### (0) Obligation:

JBC Problem based on JBC Program:
Manifest-Version: 1.0 Created-By: 1.6.0_16 (Sun Microsystems Inc.) Main-Class: PastaA10
`/** * 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 PastaA10 {    public static void main(String[] args) {        Random.args = args;        int x = Random.random();        int y = Random.random();        while (x != y) {            if (x > y) {                y++;            } else {                x++;            }        }    }}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 192 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:
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:

Integer

The ITRS R consists of the following rules:

The integer pair graph contains the following rules and edges:

(0) -> (1), if ((i14[0] < i53[0]* TRUE)∧(i53[0]* i53[1])∧(i14[0]* i14[1]))

(1) -> (0), if ((i14[1] + 1* i14[0])∧(i53[1]* i53[0]))

(1) -> (2), if ((i14[1] + 1* i14[2])∧(i53[1]* i53[2]))

(2) -> (3), if ((i14[2]* i14[3])∧(i53[2]* i53[3])∧(i14[2] > i53[2]* TRUE))

(3) -> (0), if ((i53[3] + 1* i53[0])∧(i14[3]* i14[0]))

(3) -> (2), if ((i53[3] + 1* i53[2])∧(i14[3]* i14[2]))

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:

Integer

R is empty.

The integer pair graph contains the following rules and edges:

(0) -> (1), if ((i14[0] < i53[0]* TRUE)∧(i53[0]* i53[1])∧(i14[0]* i14[1]))

(1) -> (0), if ((i14[1] + 1* i14[0])∧(i53[1]* i53[0]))

(1) -> (2), if ((i14[1] + 1* i14[2])∧(i53[1]* i53[2]))

(2) -> (3), if ((i14[2]* i14[3])∧(i53[2]* i53[3])∧(i14[2] > i53[2]* TRUE))

(3) -> (0), if ((i53[3] + 1* i53[0])∧(i14[3]* i14[0]))

(3) -> (2), if ((i53[3] + 1* i53[2])∧(i14[3]* i14[2]))

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 LOAD1069(i14, i53) → COND_LOAD1069(<(i14, i53), i14, i53) the following chains were created:
• We consider the chain LOAD1069(i14[0], i53[0]) → COND_LOAD1069(<(i14[0], i53[0]), i14[0], i53[0]), COND_LOAD1069(TRUE, i14[1], i53[1]) → LOAD1069(+(i14[1], 1), i53[1]) which results in the following constraint:

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

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

(3)    (i53[0] + [-1] + [-1]i14[0] ≥ 0 ⇒ (UIncreasing(COND_LOAD1069(<(i14[0], i53[0]), i14[0], i53[0])), ≥)∧[(-1)bni_13 + (-1)Bound*bni_13] + [bni_13]max{i14[0] + [-1]i53[0], [-1]i14[0] + i53[0]} ≥ 0∧[(-1)bso_14] + max{i14[0] + [-1]i53[0], [-1]i14[0] + i53[0]} + [-1]max{i14[0] + [-1]i53[0], [-1]i14[0] + i53[0]} ≥ 0)

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

(4)    (i53[0] + [-1] + [-1]i14[0] ≥ 0 ⇒ (UIncreasing(COND_LOAD1069(<(i14[0], i53[0]), i14[0], i53[0])), ≥)∧[(-1)bni_13 + (-1)Bound*bni_13] + [bni_13]max{i14[0] + [-1]i53[0], [-1]i14[0] + i53[0]} ≥ 0∧[(-1)bso_14] + max{i14[0] + [-1]i53[0], [-1]i14[0] + i53[0]} + [-1]max{i14[0] + [-1]i53[0], [-1]i14[0] + i53[0]} ≥ 0)

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

(5)    (i53[0] + [-1] + [-1]i14[0] ≥ 0∧[-1] + [-2]i14[0] + [2]i53[0] ≥ 0 ⇒ (UIncreasing(COND_LOAD1069(<(i14[0], i53[0]), i14[0], i53[0])), ≥)∧[(-1)bni_13 + (-1)Bound*bni_13] + [(-1)bni_13]i14[0] + [bni_13]i53[0] ≥ 0∧[(-1)bso_14] ≥ 0)

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

(6)    (i53[0] ≥ 0∧[1] + [2]i53[0] ≥ 0 ⇒ (UIncreasing(COND_LOAD1069(<(i14[0], i53[0]), i14[0], i53[0])), ≥)∧[(-1)Bound*bni_13] + [bni_13]i53[0] ≥ 0∧[(-1)bso_14] ≥ 0)

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

(7)    (i53[0] ≥ 0∧[1] + [2]i53[0] ≥ 0∧i14[0] ≥ 0 ⇒ (UIncreasing(COND_LOAD1069(<(i14[0], i53[0]), i14[0], i53[0])), ≥)∧[(-1)Bound*bni_13] + [bni_13]i53[0] ≥ 0∧[(-1)bso_14] ≥ 0)

(8)    (i53[0] ≥ 0∧[1] + [2]i53[0] ≥ 0∧i14[0] ≥ 0 ⇒ (UIncreasing(COND_LOAD1069(<(i14[0], i53[0]), i14[0], i53[0])), ≥)∧[(-1)Bound*bni_13] + [bni_13]i53[0] ≥ 0∧[(-1)bso_14] ≥ 0)

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

(9)    (i53[0] ≥ 0∧i14[0] ≥ 0∧i53[0] ≥ 0 ⇒ (UIncreasing(COND_LOAD1069(<(i14[0], i53[0]), i14[0], i53[0])), ≥)∧[(-1)Bound*bni_13] + [bni_13]i53[0] ≥ 0∧[(-1)bso_14] ≥ 0)

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

(10)    (i53[0] ≥ 0∧i14[0] ≥ 0∧i53[0] ≥ 0 ⇒ (UIncreasing(COND_LOAD1069(<(i14[0], i53[0]), i14[0], i53[0])), ≥)∧[(-1)Bound*bni_13] + [bni_13]i53[0] ≥ 0∧[(-1)bso_14] ≥ 0)

For Pair COND_LOAD1069(TRUE, i14, i53) → LOAD1069(+(i14, 1), i53) the following chains were created:

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

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

(13)    (i53[0] + [-2] + [-1]i14[1] ≥ 0 ⇒ (UIncreasing(LOAD1069(+(i14[1]1, 1), i53[1]1)), ≥)∧[(-1)bni_15 + (-1)Bound*bni_15] + [bni_15]max{[1] + i14[1] + [-1]i53[0], [-1] + [-1]i14[1] + i53[0]} ≥ 0∧[(-1)bso_16] + max{[1] + i14[1] + [-1]i53[0], [-1] + [-1]i14[1] + i53[0]} + [-1]max{[2] + i14[1] + [-1]i53[0], [-2] + [-1]i14[1] + i53[0]} ≥ 0)

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

(14)    (i53[0] + [-2] + [-1]i14[1] ≥ 0 ⇒ (UIncreasing(LOAD1069(+(i14[1]1, 1), i53[1]1)), ≥)∧[(-1)bni_15 + (-1)Bound*bni_15] + [bni_15]max{[1] + i14[1] + [-1]i53[0], [-1] + [-1]i14[1] + i53[0]} ≥ 0∧[(-1)bso_16] + max{[1] + i14[1] + [-1]i53[0], [-1] + [-1]i14[1] + i53[0]} + [-1]max{[2] + i14[1] + [-1]i53[0], [-2] + [-1]i14[1] + i53[0]} ≥ 0)

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

(15)    (i53[0] + [-2] + [-1]i14[1] ≥ 0∧[-3] + [-2]i14[1] + [2]i53[0] ≥ 0∧[4] + [2]i14[1] + [-2]i53[0] ≥ 0 ⇒ (UIncreasing(LOAD1069(+(i14[1]1, 1), i53[1]1)), ≥)∧[(-2)bni_15 + (-1)Bound*bni_15] + [(-1)bni_15]i14[1] + [bni_15]i53[0] ≥ 0∧[-3 + (-1)bso_16] + [-2]i14[1] + [2]i53[0] ≥ 0)

(16)    (i53[0] + [-2] + [-1]i14[1] ≥ 0∧[-3] + [-2]i14[1] + [2]i53[0] ≥ 0∧[-5] + [-2]i14[1] + [2]i53[0] ≥ 0 ⇒ (UIncreasing(LOAD1069(+(i14[1]1, 1), i53[1]1)), ≥)∧[(-2)bni_15 + (-1)Bound*bni_15] + [(-1)bni_15]i14[1] + [bni_15]i53[0] ≥ 0∧[1 + (-1)bso_16] ≥ 0)

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

(17)    (i53[0] ≥ 0∧[1] + [2]i53[0] ≥ 0∧[-2]i53[0] ≥ 0 ⇒ (UIncreasing(LOAD1069(+(i14[1]1, 1), i53[1]1)), ≥)∧[(-1)Bound*bni_15] + [bni_15]i53[0] ≥ 0∧[1 + (-1)bso_16] + [2]i53[0] ≥ 0)

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

(18)    (i53[0] ≥ 0∧[1] + [2]i53[0] ≥ 0∧[-1] + [2]i53[0] ≥ 0 ⇒ (UIncreasing(LOAD1069(+(i14[1]1, 1), i53[1]1)), ≥)∧[(-1)Bound*bni_15] + [bni_15]i53[0] ≥ 0∧[1 + (-1)bso_16] ≥ 0)

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

(19)    (0 ≥ 0∧[1] ≥ 0∧0 ≥ 0 ⇒ (UIncreasing(LOAD1069(+(i14[1]1, 1), i53[1]1)), ≥)∧[(-1)Bound*bni_15] ≥ 0∧[1 + (-1)bso_16] ≥ 0)

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

(20)    (0 ≥ 0∧[1] ≥ 0∧0 ≥ 0∧i14[1] ≥ 0 ⇒ (UIncreasing(LOAD1069(+(i14[1]1, 1), i53[1]1)), ≥)∧[(-1)Bound*bni_15] ≥ 0∧[1 + (-1)bso_16] ≥ 0)

(21)    (0 ≥ 0∧[1] ≥ 0∧0 ≥ 0∧i14[1] ≥ 0 ⇒ (UIncreasing(LOAD1069(+(i14[1]1, 1), i53[1]1)), ≥)∧[(-1)Bound*bni_15] ≥ 0∧[1 + (-1)bso_16] ≥ 0)

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

(22)    (i53[0] ≥ 0∧[1] + [2]i53[0] ≥ 0∧[-1] + [2]i53[0] ≥ 0∧i14[1] ≥ 0 ⇒ (UIncreasing(LOAD1069(+(i14[1]1, 1), i53[1]1)), ≥)∧[(-1)Bound*bni_15] + [bni_15]i53[0] ≥ 0∧[1 + (-1)bso_16] ≥ 0)

(23)    (i53[0] ≥ 0∧[1] + [2]i53[0] ≥ 0∧[-1] + [2]i53[0] ≥ 0∧i14[1] ≥ 0 ⇒ (UIncreasing(LOAD1069(+(i14[1]1, 1), i53[1]1)), ≥)∧[(-1)Bound*bni_15] + [bni_15]i53[0] ≥ 0∧[1 + (-1)bso_16] ≥ 0)

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

(24)    (i53[0] ≥ 0∧i14[1] ≥ 0∧i53[0] ≥ 0∧i53[0] ≥ 0 ⇒ (UIncreasing(LOAD1069(+(i14[1]1, 1), i53[1]1)), ≥)∧[(-1)Bound*bni_15] + [bni_15]i53[0] ≥ 0∧[1 + (-1)bso_16] ≥ 0)

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

(25)    (i53[0] ≥ 0∧i14[1] ≥ 0∧i53[0] ≥ 0∧i53[0] ≥ 0 ⇒ (UIncreasing(LOAD1069(+(i14[1]1, 1), i53[1]1)), ≥)∧[(-1)Bound*bni_15] + [bni_15]i53[0] ≥ 0∧[1 + (-1)bso_16] ≥ 0)

We simplified constraint (26) using rule (III) which results in the following new constraint:

We simplified constraint (27) using rule (POLY_CONSTRAINTS) which results in the following new constraint:

(28)    (i53[3] + [-1]i14[0] ≥ 0 ⇒ (UIncreasing(LOAD1069(+(i14[1], 1), i53[1])), ≥)∧[(-1)bni_15 + (-1)Bound*bni_15] + [bni_15]max{[-1] + i14[0] + [-1]i53[3], [1] + [-1]i14[0] + i53[3]} ≥ 0∧[(-1)bso_16] + max{[-1] + i14[0] + [-1]i53[3], [1] + [-1]i14[0] + i53[3]} + [-1]max{i14[0] + [-1]i53[3], [-1]i14[0] + i53[3]} ≥ 0)

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

(29)    (i53[3] + [-1]i14[0] ≥ 0 ⇒ (UIncreasing(LOAD1069(+(i14[1], 1), i53[1])), ≥)∧[(-1)bni_15 + (-1)Bound*bni_15] + [bni_15]max{[-1] + i14[0] + [-1]i53[3], [1] + [-1]i14[0] + i53[3]} ≥ 0∧[(-1)bso_16] + max{[-1] + i14[0] + [-1]i53[3], [1] + [-1]i14[0] + i53[3]} + [-1]max{i14[0] + [-1]i53[3], [-1]i14[0] + i53[3]} ≥ 0)

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

(30)    (i53[3] + [-1]i14[0] ≥ 0∧[1] + [-2]i14[0] + [2]i53[3] ≥ 0∧[2]i14[0] + [-2]i53[3] ≥ 0 ⇒ (UIncreasing(LOAD1069(+(i14[1], 1), i53[1])), ≥)∧[(-1)Bound*bni_15] + [(-1)bni_15]i14[0] + [bni_15]i53[3] ≥ 0∧[1 + (-1)bso_16] + [-2]i14[0] + [2]i53[3] ≥ 0)

(31)    (i53[3] + [-1]i14[0] ≥ 0∧[1] + [-2]i14[0] + [2]i53[3] ≥ 0∧[-1] + [-2]i14[0] + [2]i53[3] ≥ 0 ⇒ (UIncreasing(LOAD1069(+(i14[1], 1), i53[1])), ≥)∧[(-1)Bound*bni_15] + [(-1)bni_15]i14[0] + [bni_15]i53[3] ≥ 0∧[1 + (-1)bso_16] ≥ 0)

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

(32)    (i53[3] ≥ 0∧[1] + [2]i53[3] ≥ 0∧[-2]i53[3] ≥ 0 ⇒ (UIncreasing(LOAD1069(+(i14[1], 1), i53[1])), ≥)∧[(-1)Bound*bni_15] + [bni_15]i53[3] ≥ 0∧[1 + (-1)bso_16] + [2]i53[3] ≥ 0)

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

(33)    (i53[3] ≥ 0∧[1] + [2]i53[3] ≥ 0∧[-1] + [2]i53[3] ≥ 0 ⇒ (UIncreasing(LOAD1069(+(i14[1], 1), i53[1])), ≥)∧[(-1)Bound*bni_15] + [bni_15]i53[3] ≥ 0∧[1 + (-1)bso_16] ≥ 0)

We simplified constraint (32) using rule (IDP_SMT_SPLIT) which results in the following new constraint:

(34)    (0 ≥ 0∧[1] ≥ 0∧0 ≥ 0 ⇒ (UIncreasing(LOAD1069(+(i14[1], 1), i53[1])), ≥)∧[(-1)Bound*bni_15] ≥ 0∧[1 + (-1)bso_16] ≥ 0)

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

(35)    (0 ≥ 0∧[1] ≥ 0∧0 ≥ 0∧i14[0] ≥ 0 ⇒ (UIncreasing(LOAD1069(+(i14[1], 1), i53[1])), ≥)∧[(-1)Bound*bni_15] ≥ 0∧[1 + (-1)bso_16] ≥ 0)

(36)    (0 ≥ 0∧[1] ≥ 0∧0 ≥ 0∧i14[0] ≥ 0 ⇒ (UIncreasing(LOAD1069(+(i14[1], 1), i53[1])), ≥)∧[(-1)Bound*bni_15] ≥ 0∧[1 + (-1)bso_16] ≥ 0)

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

(37)    (i53[3] ≥ 0∧[1] + [2]i53[3] ≥ 0∧[-1] + [2]i53[3] ≥ 0∧i14[0] ≥ 0 ⇒ (UIncreasing(LOAD1069(+(i14[1], 1), i53[1])), ≥)∧[(-1)Bound*bni_15] + [bni_15]i53[3] ≥ 0∧[1 + (-1)bso_16] ≥ 0)

(38)    (i53[3] ≥ 0∧[1] + [2]i53[3] ≥ 0∧[-1] + [2]i53[3] ≥ 0∧i14[0] ≥ 0 ⇒ (UIncreasing(LOAD1069(+(i14[1], 1), i53[1])), ≥)∧[(-1)Bound*bni_15] + [bni_15]i53[3] ≥ 0∧[1 + (-1)bso_16] ≥ 0)

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

(39)    (i53[3] ≥ 0∧i14[0] ≥ 0∧i53[3] ≥ 0∧i53[3] ≥ 0 ⇒ (UIncreasing(LOAD1069(+(i14[1], 1), i53[1])), ≥)∧[(-1)Bound*bni_15] + [bni_15]i53[3] ≥ 0∧[1 + (-1)bso_16] ≥ 0)

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

(40)    (i53[3] ≥ 0∧i14[0] ≥ 0∧i53[3] ≥ 0∧i53[3] ≥ 0 ⇒ (UIncreasing(LOAD1069(+(i14[1], 1), i53[1])), ≥)∧[(-1)Bound*bni_15] + [bni_15]i53[3] ≥ 0∧[1 + (-1)bso_16] ≥ 0)

For Pair LOAD1069(i14, i53) → COND_LOAD10691(>(i14, i53), i14, i53) the following chains were created:
• We consider the chain LOAD1069(i14[2], i53[2]) → COND_LOAD10691(>(i14[2], i53[2]), i14[2], i53[2]), COND_LOAD10691(TRUE, i14[3], i53[3]) → LOAD1069(i14[3], +(i53[3], 1)) which results in the following constraint:

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

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

(43)    (i14[2] + [-1] + [-1]i53[2] ≥ 0 ⇒ (UIncreasing(COND_LOAD10691(>(i14[2], i53[2]), i14[2], i53[2])), ≥)∧[(-1)bni_17 + (-1)Bound*bni_17] + [bni_17]max{i14[2] + [-1]i53[2], [-1]i14[2] + i53[2]} ≥ 0∧[(-1)bso_18] + max{i14[2] + [-1]i53[2], [-1]i14[2] + i53[2]} + [-1]max{i14[2] + [-1]i53[2], [-1]i14[2] + i53[2]} ≥ 0)

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

(44)    (i14[2] + [-1] + [-1]i53[2] ≥ 0 ⇒ (UIncreasing(COND_LOAD10691(>(i14[2], i53[2]), i14[2], i53[2])), ≥)∧[(-1)bni_17 + (-1)Bound*bni_17] + [bni_17]max{i14[2] + [-1]i53[2], [-1]i14[2] + i53[2]} ≥ 0∧[(-1)bso_18] + max{i14[2] + [-1]i53[2], [-1]i14[2] + i53[2]} + [-1]max{i14[2] + [-1]i53[2], [-1]i14[2] + i53[2]} ≥ 0)

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

(45)    (i14[2] + [-1] + [-1]i53[2] ≥ 0∧[2]i14[2] + [-2]i53[2] ≥ 0 ⇒ (UIncreasing(COND_LOAD10691(>(i14[2], i53[2]), i14[2], i53[2])), ≥)∧[(-1)bni_17 + (-1)Bound*bni_17] + [bni_17]i14[2] + [(-1)bni_17]i53[2] ≥ 0∧[(-1)bso_18] ≥ 0)

We simplified constraint (45) using rule (IDP_SMT_SPLIT) which results in the following new constraint:

(46)    (i14[2] ≥ 0∧[2] + [2]i14[2] ≥ 0 ⇒ (UIncreasing(COND_LOAD10691(>(i14[2], i53[2]), i14[2], i53[2])), ≥)∧[(-1)Bound*bni_17] + [bni_17]i14[2] ≥ 0∧[(-1)bso_18] ≥ 0)

We simplified constraint (46) using rule (IDP_SMT_SPLIT) which results in the following new constraints:

(47)    (i14[2] ≥ 0∧[2] + [2]i14[2] ≥ 0∧i53[2] ≥ 0 ⇒ (UIncreasing(COND_LOAD10691(>(i14[2], i53[2]), i14[2], i53[2])), ≥)∧[(-1)Bound*bni_17] + [bni_17]i14[2] ≥ 0∧[(-1)bso_18] ≥ 0)

(48)    (i14[2] ≥ 0∧[2] + [2]i14[2] ≥ 0∧i53[2] ≥ 0 ⇒ (UIncreasing(COND_LOAD10691(>(i14[2], i53[2]), i14[2], i53[2])), ≥)∧[(-1)Bound*bni_17] + [bni_17]i14[2] ≥ 0∧[(-1)bso_18] ≥ 0)

We simplified constraint (47) using rule (IDP_POLY_GCD) which results in the following new constraint:

(49)    (i14[2] ≥ 0∧i53[2] ≥ 0∧[1] + i14[2] ≥ 0 ⇒ (UIncreasing(COND_LOAD10691(>(i14[2], i53[2]), i14[2], i53[2])), ≥)∧[(-1)Bound*bni_17] + [bni_17]i14[2] ≥ 0∧[(-1)bso_18] ≥ 0)

We simplified constraint (48) using rule (IDP_POLY_GCD) which results in the following new constraint:

(50)    (i14[2] ≥ 0∧i53[2] ≥ 0∧[1] + i14[2] ≥ 0 ⇒ (UIncreasing(COND_LOAD10691(>(i14[2], i53[2]), i14[2], i53[2])), ≥)∧[(-1)Bound*bni_17] + [bni_17]i14[2] ≥ 0∧[(-1)bso_18] ≥ 0)

For Pair COND_LOAD10691(TRUE, i14, i53) → LOAD1069(i14, +(i53, 1)) the following chains were created:

We simplified constraint (51) using rule (III) which results in the following new constraint:

We simplified constraint (52) using rule (POLY_CONSTRAINTS) which results in the following new constraint:

(53)    (i14[1] + [-1]i53[2] ≥ 0 ⇒ (UIncreasing(LOAD1069(i14[3], +(i53[3], 1))), ≥)∧[(-1)bni_19 + (-1)Bound*bni_19] + [bni_19]max{[1] + i14[1] + [-1]i53[2], [-1] + [-1]i14[1] + i53[2]} ≥ 0∧[(-1)bso_20] + max{[1] + i14[1] + [-1]i53[2], [-1] + [-1]i14[1] + i53[2]} + [-1]max{i14[1] + [-1]i53[2], [-1]i14[1] + i53[2]} ≥ 0)

We simplified constraint (53) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:

(54)    (i14[1] + [-1]i53[2] ≥ 0 ⇒ (UIncreasing(LOAD1069(i14[3], +(i53[3], 1))), ≥)∧[(-1)bni_19 + (-1)Bound*bni_19] + [bni_19]max{[1] + i14[1] + [-1]i53[2], [-1] + [-1]i14[1] + i53[2]} ≥ 0∧[(-1)bso_20] + max{[1] + i14[1] + [-1]i53[2], [-1] + [-1]i14[1] + i53[2]} + [-1]max{i14[1] + [-1]i53[2], [-1]i14[1] + i53[2]} ≥ 0)

We simplified constraint (54) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:

(55)    (i14[1] + [-1]i53[2] ≥ 0∧[2] + [2]i14[1] + [-2]i53[2] ≥ 0∧[2]i14[1] + [-2]i53[2] ≥ 0 ⇒ (UIncreasing(LOAD1069(i14[3], +(i53[3], 1))), ≥)∧[(-1)Bound*bni_19] + [bni_19]i14[1] + [(-1)bni_19]i53[2] ≥ 0∧[1 + (-1)bso_20] ≥ 0)

We simplified constraint (55) using rule (IDP_SMT_SPLIT) which results in the following new constraint:

(56)    (i14[1] ≥ 0∧[2] + [2]i14[1] ≥ 0∧[2]i14[1] ≥ 0 ⇒ (UIncreasing(LOAD1069(i14[3], +(i53[3], 1))), ≥)∧[(-1)Bound*bni_19] + [bni_19]i14[1] ≥ 0∧[1 + (-1)bso_20] ≥ 0)

We simplified constraint (56) using rule (IDP_SMT_SPLIT) which results in the following new constraints:

(57)    (i14[1] ≥ 0∧[2] + [2]i14[1] ≥ 0∧[2]i14[1] ≥ 0∧i53[2] ≥ 0 ⇒ (UIncreasing(LOAD1069(i14[3], +(i53[3], 1))), ≥)∧[(-1)Bound*bni_19] + [bni_19]i14[1] ≥ 0∧[1 + (-1)bso_20] ≥ 0)

(58)    (i14[1] ≥ 0∧[2] + [2]i14[1] ≥ 0∧[2]i14[1] ≥ 0∧i53[2] ≥ 0 ⇒ (UIncreasing(LOAD1069(i14[3], +(i53[3], 1))), ≥)∧[(-1)Bound*bni_19] + [bni_19]i14[1] ≥ 0∧[1 + (-1)bso_20] ≥ 0)

We simplified constraint (57) using rule (IDP_POLY_GCD) which results in the following new constraint:

(59)    (i14[1] ≥ 0∧i53[2] ≥ 0∧[1] + i14[1] ≥ 0∧i14[1] ≥ 0 ⇒ (UIncreasing(LOAD1069(i14[3], +(i53[3], 1))), ≥)∧[(-1)Bound*bni_19] + [bni_19]i14[1] ≥ 0∧[1 + (-1)bso_20] ≥ 0)

We simplified constraint (58) using rule (IDP_POLY_GCD) which results in the following new constraint:

(60)    (i14[1] ≥ 0∧i53[2] ≥ 0∧[1] + i14[1] ≥ 0∧i14[1] ≥ 0 ⇒ (UIncreasing(LOAD1069(i14[3], +(i53[3], 1))), ≥)∧[(-1)Bound*bni_19] + [bni_19]i14[1] ≥ 0∧[1 + (-1)bso_20] ≥ 0)

We simplified constraint (61) using rule (III) which results in the following new constraint:

We simplified constraint (62) using rule (POLY_CONSTRAINTS) which results in the following new constraint:

(63)    (i14[2] + [-2] + [-1]i53[3] ≥ 0 ⇒ (UIncreasing(LOAD1069(i14[3]1, +(i53[3]1, 1))), ≥)∧[(-1)bni_19 + (-1)Bound*bni_19] + [bni_19]max{[-1] + i14[2] + [-1]i53[3], [1] + [-1]i14[2] + i53[3]} ≥ 0∧[(-1)bso_20] + max{[-1] + i14[2] + [-1]i53[3], [1] + [-1]i14[2] + i53[3]} + [-1]max{[-2] + i14[2] + [-1]i53[3], [2] + [-1]i14[2] + i53[3]} ≥ 0)

We simplified constraint (63) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:

(64)    (i14[2] + [-2] + [-1]i53[3] ≥ 0 ⇒ (UIncreasing(LOAD1069(i14[3]1, +(i53[3]1, 1))), ≥)∧[(-1)bni_19 + (-1)Bound*bni_19] + [bni_19]max{[-1] + i14[2] + [-1]i53[3], [1] + [-1]i14[2] + i53[3]} ≥ 0∧[(-1)bso_20] + max{[-1] + i14[2] + [-1]i53[3], [1] + [-1]i14[2] + i53[3]} + [-1]max{[-2] + i14[2] + [-1]i53[3], [2] + [-1]i14[2] + i53[3]} ≥ 0)

We simplified constraint (64) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:

(65)    (i14[2] + [-2] + [-1]i53[3] ≥ 0∧[-2] + [2]i14[2] + [-2]i53[3] ≥ 0∧[-4] + [2]i14[2] + [-2]i53[3] ≥ 0 ⇒ (UIncreasing(LOAD1069(i14[3]1, +(i53[3]1, 1))), ≥)∧[(-2)bni_19 + (-1)Bound*bni_19] + [bni_19]i14[2] + [(-1)bni_19]i53[3] ≥ 0∧[1 + (-1)bso_20] ≥ 0)

We simplified constraint (65) using rule (IDP_SMT_SPLIT) which results in the following new constraint:

(66)    (i14[2] ≥ 0∧[2] + [2]i14[2] ≥ 0∧[2]i14[2] ≥ 0 ⇒ (UIncreasing(LOAD1069(i14[3]1, +(i53[3]1, 1))), ≥)∧[(-1)Bound*bni_19] + [bni_19]i14[2] ≥ 0∧[1 + (-1)bso_20] ≥ 0)

We simplified constraint (66) using rule (IDP_SMT_SPLIT) which results in the following new constraints:

(67)    (i14[2] ≥ 0∧[2] + [2]i14[2] ≥ 0∧[2]i14[2] ≥ 0∧i53[3] ≥ 0 ⇒ (UIncreasing(LOAD1069(i14[3]1, +(i53[3]1, 1))), ≥)∧[(-1)Bound*bni_19] + [bni_19]i14[2] ≥ 0∧[1 + (-1)bso_20] ≥ 0)

(68)    (i14[2] ≥ 0∧[2] + [2]i14[2] ≥ 0∧[2]i14[2] ≥ 0∧i53[3] ≥ 0 ⇒ (UIncreasing(LOAD1069(i14[3]1, +(i53[3]1, 1))), ≥)∧[(-1)Bound*bni_19] + [bni_19]i14[2] ≥ 0∧[1 + (-1)bso_20] ≥ 0)

We simplified constraint (67) using rule (IDP_POLY_GCD) which results in the following new constraint:

(69)    (i14[2] ≥ 0∧i53[3] ≥ 0∧[1] + i14[2] ≥ 0∧i14[2] ≥ 0 ⇒ (UIncreasing(LOAD1069(i14[3]1, +(i53[3]1, 1))), ≥)∧[(-1)Bound*bni_19] + [bni_19]i14[2] ≥ 0∧[1 + (-1)bso_20] ≥ 0)

We simplified constraint (68) using rule (IDP_POLY_GCD) which results in the following new constraint:

(70)    (i14[2] ≥ 0∧i53[3] ≥ 0∧[1] + i14[2] ≥ 0∧i14[2] ≥ 0 ⇒ (UIncreasing(LOAD1069(i14[3]1, +(i53[3]1, 1))), ≥)∧[(-1)Bound*bni_19] + [bni_19]i14[2] ≥ 0∧[1 + (-1)bso_20] ≥ 0)

To summarize, we get the following constraints P for the following pairs.
• (i53[0] ≥ 0∧i14[0] ≥ 0∧i53[0] ≥ 0 ⇒ (UIncreasing(COND_LOAD1069(<(i14[0], i53[0]), i14[0], i53[0])), ≥)∧[(-1)Bound*bni_13] + [bni_13]i53[0] ≥ 0∧[(-1)bso_14] ≥ 0)
• (i53[0] ≥ 0∧i14[0] ≥ 0∧i53[0] ≥ 0 ⇒ (UIncreasing(COND_LOAD1069(<(i14[0], i53[0]), i14[0], i53[0])), ≥)∧[(-1)Bound*bni_13] + [bni_13]i53[0] ≥ 0∧[(-1)bso_14] ≥ 0)

• (0 ≥ 0∧[1] ≥ 0∧0 ≥ 0∧i14[1] ≥ 0 ⇒ (UIncreasing(LOAD1069(+(i14[1]1, 1), i53[1]1)), ≥)∧[(-1)Bound*bni_15] ≥ 0∧[1 + (-1)bso_16] ≥ 0)
• (0 ≥ 0∧[1] ≥ 0∧0 ≥ 0∧i14[1] ≥ 0 ⇒ (UIncreasing(LOAD1069(+(i14[1]1, 1), i53[1]1)), ≥)∧[(-1)Bound*bni_15] ≥ 0∧[1 + (-1)bso_16] ≥ 0)
• (i53[0] ≥ 0∧i14[1] ≥ 0∧i53[0] ≥ 0∧i53[0] ≥ 0 ⇒ (UIncreasing(LOAD1069(+(i14[1]1, 1), i53[1]1)), ≥)∧[(-1)Bound*bni_15] + [bni_15]i53[0] ≥ 0∧[1 + (-1)bso_16] ≥ 0)
• (i53[0] ≥ 0∧i14[1] ≥ 0∧i53[0] ≥ 0∧i53[0] ≥ 0 ⇒ (UIncreasing(LOAD1069(+(i14[1]1, 1), i53[1]1)), ≥)∧[(-1)Bound*bni_15] + [bni_15]i53[0] ≥ 0∧[1 + (-1)bso_16] ≥ 0)
• (0 ≥ 0∧[1] ≥ 0∧0 ≥ 0∧i14[0] ≥ 0 ⇒ (UIncreasing(LOAD1069(+(i14[1], 1), i53[1])), ≥)∧[(-1)Bound*bni_15] ≥ 0∧[1 + (-1)bso_16] ≥ 0)
• (0 ≥ 0∧[1] ≥ 0∧0 ≥ 0∧i14[0] ≥ 0 ⇒ (UIncreasing(LOAD1069(+(i14[1], 1), i53[1])), ≥)∧[(-1)Bound*bni_15] ≥ 0∧[1 + (-1)bso_16] ≥ 0)
• (i53[3] ≥ 0∧i14[0] ≥ 0∧i53[3] ≥ 0∧i53[3] ≥ 0 ⇒ (UIncreasing(LOAD1069(+(i14[1], 1), i53[1])), ≥)∧[(-1)Bound*bni_15] + [bni_15]i53[3] ≥ 0∧[1 + (-1)bso_16] ≥ 0)
• (i53[3] ≥ 0∧i14[0] ≥ 0∧i53[3] ≥ 0∧i53[3] ≥ 0 ⇒ (UIncreasing(LOAD1069(+(i14[1], 1), i53[1])), ≥)∧[(-1)Bound*bni_15] + [bni_15]i53[3] ≥ 0∧[1 + (-1)bso_16] ≥ 0)

• (i14[2] ≥ 0∧i53[2] ≥ 0∧[1] + i14[2] ≥ 0 ⇒ (UIncreasing(COND_LOAD10691(>(i14[2], i53[2]), i14[2], i53[2])), ≥)∧[(-1)Bound*bni_17] + [bni_17]i14[2] ≥ 0∧[(-1)bso_18] ≥ 0)
• (i14[2] ≥ 0∧i53[2] ≥ 0∧[1] + i14[2] ≥ 0 ⇒ (UIncreasing(COND_LOAD10691(>(i14[2], i53[2]), i14[2], i53[2])), ≥)∧[(-1)Bound*bni_17] + [bni_17]i14[2] ≥ 0∧[(-1)bso_18] ≥ 0)

• (i14[1] ≥ 0∧i53[2] ≥ 0∧[1] + i14[1] ≥ 0∧i14[1] ≥ 0 ⇒ (UIncreasing(LOAD1069(i14[3], +(i53[3], 1))), ≥)∧[(-1)Bound*bni_19] + [bni_19]i14[1] ≥ 0∧[1 + (-1)bso_20] ≥ 0)
• (i14[1] ≥ 0∧i53[2] ≥ 0∧[1] + i14[1] ≥ 0∧i14[1] ≥ 0 ⇒ (UIncreasing(LOAD1069(i14[3], +(i53[3], 1))), ≥)∧[(-1)Bound*bni_19] + [bni_19]i14[1] ≥ 0∧[1 + (-1)bso_20] ≥ 0)
• (i14[2] ≥ 0∧i53[3] ≥ 0∧[1] + i14[2] ≥ 0∧i14[2] ≥ 0 ⇒ (UIncreasing(LOAD1069(i14[3]1, +(i53[3]1, 1))), ≥)∧[(-1)Bound*bni_19] + [bni_19]i14[2] ≥ 0∧[1 + (-1)bso_20] ≥ 0)
• (i14[2] ≥ 0∧i53[3] ≥ 0∧[1] + i14[2] ≥ 0∧i14[2] ≥ 0 ⇒ (UIncreasing(LOAD1069(i14[3]1, +(i53[3]1, 1))), ≥)∧[(-1)Bound*bni_19] + [bni_19]i14[2] ≥ 0∧[1 + (-1)bso_20] ≥ 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(LOAD1069(x1, x2)) = [-1] + max{x1 + [-1]x2, [-1]x1 + x2}
POL(COND_LOAD1069(x1, x2, x3)) = [-1] + max{x2 + [-1]x3, [-1]x2 + x3}
POL(<(x1, x2)) = [-1]
POL(+(x1, x2)) = x1 + x2
POL(1) = [1]
POL(COND_LOAD10691(x1, x2, x3)) = [-1] + max{x2 + [-1]x3, [-1]x2 + x3}
POL(>(x1, x2)) = [-1]

The following pairs are in P>:

The following pairs are in Pbound:

The following pairs are in P:

There are no usable rules.

### (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:

Integer

R is empty.

The integer pair graph contains the following rules and edges:

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 2 less nodes.

### (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: