### (0) Obligation:

JBC Problem based on JBC Program:
Manifest-Version: 1.0 Created-By: 1.6.0_16 (Sun Microsystems Inc.) Main-Class: PastaB17
`/** * 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 PastaB17 {    public static void main(String[] args) {        Random.args = args;        int x = Random.random();        int y = Random.random();        int z = Random.random();        while (x > z) {            while (y > z) {                y--;            }            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 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:
Load1003(i8, i23, i56) → Cond_Load1003(i23 > i56 && i8 > i56, i8, i23, i56)
Load1003(i8, i23, i56) → Cond_Load10031(i23 <= i56 && i8 > i56, i8, i23, i56)
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, Boolean

The ITRS R consists of the following rules:
Load1003(i8, i23, i56) → Cond_Load1003(i23 > i56 && i8 > i56, i8, i23, i56)
Load1003(i8, i23, i56) → Cond_Load10031(i23 <= i56 && i8 > i56, i8, i23, i56)

The integer pair graph contains the following rules and edges:
(2): LOAD1003(i8[2], i23[2], i56[2]) → COND_LOAD1003(i23[2] > i56[2] && i8[2] > i56[2], i8[2], i23[2], i56[2])
(6): LOAD1003(i8[6], i23[6], i56[6]) → COND_LOAD10031(i23[6] <= i56[6] && i8[6] > i56[6], i8[6], i23[6], i56[6])

(0) -> (1), if ((i8[0]* i8[1])∧(i56[0]* i56[1])∧(i23[0] > i56[0]* TRUE)∧(i23[0]* i23[1]))

(1) -> (0), if ((i8[1]* i8[0])∧(i23[1] + -1* i23[0])∧(i56[1]* i56[0]))

(1) -> (4), if ((i23[1] + -1* i23[4])∧(i56[1]* i56[4])∧(i8[1]* i8[4]))

(2) -> (3), if ((i23[2] > i56[2] && i8[2] > i56[2]* TRUE)∧(i23[2]* i23[3])∧(i56[2]* i56[3])∧(i8[2]* i8[3]))

(3) -> (0), if ((i8[3]* i8[0])∧(i23[3] + -1* i23[0])∧(i56[3]* i56[0]))

(3) -> (4), if ((i23[3] + -1* i23[4])∧(i56[3]* i56[4])∧(i8[3]* i8[4]))

(4) -> (5), if ((i8[4]* i8[5])∧(i23[4] <= i56[4]* TRUE)∧(i56[4]* i56[5])∧(i23[4]* i23[5]))

(5) -> (2), if ((i56[5]* i56[2])∧(i23[5]* i23[2])∧(i8[5] + -1* i8[2]))

(5) -> (6), if ((i23[5]* i23[6])∧(i8[5] + -1* i8[6])∧(i56[5]* i56[6]))

(6) -> (7), if ((i23[6] <= i56[6] && i8[6] > i56[6]* TRUE)∧(i56[6]* i56[7])∧(i23[6]* i23[7])∧(i8[6]* i8[7]))

(7) -> (2), if ((i56[7]* i56[2])∧(i8[7] + -1* i8[2])∧(i23[7]* i23[2]))

(7) -> (6), if ((i8[7] + -1* i8[6])∧(i56[7]* i56[6])∧(i23[7]* i23[6]))

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, Boolean

R is empty.

The integer pair graph contains the following rules and edges:
(2): LOAD1003(i8[2], i23[2], i56[2]) → COND_LOAD1003(i23[2] > i56[2] && i8[2] > i56[2], i8[2], i23[2], i56[2])
(6): LOAD1003(i8[6], i23[6], i56[6]) → COND_LOAD10031(i23[6] <= i56[6] && i8[6] > i56[6], i8[6], i23[6], i56[6])

(0) -> (1), if ((i8[0]* i8[1])∧(i56[0]* i56[1])∧(i23[0] > i56[0]* TRUE)∧(i23[0]* i23[1]))

(1) -> (0), if ((i8[1]* i8[0])∧(i23[1] + -1* i23[0])∧(i56[1]* i56[0]))

(1) -> (4), if ((i23[1] + -1* i23[4])∧(i56[1]* i56[4])∧(i8[1]* i8[4]))

(2) -> (3), if ((i23[2] > i56[2] && i8[2] > i56[2]* TRUE)∧(i23[2]* i23[3])∧(i56[2]* i56[3])∧(i8[2]* i8[3]))

(3) -> (0), if ((i8[3]* i8[0])∧(i23[3] + -1* i23[0])∧(i56[3]* i56[0]))

(3) -> (4), if ((i23[3] + -1* i23[4])∧(i56[3]* i56[4])∧(i8[3]* i8[4]))

(4) -> (5), if ((i8[4]* i8[5])∧(i23[4] <= i56[4]* TRUE)∧(i56[4]* i56[5])∧(i23[4]* i23[5]))

(5) -> (2), if ((i56[5]* i56[2])∧(i23[5]* i23[2])∧(i8[5] + -1* i8[2]))

(5) -> (6), if ((i23[5]* i23[6])∧(i8[5] + -1* i8[6])∧(i56[5]* i56[6]))

(6) -> (7), if ((i23[6] <= i56[6] && i8[6] > i56[6]* TRUE)∧(i56[6]* i56[7])∧(i23[6]* i23[7])∧(i8[6]* i8[7]))

(7) -> (2), if ((i56[7]* i56[2])∧(i8[7] + -1* i8[2])∧(i23[7]* i23[2]))

(7) -> (6), if ((i8[7] + -1* i8[6])∧(i56[7]* i56[6])∧(i23[7]* i23[6]))

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 LOAD1082(i8, i23, i56) → COND_LOAD1082(>(i23, i56), i8, i23, i56) the following chains were created:
• We consider the chain LOAD1082(i8[0], i23[0], i56[0]) → COND_LOAD1082(>(i23[0], i56[0]), i8[0], i23[0], i56[0]), COND_LOAD1082(TRUE, i8[1], i23[1], i56[1]) → LOAD1082(i8[1], +(i23[1], -1), i56[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)    (i23[0] + [-1] + [-1]i56[0] ≥ 0 ⇒ (UIncreasing(COND_LOAD1082(>(i23[0], i56[0]), i8[0], i23[0], i56[0])), ≥)∧[(-1)bni_34 + (-1)Bound*bni_34] + [(-1)bni_34]i56[0] + [bni_34]i8[0] ≥ 0∧[(-1)bso_35] ≥ 0)

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

(4)    (i23[0] + [-1] + [-1]i56[0] ≥ 0 ⇒ (UIncreasing(COND_LOAD1082(>(i23[0], i56[0]), i8[0], i23[0], i56[0])), ≥)∧[(-1)bni_34 + (-1)Bound*bni_34] + [(-1)bni_34]i56[0] + [bni_34]i8[0] ≥ 0∧[(-1)bso_35] ≥ 0)

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

(5)    (i23[0] + [-1] + [-1]i56[0] ≥ 0 ⇒ (UIncreasing(COND_LOAD1082(>(i23[0], i56[0]), i8[0], i23[0], i56[0])), ≥)∧[(-1)bni_34 + (-1)Bound*bni_34] + [(-1)bni_34]i56[0] + [bni_34]i8[0] ≥ 0∧[(-1)bso_35] ≥ 0)

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

(6)    (i23[0] + [-1] + [-1]i56[0] ≥ 0 ⇒ (UIncreasing(COND_LOAD1082(>(i23[0], i56[0]), i8[0], i23[0], i56[0])), ≥)∧[bni_34] = 0∧[(-1)bni_34 + (-1)Bound*bni_34] + [(-1)bni_34]i56[0] ≥ 0∧0 = 0∧[(-1)bso_35] ≥ 0)

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

(7)    (i23[0] ≥ 0 ⇒ (UIncreasing(COND_LOAD1082(>(i23[0], i56[0]), i8[0], i23[0], i56[0])), ≥)∧[bni_34] = 0∧[(-1)bni_34 + (-1)Bound*bni_34] + [(-1)bni_34]i56[0] ≥ 0∧0 = 0∧[(-1)bso_35] ≥ 0)

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

(8)    (i23[0] ≥ 0∧i56[0] ≥ 0 ⇒ (UIncreasing(COND_LOAD1082(>(i23[0], i56[0]), i8[0], i23[0], i56[0])), ≥)∧[bni_34] = 0∧[(-1)bni_34 + (-1)Bound*bni_34] + [(-1)bni_34]i56[0] ≥ 0∧0 = 0∧[(-1)bso_35] ≥ 0)

(9)    (i23[0] ≥ 0∧i56[0] ≥ 0 ⇒ (UIncreasing(COND_LOAD1082(>(i23[0], i56[0]), i8[0], i23[0], i56[0])), ≥)∧[bni_34] = 0∧[(-1)bni_34 + (-1)Bound*bni_34] + [bni_34]i56[0] ≥ 0∧0 = 0∧[(-1)bso_35] ≥ 0)

For Pair COND_LOAD1082(TRUE, i8, i23, i56) → LOAD1082(i8, +(i23, -1), i56) the following chains were created:
• We consider the chain LOAD1082(i8[0], i23[0], i56[0]) → COND_LOAD1082(>(i23[0], i56[0]), i8[0], i23[0], i56[0]), COND_LOAD1082(TRUE, i8[1], i23[1], i56[1]) → LOAD1082(i8[1], +(i23[1], -1), i56[1]), LOAD1082(i8[0], i23[0], i56[0]) → COND_LOAD1082(>(i23[0], i56[0]), i8[0], i23[0], i56[0]) which results in the following constraint:

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

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

(12)    (i23[0] + [-1] + [-1]i56[0] ≥ 0 ⇒ (UIncreasing(LOAD1082(i8[1], +(i23[1], -1), i56[1])), ≥)∧[(-1)bni_36 + (-1)Bound*bni_36] + [(-1)bni_36]i56[0] + [bni_36]i8[0] ≥ 0∧[(-1)bso_37] ≥ 0)

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

(13)    (i23[0] + [-1] + [-1]i56[0] ≥ 0 ⇒ (UIncreasing(LOAD1082(i8[1], +(i23[1], -1), i56[1])), ≥)∧[(-1)bni_36 + (-1)Bound*bni_36] + [(-1)bni_36]i56[0] + [bni_36]i8[0] ≥ 0∧[(-1)bso_37] ≥ 0)

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

(14)    (i23[0] + [-1] + [-1]i56[0] ≥ 0 ⇒ (UIncreasing(LOAD1082(i8[1], +(i23[1], -1), i56[1])), ≥)∧[(-1)bni_36 + (-1)Bound*bni_36] + [(-1)bni_36]i56[0] + [bni_36]i8[0] ≥ 0∧[(-1)bso_37] ≥ 0)

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

(15)    (i23[0] + [-1] + [-1]i56[0] ≥ 0 ⇒ (UIncreasing(LOAD1082(i8[1], +(i23[1], -1), i56[1])), ≥)∧[bni_36] = 0∧[(-1)bni_36 + (-1)Bound*bni_36] + [(-1)bni_36]i56[0] ≥ 0∧0 = 0∧[(-1)bso_37] ≥ 0)

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

(16)    (i23[0] ≥ 0 ⇒ (UIncreasing(LOAD1082(i8[1], +(i23[1], -1), i56[1])), ≥)∧[bni_36] = 0∧[(-1)bni_36 + (-1)Bound*bni_36] + [(-1)bni_36]i56[0] ≥ 0∧0 = 0∧[(-1)bso_37] ≥ 0)

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

(17)    (i23[0] ≥ 0∧i56[0] ≥ 0 ⇒ (UIncreasing(LOAD1082(i8[1], +(i23[1], -1), i56[1])), ≥)∧[bni_36] = 0∧[(-1)bni_36 + (-1)Bound*bni_36] + [(-1)bni_36]i56[0] ≥ 0∧0 = 0∧[(-1)bso_37] ≥ 0)

(18)    (i23[0] ≥ 0∧i56[0] ≥ 0 ⇒ (UIncreasing(LOAD1082(i8[1], +(i23[1], -1), i56[1])), ≥)∧[bni_36] = 0∧[(-1)bni_36 + (-1)Bound*bni_36] + [bni_36]i56[0] ≥ 0∧0 = 0∧[(-1)bso_37] ≥ 0)

• We consider the chain LOAD1082(i8[0], i23[0], i56[0]) → COND_LOAD1082(>(i23[0], i56[0]), i8[0], i23[0], i56[0]), COND_LOAD1082(TRUE, i8[1], i23[1], i56[1]) → LOAD1082(i8[1], +(i23[1], -1), i56[1]), LOAD1082(i8[4], i23[4], i56[4]) → COND_LOAD10821(<=(i23[4], i56[4]), i8[4], i23[4], i56[4]) which results in the following constraint:

We simplified constraint (19) using rules (III), (IV) which results in the following new constraint:

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

(21)    (i23[0] + [-1] + [-1]i56[0] ≥ 0 ⇒ (UIncreasing(LOAD1082(i8[1], +(i23[1], -1), i56[1])), ≥)∧[(-1)bni_36 + (-1)Bound*bni_36] + [(-1)bni_36]i56[0] + [bni_36]i8[0] ≥ 0∧[(-1)bso_37] ≥ 0)

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

(22)    (i23[0] + [-1] + [-1]i56[0] ≥ 0 ⇒ (UIncreasing(LOAD1082(i8[1], +(i23[1], -1), i56[1])), ≥)∧[(-1)bni_36 + (-1)Bound*bni_36] + [(-1)bni_36]i56[0] + [bni_36]i8[0] ≥ 0∧[(-1)bso_37] ≥ 0)

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

(23)    (i23[0] + [-1] + [-1]i56[0] ≥ 0 ⇒ (UIncreasing(LOAD1082(i8[1], +(i23[1], -1), i56[1])), ≥)∧[(-1)bni_36 + (-1)Bound*bni_36] + [(-1)bni_36]i56[0] + [bni_36]i8[0] ≥ 0∧[(-1)bso_37] ≥ 0)

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

(24)    (i23[0] + [-1] + [-1]i56[0] ≥ 0 ⇒ (UIncreasing(LOAD1082(i8[1], +(i23[1], -1), i56[1])), ≥)∧[bni_36] = 0∧[(-1)bni_36 + (-1)Bound*bni_36] + [(-1)bni_36]i56[0] ≥ 0∧0 = 0∧[(-1)bso_37] ≥ 0)

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

(25)    (i23[0] ≥ 0 ⇒ (UIncreasing(LOAD1082(i8[1], +(i23[1], -1), i56[1])), ≥)∧[bni_36] = 0∧[(-1)bni_36 + (-1)Bound*bni_36] + [(-1)bni_36]i56[0] ≥ 0∧0 = 0∧[(-1)bso_37] ≥ 0)

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

(26)    (i23[0] ≥ 0∧i56[0] ≥ 0 ⇒ (UIncreasing(LOAD1082(i8[1], +(i23[1], -1), i56[1])), ≥)∧[bni_36] = 0∧[(-1)bni_36 + (-1)Bound*bni_36] + [(-1)bni_36]i56[0] ≥ 0∧0 = 0∧[(-1)bso_37] ≥ 0)

(27)    (i23[0] ≥ 0∧i56[0] ≥ 0 ⇒ (UIncreasing(LOAD1082(i8[1], +(i23[1], -1), i56[1])), ≥)∧[bni_36] = 0∧[(-1)bni_36 + (-1)Bound*bni_36] + [bni_36]i56[0] ≥ 0∧0 = 0∧[(-1)bso_37] ≥ 0)

For Pair LOAD1003(i8, i23, i56) → COND_LOAD1003(&&(>(i23, i56), >(i8, i56)), i8, i23, i56) the following chains were created:
• We consider the chain LOAD1003(i8[2], i23[2], i56[2]) → COND_LOAD1003(&&(>(i23[2], i56[2]), >(i8[2], i56[2])), i8[2], i23[2], i56[2]), COND_LOAD1003(TRUE, i8[3], i23[3], i56[3]) → LOAD1082(i8[3], +(i23[3], -1), i56[3]) which results in the following constraint:

We simplified constraint (28) using rules (IV), (IDP_BOOLEAN) which results in the following new constraint:

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

(30)    (i23[2] + [-1] + [-1]i56[2] ≥ 0∧i8[2] + [-1] + [-1]i56[2] ≥ 0 ⇒ (UIncreasing(COND_LOAD1003(&&(>(i23[2], i56[2]), >(i8[2], i56[2])), i8[2], i23[2], i56[2])), ≥)∧[(-1)bni_38 + (-1)Bound*bni_38] + [(-1)bni_38]i56[2] + [bni_38]i8[2] ≥ 0∧[(-1)bso_39] ≥ 0)

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

(31)    (i23[2] + [-1] + [-1]i56[2] ≥ 0∧i8[2] + [-1] + [-1]i56[2] ≥ 0 ⇒ (UIncreasing(COND_LOAD1003(&&(>(i23[2], i56[2]), >(i8[2], i56[2])), i8[2], i23[2], i56[2])), ≥)∧[(-1)bni_38 + (-1)Bound*bni_38] + [(-1)bni_38]i56[2] + [bni_38]i8[2] ≥ 0∧[(-1)bso_39] ≥ 0)

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

(32)    (i23[2] + [-1] + [-1]i56[2] ≥ 0∧i8[2] + [-1] + [-1]i56[2] ≥ 0 ⇒ (UIncreasing(COND_LOAD1003(&&(>(i23[2], i56[2]), >(i8[2], i56[2])), i8[2], i23[2], i56[2])), ≥)∧[(-1)bni_38 + (-1)Bound*bni_38] + [(-1)bni_38]i56[2] + [bni_38]i8[2] ≥ 0∧[(-1)bso_39] ≥ 0)

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

(33)    (i23[2] ≥ 0∧i8[2] + [-1] + [-1]i56[2] ≥ 0 ⇒ (UIncreasing(COND_LOAD1003(&&(>(i23[2], i56[2]), >(i8[2], i56[2])), i8[2], i23[2], i56[2])), ≥)∧[(-1)bni_38 + (-1)Bound*bni_38] + [(-1)bni_38]i56[2] + [bni_38]i8[2] ≥ 0∧[(-1)bso_39] ≥ 0)

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

(34)    (i23[2] ≥ 0∧i56[2] ≥ 0 ⇒ (UIncreasing(COND_LOAD1003(&&(>(i23[2], i56[2]), >(i8[2], i56[2])), i8[2], i23[2], i56[2])), ≥)∧[(-1)Bound*bni_38] + [bni_38]i56[2] ≥ 0∧[(-1)bso_39] ≥ 0)

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

(35)    (i23[2] ≥ 0∧i56[2] ≥ 0∧i8[2] ≥ 0 ⇒ (UIncreasing(COND_LOAD1003(&&(>(i23[2], i56[2]), >(i8[2], i56[2])), i8[2], i23[2], i56[2])), ≥)∧[(-1)Bound*bni_38] + [bni_38]i56[2] ≥ 0∧[(-1)bso_39] ≥ 0)

(36)    (i23[2] ≥ 0∧i56[2] ≥ 0∧i8[2] ≥ 0 ⇒ (UIncreasing(COND_LOAD1003(&&(>(i23[2], i56[2]), >(i8[2], i56[2])), i8[2], i23[2], i56[2])), ≥)∧[(-1)Bound*bni_38] + [bni_38]i56[2] ≥ 0∧[(-1)bso_39] ≥ 0)

For Pair COND_LOAD1003(TRUE, i8, i23, i56) → LOAD1082(i8, +(i23, -1), i56) the following chains were created:
• We consider the chain LOAD1003(i8[2], i23[2], i56[2]) → COND_LOAD1003(&&(>(i23[2], i56[2]), >(i8[2], i56[2])), i8[2], i23[2], i56[2]), COND_LOAD1003(TRUE, i8[3], i23[3], i56[3]) → LOAD1082(i8[3], +(i23[3], -1), i56[3]), LOAD1082(i8[0], i23[0], i56[0]) → COND_LOAD1082(>(i23[0], i56[0]), i8[0], i23[0], i56[0]) which results in the following constraint:

We simplified constraint (37) using rules (III), (IV), (IDP_BOOLEAN) which results in the following new constraint:

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

(39)    (i23[2] + [-1] + [-1]i56[2] ≥ 0∧i8[2] + [-1] + [-1]i56[2] ≥ 0 ⇒ (UIncreasing(LOAD1082(i8[3], +(i23[3], -1), i56[3])), ≥)∧[(-1)bni_40 + (-1)Bound*bni_40] + [(-1)bni_40]i56[2] + [bni_40]i8[2] ≥ 0∧[(-1)bso_41] ≥ 0)

We simplified constraint (39) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:

(40)    (i23[2] + [-1] + [-1]i56[2] ≥ 0∧i8[2] + [-1] + [-1]i56[2] ≥ 0 ⇒ (UIncreasing(LOAD1082(i8[3], +(i23[3], -1), i56[3])), ≥)∧[(-1)bni_40 + (-1)Bound*bni_40] + [(-1)bni_40]i56[2] + [bni_40]i8[2] ≥ 0∧[(-1)bso_41] ≥ 0)

We simplified constraint (40) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:

(41)    (i23[2] + [-1] + [-1]i56[2] ≥ 0∧i8[2] + [-1] + [-1]i56[2] ≥ 0 ⇒ (UIncreasing(LOAD1082(i8[3], +(i23[3], -1), i56[3])), ≥)∧[(-1)bni_40 + (-1)Bound*bni_40] + [(-1)bni_40]i56[2] + [bni_40]i8[2] ≥ 0∧[(-1)bso_41] ≥ 0)

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

(42)    (i23[2] ≥ 0∧i8[2] + [-1] + [-1]i56[2] ≥ 0 ⇒ (UIncreasing(LOAD1082(i8[3], +(i23[3], -1), i56[3])), ≥)∧[(-1)bni_40 + (-1)Bound*bni_40] + [(-1)bni_40]i56[2] + [bni_40]i8[2] ≥ 0∧[(-1)bso_41] ≥ 0)

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

(43)    (i23[2] ≥ 0∧i56[2] ≥ 0 ⇒ (UIncreasing(LOAD1082(i8[3], +(i23[3], -1), i56[3])), ≥)∧[(-1)Bound*bni_40] + [bni_40]i56[2] ≥ 0∧[(-1)bso_41] ≥ 0)

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

(44)    (i23[2] ≥ 0∧i56[2] ≥ 0∧i8[2] ≥ 0 ⇒ (UIncreasing(LOAD1082(i8[3], +(i23[3], -1), i56[3])), ≥)∧[(-1)Bound*bni_40] + [bni_40]i56[2] ≥ 0∧[(-1)bso_41] ≥ 0)

(45)    (i23[2] ≥ 0∧i56[2] ≥ 0∧i8[2] ≥ 0 ⇒ (UIncreasing(LOAD1082(i8[3], +(i23[3], -1), i56[3])), ≥)∧[(-1)Bound*bni_40] + [bni_40]i56[2] ≥ 0∧[(-1)bso_41] ≥ 0)

• We consider the chain LOAD1003(i8[2], i23[2], i56[2]) → COND_LOAD1003(&&(>(i23[2], i56[2]), >(i8[2], i56[2])), i8[2], i23[2], i56[2]), COND_LOAD1003(TRUE, i8[3], i23[3], i56[3]) → LOAD1082(i8[3], +(i23[3], -1), i56[3]), LOAD1082(i8[4], i23[4], i56[4]) → COND_LOAD10821(<=(i23[4], i56[4]), i8[4], i23[4], i56[4]) which results in the following constraint:

We simplified constraint (46) using rules (III), (IV), (IDP_BOOLEAN) which results in the following new constraint:

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

(48)    (i23[2] + [-1] + [-1]i56[2] ≥ 0∧i8[2] + [-1] + [-1]i56[2] ≥ 0 ⇒ (UIncreasing(LOAD1082(i8[3], +(i23[3], -1), i56[3])), ≥)∧[(-1)bni_40 + (-1)Bound*bni_40] + [(-1)bni_40]i56[2] + [bni_40]i8[2] ≥ 0∧[(-1)bso_41] ≥ 0)

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

(49)    (i23[2] + [-1] + [-1]i56[2] ≥ 0∧i8[2] + [-1] + [-1]i56[2] ≥ 0 ⇒ (UIncreasing(LOAD1082(i8[3], +(i23[3], -1), i56[3])), ≥)∧[(-1)bni_40 + (-1)Bound*bni_40] + [(-1)bni_40]i56[2] + [bni_40]i8[2] ≥ 0∧[(-1)bso_41] ≥ 0)

We simplified constraint (49) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:

(50)    (i23[2] + [-1] + [-1]i56[2] ≥ 0∧i8[2] + [-1] + [-1]i56[2] ≥ 0 ⇒ (UIncreasing(LOAD1082(i8[3], +(i23[3], -1), i56[3])), ≥)∧[(-1)bni_40 + (-1)Bound*bni_40] + [(-1)bni_40]i56[2] + [bni_40]i8[2] ≥ 0∧[(-1)bso_41] ≥ 0)

We simplified constraint (50) using rule (IDP_SMT_SPLIT) which results in the following new constraint:

(51)    (i23[2] ≥ 0∧i8[2] + [-1] + [-1]i56[2] ≥ 0 ⇒ (UIncreasing(LOAD1082(i8[3], +(i23[3], -1), i56[3])), ≥)∧[(-1)bni_40 + (-1)Bound*bni_40] + [(-1)bni_40]i56[2] + [bni_40]i8[2] ≥ 0∧[(-1)bso_41] ≥ 0)

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

(52)    (i23[2] ≥ 0∧i56[2] ≥ 0 ⇒ (UIncreasing(LOAD1082(i8[3], +(i23[3], -1), i56[3])), ≥)∧[(-1)Bound*bni_40] + [bni_40]i56[2] ≥ 0∧[(-1)bso_41] ≥ 0)

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

(53)    (i23[2] ≥ 0∧i56[2] ≥ 0∧i8[2] ≥ 0 ⇒ (UIncreasing(LOAD1082(i8[3], +(i23[3], -1), i56[3])), ≥)∧[(-1)Bound*bni_40] + [bni_40]i56[2] ≥ 0∧[(-1)bso_41] ≥ 0)

(54)    (i23[2] ≥ 0∧i56[2] ≥ 0∧i8[2] ≥ 0 ⇒ (UIncreasing(LOAD1082(i8[3], +(i23[3], -1), i56[3])), ≥)∧[(-1)Bound*bni_40] + [bni_40]i56[2] ≥ 0∧[(-1)bso_41] ≥ 0)

For Pair LOAD1082(i8, i23, i56) → COND_LOAD10821(<=(i23, i56), i8, i23, i56) the following chains were created:
• We consider the chain LOAD1082(i8[4], i23[4], i56[4]) → COND_LOAD10821(<=(i23[4], i56[4]), i8[4], i23[4], i56[4]), COND_LOAD10821(TRUE, i8[5], i23[5], i56[5]) → LOAD1003(+(i8[5], -1), i23[5], i56[5]) which results in the following constraint:

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

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

(57)    (i56[4] + [-1]i23[4] ≥ 0 ⇒ (UIncreasing(COND_LOAD10821(<=(i23[4], i56[4]), i8[4], i23[4], i56[4])), ≥)∧[(-1)bni_42 + (-1)Bound*bni_42] + [(-1)bni_42]i56[4] + [bni_42]i8[4] ≥ 0∧[(-1)bso_43] ≥ 0)

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

(58)    (i56[4] + [-1]i23[4] ≥ 0 ⇒ (UIncreasing(COND_LOAD10821(<=(i23[4], i56[4]), i8[4], i23[4], i56[4])), ≥)∧[(-1)bni_42 + (-1)Bound*bni_42] + [(-1)bni_42]i56[4] + [bni_42]i8[4] ≥ 0∧[(-1)bso_43] ≥ 0)

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

(59)    (i56[4] + [-1]i23[4] ≥ 0 ⇒ (UIncreasing(COND_LOAD10821(<=(i23[4], i56[4]), i8[4], i23[4], i56[4])), ≥)∧[(-1)bni_42 + (-1)Bound*bni_42] + [(-1)bni_42]i56[4] + [bni_42]i8[4] ≥ 0∧[(-1)bso_43] ≥ 0)

We simplified constraint (59) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint:

(60)    (i56[4] + [-1]i23[4] ≥ 0 ⇒ (UIncreasing(COND_LOAD10821(<=(i23[4], i56[4]), i8[4], i23[4], i56[4])), ≥)∧[bni_42] = 0∧[(-1)bni_42 + (-1)Bound*bni_42] + [(-1)bni_42]i56[4] ≥ 0∧0 = 0∧[(-1)bso_43] ≥ 0)

We simplified constraint (60) using rule (IDP_SMT_SPLIT) which results in the following new constraint:

(61)    (i56[4] ≥ 0 ⇒ (UIncreasing(COND_LOAD10821(<=(i23[4], i56[4]), i8[4], i23[4], i56[4])), ≥)∧[bni_42] = 0∧[(-1)bni_42 + (-1)Bound*bni_42] + [(-1)bni_42]i23[4] + [(-1)bni_42]i56[4] ≥ 0∧0 = 0∧[(-1)bso_43] ≥ 0)

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

(62)    (i56[4] ≥ 0∧i23[4] ≥ 0 ⇒ (UIncreasing(COND_LOAD10821(<=(i23[4], i56[4]), i8[4], i23[4], i56[4])), ≥)∧[bni_42] = 0∧[(-1)bni_42 + (-1)Bound*bni_42] + [(-1)bni_42]i23[4] + [(-1)bni_42]i56[4] ≥ 0∧0 = 0∧[(-1)bso_43] ≥ 0)

(63)    (i56[4] ≥ 0∧i23[4] ≥ 0 ⇒ (UIncreasing(COND_LOAD10821(<=(i23[4], i56[4]), i8[4], i23[4], i56[4])), ≥)∧[bni_42] = 0∧[(-1)bni_42 + (-1)Bound*bni_42] + [bni_42]i23[4] + [(-1)bni_42]i56[4] ≥ 0∧0 = 0∧[(-1)bso_43] ≥ 0)

For Pair COND_LOAD10821(TRUE, i8, i23, i56) → LOAD1003(+(i8, -1), i23, i56) the following chains were created:
• We consider the chain LOAD1082(i8[4], i23[4], i56[4]) → COND_LOAD10821(<=(i23[4], i56[4]), i8[4], i23[4], i56[4]), COND_LOAD10821(TRUE, i8[5], i23[5], i56[5]) → LOAD1003(+(i8[5], -1), i23[5], i56[5]), LOAD1003(i8[2], i23[2], i56[2]) → COND_LOAD1003(&&(>(i23[2], i56[2]), >(i8[2], i56[2])), i8[2], i23[2], i56[2]) which results in the following constraint:

We simplified constraint (64) using rules (III), (IV) which results in the following new constraint:

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

(66)    (i56[4] + [-1]i23[4] ≥ 0 ⇒ (UIncreasing(LOAD1003(+(i8[5], -1), i23[5], i56[5])), ≥)∧[(-1)bni_44 + (-1)Bound*bni_44] + [(-1)bni_44]i56[4] + [bni_44]i8[4] ≥ 0∧[1 + (-1)bso_45] ≥ 0)

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

(67)    (i56[4] + [-1]i23[4] ≥ 0 ⇒ (UIncreasing(LOAD1003(+(i8[5], -1), i23[5], i56[5])), ≥)∧[(-1)bni_44 + (-1)Bound*bni_44] + [(-1)bni_44]i56[4] + [bni_44]i8[4] ≥ 0∧[1 + (-1)bso_45] ≥ 0)

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

(68)    (i56[4] + [-1]i23[4] ≥ 0 ⇒ (UIncreasing(LOAD1003(+(i8[5], -1), i23[5], i56[5])), ≥)∧[(-1)bni_44 + (-1)Bound*bni_44] + [(-1)bni_44]i56[4] + [bni_44]i8[4] ≥ 0∧[1 + (-1)bso_45] ≥ 0)

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

(69)    (i56[4] + [-1]i23[4] ≥ 0 ⇒ (UIncreasing(LOAD1003(+(i8[5], -1), i23[5], i56[5])), ≥)∧[bni_44] = 0∧[(-1)bni_44 + (-1)Bound*bni_44] + [(-1)bni_44]i56[4] ≥ 0∧0 = 0∧[1 + (-1)bso_45] ≥ 0)

We simplified constraint (69) using rule (IDP_SMT_SPLIT) which results in the following new constraint:

(70)    (i56[4] ≥ 0 ⇒ (UIncreasing(LOAD1003(+(i8[5], -1), i23[5], i56[5])), ≥)∧[bni_44] = 0∧[(-1)bni_44 + (-1)Bound*bni_44] + [(-1)bni_44]i23[4] + [(-1)bni_44]i56[4] ≥ 0∧0 = 0∧[1 + (-1)bso_45] ≥ 0)

We simplified constraint (70) using rule (IDP_SMT_SPLIT) which results in the following new constraints:

(71)    (i56[4] ≥ 0∧i23[4] ≥ 0 ⇒ (UIncreasing(LOAD1003(+(i8[5], -1), i23[5], i56[5])), ≥)∧[bni_44] = 0∧[(-1)bni_44 + (-1)Bound*bni_44] + [bni_44]i23[4] + [(-1)bni_44]i56[4] ≥ 0∧0 = 0∧[1 + (-1)bso_45] ≥ 0)

(72)    (i56[4] ≥ 0∧i23[4] ≥ 0 ⇒ (UIncreasing(LOAD1003(+(i8[5], -1), i23[5], i56[5])), ≥)∧[bni_44] = 0∧[(-1)bni_44 + (-1)Bound*bni_44] + [(-1)bni_44]i23[4] + [(-1)bni_44]i56[4] ≥ 0∧0 = 0∧[1 + (-1)bso_45] ≥ 0)

• We consider the chain LOAD1082(i8[4], i23[4], i56[4]) → COND_LOAD10821(<=(i23[4], i56[4]), i8[4], i23[4], i56[4]), COND_LOAD10821(TRUE, i8[5], i23[5], i56[5]) → LOAD1003(+(i8[5], -1), i23[5], i56[5]), LOAD1003(i8[6], i23[6], i56[6]) → COND_LOAD10031(&&(<=(i23[6], i56[6]), >(i8[6], i56[6])), i8[6], i23[6], i56[6]) which results in the following constraint:

We simplified constraint (73) using rules (III), (IV) which results in the following new constraint:

We simplified constraint (74) using rule (POLY_CONSTRAINTS) which results in the following new constraint:

(75)    (i56[4] + [-1]i23[4] ≥ 0 ⇒ (UIncreasing(LOAD1003(+(i8[5], -1), i23[5], i56[5])), ≥)∧[(-1)bni_44 + (-1)Bound*bni_44] + [(-1)bni_44]i56[4] + [bni_44]i8[4] ≥ 0∧[1 + (-1)bso_45] ≥ 0)

We simplified constraint (75) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:

(76)    (i56[4] + [-1]i23[4] ≥ 0 ⇒ (UIncreasing(LOAD1003(+(i8[5], -1), i23[5], i56[5])), ≥)∧[(-1)bni_44 + (-1)Bound*bni_44] + [(-1)bni_44]i56[4] + [bni_44]i8[4] ≥ 0∧[1 + (-1)bso_45] ≥ 0)

We simplified constraint (76) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:

(77)    (i56[4] + [-1]i23[4] ≥ 0 ⇒ (UIncreasing(LOAD1003(+(i8[5], -1), i23[5], i56[5])), ≥)∧[(-1)bni_44 + (-1)Bound*bni_44] + [(-1)bni_44]i56[4] + [bni_44]i8[4] ≥ 0∧[1 + (-1)bso_45] ≥ 0)

We simplified constraint (77) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint:

(78)    (i56[4] + [-1]i23[4] ≥ 0 ⇒ (UIncreasing(LOAD1003(+(i8[5], -1), i23[5], i56[5])), ≥)∧[bni_44] = 0∧[(-1)bni_44 + (-1)Bound*bni_44] + [(-1)bni_44]i56[4] ≥ 0∧0 = 0∧[1 + (-1)bso_45] ≥ 0)

We simplified constraint (78) using rule (IDP_SMT_SPLIT) which results in the following new constraint:

(79)    (i56[4] ≥ 0 ⇒ (UIncreasing(LOAD1003(+(i8[5], -1), i23[5], i56[5])), ≥)∧[bni_44] = 0∧[(-1)bni_44 + (-1)Bound*bni_44] + [(-1)bni_44]i23[4] + [(-1)bni_44]i56[4] ≥ 0∧0 = 0∧[1 + (-1)bso_45] ≥ 0)

We simplified constraint (79) using rule (IDP_SMT_SPLIT) which results in the following new constraints:

(80)    (i56[4] ≥ 0∧i23[4] ≥ 0 ⇒ (UIncreasing(LOAD1003(+(i8[5], -1), i23[5], i56[5])), ≥)∧[bni_44] = 0∧[(-1)bni_44 + (-1)Bound*bni_44] + [bni_44]i23[4] + [(-1)bni_44]i56[4] ≥ 0∧0 = 0∧[1 + (-1)bso_45] ≥ 0)

(81)    (i56[4] ≥ 0∧i23[4] ≥ 0 ⇒ (UIncreasing(LOAD1003(+(i8[5], -1), i23[5], i56[5])), ≥)∧[bni_44] = 0∧[(-1)bni_44 + (-1)Bound*bni_44] + [(-1)bni_44]i23[4] + [(-1)bni_44]i56[4] ≥ 0∧0 = 0∧[1 + (-1)bso_45] ≥ 0)

For Pair LOAD1003(i8, i23, i56) → COND_LOAD10031(&&(<=(i23, i56), >(i8, i56)), i8, i23, i56) the following chains were created:
• We consider the chain LOAD1003(i8[6], i23[6], i56[6]) → COND_LOAD10031(&&(<=(i23[6], i56[6]), >(i8[6], i56[6])), i8[6], i23[6], i56[6]), COND_LOAD10031(TRUE, i8[7], i23[7], i56[7]) → LOAD1003(+(i8[7], -1), i23[7], i56[7]) which results in the following constraint:

We simplified constraint (82) using rules (IV), (IDP_BOOLEAN) which results in the following new constraint:

We simplified constraint (83) using rule (POLY_CONSTRAINTS) which results in the following new constraint:

(84)    (i56[6] + [-1]i23[6] ≥ 0∧i8[6] + [-1] + [-1]i56[6] ≥ 0 ⇒ (UIncreasing(COND_LOAD10031(&&(<=(i23[6], i56[6]), >(i8[6], i56[6])), i8[6], i23[6], i56[6])), ≥)∧[(-1)bni_46 + (-1)Bound*bni_46] + [(-1)bni_46]i56[6] + [bni_46]i8[6] ≥ 0∧[(-1)bso_47] ≥ 0)

We simplified constraint (84) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:

(85)    (i56[6] + [-1]i23[6] ≥ 0∧i8[6] + [-1] + [-1]i56[6] ≥ 0 ⇒ (UIncreasing(COND_LOAD10031(&&(<=(i23[6], i56[6]), >(i8[6], i56[6])), i8[6], i23[6], i56[6])), ≥)∧[(-1)bni_46 + (-1)Bound*bni_46] + [(-1)bni_46]i56[6] + [bni_46]i8[6] ≥ 0∧[(-1)bso_47] ≥ 0)

We simplified constraint (85) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:

(86)    (i56[6] + [-1]i23[6] ≥ 0∧i8[6] + [-1] + [-1]i56[6] ≥ 0 ⇒ (UIncreasing(COND_LOAD10031(&&(<=(i23[6], i56[6]), >(i8[6], i56[6])), i8[6], i23[6], i56[6])), ≥)∧[(-1)bni_46 + (-1)Bound*bni_46] + [(-1)bni_46]i56[6] + [bni_46]i8[6] ≥ 0∧[(-1)bso_47] ≥ 0)

We simplified constraint (86) using rule (IDP_SMT_SPLIT) which results in the following new constraint:

(87)    (i56[6] ≥ 0∧i8[6] + [-1] + [-1]i23[6] + [-1]i56[6] ≥ 0 ⇒ (UIncreasing(COND_LOAD10031(&&(<=(i23[6], i56[6]), >(i8[6], i56[6])), i8[6], i23[6], i56[6])), ≥)∧[(-1)bni_46 + (-1)Bound*bni_46] + [(-1)bni_46]i23[6] + [(-1)bni_46]i56[6] + [bni_46]i8[6] ≥ 0∧[(-1)bso_47] ≥ 0)

We simplified constraint (87) using rule (IDP_SMT_SPLIT) which results in the following new constraint:

(88)    (i56[6] ≥ 0∧i23[6] ≥ 0 ⇒ (UIncreasing(COND_LOAD10031(&&(<=(i23[6], i56[6]), >(i8[6], i56[6])), i8[6], i23[6], i56[6])), ≥)∧[(-1)Bound*bni_46] + [bni_46]i23[6] ≥ 0∧[(-1)bso_47] ≥ 0)

We simplified constraint (88) using rule (IDP_SMT_SPLIT) which results in the following new constraints:

(89)    (i56[6] ≥ 0∧i23[6] ≥ 0∧i8[6] ≥ 0 ⇒ (UIncreasing(COND_LOAD10031(&&(<=(i23[6], i56[6]), >(i8[6], i56[6])), i8[6], i23[6], i56[6])), ≥)∧[(-1)Bound*bni_46] + [bni_46]i23[6] ≥ 0∧[(-1)bso_47] ≥ 0)

(90)    (i56[6] ≥ 0∧i23[6] ≥ 0∧i8[6] ≥ 0 ⇒ (UIncreasing(COND_LOAD10031(&&(<=(i23[6], i56[6]), >(i8[6], i56[6])), i8[6], i23[6], i56[6])), ≥)∧[(-1)Bound*bni_46] + [bni_46]i23[6] ≥ 0∧[(-1)bso_47] ≥ 0)

For Pair COND_LOAD10031(TRUE, i8, i23, i56) → LOAD1003(+(i8, -1), i23, i56) the following chains were created:
• We consider the chain LOAD1003(i8[6], i23[6], i56[6]) → COND_LOAD10031(&&(<=(i23[6], i56[6]), >(i8[6], i56[6])), i8[6], i23[6], i56[6]), COND_LOAD10031(TRUE, i8[7], i23[7], i56[7]) → LOAD1003(+(i8[7], -1), i23[7], i56[7]), LOAD1003(i8[2], i23[2], i56[2]) → COND_LOAD1003(&&(>(i23[2], i56[2]), >(i8[2], i56[2])), i8[2], i23[2], i56[2]) which results in the following constraint:

We simplified constraint (91) using rules (III), (IV), (IDP_BOOLEAN) which results in the following new constraint:

We simplified constraint (92) using rule (POLY_CONSTRAINTS) which results in the following new constraint:

(93)    (i56[6] + [-1]i23[6] ≥ 0∧i8[6] + [-1] + [-1]i56[6] ≥ 0 ⇒ (UIncreasing(LOAD1003(+(i8[7], -1), i23[7], i56[7])), ≥)∧[(-1)bni_48 + (-1)Bound*bni_48] + [(-1)bni_48]i56[6] + [bni_48]i8[6] ≥ 0∧[1 + (-1)bso_49] ≥ 0)

We simplified constraint (93) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:

(94)    (i56[6] + [-1]i23[6] ≥ 0∧i8[6] + [-1] + [-1]i56[6] ≥ 0 ⇒ (UIncreasing(LOAD1003(+(i8[7], -1), i23[7], i56[7])), ≥)∧[(-1)bni_48 + (-1)Bound*bni_48] + [(-1)bni_48]i56[6] + [bni_48]i8[6] ≥ 0∧[1 + (-1)bso_49] ≥ 0)

We simplified constraint (94) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:

(95)    (i56[6] + [-1]i23[6] ≥ 0∧i8[6] + [-1] + [-1]i56[6] ≥ 0 ⇒ (UIncreasing(LOAD1003(+(i8[7], -1), i23[7], i56[7])), ≥)∧[(-1)bni_48 + (-1)Bound*bni_48] + [(-1)bni_48]i56[6] + [bni_48]i8[6] ≥ 0∧[1 + (-1)bso_49] ≥ 0)

We simplified constraint (95) using rule (IDP_SMT_SPLIT) which results in the following new constraint:

(96)    (i56[6] ≥ 0∧i8[6] + [-1] + [-1]i23[6] + [-1]i56[6] ≥ 0 ⇒ (UIncreasing(LOAD1003(+(i8[7], -1), i23[7], i56[7])), ≥)∧[(-1)bni_48 + (-1)Bound*bni_48] + [(-1)bni_48]i23[6] + [(-1)bni_48]i56[6] + [bni_48]i8[6] ≥ 0∧[1 + (-1)bso_49] ≥ 0)

We simplified constraint (96) using rule (IDP_SMT_SPLIT) which results in the following new constraint:

(97)    (i56[6] ≥ 0∧i23[6] ≥ 0 ⇒ (UIncreasing(LOAD1003(+(i8[7], -1), i23[7], i56[7])), ≥)∧[(-1)Bound*bni_48] + [bni_48]i23[6] ≥ 0∧[1 + (-1)bso_49] ≥ 0)

We simplified constraint (97) using rule (IDP_SMT_SPLIT) which results in the following new constraints:

(98)    (i56[6] ≥ 0∧i23[6] ≥ 0∧i8[6] ≥ 0 ⇒ (UIncreasing(LOAD1003(+(i8[7], -1), i23[7], i56[7])), ≥)∧[(-1)Bound*bni_48] + [bni_48]i23[6] ≥ 0∧[1 + (-1)bso_49] ≥ 0)

(99)    (i56[6] ≥ 0∧i23[6] ≥ 0∧i8[6] ≥ 0 ⇒ (UIncreasing(LOAD1003(+(i8[7], -1), i23[7], i56[7])), ≥)∧[(-1)Bound*bni_48] + [bni_48]i23[6] ≥ 0∧[1 + (-1)bso_49] ≥ 0)

• We consider the chain LOAD1003(i8[6], i23[6], i56[6]) → COND_LOAD10031(&&(<=(i23[6], i56[6]), >(i8[6], i56[6])), i8[6], i23[6], i56[6]), COND_LOAD10031(TRUE, i8[7], i23[7], i56[7]) → LOAD1003(+(i8[7], -1), i23[7], i56[7]), LOAD1003(i8[6], i23[6], i56[6]) → COND_LOAD10031(&&(<=(i23[6], i56[6]), >(i8[6], i56[6])), i8[6], i23[6], i56[6]) which results in the following constraint:

We simplified constraint (100) using rules (III), (IV), (IDP_BOOLEAN) which results in the following new constraint:

We simplified constraint (101) using rule (POLY_CONSTRAINTS) which results in the following new constraint:

(102)    (i56[6] + [-1]i23[6] ≥ 0∧i8[6] + [-1] + [-1]i56[6] ≥ 0 ⇒ (UIncreasing(LOAD1003(+(i8[7], -1), i23[7], i56[7])), ≥)∧[(-1)bni_48 + (-1)Bound*bni_48] + [(-1)bni_48]i56[6] + [bni_48]i8[6] ≥ 0∧[1 + (-1)bso_49] ≥ 0)

We simplified constraint (102) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:

(103)    (i56[6] + [-1]i23[6] ≥ 0∧i8[6] + [-1] + [-1]i56[6] ≥ 0 ⇒ (UIncreasing(LOAD1003(+(i8[7], -1), i23[7], i56[7])), ≥)∧[(-1)bni_48 + (-1)Bound*bni_48] + [(-1)bni_48]i56[6] + [bni_48]i8[6] ≥ 0∧[1 + (-1)bso_49] ≥ 0)

We simplified constraint (103) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:

(104)    (i56[6] + [-1]i23[6] ≥ 0∧i8[6] + [-1] + [-1]i56[6] ≥ 0 ⇒ (UIncreasing(LOAD1003(+(i8[7], -1), i23[7], i56[7])), ≥)∧[(-1)bni_48 + (-1)Bound*bni_48] + [(-1)bni_48]i56[6] + [bni_48]i8[6] ≥ 0∧[1 + (-1)bso_49] ≥ 0)

We simplified constraint (104) using rule (IDP_SMT_SPLIT) which results in the following new constraint:

(105)    (i56[6] ≥ 0∧i8[6] + [-1] + [-1]i23[6] + [-1]i56[6] ≥ 0 ⇒ (UIncreasing(LOAD1003(+(i8[7], -1), i23[7], i56[7])), ≥)∧[(-1)bni_48 + (-1)Bound*bni_48] + [(-1)bni_48]i23[6] + [(-1)bni_48]i56[6] + [bni_48]i8[6] ≥ 0∧[1 + (-1)bso_49] ≥ 0)

We simplified constraint (105) using rule (IDP_SMT_SPLIT) which results in the following new constraint:

(106)    (i56[6] ≥ 0∧i23[6] ≥ 0 ⇒ (UIncreasing(LOAD1003(+(i8[7], -1), i23[7], i56[7])), ≥)∧[(-1)Bound*bni_48] + [bni_48]i23[6] ≥ 0∧[1 + (-1)bso_49] ≥ 0)

We simplified constraint (106) using rule (IDP_SMT_SPLIT) which results in the following new constraints:

(107)    (i56[6] ≥ 0∧i23[6] ≥ 0∧i8[6] ≥ 0 ⇒ (UIncreasing(LOAD1003(+(i8[7], -1), i23[7], i56[7])), ≥)∧[(-1)Bound*bni_48] + [bni_48]i23[6] ≥ 0∧[1 + (-1)bso_49] ≥ 0)

(108)    (i56[6] ≥ 0∧i23[6] ≥ 0∧i8[6] ≥ 0 ⇒ (UIncreasing(LOAD1003(+(i8[7], -1), i23[7], i56[7])), ≥)∧[(-1)Bound*bni_48] + [bni_48]i23[6] ≥ 0∧[1 + (-1)bso_49] ≥ 0)

To summarize, we get the following constraints P for the following pairs.
• (i23[0] ≥ 0∧i56[0] ≥ 0 ⇒ (UIncreasing(COND_LOAD1082(>(i23[0], i56[0]), i8[0], i23[0], i56[0])), ≥)∧[bni_34] = 0∧[(-1)bni_34 + (-1)Bound*bni_34] + [(-1)bni_34]i56[0] ≥ 0∧0 = 0∧[(-1)bso_35] ≥ 0)
• (i23[0] ≥ 0∧i56[0] ≥ 0 ⇒ (UIncreasing(COND_LOAD1082(>(i23[0], i56[0]), i8[0], i23[0], i56[0])), ≥)∧[bni_34] = 0∧[(-1)bni_34 + (-1)Bound*bni_34] + [bni_34]i56[0] ≥ 0∧0 = 0∧[(-1)bso_35] ≥ 0)

• (i23[0] ≥ 0∧i56[0] ≥ 0 ⇒ (UIncreasing(LOAD1082(i8[1], +(i23[1], -1), i56[1])), ≥)∧[bni_36] = 0∧[(-1)bni_36 + (-1)Bound*bni_36] + [(-1)bni_36]i56[0] ≥ 0∧0 = 0∧[(-1)bso_37] ≥ 0)
• (i23[0] ≥ 0∧i56[0] ≥ 0 ⇒ (UIncreasing(LOAD1082(i8[1], +(i23[1], -1), i56[1])), ≥)∧[bni_36] = 0∧[(-1)bni_36 + (-1)Bound*bni_36] + [bni_36]i56[0] ≥ 0∧0 = 0∧[(-1)bso_37] ≥ 0)
• (i23[0] ≥ 0∧i56[0] ≥ 0 ⇒ (UIncreasing(LOAD1082(i8[1], +(i23[1], -1), i56[1])), ≥)∧[bni_36] = 0∧[(-1)bni_36 + (-1)Bound*bni_36] + [(-1)bni_36]i56[0] ≥ 0∧0 = 0∧[(-1)bso_37] ≥ 0)
• (i23[0] ≥ 0∧i56[0] ≥ 0 ⇒ (UIncreasing(LOAD1082(i8[1], +(i23[1], -1), i56[1])), ≥)∧[bni_36] = 0∧[(-1)bni_36 + (-1)Bound*bni_36] + [bni_36]i56[0] ≥ 0∧0 = 0∧[(-1)bso_37] ≥ 0)

• LOAD1003(i8, i23, i56) → COND_LOAD1003(&&(>(i23, i56), >(i8, i56)), i8, i23, i56)
• (i23[2] ≥ 0∧i56[2] ≥ 0∧i8[2] ≥ 0 ⇒ (UIncreasing(COND_LOAD1003(&&(>(i23[2], i56[2]), >(i8[2], i56[2])), i8[2], i23[2], i56[2])), ≥)∧[(-1)Bound*bni_38] + [bni_38]i56[2] ≥ 0∧[(-1)bso_39] ≥ 0)
• (i23[2] ≥ 0∧i56[2] ≥ 0∧i8[2] ≥ 0 ⇒ (UIncreasing(COND_LOAD1003(&&(>(i23[2], i56[2]), >(i8[2], i56[2])), i8[2], i23[2], i56[2])), ≥)∧[(-1)Bound*bni_38] + [bni_38]i56[2] ≥ 0∧[(-1)bso_39] ≥ 0)

• (i23[2] ≥ 0∧i56[2] ≥ 0∧i8[2] ≥ 0 ⇒ (UIncreasing(LOAD1082(i8[3], +(i23[3], -1), i56[3])), ≥)∧[(-1)Bound*bni_40] + [bni_40]i56[2] ≥ 0∧[(-1)bso_41] ≥ 0)
• (i23[2] ≥ 0∧i56[2] ≥ 0∧i8[2] ≥ 0 ⇒ (UIncreasing(LOAD1082(i8[3], +(i23[3], -1), i56[3])), ≥)∧[(-1)Bound*bni_40] + [bni_40]i56[2] ≥ 0∧[(-1)bso_41] ≥ 0)
• (i23[2] ≥ 0∧i56[2] ≥ 0∧i8[2] ≥ 0 ⇒ (UIncreasing(LOAD1082(i8[3], +(i23[3], -1), i56[3])), ≥)∧[(-1)Bound*bni_40] + [bni_40]i56[2] ≥ 0∧[(-1)bso_41] ≥ 0)
• (i23[2] ≥ 0∧i56[2] ≥ 0∧i8[2] ≥ 0 ⇒ (UIncreasing(LOAD1082(i8[3], +(i23[3], -1), i56[3])), ≥)∧[(-1)Bound*bni_40] + [bni_40]i56[2] ≥ 0∧[(-1)bso_41] ≥ 0)

• (i56[4] ≥ 0∧i23[4] ≥ 0 ⇒ (UIncreasing(COND_LOAD10821(<=(i23[4], i56[4]), i8[4], i23[4], i56[4])), ≥)∧[bni_42] = 0∧[(-1)bni_42 + (-1)Bound*bni_42] + [(-1)bni_42]i23[4] + [(-1)bni_42]i56[4] ≥ 0∧0 = 0∧[(-1)bso_43] ≥ 0)
• (i56[4] ≥ 0∧i23[4] ≥ 0 ⇒ (UIncreasing(COND_LOAD10821(<=(i23[4], i56[4]), i8[4], i23[4], i56[4])), ≥)∧[bni_42] = 0∧[(-1)bni_42 + (-1)Bound*bni_42] + [bni_42]i23[4] + [(-1)bni_42]i56[4] ≥ 0∧0 = 0∧[(-1)bso_43] ≥ 0)

• (i56[4] ≥ 0∧i23[4] ≥ 0 ⇒ (UIncreasing(LOAD1003(+(i8[5], -1), i23[5], i56[5])), ≥)∧[bni_44] = 0∧[(-1)bni_44 + (-1)Bound*bni_44] + [bni_44]i23[4] + [(-1)bni_44]i56[4] ≥ 0∧0 = 0∧[1 + (-1)bso_45] ≥ 0)
• (i56[4] ≥ 0∧i23[4] ≥ 0 ⇒ (UIncreasing(LOAD1003(+(i8[5], -1), i23[5], i56[5])), ≥)∧[bni_44] = 0∧[(-1)bni_44 + (-1)Bound*bni_44] + [(-1)bni_44]i23[4] + [(-1)bni_44]i56[4] ≥ 0∧0 = 0∧[1 + (-1)bso_45] ≥ 0)
• (i56[4] ≥ 0∧i23[4] ≥ 0 ⇒ (UIncreasing(LOAD1003(+(i8[5], -1), i23[5], i56[5])), ≥)∧[bni_44] = 0∧[(-1)bni_44 + (-1)Bound*bni_44] + [bni_44]i23[4] + [(-1)bni_44]i56[4] ≥ 0∧0 = 0∧[1 + (-1)bso_45] ≥ 0)
• (i56[4] ≥ 0∧i23[4] ≥ 0 ⇒ (UIncreasing(LOAD1003(+(i8[5], -1), i23[5], i56[5])), ≥)∧[bni_44] = 0∧[(-1)bni_44 + (-1)Bound*bni_44] + [(-1)bni_44]i23[4] + [(-1)bni_44]i56[4] ≥ 0∧0 = 0∧[1 + (-1)bso_45] ≥ 0)

• LOAD1003(i8, i23, i56) → COND_LOAD10031(&&(<=(i23, i56), >(i8, i56)), i8, i23, i56)
• (i56[6] ≥ 0∧i23[6] ≥ 0∧i8[6] ≥ 0 ⇒ (UIncreasing(COND_LOAD10031(&&(<=(i23[6], i56[6]), >(i8[6], i56[6])), i8[6], i23[6], i56[6])), ≥)∧[(-1)Bound*bni_46] + [bni_46]i23[6] ≥ 0∧[(-1)bso_47] ≥ 0)
• (i56[6] ≥ 0∧i23[6] ≥ 0∧i8[6] ≥ 0 ⇒ (UIncreasing(COND_LOAD10031(&&(<=(i23[6], i56[6]), >(i8[6], i56[6])), i8[6], i23[6], i56[6])), ≥)∧[(-1)Bound*bni_46] + [bni_46]i23[6] ≥ 0∧[(-1)bso_47] ≥ 0)

• (i56[6] ≥ 0∧i23[6] ≥ 0∧i8[6] ≥ 0 ⇒ (UIncreasing(LOAD1003(+(i8[7], -1), i23[7], i56[7])), ≥)∧[(-1)Bound*bni_48] + [bni_48]i23[6] ≥ 0∧[1 + (-1)bso_49] ≥ 0)
• (i56[6] ≥ 0∧i23[6] ≥ 0∧i8[6] ≥ 0 ⇒ (UIncreasing(LOAD1003(+(i8[7], -1), i23[7], i56[7])), ≥)∧[(-1)Bound*bni_48] + [bni_48]i23[6] ≥ 0∧[1 + (-1)bso_49] ≥ 0)
• (i56[6] ≥ 0∧i23[6] ≥ 0∧i8[6] ≥ 0 ⇒ (UIncreasing(LOAD1003(+(i8[7], -1), i23[7], i56[7])), ≥)∧[(-1)Bound*bni_48] + [bni_48]i23[6] ≥ 0∧[1 + (-1)bso_49] ≥ 0)
• (i56[6] ≥ 0∧i23[6] ≥ 0∧i8[6] ≥ 0 ⇒ (UIncreasing(LOAD1003(+(i8[7], -1), i23[7], i56[7])), ≥)∧[(-1)Bound*bni_48] + [bni_48]i23[6] ≥ 0∧[1 + (-1)bso_49] ≥ 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) = [3]
POL(LOAD1082(x1, x2, x3)) = [-1] + [-1]x3 + x1
POL(COND_LOAD1082(x1, x2, x3, x4)) = [-1] + [-1]x4 + x2
POL(>(x1, x2)) = 0
POL(+(x1, x2)) = x1 + x2
POL(-1) = [-1]
POL(LOAD1003(x1, x2, x3)) = [-1] + [-1]x3 + x1
POL(COND_LOAD1003(x1, x2, x3, x4)) = [-1] + [-1]x4 + x2
POL(&&(x1, x2)) = [-1]
POL(COND_LOAD10821(x1, x2, x3, x4)) = [-1] + [-1]x4 + x2
POL(<=(x1, x2)) = [-1]
POL(COND_LOAD10031(x1, x2, x3, x4)) = [-1] + [-1]x4 + x2

The following pairs are in P>:

The following pairs are in Pbound:

The following pairs are in P:

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

### (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, Boolean

R is empty.

The integer pair graph contains the following rules and edges:
(2): LOAD1003(i8[2], i23[2], i56[2]) → COND_LOAD1003(i23[2] > i56[2] && i8[2] > i56[2], i8[2], i23[2], i56[2])
(6): LOAD1003(i8[6], i23[6], i56[6]) → COND_LOAD10031(i23[6] <= i56[6] && i8[6] > i56[6], i8[6], i23[6], i56[6])

(1) -> (0), if ((i8[1]* i8[0])∧(i23[1] + -1* i23[0])∧(i56[1]* i56[0]))

(3) -> (0), if ((i8[3]* i8[0])∧(i23[3] + -1* i23[0])∧(i56[3]* i56[0]))

(0) -> (1), if ((i8[0]* i8[1])∧(i56[0]* i56[1])∧(i23[0] > i56[0]* TRUE)∧(i23[0]* i23[1]))

(2) -> (3), if ((i23[2] > i56[2] && i8[2] > i56[2]* TRUE)∧(i23[2]* i23[3])∧(i56[2]* i56[3])∧(i8[2]* i8[3]))

(1) -> (4), if ((i23[1] + -1* i23[4])∧(i56[1]* i56[4])∧(i8[1]* i8[4]))

(3) -> (4), if ((i23[3] + -1* i23[4])∧(i56[3]* i56[4])∧(i8[3]* i8[4]))

The set Q consists of the following terms:

### (12) IDependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 4 less nodes.

### (13) 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) -> (0), if ((i8[1]* i8[0])∧(i23[1] + -1* i23[0])∧(i56[1]* i56[0]))

(0) -> (1), if ((i8[0]* i8[1])∧(i56[0]* i56[1])∧(i23[0] > i56[0]* TRUE)∧(i23[0]* i23[1]))

The set Q consists of the following terms:

### (14) 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_LOAD1082(TRUE, i8[1], i23[1], i56[1]) → LOAD1082(i8[1], +(i23[1], -1), i56[1]) the following chains were created:
• We consider the chain LOAD1082(i8[0], i23[0], i56[0]) → COND_LOAD1082(>(i23[0], i56[0]), i8[0], i23[0], i56[0]), COND_LOAD1082(TRUE, i8[1], i23[1], i56[1]) → LOAD1082(i8[1], +(i23[1], -1), i56[1]), LOAD1082(i8[0], i23[0], i56[0]) → COND_LOAD1082(>(i23[0], i56[0]), i8[0], i23[0], i56[0]) which results in the following constraint:

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

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

(3)    (i23[0] + [-1] + [-1]i56[0] ≥ 0 ⇒ (UIncreasing(LOAD1082(i8[1], +(i23[1], -1), i56[1])), ≥)∧[(-1)bni_13 + (-1)Bound*bni_13] + [(-1)bni_13]i56[0] + [bni_13]i23[0] ≥ 0∧[1 + (-1)bso_14] ≥ 0)

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

(4)    (i23[0] + [-1] + [-1]i56[0] ≥ 0 ⇒ (UIncreasing(LOAD1082(i8[1], +(i23[1], -1), i56[1])), ≥)∧[(-1)bni_13 + (-1)Bound*bni_13] + [(-1)bni_13]i56[0] + [bni_13]i23[0] ≥ 0∧[1 + (-1)bso_14] ≥ 0)

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

(5)    (i23[0] + [-1] + [-1]i56[0] ≥ 0 ⇒ (UIncreasing(LOAD1082(i8[1], +(i23[1], -1), i56[1])), ≥)∧[(-1)bni_13 + (-1)Bound*bni_13] + [(-1)bni_13]i56[0] + [bni_13]i23[0] ≥ 0∧[1 + (-1)bso_14] ≥ 0)

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

(6)    (i23[0] + [-1] + [-1]i56[0] ≥ 0 ⇒ (UIncreasing(LOAD1082(i8[1], +(i23[1], -1), i56[1])), ≥)∧0 = 0∧[(-1)bni_13 + (-1)Bound*bni_13] + [(-1)bni_13]i56[0] + [bni_13]i23[0] ≥ 0∧0 = 0∧[1 + (-1)bso_14] ≥ 0)

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

(7)    (i23[0] ≥ 0 ⇒ (UIncreasing(LOAD1082(i8[1], +(i23[1], -1), i56[1])), ≥)∧0 = 0∧[(-1)Bound*bni_13] + [bni_13]i23[0] ≥ 0∧0 = 0∧[1 + (-1)bso_14] ≥ 0)

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

(8)    (i23[0] ≥ 0∧i56[0] ≥ 0 ⇒ (UIncreasing(LOAD1082(i8[1], +(i23[1], -1), i56[1])), ≥)∧0 = 0∧[(-1)Bound*bni_13] + [bni_13]i23[0] ≥ 0∧0 = 0∧[1 + (-1)bso_14] ≥ 0)

(9)    (i23[0] ≥ 0∧i56[0] ≥ 0 ⇒ (UIncreasing(LOAD1082(i8[1], +(i23[1], -1), i56[1])), ≥)∧0 = 0∧[(-1)Bound*bni_13] + [bni_13]i23[0] ≥ 0∧0 = 0∧[1 + (-1)bso_14] ≥ 0)

For Pair LOAD1082(i8[0], i23[0], i56[0]) → COND_LOAD1082(>(i23[0], i56[0]), i8[0], i23[0], i56[0]) the following chains were created:
• We consider the chain LOAD1082(i8[0], i23[0], i56[0]) → COND_LOAD1082(>(i23[0], i56[0]), i8[0], i23[0], i56[0]), COND_LOAD1082(TRUE, i8[1], i23[1], i56[1]) → LOAD1082(i8[1], +(i23[1], -1), i56[1]) which results in the following constraint:

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

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

(12)    (i23[0] + [-1] + [-1]i56[0] ≥ 0 ⇒ (UIncreasing(COND_LOAD1082(>(i23[0], i56[0]), i8[0], i23[0], i56[0])), ≥)∧[(-1)bni_15 + (-1)Bound*bni_15] + [(-1)bni_15]i56[0] + [bni_15]i23[0] ≥ 0∧[(-1)bso_16] ≥ 0)

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

(13)    (i23[0] + [-1] + [-1]i56[0] ≥ 0 ⇒ (UIncreasing(COND_LOAD1082(>(i23[0], i56[0]), i8[0], i23[0], i56[0])), ≥)∧[(-1)bni_15 + (-1)Bound*bni_15] + [(-1)bni_15]i56[0] + [bni_15]i23[0] ≥ 0∧[(-1)bso_16] ≥ 0)

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

(14)    (i23[0] + [-1] + [-1]i56[0] ≥ 0 ⇒ (UIncreasing(COND_LOAD1082(>(i23[0], i56[0]), i8[0], i23[0], i56[0])), ≥)∧[(-1)bni_15 + (-1)Bound*bni_15] + [(-1)bni_15]i56[0] + [bni_15]i23[0] ≥ 0∧[(-1)bso_16] ≥ 0)

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

(15)    (i23[0] + [-1] + [-1]i56[0] ≥ 0 ⇒ (UIncreasing(COND_LOAD1082(>(i23[0], i56[0]), i8[0], i23[0], i56[0])), ≥)∧0 = 0∧[(-1)bni_15 + (-1)Bound*bni_15] + [(-1)bni_15]i56[0] + [bni_15]i23[0] ≥ 0∧0 = 0∧[(-1)bso_16] ≥ 0)

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

(16)    (i23[0] ≥ 0 ⇒ (UIncreasing(COND_LOAD1082(>(i23[0], i56[0]), i8[0], i23[0], i56[0])), ≥)∧0 = 0∧[(-1)Bound*bni_15] + [bni_15]i23[0] ≥ 0∧0 = 0∧[(-1)bso_16] ≥ 0)

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

(17)    (i23[0] ≥ 0∧i56[0] ≥ 0 ⇒ (UIncreasing(COND_LOAD1082(>(i23[0], i56[0]), i8[0], i23[0], i56[0])), ≥)∧0 = 0∧[(-1)Bound*bni_15] + [bni_15]i23[0] ≥ 0∧0 = 0∧[(-1)bso_16] ≥ 0)

(18)    (i23[0] ≥ 0∧i56[0] ≥ 0 ⇒ (UIncreasing(COND_LOAD1082(>(i23[0], i56[0]), i8[0], i23[0], i56[0])), ≥)∧0 = 0∧[(-1)Bound*bni_15] + [bni_15]i23[0] ≥ 0∧0 = 0∧[(-1)bso_16] ≥ 0)

To summarize, we get the following constraints P for the following pairs.
• (i23[0] ≥ 0∧i56[0] ≥ 0 ⇒ (UIncreasing(LOAD1082(i8[1], +(i23[1], -1), i56[1])), ≥)∧0 = 0∧[(-1)Bound*bni_13] + [bni_13]i23[0] ≥ 0∧0 = 0∧[1 + (-1)bso_14] ≥ 0)
• (i23[0] ≥ 0∧i56[0] ≥ 0 ⇒ (UIncreasing(LOAD1082(i8[1], +(i23[1], -1), i56[1])), ≥)∧0 = 0∧[(-1)Bound*bni_13] + [bni_13]i23[0] ≥ 0∧0 = 0∧[1 + (-1)bso_14] ≥ 0)

• (i23[0] ≥ 0∧i56[0] ≥ 0 ⇒ (UIncreasing(COND_LOAD1082(>(i23[0], i56[0]), i8[0], i23[0], i56[0])), ≥)∧0 = 0∧[(-1)Bound*bni_15] + [bni_15]i23[0] ≥ 0∧0 = 0∧[(-1)bso_16] ≥ 0)
• (i23[0] ≥ 0∧i56[0] ≥ 0 ⇒ (UIncreasing(COND_LOAD1082(>(i23[0], i56[0]), i8[0], i23[0], i56[0])), ≥)∧0 = 0∧[(-1)Bound*bni_15] + [bni_15]i23[0] ≥ 0∧0 = 0∧[(-1)bso_16] ≥ 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_LOAD1082(x1, x2, x3, x4)) = [-1] + [-1]x4 + x3
POL(LOAD1082(x1, x2, x3)) = [-1] + [-1]x3 + x2
POL(+(x1, x2)) = x1 + x2
POL(-1) = [-1]
POL(>(x1, x2)) = 0

The following pairs are in P>:

The following pairs are in Pbound:

The following pairs are in P:

There are no usable rules.

### (16) 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:

### (17) IDependencyGraphProof (EQUIVALENT transformation)

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

### (19) 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:

### (20) IDependencyGraphProof (EQUIVALENT transformation)

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

### (22) 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) -> (0), if ((i8[1]* i8[0])∧(i23[1] + -1* i23[0])∧(i56[1]* i56[0]))

(0) -> (1), if ((i8[0]* i8[1])∧(i56[0]* i56[1])∧(i23[0] > i56[0]* TRUE)∧(i23[0]* i23[1]))

(1) -> (4), if ((i23[1] + -1* i23[4])∧(i56[1]* i56[4])∧(i8[1]* i8[4]))

(4) -> (5), if ((i8[4]* i8[5])∧(i23[4] <= i56[4]* TRUE)∧(i56[4]* i56[5])∧(i23[4]* i23[5]))

The set Q consists of the following terms:

### (23) IDependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 2 less nodes.

### (24) 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) -> (0), if ((i8[1]* i8[0])∧(i23[1] + -1* i23[0])∧(i56[1]* i56[0]))

(0) -> (1), if ((i8[0]* i8[1])∧(i56[0]* i56[1])∧(i23[0] > i56[0]* TRUE)∧(i23[0]* i23[1]))

The set Q consists of the following terms:

### (25) 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_LOAD1082(TRUE, i8[1], i23[1], i56[1]) → LOAD1082(i8[1], +(i23[1], -1), i56[1]) the following chains were created:
• We consider the chain LOAD1082(i8[0], i23[0], i56[0]) → COND_LOAD1082(>(i23[0], i56[0]), i8[0], i23[0], i56[0]), COND_LOAD1082(TRUE, i8[1], i23[1], i56[1]) → LOAD1082(i8[1], +(i23[1], -1), i56[1]), LOAD1082(i8[0], i23[0], i56[0]) → COND_LOAD1082(>(i23[0], i56[0]), i8[0], i23[0], i56[0]) which results in the following constraint:

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

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

(3)    (i23[0] + [-1] + [-1]i56[0] ≥ 0 ⇒ (UIncreasing(LOAD1082(i8[1], +(i23[1], -1), i56[1])), ≥)∧[(-1)bni_13 + (-1)Bound*bni_13] + [(-1)bni_13]i56[0] + [bni_13]i23[0] ≥ 0∧[1 + (-1)bso_14] ≥ 0)

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

(4)    (i23[0] + [-1] + [-1]i56[0] ≥ 0 ⇒ (UIncreasing(LOAD1082(i8[1], +(i23[1], -1), i56[1])), ≥)∧[(-1)bni_13 + (-1)Bound*bni_13] + [(-1)bni_13]i56[0] + [bni_13]i23[0] ≥ 0∧[1 + (-1)bso_14] ≥ 0)

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

(5)    (i23[0] + [-1] + [-1]i56[0] ≥ 0 ⇒ (UIncreasing(LOAD1082(i8[1], +(i23[1], -1), i56[1])), ≥)∧[(-1)bni_13 + (-1)Bound*bni_13] + [(-1)bni_13]i56[0] + [bni_13]i23[0] ≥ 0∧[1 + (-1)bso_14] ≥ 0)

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

(6)    (i23[0] + [-1] + [-1]i56[0] ≥ 0 ⇒ (UIncreasing(LOAD1082(i8[1], +(i23[1], -1), i56[1])), ≥)∧0 = 0∧[(-1)bni_13 + (-1)Bound*bni_13] + [(-1)bni_13]i56[0] + [bni_13]i23[0] ≥ 0∧0 = 0∧[1 + (-1)bso_14] ≥ 0)

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

(7)    (i23[0] ≥ 0 ⇒ (UIncreasing(LOAD1082(i8[1], +(i23[1], -1), i56[1])), ≥)∧0 = 0∧[(-1)Bound*bni_13] + [bni_13]i23[0] ≥ 0∧0 = 0∧[1 + (-1)bso_14] ≥ 0)

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

(8)    (i23[0] ≥ 0∧i56[0] ≥ 0 ⇒ (UIncreasing(LOAD1082(i8[1], +(i23[1], -1), i56[1])), ≥)∧0 = 0∧[(-1)Bound*bni_13] + [bni_13]i23[0] ≥ 0∧0 = 0∧[1 + (-1)bso_14] ≥ 0)

(9)    (i23[0] ≥ 0∧i56[0] ≥ 0 ⇒ (UIncreasing(LOAD1082(i8[1], +(i23[1], -1), i56[1])), ≥)∧0 = 0∧[(-1)Bound*bni_13] + [bni_13]i23[0] ≥ 0∧0 = 0∧[1 + (-1)bso_14] ≥ 0)

For Pair LOAD1082(i8[0], i23[0], i56[0]) → COND_LOAD1082(>(i23[0], i56[0]), i8[0], i23[0], i56[0]) the following chains were created:
• We consider the chain LOAD1082(i8[0], i23[0], i56[0]) → COND_LOAD1082(>(i23[0], i56[0]), i8[0], i23[0], i56[0]), COND_LOAD1082(TRUE, i8[1], i23[1], i56[1]) → LOAD1082(i8[1], +(i23[1], -1), i56[1]) which results in the following constraint:

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

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

(12)    (i23[0] + [-1] + [-1]i56[0] ≥ 0 ⇒ (UIncreasing(COND_LOAD1082(>(i23[0], i56[0]), i8[0], i23[0], i56[0])), ≥)∧[(-1)bni_15 + (-1)Bound*bni_15] + [(-1)bni_15]i56[0] + [bni_15]i23[0] ≥ 0∧[(-1)bso_16] ≥ 0)

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

(13)    (i23[0] + [-1] + [-1]i56[0] ≥ 0 ⇒ (UIncreasing(COND_LOAD1082(>(i23[0], i56[0]), i8[0], i23[0], i56[0])), ≥)∧[(-1)bni_15 + (-1)Bound*bni_15] + [(-1)bni_15]i56[0] + [bni_15]i23[0] ≥ 0∧[(-1)bso_16] ≥ 0)

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

(14)    (i23[0] + [-1] + [-1]i56[0] ≥ 0 ⇒ (UIncreasing(COND_LOAD1082(>(i23[0], i56[0]), i8[0], i23[0], i56[0])), ≥)∧[(-1)bni_15 + (-1)Bound*bni_15] + [(-1)bni_15]i56[0] + [bni_15]i23[0] ≥ 0∧[(-1)bso_16] ≥ 0)

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

(15)    (i23[0] + [-1] + [-1]i56[0] ≥ 0 ⇒ (UIncreasing(COND_LOAD1082(>(i23[0], i56[0]), i8[0], i23[0], i56[0])), ≥)∧0 = 0∧[(-1)bni_15 + (-1)Bound*bni_15] + [(-1)bni_15]i56[0] + [bni_15]i23[0] ≥ 0∧0 = 0∧[(-1)bso_16] ≥ 0)

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

(16)    (i23[0] ≥ 0 ⇒ (UIncreasing(COND_LOAD1082(>(i23[0], i56[0]), i8[0], i23[0], i56[0])), ≥)∧0 = 0∧[(-1)Bound*bni_15] + [bni_15]i23[0] ≥ 0∧0 = 0∧[(-1)bso_16] ≥ 0)

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

(17)    (i23[0] ≥ 0∧i56[0] ≥ 0 ⇒ (UIncreasing(COND_LOAD1082(>(i23[0], i56[0]), i8[0], i23[0], i56[0])), ≥)∧0 = 0∧[(-1)Bound*bni_15] + [bni_15]i23[0] ≥ 0∧0 = 0∧[(-1)bso_16] ≥ 0)

(18)    (i23[0] ≥ 0∧i56[0] ≥ 0 ⇒ (UIncreasing(COND_LOAD1082(>(i23[0], i56[0]), i8[0], i23[0], i56[0])), ≥)∧0 = 0∧[(-1)Bound*bni_15] + [bni_15]i23[0] ≥ 0∧0 = 0∧[(-1)bso_16] ≥ 0)

To summarize, we get the following constraints P for the following pairs.
• (i23[0] ≥ 0∧i56[0] ≥ 0 ⇒ (UIncreasing(LOAD1082(i8[1], +(i23[1], -1), i56[1])), ≥)∧0 = 0∧[(-1)Bound*bni_13] + [bni_13]i23[0] ≥ 0∧0 = 0∧[1 + (-1)bso_14] ≥ 0)
• (i23[0] ≥ 0∧i56[0] ≥ 0 ⇒ (UIncreasing(LOAD1082(i8[1], +(i23[1], -1), i56[1])), ≥)∧0 = 0∧[(-1)Bound*bni_13] + [bni_13]i23[0] ≥ 0∧0 = 0∧[1 + (-1)bso_14] ≥ 0)

• (i23[0] ≥ 0∧i56[0] ≥ 0 ⇒ (UIncreasing(COND_LOAD1082(>(i23[0], i56[0]), i8[0], i23[0], i56[0])), ≥)∧0 = 0∧[(-1)Bound*bni_15] + [bni_15]i23[0] ≥ 0∧0 = 0∧[(-1)bso_16] ≥ 0)
• (i23[0] ≥ 0∧i56[0] ≥ 0 ⇒ (UIncreasing(COND_LOAD1082(>(i23[0], i56[0]), i8[0], i23[0], i56[0])), ≥)∧0 = 0∧[(-1)Bound*bni_15] + [bni_15]i23[0] ≥ 0∧0 = 0∧[(-1)bso_16] ≥ 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_LOAD1082(x1, x2, x3, x4)) = [-1] + [-1]x4 + x3
POL(LOAD1082(x1, x2, x3)) = [-1] + [-1]x3 + x2
POL(+(x1, x2)) = x1 + x2
POL(-1) = [-1]
POL(>(x1, x2)) = 0

The following pairs are in P>:

The following pairs are in Pbound:

The following pairs are in P:

There are no usable rules.

### (27) 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:

### (28) IDependencyGraphProof (EQUIVALENT transformation)

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

### (30) 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: