒ (UIncreasing(LOAD1620(i14[1], i94[1], +(i87[1], i94[1]))), ≥)∧[(-1)bni_17 + (-1)Bound*bni_17] + [(-1)bni_17]i87[0] + [(-1)bni_17]i94[0] + [bni_17]i14[0] ≥ 0∧[(-1)bso_18] ≥ 0)

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

(14)    (i14[0] + [-1]i87[0] ≥ 0∧i94[0] + [-1] ≥ 0 ⇒ (UIncreasing(LOAD1620(i14[1], i94[1], +(i87[1], i94[1]))), ≥)∧[(-1)bni_17 + (-1)Bound*bni_17] + [(-1)bni_17]i87[0] + [(-1)bni_17]i94[0] + [bni_17]i14[0] ≥ 0∧[(-1)bso_18] ≥ 0)

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

(15)    (i14[0] ≥ 0∧i94[0] + [-1] ≥ 0 ⇒ (UIncreasing(LOAD1620(i14[1], i94[1], +(i87[1], i94[1]))), ≥)∧[(-1)bni_17 + (-1)Bound*bni_17] + [(-1)bni_17]i94[0] + [bni_17]i14[0] ≥ 0∧[(-1)bso_18] ≥ 0)

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

(16)    (i14[0] ≥ 0∧i94[0] ≥ 0 ⇒ (UIncreasing(LOAD1620(i14[1], i94[1], +(i87[1], i94[1]))), ≥)∧[(-2)bni_17 + (-1)Bound*bni_17] + [(-1)bni_17]i94[0] + [bni_17]i14[0] ≥ 0∧[(-1)bso_18] ≥ 0)

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

(17)    (i14[0] ≥ 0∧i94[0] ≥ 0∧i87[0] ≥ 0 ⇒ (UIncreasing(LOAD1620(i14[1], i94[1], +(i87[1], i94[1]))), ≥)∧[(-2)bni_17 + (-1)Bound*bni_17] + [(-1)bni_17]i94[0] + [bni_17]i14[0] ≥ 0∧[(-1)bso_18] ≥ 0)

(18)    (i14[0] ≥ 0∧i94[0] ≥ 0∧i87[0] ≥ 0 ⇒ (UIncreasing(LOAD1620(i14[1], i94[1], +(i87[1], i94[1]))), ≥)∧[(-2)bni_17 + (-1)Bound*bni_17] + [(-1)bni_17]i94[0] + [bni_17]i14[0] ≥ 0∧[(-1)bso_18] ≥ 0)

To summarize, we get the following constraints P for the following pairs.
• LOAD1620(i14, i94, i87) → COND_LOAD1620(&&(>=(i14, i87), >(i94, 0)), i14, i94, i87)
• (i14[0] ≥ 0∧i94[0] ≥ 0∧i87[0] ≥ 0 ⇒ (UIncreasing(COND_LOAD1620(&&(>=(i14[0], i87[0]), >(i94[0], 0)), i14[0], i94[0], i87[0])), ≥)∧[(-1)bni_15 + (-1)Bound*bni_15] + [bni_15]i14[0] ≥ 0∧[1 + (-1)bso_16] + i94[0] ≥ 0)
• (i14[0] ≥ 0∧i94[0] ≥ 0∧i87[0] ≥ 0 ⇒ (UIncreasing(COND_LOAD1620(&&(>=(i14[0], i87[0]), >(i94[0], 0)), i14[0], i94[0], i87[0])), ≥)∧[(-1)bni_15 + (-1)Bound*bni_15] + [bni_15]i14[0] ≥ 0∧[1 + (-1)bso_16] + i94[0] ≥ 0)

• COND_LOAD1620(TRUE, i14, i94, i87) → LOAD1620(i14, i94, +(i87, i94))
• (i14[0] ≥ 0∧i94[0] ≥ 0∧i87[0] ≥ 0 ⇒ (UIncreasing(LOAD1620(i14[1], i94[1], +(i87[1], i94[1]))), ≥)∧[(-2)bni_17 + (-1)Bound*bni_17] + [(-1)bni_17]i94[0] + [bni_17]i14[0] ≥ 0∧[(-1)bso_18] ≥ 0)
• (i14[0] ≥ 0∧i94[0] ≥ 0∧i87[0] ≥ 0 ⇒ (UIncreasing(LOAD1620(i14[1], i94[1], +(i87[1], i94[1]))), ≥)∧[(-2)bni_17 + (-1)Bound*bni_17] + [(-1)bni_17]i94[0] + [bni_17]i14[0] ≥ 0∧[(-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) = [1]
POL(FALSE) = [3]
POL(LOAD1620(x1, x2, x3)) = [-1] + [-1]x3 + x1
POL(COND_LOAD1620(x1, x2, x3, x4)) = [-1]x4 + [-1]x3 + x2 + [-1]x1
POL(&&(x1, x2)) = [1]
POL(>=(x1, x2)) = [-1]
POL(>(x1, x2)) = [-1]
POL(0) = 0
POL(+(x1, x2)) = x1 + x2

The following pairs are in P>:

LOAD1620(i14[0], i94[0], i87[0]) → COND_LOAD1620(&&(>=(i14[0], i87[0]), >(i94[0], 0)), i14[0], i94[0], i87[0])

The following pairs are in Pbound:

LOAD1620(i14[0], i94[0], i87[0]) → COND_LOAD1620(&&(>=(i14[0], i87[0]), >(i94[0], 0)), i14[0], i94[0], i87[0])

The following pairs are in P:

COND_LOAD1620(TRUE, i14[1], i94[1], i87[1]) → LOAD1620(i14[1], i94[1], +(i87[1], i94[1]))

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
FALSE1&&(FALSE, FALSE)1

### (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:
(1): COND_LOAD1620(TRUE, i14[1], i94[1], i87[1]) → LOAD1620(i14[1], i94[1], i87[1] + i94[1])

The set Q consists of the following terms: