### (0) Obligation:

JBC Problem based on JBC Program:
Manifest-Version: 1.0 Created-By: 1.6.0_16 (Sun Microsystems Inc.) Main-Class: PastaB12
`/** * 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 PastaB12 {    public static void main(String[] args) {        Random.args = args;        int x = Random.random();        int y = Random.random();        while (x > 0 || y > 0) {            if (x > 0) {                x--;            } else if (y > 0) {                y--;            } else {                continue;            }        }    }}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:
PastaB12.main([Ljava/lang/String;)V: Graph of 165 nodes with 1 SCC.

### (3) FIGtoITRSProof (SOUND transformation)

Transformed FIGraph SCCs to IDPs. Logs:

Log for SCC 0:

Generated 20 rules for P and 2 rules for R.

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

Filtered ground terms:

576_0_main_GT(x1, x2, x3, x4) → 576_0_main_GT(x2, x3, x4)
Cond_576_0_main_GT1(x1, x2, x3, x4, x5) → Cond_576_0_main_GT1(x1, x3, x4, x5)
Cond_576_0_main_GT(x1, x2, x3, x4, x5) → Cond_576_0_main_GT(x1, x4)

Filtered duplicate args:

576_0_main_GT(x1, x2, x3) → 576_0_main_GT(x2, x3)
Cond_576_0_main_GT1(x1, x2, x3, x4) → Cond_576_0_main_GT1(x1, x3, x4)

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

Finished conversion. Obtained 2 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): 576_0_MAIN_GT(x1[0], 0) → COND_576_0_MAIN_GT(x1[0] > 0, x1[0], 0)
(1): COND_576_0_MAIN_GT(TRUE, x1[1], 0) → 576_0_MAIN_GT(x1[1] + -1, 0)
(2): 576_0_MAIN_GT(x1[2], x0[2]) → COND_576_0_MAIN_GT1(x0[2] > 0, x1[2], x0[2])
(3): COND_576_0_MAIN_GT1(TRUE, x1[3], x0[3]) → 576_0_MAIN_GT(x1[3], x0[3] + -1)

(0) -> (1), if ((x1[0] > 0* TRUE)∧(x1[0]* x1[1]))

(1) -> (0), if (x1[1] + -1* x1[0])

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

(2) -> (3), if ((x0[2] > 0* TRUE)∧(x1[2]* x1[3])∧(x0[2]* x0[3]))

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

(3) -> (2), if ((x1[3]* x1[2])∧(x0[3] + -1* x0[2]))

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 576_0_MAIN_GT(x1, 0) → COND_576_0_MAIN_GT(>(x1, 0), x1, 0) the following chains were created:
• We consider the chain 576_0_MAIN_GT(x1[0], 0) → COND_576_0_MAIN_GT(>(x1[0], 0), x1[0], 0), COND_576_0_MAIN_GT(TRUE, x1[1], 0) → 576_0_MAIN_GT(+(x1[1], -1), 0) which results in the following constraint:

(1)    (>(x1[0], 0)=TRUEx1[0]=x1[1]576_0_MAIN_GT(x1[0], 0)≥NonInfC∧576_0_MAIN_GT(x1[0], 0)≥COND_576_0_MAIN_GT(>(x1[0], 0), x1[0], 0)∧(UIncreasing(COND_576_0_MAIN_GT(>(x1[0], 0), x1[0], 0)), ≥))

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

(2)    (>(x1[0], 0)=TRUE576_0_MAIN_GT(x1[0], 0)≥NonInfC∧576_0_MAIN_GT(x1[0], 0)≥COND_576_0_MAIN_GT(>(x1[0], 0), x1[0], 0)∧(UIncreasing(COND_576_0_MAIN_GT(>(x1[0], 0), x1[0], 0)), ≥))

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

(3)    (x1[0] + [-1] ≥ 0 ⇒ (UIncreasing(COND_576_0_MAIN_GT(>(x1[0], 0), x1[0], 0)), ≥)∧[(-1)bni_10 + (-1)Bound*bni_10] ≥ 0∧[(-1)bso_11] ≥ 0)

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

(4)    (x1[0] + [-1] ≥ 0 ⇒ (UIncreasing(COND_576_0_MAIN_GT(>(x1[0], 0), x1[0], 0)), ≥)∧[(-1)bni_10 + (-1)Bound*bni_10] ≥ 0∧[(-1)bso_11] ≥ 0)

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

(5)    (x1[0] + [-1] ≥ 0 ⇒ (UIncreasing(COND_576_0_MAIN_GT(>(x1[0], 0), x1[0], 0)), ≥)∧[(-1)bni_10 + (-1)Bound*bni_10] ≥ 0∧[(-1)bso_11] ≥ 0)

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

(6)    (x1[0] ≥ 0 ⇒ (UIncreasing(COND_576_0_MAIN_GT(>(x1[0], 0), x1[0], 0)), ≥)∧[(-1)bni_10 + (-1)Bound*bni_10] ≥ 0∧[(-1)bso_11] ≥ 0)

For Pair COND_576_0_MAIN_GT(TRUE, x1, 0) → 576_0_MAIN_GT(+(x1, -1), 0) the following chains were created:
• We consider the chain COND_576_0_MAIN_GT(TRUE, x1[1], 0) → 576_0_MAIN_GT(+(x1[1], -1), 0) which results in the following constraint:

(7)    (COND_576_0_MAIN_GT(TRUE, x1[1], 0)≥NonInfC∧COND_576_0_MAIN_GT(TRUE, x1[1], 0)≥576_0_MAIN_GT(+(x1[1], -1), 0)∧(UIncreasing(576_0_MAIN_GT(+(x1[1], -1), 0)), ≥))

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

(8)    ((UIncreasing(576_0_MAIN_GT(+(x1[1], -1), 0)), ≥)∧[(-1)bso_13] ≥ 0)

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

(9)    ((UIncreasing(576_0_MAIN_GT(+(x1[1], -1), 0)), ≥)∧[(-1)bso_13] ≥ 0)

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

(10)    ((UIncreasing(576_0_MAIN_GT(+(x1[1], -1), 0)), ≥)∧[(-1)bso_13] ≥ 0)

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

(11)    ((UIncreasing(576_0_MAIN_GT(+(x1[1], -1), 0)), ≥)∧0 = 0∧[(-1)bso_13] ≥ 0)

For Pair 576_0_MAIN_GT(x1, x0) → COND_576_0_MAIN_GT1(>(x0, 0), x1, x0) the following chains were created:
• We consider the chain 576_0_MAIN_GT(x1[2], x0[2]) → COND_576_0_MAIN_GT1(>(x0[2], 0), x1[2], x0[2]), COND_576_0_MAIN_GT1(TRUE, x1[3], x0[3]) → 576_0_MAIN_GT(x1[3], +(x0[3], -1)) which results in the following constraint:

(12)    (>(x0[2], 0)=TRUEx1[2]=x1[3]x0[2]=x0[3]576_0_MAIN_GT(x1[2], x0[2])≥NonInfC∧576_0_MAIN_GT(x1[2], x0[2])≥COND_576_0_MAIN_GT1(>(x0[2], 0), x1[2], x0[2])∧(UIncreasing(COND_576_0_MAIN_GT1(>(x0[2], 0), x1[2], x0[2])), ≥))

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

(13)    (>(x0[2], 0)=TRUE576_0_MAIN_GT(x1[2], x0[2])≥NonInfC∧576_0_MAIN_GT(x1[2], x0[2])≥COND_576_0_MAIN_GT1(>(x0[2], 0), x1[2], x0[2])∧(UIncreasing(COND_576_0_MAIN_GT1(>(x0[2], 0), x1[2], x0[2])), ≥))

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

(14)    (x0[2] + [-1] ≥ 0 ⇒ (UIncreasing(COND_576_0_MAIN_GT1(>(x0[2], 0), x1[2], x0[2])), ≥)∧[(-1)bni_14 + (-1)Bound*bni_14] + [bni_14]x0[2] ≥ 0∧[(-1)bso_15] ≥ 0)

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

(15)    (x0[2] + [-1] ≥ 0 ⇒ (UIncreasing(COND_576_0_MAIN_GT1(>(x0[2], 0), x1[2], x0[2])), ≥)∧[(-1)bni_14 + (-1)Bound*bni_14] + [bni_14]x0[2] ≥ 0∧[(-1)bso_15] ≥ 0)

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

(16)    (x0[2] + [-1] ≥ 0 ⇒ (UIncreasing(COND_576_0_MAIN_GT1(>(x0[2], 0), x1[2], x0[2])), ≥)∧[(-1)bni_14 + (-1)Bound*bni_14] + [bni_14]x0[2] ≥ 0∧[(-1)bso_15] ≥ 0)

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

(17)    (x0[2] + [-1] ≥ 0 ⇒ (UIncreasing(COND_576_0_MAIN_GT1(>(x0[2], 0), x1[2], x0[2])), ≥)∧0 = 0∧[(-1)bni_14 + (-1)Bound*bni_14] + [bni_14]x0[2] ≥ 0∧0 = 0∧[(-1)bso_15] ≥ 0)

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

(18)    (x0[2] ≥ 0 ⇒ (UIncreasing(COND_576_0_MAIN_GT1(>(x0[2], 0), x1[2], x0[2])), ≥)∧0 = 0∧[(-1)Bound*bni_14] + [bni_14]x0[2] ≥ 0∧0 = 0∧[(-1)bso_15] ≥ 0)

For Pair COND_576_0_MAIN_GT1(TRUE, x1, x0) → 576_0_MAIN_GT(x1, +(x0, -1)) the following chains were created:
• We consider the chain COND_576_0_MAIN_GT1(TRUE, x1[3], x0[3]) → 576_0_MAIN_GT(x1[3], +(x0[3], -1)) which results in the following constraint:

(19)    (COND_576_0_MAIN_GT1(TRUE, x1[3], x0[3])≥NonInfC∧COND_576_0_MAIN_GT1(TRUE, x1[3], x0[3])≥576_0_MAIN_GT(x1[3], +(x0[3], -1))∧(UIncreasing(576_0_MAIN_GT(x1[3], +(x0[3], -1))), ≥))

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

(20)    ((UIncreasing(576_0_MAIN_GT(x1[3], +(x0[3], -1))), ≥)∧[1 + (-1)bso_17] ≥ 0)

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

(21)    ((UIncreasing(576_0_MAIN_GT(x1[3], +(x0[3], -1))), ≥)∧[1 + (-1)bso_17] ≥ 0)

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

(22)    ((UIncreasing(576_0_MAIN_GT(x1[3], +(x0[3], -1))), ≥)∧[1 + (-1)bso_17] ≥ 0)

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

(23)    ((UIncreasing(576_0_MAIN_GT(x1[3], +(x0[3], -1))), ≥)∧0 = 0∧0 = 0∧[1 + (-1)bso_17] ≥ 0)

To summarize, we get the following constraints P for the following pairs.
• 576_0_MAIN_GT(x1, 0) → COND_576_0_MAIN_GT(>(x1, 0), x1, 0)
• (x1[0] ≥ 0 ⇒ (UIncreasing(COND_576_0_MAIN_GT(>(x1[0], 0), x1[0], 0)), ≥)∧[(-1)bni_10 + (-1)Bound*bni_10] ≥ 0∧[(-1)bso_11] ≥ 0)

• COND_576_0_MAIN_GT(TRUE, x1, 0) → 576_0_MAIN_GT(+(x1, -1), 0)
• ((UIncreasing(576_0_MAIN_GT(+(x1[1], -1), 0)), ≥)∧0 = 0∧[(-1)bso_13] ≥ 0)

• 576_0_MAIN_GT(x1, x0) → COND_576_0_MAIN_GT1(>(x0, 0), x1, x0)
• (x0[2] ≥ 0 ⇒ (UIncreasing(COND_576_0_MAIN_GT1(>(x0[2], 0), x1[2], x0[2])), ≥)∧0 = 0∧[(-1)Bound*bni_14] + [bni_14]x0[2] ≥ 0∧0 = 0∧[(-1)bso_15] ≥ 0)

• COND_576_0_MAIN_GT1(TRUE, x1, x0) → 576_0_MAIN_GT(x1, +(x0, -1))
• ((UIncreasing(576_0_MAIN_GT(x1[3], +(x0[3], -1))), ≥)∧0 = 0∧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(576_0_MAIN_GT(x1, x2)) = [-1] + x2
POL(0) = 0
POL(COND_576_0_MAIN_GT(x1, x2, x3)) = [-1]
POL(>(x1, x2)) = [-1]
POL(+(x1, x2)) = x1 + x2
POL(-1) = [-1]
POL(COND_576_0_MAIN_GT1(x1, x2, x3)) = [-1] + x3

The following pairs are in P>:

COND_576_0_MAIN_GT1(TRUE, x1[3], x0[3]) → 576_0_MAIN_GT(x1[3], +(x0[3], -1))

The following pairs are in Pbound:

576_0_MAIN_GT(x1[0], 0) → COND_576_0_MAIN_GT(>(x1[0], 0), x1[0], 0)
576_0_MAIN_GT(x1[2], x0[2]) → COND_576_0_MAIN_GT1(>(x0[2], 0), x1[2], x0[2])

The following pairs are in P:

576_0_MAIN_GT(x1[0], 0) → COND_576_0_MAIN_GT(>(x1[0], 0), x1[0], 0)
COND_576_0_MAIN_GT(TRUE, x1[1], 0) → 576_0_MAIN_GT(+(x1[1], -1), 0)
576_0_MAIN_GT(x1[2], x0[2]) → COND_576_0_MAIN_GT1(>(x0[2], 0), x1[2], x0[2])

There are no usable rules.

### (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): 576_0_MAIN_GT(x1[0], 0) → COND_576_0_MAIN_GT(x1[0] > 0, x1[0], 0)
(1): COND_576_0_MAIN_GT(TRUE, x1[1], 0) → 576_0_MAIN_GT(x1[1] + -1, 0)
(2): 576_0_MAIN_GT(x1[2], x0[2]) → COND_576_0_MAIN_GT1(x0[2] > 0, x1[2], x0[2])

(1) -> (0), if (x1[1] + -1* x1[0])

(0) -> (1), if ((x1[0] > 0* TRUE)∧(x1[0]* x1[1]))

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

The set Q is empty.

### (8) IDependencyGraphProof (EQUIVALENT transformation)

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

### (9) 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_576_0_MAIN_GT(TRUE, x1[1], 0) → 576_0_MAIN_GT(x1[1] + -1, 0)
(0): 576_0_MAIN_GT(x1[0], 0) → COND_576_0_MAIN_GT(x1[0] > 0, x1[0], 0)

(1) -> (0), if (x1[1] + -1* x1[0])

(0) -> (1), if ((x1[0] > 0* TRUE)∧(x1[0]* x1[1]))

The set Q is empty.

### (10) 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 COND_576_0_MAIN_GT(TRUE, x1[1], 0) → 576_0_MAIN_GT(+(x1[1], -1), 0) the following chains were created:
• We consider the chain COND_576_0_MAIN_GT(TRUE, x1[1], 0) → 576_0_MAIN_GT(+(x1[1], -1), 0) which results in the following constraint:

(1)    (COND_576_0_MAIN_GT(TRUE, x1[1], 0)≥NonInfC∧COND_576_0_MAIN_GT(TRUE, x1[1], 0)≥576_0_MAIN_GT(+(x1[1], -1), 0)∧(UIncreasing(576_0_MAIN_GT(+(x1[1], -1), 0)), ≥))

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

(2)    ((UIncreasing(576_0_MAIN_GT(+(x1[1], -1), 0)), ≥)∧[2 + (-1)bso_7] ≥ 0)

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

(3)    ((UIncreasing(576_0_MAIN_GT(+(x1[1], -1), 0)), ≥)∧[2 + (-1)bso_7] ≥ 0)

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

(4)    ((UIncreasing(576_0_MAIN_GT(+(x1[1], -1), 0)), ≥)∧[2 + (-1)bso_7] ≥ 0)

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

(5)    ((UIncreasing(576_0_MAIN_GT(+(x1[1], -1), 0)), ≥)∧0 = 0∧[2 + (-1)bso_7] ≥ 0)

For Pair 576_0_MAIN_GT(x1[0], 0) → COND_576_0_MAIN_GT(>(x1[0], 0), x1[0], 0) the following chains were created:
• We consider the chain 576_0_MAIN_GT(x1[0], 0) → COND_576_0_MAIN_GT(>(x1[0], 0), x1[0], 0), COND_576_0_MAIN_GT(TRUE, x1[1], 0) → 576_0_MAIN_GT(+(x1[1], -1), 0) which results in the following constraint:

(6)    (>(x1[0], 0)=TRUEx1[0]=x1[1]576_0_MAIN_GT(x1[0], 0)≥NonInfC∧576_0_MAIN_GT(x1[0], 0)≥COND_576_0_MAIN_GT(>(x1[0], 0), x1[0], 0)∧(UIncreasing(COND_576_0_MAIN_GT(>(x1[0], 0), x1[0], 0)), ≥))

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

(7)    (>(x1[0], 0)=TRUE576_0_MAIN_GT(x1[0], 0)≥NonInfC∧576_0_MAIN_GT(x1[0], 0)≥COND_576_0_MAIN_GT(>(x1[0], 0), x1[0], 0)∧(UIncreasing(COND_576_0_MAIN_GT(>(x1[0], 0), x1[0], 0)), ≥))

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

(8)    (x1[0] + [-1] ≥ 0 ⇒ (UIncreasing(COND_576_0_MAIN_GT(>(x1[0], 0), x1[0], 0)), ≥)∧[(-1)bni_8 + (-1)Bound*bni_8] + [(2)bni_8]x1[0] ≥ 0∧[(-1)bso_9] ≥ 0)

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

(9)    (x1[0] + [-1] ≥ 0 ⇒ (UIncreasing(COND_576_0_MAIN_GT(>(x1[0], 0), x1[0], 0)), ≥)∧[(-1)bni_8 + (-1)Bound*bni_8] + [(2)bni_8]x1[0] ≥ 0∧[(-1)bso_9] ≥ 0)

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

(10)    (x1[0] + [-1] ≥ 0 ⇒ (UIncreasing(COND_576_0_MAIN_GT(>(x1[0], 0), x1[0], 0)), ≥)∧[(-1)bni_8 + (-1)Bound*bni_8] + [(2)bni_8]x1[0] ≥ 0∧[(-1)bso_9] ≥ 0)

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

(11)    (x1[0] ≥ 0 ⇒ (UIncreasing(COND_576_0_MAIN_GT(>(x1[0], 0), x1[0], 0)), ≥)∧[bni_8 + (-1)Bound*bni_8] + [(2)bni_8]x1[0] ≥ 0∧[(-1)bso_9] ≥ 0)

To summarize, we get the following constraints P for the following pairs.
• COND_576_0_MAIN_GT(TRUE, x1[1], 0) → 576_0_MAIN_GT(+(x1[1], -1), 0)
• ((UIncreasing(576_0_MAIN_GT(+(x1[1], -1), 0)), ≥)∧0 = 0∧[2 + (-1)bso_7] ≥ 0)

• 576_0_MAIN_GT(x1[0], 0) → COND_576_0_MAIN_GT(>(x1[0], 0), x1[0], 0)
• (x1[0] ≥ 0 ⇒ (UIncreasing(COND_576_0_MAIN_GT(>(x1[0], 0), x1[0], 0)), ≥)∧[bni_8 + (-1)Bound*bni_8] + [(2)bni_8]x1[0] ≥ 0∧[(-1)bso_9] ≥ 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_576_0_MAIN_GT(x1, x2, x3)) = [-1] + [2]x2
POL(0) = 0
POL(576_0_MAIN_GT(x1, x2)) = [-1] + [2]x1
POL(+(x1, x2)) = x1 + x2
POL(-1) = [-1]
POL(>(x1, x2)) = [2]

The following pairs are in P>:

COND_576_0_MAIN_GT(TRUE, x1[1], 0) → 576_0_MAIN_GT(+(x1[1], -1), 0)

The following pairs are in Pbound:

576_0_MAIN_GT(x1[0], 0) → COND_576_0_MAIN_GT(>(x1[0], 0), x1[0], 0)

The following pairs are in P:

576_0_MAIN_GT(x1[0], 0) → COND_576_0_MAIN_GT(>(x1[0], 0), x1[0], 0)

There are no usable rules.

### (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:

Integer

R is empty.

The integer pair graph contains the following rules and edges:
(0): 576_0_MAIN_GT(x1[0], 0) → COND_576_0_MAIN_GT(x1[0] > 0, x1[0], 0)

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.

### (15) 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_576_0_MAIN_GT(TRUE, x1[1], 0) → 576_0_MAIN_GT(x1[1] + -1, 0)

The set Q is empty.

### (16) IDependencyGraphProof (EQUIVALENT transformation)

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

### (18) 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_576_0_MAIN_GT(TRUE, x1[1], 0) → 576_0_MAIN_GT(x1[1] + -1, 0)
(3): COND_576_0_MAIN_GT1(TRUE, x1[3], x0[3]) → 576_0_MAIN_GT(x1[3], x0[3] + -1)

The set Q is empty.

### (19) IDependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 2 less nodes.