### (0) Obligation:

JBC Problem based on JBC Program:
`No human-readable program information known.`

Manifest-Version: 1.0 Created-By: 1.6.0_16 (Sun Microsystems Inc.) Main-Class: PastaB10

### (1) JBC2FIG (SOUND transformation)

Constructed FIGraph.

### (2) Obligation:

FIGraph based on JBC Program:
Graph of 174 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:
Load431(i84, i48) → Cond_Load4312(i84 > 0 && i84 + i48 > 0, i84, i48)
The set Q consists of the following terms:

### (5) GroundTermsRemoverProof (EQUIVALENT transformation)

Some arguments are removed because they always contain the same ground term.
We removed the following ground terms:
• 0

We removed arguments according to the following replacements:

### (6) 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:
Load431(i84, i48) → Cond_Load4312(i84 > 0 && i84 + i48 > 0, i84, i48)
The set Q consists of the following terms:

### (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

The ITRS R consists of the following rules:
Load431(i84, i48) → Cond_Load4312(i84 > 0 && i84 + i48 > 0, i84, i48)

The integer pair graph contains the following rules and edges:
(4): LOAD431(i84[4], i48[4]) → COND_LOAD4312(i84[4] > 0 && i84[4] + i48[4] > 0, i84[4], i48[4])

(0) -> (1), if ((0 > 0* TRUE))

(1) -> (0), if true

(1) -> (2), if (0* i92[2])

(1) -> (4), if ((0* i84[4])∧(0* i48[4]))

(2) -> (3), if ((i92[2]* i92[3])∧(i92[2] > 0* TRUE))

(3) -> (0), if (i92[3] + -1* 0)

(3) -> (2), if (i92[3] + -1* i92[2])

(3) -> (4), if ((i92[3] + -1* i48[4])∧(0* i84[4]))

(4) -> (5), if ((i84[4]* i84[5])∧(i84[4] > 0 && i84[4] + i48[4] > 0* TRUE)∧(i48[4]* i48[5]))

(5) -> (0), if ((i48[5]* 0)∧(i84[5] + -1* 0))

(5) -> (2), if ((i84[5] + -1* 0)∧(i48[5]* i92[2]))

(5) -> (4), if ((i48[5]* i48[4])∧(i84[5] + -1* i84[4]))

The set Q consists of the following terms:

### (9) 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.

### (10) Obligation:

IDP problem:
The following function symbols are pre-defined:
 != ~ Neq: (Integer, Integer) -> Boolean * ~ Mul: (Integer, Integer) -> Integer >= ~ Ge: (Integer, Integer) -> Boolean -1 ~ UnaryMinus: (Integer) -> Integer | ~ Bwor: (Integer, Integer) -> Integer / ~ Div: (Integer, Integer) -> Integer = ~ Eq: (Integer, Integer) -> Boolean ~ Bwxor: (Integer, Integer) -> Integer || ~ Lor: (Boolean, Boolean) -> Boolean ! ~ Lnot: (Boolean) -> Boolean < ~ Lt: (Integer, Integer) -> Boolean - ~ Sub: (Integer, Integer) -> Integer <= ~ Le: (Integer, Integer) -> Boolean > ~ Gt: (Integer, Integer) -> Boolean ~ ~ Bwnot: (Integer) -> Integer % ~ Mod: (Integer, Integer) -> Integer & ~ Bwand: (Integer, Integer) -> Integer + ~ Add: (Integer, Integer) -> Integer && ~ Land: (Boolean, Boolean) -> Boolean

The following domains are used:

Integer, Boolean

R is empty.

The integer pair graph contains the following rules and edges:
(4): LOAD431(i84[4], i48[4]) → COND_LOAD4312(i84[4] > 0 && i84[4] + i48[4] > 0, i84[4], i48[4])

(0) -> (1), if ((0 > 0* TRUE))

(1) -> (0), if true

(1) -> (2), if (0* i92[2])

(1) -> (4), if ((0* i84[4])∧(0* i48[4]))

(2) -> (3), if ((i92[2]* i92[3])∧(i92[2] > 0* TRUE))

(3) -> (0), if (i92[3] + -1* 0)

(3) -> (2), if (i92[3] + -1* i92[2])

(3) -> (4), if ((i92[3] + -1* i48[4])∧(0* i84[4]))

(4) -> (5), if ((i84[4]* i84[5])∧(i84[4] > 0 && i84[4] + i48[4] > 0* TRUE)∧(i48[4]* i48[5]))

(5) -> (0), if ((i48[5]* 0)∧(i84[5] + -1* 0))

(5) -> (2), if ((i84[5] + -1* 0)∧(i48[5]* i92[2]))

(5) -> (4), if ((i48[5]* i48[4])∧(i84[5] + -1* i84[4]))

The set Q consists of the following terms:

### (11) 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 LOAD431(0, 0) → COND_LOAD431(>(0, 0)) the following chains were created:

We solved constraint (1) using rules (I), (II), (IDP_CONSTANT_FOLD).

• We consider the chain COND_LOAD431(TRUE) → LOAD431(0, 0) which results in the following constraint:

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

(3)    ((UIncreasing(LOAD431(0, 0)), ≥)∧[(-1)bso_13] ≥ 0)

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

(4)    ((UIncreasing(LOAD431(0, 0)), ≥)∧[(-1)bso_13] ≥ 0)

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

(5)    ((UIncreasing(LOAD431(0, 0)), ≥)∧[(-1)bso_13] ≥ 0)

For Pair LOAD431(0, i92) → COND_LOAD4311(>(i92, 0), i92) the following chains were created:
• We consider the chain LOAD431(0, i92[2]) → COND_LOAD4311(>(i92[2], 0), i92[2]), COND_LOAD4311(TRUE, i92[3]) → LOAD431(0, +(i92[3], -1)) which results in the following constraint:

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

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

(8)    (i92[2] + [-1] ≥ 0 ⇒ (UIncreasing(COND_LOAD4311(>(i92[2], 0), i92[2])), ≥)∧[(-1)bni_14 + (-1)Bound*bni_14] ≥ 0∧[(-1)bso_15] ≥ 0)

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

(9)    (i92[2] + [-1] ≥ 0 ⇒ (UIncreasing(COND_LOAD4311(>(i92[2], 0), i92[2])), ≥)∧[(-1)bni_14 + (-1)Bound*bni_14] ≥ 0∧[(-1)bso_15] ≥ 0)

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

(10)    (i92[2] + [-1] ≥ 0 ⇒ (UIncreasing(COND_LOAD4311(>(i92[2], 0), i92[2])), ≥)∧[(-1)bni_14 + (-1)Bound*bni_14] ≥ 0∧[(-1)bso_15] ≥ 0)

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

(11)    (i92[2] ≥ 0 ⇒ (UIncreasing(COND_LOAD4311(>(i92[2], 0), i92[2])), ≥)∧[(-1)bni_14 + (-1)Bound*bni_14] ≥ 0∧[(-1)bso_15] ≥ 0)

For Pair COND_LOAD4311(TRUE, i92) → LOAD431(0, +(i92, -1)) the following chains were created:
• We consider the chain COND_LOAD4311(TRUE, i92[3]) → LOAD431(0, +(i92[3], -1)) which results in the following constraint:

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

(13)    ((UIncreasing(LOAD431(0, +(i92[3], -1))), ≥)∧[(-1)bso_17] ≥ 0)

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

(14)    ((UIncreasing(LOAD431(0, +(i92[3], -1))), ≥)∧[(-1)bso_17] ≥ 0)

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

(15)    ((UIncreasing(LOAD431(0, +(i92[3], -1))), ≥)∧[(-1)bso_17] ≥ 0)

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

(16)    ((UIncreasing(LOAD431(0, +(i92[3], -1))), ≥)∧0 = 0∧[(-1)bso_17] ≥ 0)

For Pair LOAD431(i84, i48) → COND_LOAD4312(&&(>(i84, 0), >(+(i84, i48), 0)), i84, i48) the following chains were created:
• We consider the chain LOAD431(i84[4], i48[4]) → COND_LOAD4312(&&(>(i84[4], 0), >(+(i84[4], i48[4]), 0)), i84[4], i48[4]), COND_LOAD4312(TRUE, i84[5], i48[5]) → LOAD431(+(i84[5], -1), i48[5]) which results in the following constraint:

We simplified constraint (17) using rules (IV), (IDP_BOOLEAN) which results in the following new constraint:

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

(19)    (i84[4] + [-1] ≥ 0∧i84[4] + [-1] + i48[4] ≥ 0 ⇒ (UIncreasing(COND_LOAD4312(&&(>(i84[4], 0), >(+(i84[4], i48[4]), 0)), i84[4], i48[4])), ≥)∧[(-1)bni_18 + (-1)Bound*bni_18] + [bni_18]i84[4] ≥ 0∧[(-1)bso_19] ≥ 0)

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

(20)    (i84[4] + [-1] ≥ 0∧i84[4] + [-1] + i48[4] ≥ 0 ⇒ (UIncreasing(COND_LOAD4312(&&(>(i84[4], 0), >(+(i84[4], i48[4]), 0)), i84[4], i48[4])), ≥)∧[(-1)bni_18 + (-1)Bound*bni_18] + [bni_18]i84[4] ≥ 0∧[(-1)bso_19] ≥ 0)

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

(21)    (i84[4] + [-1] ≥ 0∧i84[4] + [-1] + i48[4] ≥ 0 ⇒ (UIncreasing(COND_LOAD4312(&&(>(i84[4], 0), >(+(i84[4], i48[4]), 0)), i84[4], i48[4])), ≥)∧[(-1)bni_18 + (-1)Bound*bni_18] + [bni_18]i84[4] ≥ 0∧[(-1)bso_19] ≥ 0)

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

(22)    (i84[4] ≥ 0∧i84[4] + i48[4] ≥ 0 ⇒ (UIncreasing(COND_LOAD4312(&&(>(i84[4], 0), >(+(i84[4], i48[4]), 0)), i84[4], i48[4])), ≥)∧[(-1)Bound*bni_18] + [bni_18]i84[4] ≥ 0∧[(-1)bso_19] ≥ 0)

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

(23)    (i84[4] ≥ 0∧i84[4] + i48[4] ≥ 0∧i48[4] ≥ 0 ⇒ (UIncreasing(COND_LOAD4312(&&(>(i84[4], 0), >(+(i84[4], i48[4]), 0)), i84[4], i48[4])), ≥)∧[(-1)Bound*bni_18] + [bni_18]i84[4] ≥ 0∧[(-1)bso_19] ≥ 0)

(24)    (i84[4] ≥ 0∧i84[4] + [-1]i48[4] ≥ 0∧i48[4] ≥ 0 ⇒ (UIncreasing(COND_LOAD4312(&&(>(i84[4], 0), >(+(i84[4], i48[4]), 0)), i84[4], i48[4])), ≥)∧[(-1)Bound*bni_18] + [bni_18]i84[4] ≥ 0∧[(-1)bso_19] ≥ 0)

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

(25)    (i48[4] + i84[4] ≥ 0∧i84[4] ≥ 0∧i48[4] ≥ 0 ⇒ (UIncreasing(COND_LOAD4312(&&(>(i84[4], 0), >(+(i84[4], i48[4]), 0)), i84[4], i48[4])), ≥)∧[(-1)Bound*bni_18] + [bni_18]i48[4] + [bni_18]i84[4] ≥ 0∧[(-1)bso_19] ≥ 0)

For Pair COND_LOAD4312(TRUE, i84, i48) → LOAD431(+(i84, -1), i48) the following chains were created:
• We consider the chain COND_LOAD4312(TRUE, i84[5], i48[5]) → LOAD431(+(i84[5], -1), i48[5]) which results in the following constraint:

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

(27)    ((UIncreasing(LOAD431(+(i84[5], -1), i48[5])), ≥)∧[1 + (-1)bso_21] ≥ 0)

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

(28)    ((UIncreasing(LOAD431(+(i84[5], -1), i48[5])), ≥)∧[1 + (-1)bso_21] ≥ 0)

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

(29)    ((UIncreasing(LOAD431(+(i84[5], -1), i48[5])), ≥)∧[1 + (-1)bso_21] ≥ 0)

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

(30)    ((UIncreasing(LOAD431(+(i84[5], -1), i48[5])), ≥)∧0 = 0∧0 = 0∧[1 + (-1)bso_21] ≥ 0)

To summarize, we get the following constraints P for the following pairs.

• ((UIncreasing(LOAD431(0, 0)), ≥)∧[(-1)bso_13] ≥ 0)

• (i92[2] ≥ 0 ⇒ (UIncreasing(COND_LOAD4311(>(i92[2], 0), i92[2])), ≥)∧[(-1)bni_14 + (-1)Bound*bni_14] ≥ 0∧[(-1)bso_15] ≥ 0)

• ((UIncreasing(LOAD431(0, +(i92[3], -1))), ≥)∧0 = 0∧[(-1)bso_17] ≥ 0)

• (i84[4] ≥ 0∧i84[4] + i48[4] ≥ 0∧i48[4] ≥ 0 ⇒ (UIncreasing(COND_LOAD4312(&&(>(i84[4], 0), >(+(i84[4], i48[4]), 0)), i84[4], i48[4])), ≥)∧[(-1)Bound*bni_18] + [bni_18]i84[4] ≥ 0∧[(-1)bso_19] ≥ 0)
• (i48[4] + i84[4] ≥ 0∧i84[4] ≥ 0∧i48[4] ≥ 0 ⇒ (UIncreasing(COND_LOAD4312(&&(>(i84[4], 0), >(+(i84[4], i48[4]), 0)), i84[4], i48[4])), ≥)∧[(-1)Bound*bni_18] + [bni_18]i48[4] + [bni_18]i84[4] ≥ 0∧[(-1)bso_19] ≥ 0)

• ((UIncreasing(LOAD431(+(i84[5], -1), i48[5])), ≥)∧0 = 0∧0 = 0∧[1 + (-1)bso_21] ≥ 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(LOAD431(x1, x2)) = [-1] + x1
POL(0) = 0
POL(>(x1, x2)) = [-1]
POL(+(x1, x2)) = x1 + x2
POL(-1) = [-1]
POL(COND_LOAD4312(x1, x2, x3)) = [-1] + x2
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.

### (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, Boolean

R is empty.

The integer pair graph contains the following rules and edges:
(4): LOAD431(i84[4], i48[4]) → COND_LOAD4312(i84[4] > 0 && i84[4] + i48[4] > 0, i84[4], i48[4])

(1) -> (2), if (0* i92[2])

(3) -> (2), if (i92[3] + -1* i92[2])

(2) -> (3), if ((i92[2]* i92[3])∧(i92[2] > 0* TRUE))

(1) -> (4), if ((0* i84[4])∧(0* i48[4]))

(3) -> (4), if ((i92[3] + -1* i48[4])∧(0* i84[4]))

The set Q consists of the following terms:

### (14) IDependencyGraphProof (EQUIVALENT transformation)

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

### (15) Obligation:

IDP problem:
The following function symbols are pre-defined:
 != ~ Neq: (Integer, Integer) -> Boolean * ~ Mul: (Integer, Integer) -> Integer >= ~ Ge: (Integer, Integer) -> Boolean -1 ~ UnaryMinus: (Integer) -> Integer | ~ Bwor: (Integer, Integer) -> Integer / ~ Div: (Integer, Integer) -> Integer = ~ Eq: (Integer, Integer) -> Boolean ~ Bwxor: (Integer, Integer) -> Integer || ~ Lor: (Boolean, Boolean) -> Boolean ! ~ Lnot: (Boolean) -> Boolean < ~ Lt: (Integer, Integer) -> Boolean - ~ Sub: (Integer, Integer) -> Integer <= ~ Le: (Integer, Integer) -> Boolean > ~ Gt: (Integer, Integer) -> Boolean ~ ~ Bwnot: (Integer) -> Integer % ~ Mod: (Integer, Integer) -> Integer & ~ Bwand: (Integer, Integer) -> Integer + ~ Add: (Integer, Integer) -> Integer && ~ Land: (Boolean, Boolean) -> Boolean

The following domains are used:

Integer

R is empty.

The integer pair graph contains the following rules and edges:

(3) -> (2), if (i92[3] + -1* i92[2])

(2) -> (3), if ((i92[2]* i92[3])∧(i92[2] > 0* TRUE))

The set Q consists of the following terms:

### (16) 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_LOAD4311(TRUE, i92[3]) → LOAD431(0, +(i92[3], -1)) the following chains were created:
• We consider the chain COND_LOAD4311(TRUE, i92[3]) → LOAD431(0, +(i92[3], -1)) which results in the following constraint:

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

(2)    ((UIncreasing(LOAD431(0, +(i92[3], -1))), ≥)∧[1 + (-1)bso_7] ≥ 0)

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

(3)    ((UIncreasing(LOAD431(0, +(i92[3], -1))), ≥)∧[1 + (-1)bso_7] ≥ 0)

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

(4)    ((UIncreasing(LOAD431(0, +(i92[3], -1))), ≥)∧[1 + (-1)bso_7] ≥ 0)

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

(5)    ((UIncreasing(LOAD431(0, +(i92[3], -1))), ≥)∧0 = 0∧[1 + (-1)bso_7] ≥ 0)

For Pair LOAD431(0, i92[2]) → COND_LOAD4311(>(i92[2], 0), i92[2]) the following chains were created:
• We consider the chain LOAD431(0, i92[2]) → COND_LOAD4311(>(i92[2], 0), i92[2]), COND_LOAD4311(TRUE, i92[3]) → LOAD431(0, +(i92[3], -1)) which results in the following constraint:

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

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

(8)    (i92[2] + [-1] ≥ 0 ⇒ (UIncreasing(COND_LOAD4311(>(i92[2], 0), i92[2])), ≥)∧[bni_8 + (-1)Bound*bni_8] + [bni_8]i92[2] ≥ 0∧[(-1)bso_9] ≥ 0)

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

(9)    (i92[2] + [-1] ≥ 0 ⇒ (UIncreasing(COND_LOAD4311(>(i92[2], 0), i92[2])), ≥)∧[bni_8 + (-1)Bound*bni_8] + [bni_8]i92[2] ≥ 0∧[(-1)bso_9] ≥ 0)

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

(10)    (i92[2] + [-1] ≥ 0 ⇒ (UIncreasing(COND_LOAD4311(>(i92[2], 0), i92[2])), ≥)∧[bni_8 + (-1)Bound*bni_8] + [bni_8]i92[2] ≥ 0∧[(-1)bso_9] ≥ 0)

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

(11)    (i92[2] ≥ 0 ⇒ (UIncreasing(COND_LOAD4311(>(i92[2], 0), i92[2])), ≥)∧[(2)bni_8 + (-1)Bound*bni_8] + [bni_8]i92[2] ≥ 0∧[(-1)bso_9] ≥ 0)

To summarize, we get the following constraints P for the following pairs.
• ((UIncreasing(LOAD431(0, +(i92[3], -1))), ≥)∧0 = 0∧[1 + (-1)bso_7] ≥ 0)

• (i92[2] ≥ 0 ⇒ (UIncreasing(COND_LOAD4311(>(i92[2], 0), i92[2])), ≥)∧[(2)bni_8 + (-1)Bound*bni_8] + [bni_8]i92[2] ≥ 0∧[(-1)bso_9] ≥ 0)

The constraints for P> respective Pbound are constructed from P where we just replace every occurence of "t ≥ s" in P by "t > s" respective "t ≥ c". Here c stands for the fresh constant used for Pbound.
Using the following integer polynomial ordering the resulting constraints can be solved
Polynomial interpretation over integers[POLO]:

POL(TRUE) = 0
POL(FALSE) = 0
POL(COND_LOAD4311(x1, x2)) = [1] + x2
POL(LOAD431(x1, x2)) = [1] + x2
POL(0) = 0
POL(+(x1, x2)) = x1 + x2
POL(-1) = [-1]
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.

### (18) Obligation:

IDP problem:
The following function symbols are pre-defined:
 != ~ Neq: (Integer, Integer) -> Boolean * ~ Mul: (Integer, Integer) -> Integer >= ~ Ge: (Integer, Integer) -> Boolean -1 ~ UnaryMinus: (Integer) -> Integer | ~ Bwor: (Integer, Integer) -> Integer / ~ Div: (Integer, Integer) -> Integer = ~ Eq: (Integer, Integer) -> Boolean ~ Bwxor: (Integer, Integer) -> Integer || ~ Lor: (Boolean, Boolean) -> Boolean ! ~ Lnot: (Boolean) -> Boolean < ~ Lt: (Integer, Integer) -> Boolean - ~ Sub: (Integer, Integer) -> Integer <= ~ Le: (Integer, Integer) -> Boolean > ~ Gt: (Integer, Integer) -> Boolean ~ ~ Bwnot: (Integer) -> Integer % ~ Mod: (Integer, Integer) -> Integer & ~ Bwand: (Integer, Integer) -> Integer + ~ Add: (Integer, Integer) -> Integer && ~ Land: (Boolean, Boolean) -> Boolean

The following domains are used:

Integer

R is empty.

The integer pair graph contains the following rules and edges:

The set Q consists of the following terms:

### (19) IDependencyGraphProof (EQUIVALENT transformation)

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

### (21) 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:

### (22) IDependencyGraphProof (EQUIVALENT transformation)

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

### (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: