### (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: DivMinus2

### (1) JBC2FIG (SOUND transformation)

Constructed FIGraph.

### (2) Obligation:

FIGraph based on JBC Program:
Graph of 195 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:
Load783(i95, i88, i95, i90) → Cond_Load783(i95 > 0 && i88 >= i95, i95, i88, i95, i90)
Cond_Load783(TRUE, i95, i88, i95, i90) → Load866(i95, i95, i90, i95, i88, i95)
Load866(i95, i95, i90, i95, i100, i103) → Cond_Load866(i103 > 0, i95, i95, i90, i95, i100, i103)
Cond_Load866(TRUE, i95, i95, i90, i95, i100, i103) → Load866(i95, i95, i90, i95, i100 + -1, i103 + -1)
Load866(i95, i95, i90, i95, i100, 0) → Load783(i95, i100, i95, i90 + 1)
The set Q consists of the following terms:
Load866(x0, x0, x1, x0, x2, x3)
Cond_Load866(TRUE, x0, x0, x1, x0, x2, x3)

### (5) DuplicateArgsRemoverProof (EQUIVALENT transformation)

Some arguments are removed because they only appear as duplicates.
We removed arguments according to the following replacements:

Cond_Load866(x1, x2, x3, x4, x5, x6, x7) → Cond_Load866(x1, x4, x5, x6, x7)

### (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:
Load783(i88, i95, i90) → Cond_Load783(i95 > 0 && i88 >= i95, i88, i95, i90)
Load866(i90, i95, i100, i103) → Cond_Load866(i103 > 0, i90, i95, i100, i103)
Cond_Load866(TRUE, i90, i95, i100, i103) → Load866(i90, i95, i100 + -1, i103 + -1)
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:

Boolean, Integer

The ITRS R consists of the following rules:
Load783(i88, i95, i90) → Cond_Load783(i95 > 0 && i88 >= i95, i88, i95, i90)
Load866(i90, i95, i100, i103) → Cond_Load866(i103 > 0, i90, i95, i100, i103)
Cond_Load866(TRUE, i90, i95, i100, i103) → Load866(i90, i95, i100 + -1, i103 + -1)

The integer pair graph contains the following rules and edges:
(0): LOAD783(i88[0], i95[0], i90[0]) → COND_LOAD783(i95[0] > 0 && i88[0] >= i95[0], i88[0], i95[0], i90[0])
(2): LOAD866(i90[2], i95[2], i100[2], i103[2]) → COND_LOAD866(i103[2] > 0, i90[2], i95[2], i100[2], i103[2])
(3): COND_LOAD866(TRUE, i90[3], i95[3], i100[3], i103[3]) → LOAD866(i90[3], i95[3], i100[3] + -1, i103[3] + -1)

(0) -> (1), if ((i88[0]* i88[1])∧(i95[0] > 0 && i88[0] >= i95[0]* TRUE)∧(i90[0]* i90[1])∧(i95[0]* i95[1]))

(1) -> (2), if ((i95[1]* i95[2])∧(i90[1]* i90[2])∧(i95[1]* i103[2])∧(i88[1]* i100[2]))

(1) -> (4), if ((i95[1]* 0)∧(i88[1]* i100[4])∧(i90[1]* i90[4])∧(i95[1]* i95[4]))

(2) -> (3), if ((i100[2]* i100[3])∧(i90[2]* i90[3])∧(i103[2]* i103[3])∧(i95[2]* i95[3])∧(i103[2] > 0* TRUE))

(3) -> (2), if ((i103[3] + -1* i103[2])∧(i95[3]* i95[2])∧(i90[3]* i90[2])∧(i100[3] + -1* i100[2]))

(3) -> (4), if ((i100[3] + -1* i100[4])∧(i95[3]* i95[4])∧(i103[3] + -1* 0)∧(i90[3]* i90[4]))

(4) -> (0), if ((i100[4]* i88[0])∧(i95[4]* i95[0])∧(i90[4] + 1* i90[0]))

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:

Boolean, Integer

R is empty.

The integer pair graph contains the following rules and edges:
(0): LOAD783(i88[0], i95[0], i90[0]) → COND_LOAD783(i95[0] > 0 && i88[0] >= i95[0], i88[0], i95[0], i90[0])
(2): LOAD866(i90[2], i95[2], i100[2], i103[2]) → COND_LOAD866(i103[2] > 0, i90[2], i95[2], i100[2], i103[2])
(3): COND_LOAD866(TRUE, i90[3], i95[3], i100[3], i103[3]) → LOAD866(i90[3], i95[3], i100[3] + -1, i103[3] + -1)

(0) -> (1), if ((i88[0]* i88[1])∧(i95[0] > 0 && i88[0] >= i95[0]* TRUE)∧(i90[0]* i90[1])∧(i95[0]* i95[1]))

(1) -> (2), if ((i95[1]* i95[2])∧(i90[1]* i90[2])∧(i95[1]* i103[2])∧(i88[1]* i100[2]))

(1) -> (4), if ((i95[1]* 0)∧(i88[1]* i100[4])∧(i90[1]* i90[4])∧(i95[1]* i95[4]))

(2) -> (3), if ((i100[2]* i100[3])∧(i90[2]* i90[3])∧(i103[2]* i103[3])∧(i95[2]* i95[3])∧(i103[2] > 0* TRUE))

(3) -> (2), if ((i103[3] + -1* i103[2])∧(i95[3]* i95[2])∧(i90[3]* i90[2])∧(i100[3] + -1* i100[2]))

(3) -> (4), if ((i100[3] + -1* i100[4])∧(i95[3]* i95[4])∧(i103[3] + -1* 0)∧(i90[3]* i90[4]))

(4) -> (0), if ((i100[4]* i88[0])∧(i95[4]* i95[0])∧(i90[4] + 1* i90[0]))

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 LOAD783(i88, i95, i90) → COND_LOAD783(&&(>(i95, 0), >=(i88, i95)), i88, i95, i90) the following chains were created:
• We consider the chain LOAD783(i88[0], i95[0], i90[0]) → COND_LOAD783(&&(>(i95[0], 0), >=(i88[0], i95[0])), i88[0], i95[0], i90[0]), COND_LOAD783(TRUE, i88[1], i95[1], i90[1]) → LOAD866(i90[1], i95[1], i88[1], i95[1]) which results in the following constraint:

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

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

(3)    (i95[0] + [-1] ≥ 0∧i88[0] + [-1]i95[0] ≥ 0 ⇒ (UIncreasing(COND_LOAD783(&&(>(i95[0], 0), >=(i88[0], i95[0])), i88[0], i95[0], i90[0])), ≥)∧[(-1)bni_26 + (-1)Bound*bni_26] + [(2)bni_26]i88[0] ≥ 0∧[(-1)bso_27] + i95[0] ≥ 0)

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

(4)    (i95[0] + [-1] ≥ 0∧i88[0] + [-1]i95[0] ≥ 0 ⇒ (UIncreasing(COND_LOAD783(&&(>(i95[0], 0), >=(i88[0], i95[0])), i88[0], i95[0], i90[0])), ≥)∧[(-1)bni_26 + (-1)Bound*bni_26] + [(2)bni_26]i88[0] ≥ 0∧[(-1)bso_27] + i95[0] ≥ 0)

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

(5)    (i95[0] + [-1] ≥ 0∧i88[0] + [-1]i95[0] ≥ 0 ⇒ (UIncreasing(COND_LOAD783(&&(>(i95[0], 0), >=(i88[0], i95[0])), i88[0], i95[0], i90[0])), ≥)∧[(-1)bni_26 + (-1)Bound*bni_26] + [(2)bni_26]i88[0] ≥ 0∧[(-1)bso_27] + i95[0] ≥ 0)

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

(6)    (i95[0] + [-1] ≥ 0∧i88[0] + [-1]i95[0] ≥ 0 ⇒ (UIncreasing(COND_LOAD783(&&(>(i95[0], 0), >=(i88[0], i95[0])), i88[0], i95[0], i90[0])), ≥)∧0 = 0∧[(-1)bni_26 + (-1)Bound*bni_26] + [(2)bni_26]i88[0] ≥ 0∧0 = 0∧[(-1)bso_27] + i95[0] ≥ 0)

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

(7)    (i95[0] ≥ 0∧i88[0] + [-1] + [-1]i95[0] ≥ 0 ⇒ (UIncreasing(COND_LOAD783(&&(>(i95[0], 0), >=(i88[0], i95[0])), i88[0], i95[0], i90[0])), ≥)∧0 = 0∧[(-1)bni_26 + (-1)Bound*bni_26] + [(2)bni_26]i88[0] ≥ 0∧0 = 0∧[1 + (-1)bso_27] + i95[0] ≥ 0)

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

(8)    (i95[0] ≥ 0∧i88[0] ≥ 0 ⇒ (UIncreasing(COND_LOAD783(&&(>(i95[0], 0), >=(i88[0], i95[0])), i88[0], i95[0], i90[0])), ≥)∧0 = 0∧[bni_26 + (-1)Bound*bni_26] + [(2)bni_26]i95[0] + [(2)bni_26]i88[0] ≥ 0∧0 = 0∧[1 + (-1)bso_27] + i95[0] ≥ 0)

For Pair COND_LOAD783(TRUE, i88, i95, i90) → LOAD866(i90, i95, i88, i95) the following chains were created:
• We consider the chain COND_LOAD783(TRUE, i88[1], i95[1], i90[1]) → LOAD866(i90[1], i95[1], i88[1], i95[1]), LOAD866(i90[2], i95[2], i100[2], i103[2]) → COND_LOAD866(>(i103[2], 0), i90[2], i95[2], i100[2], i103[2]) which results in the following constraint:

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

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

(11)    ((UIncreasing(LOAD866(i90[1], i95[1], i88[1], i95[1])), ≥)∧[(-1)bso_29] ≥ 0)

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

(12)    ((UIncreasing(LOAD866(i90[1], i95[1], i88[1], i95[1])), ≥)∧[(-1)bso_29] ≥ 0)

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

(13)    ((UIncreasing(LOAD866(i90[1], i95[1], i88[1], i95[1])), ≥)∧[(-1)bso_29] ≥ 0)

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

(14)    ((UIncreasing(LOAD866(i90[1], i95[1], i88[1], i95[1])), ≥)∧0 = 0∧0 = 0∧0 = 0∧[(-1)bso_29] ≥ 0)

• We consider the chain COND_LOAD783(TRUE, i88[1], i95[1], i90[1]) → LOAD866(i90[1], i95[1], i88[1], i95[1]), LOAD866(i90[4], i95[4], i100[4], 0) → LOAD783(i100[4], i95[4], +(i90[4], 1)) which results in the following constraint:

We simplified constraint (15) using rules (III), (IV) which results in the following new constraint:

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

(17)    ((UIncreasing(LOAD866(i90[1], i95[1], i88[1], i95[1])), ≥)∧[(-1)bso_29] ≥ 0)

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

(18)    ((UIncreasing(LOAD866(i90[1], i95[1], i88[1], i95[1])), ≥)∧[(-1)bso_29] ≥ 0)

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

(19)    ((UIncreasing(LOAD866(i90[1], i95[1], i88[1], i95[1])), ≥)∧[(-1)bso_29] ≥ 0)

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

(20)    ((UIncreasing(LOAD866(i90[1], i95[1], i88[1], i95[1])), ≥)∧0 = 0∧0 = 0∧[(-1)bso_29] ≥ 0)

For Pair LOAD866(i90, i95, i100, i103) → COND_LOAD866(>(i103, 0), i90, i95, i100, i103) the following chains were created:
• We consider the chain LOAD866(i90[2], i95[2], i100[2], i103[2]) → COND_LOAD866(>(i103[2], 0), i90[2], i95[2], i100[2], i103[2]), COND_LOAD866(TRUE, i90[3], i95[3], i100[3], i103[3]) → LOAD866(i90[3], i95[3], +(i100[3], -1), +(i103[3], -1)) which results in the following constraint:

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

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

(23)    (i103[2] + [-1] ≥ 0 ⇒ (UIncreasing(COND_LOAD866(>(i103[2], 0), i90[2], i95[2], i100[2], i103[2])), ≥)∧[(-1)bni_30 + (-1)Bound*bni_30] + [(-1)bni_30]i103[2] + [(2)bni_30]i100[2] ≥ 0∧[(-1)bso_31] ≥ 0)

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

(24)    (i103[2] + [-1] ≥ 0 ⇒ (UIncreasing(COND_LOAD866(>(i103[2], 0), i90[2], i95[2], i100[2], i103[2])), ≥)∧[(-1)bni_30 + (-1)Bound*bni_30] + [(-1)bni_30]i103[2] + [(2)bni_30]i100[2] ≥ 0∧[(-1)bso_31] ≥ 0)

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

(25)    (i103[2] + [-1] ≥ 0 ⇒ (UIncreasing(COND_LOAD866(>(i103[2], 0), i90[2], i95[2], i100[2], i103[2])), ≥)∧[(-1)bni_30 + (-1)Bound*bni_30] + [(-1)bni_30]i103[2] + [(2)bni_30]i100[2] ≥ 0∧[(-1)bso_31] ≥ 0)

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

(26)    (i103[2] + [-1] ≥ 0 ⇒ (UIncreasing(COND_LOAD866(>(i103[2], 0), i90[2], i95[2], i100[2], i103[2])), ≥)∧[(2)bni_30] = 0∧0 = 0∧0 = 0∧[(-1)bni_30 + (-1)Bound*bni_30] + [(-1)bni_30]i103[2] ≥ 0∧0 = 0∧0 = 0∧0 = 0∧[(-1)bso_31] ≥ 0)

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

(27)    (i103[2] ≥ 0 ⇒ (UIncreasing(COND_LOAD866(>(i103[2], 0), i90[2], i95[2], i100[2], i103[2])), ≥)∧[(2)bni_30] = 0∧0 = 0∧0 = 0∧[(-2)bni_30 + (-1)Bound*bni_30] + [(-1)bni_30]i103[2] ≥ 0∧0 = 0∧0 = 0∧0 = 0∧[(-1)bso_31] ≥ 0)

For Pair COND_LOAD866(TRUE, i90, i95, i100, i103) → LOAD866(i90, i95, +(i100, -1), +(i103, -1)) the following chains were created:
• We consider the chain LOAD866(i90[2], i95[2], i100[2], i103[2]) → COND_LOAD866(>(i103[2], 0), i90[2], i95[2], i100[2], i103[2]), COND_LOAD866(TRUE, i90[3], i95[3], i100[3], i103[3]) → LOAD866(i90[3], i95[3], +(i100[3], -1), +(i103[3], -1)), LOAD866(i90[2], i95[2], i100[2], i103[2]) → COND_LOAD866(>(i103[2], 0), i90[2], i95[2], i100[2], i103[2]) which results in the following constraint:

(28)    (i100[2]=i100[3]i90[2]=i90[3]i103[2]=i103[3]i95[2]=i95[3]>(i103[2], 0)=TRUE+(i103[3], -1)=i103[2]1i95[3]=i95[2]1i90[3]=i90[2]1+(i100[3], -1)=i100[2]1COND_LOAD866(TRUE, i90[3], i95[3], i100[3], i103[3])≥NonInfC∧COND_LOAD866(TRUE, i90[3], i95[3], i100[3], i103[3])≥LOAD866(i90[3], i95[3], +(i100[3], -1), +(i103[3], -1))∧(UIncreasing(LOAD866(i90[3], i95[3], +(i100[3], -1), +(i103[3], -1))), ≥))

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

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

(30)    (i103[2] + [-1] ≥ 0 ⇒ (UIncreasing(LOAD866(i90[3], i95[3], +(i100[3], -1), +(i103[3], -1))), ≥)∧[(-1)bni_32 + (-1)Bound*bni_32] + [(-1)bni_32]i103[2] + [(2)bni_32]i100[2] ≥ 0∧[1 + (-1)bso_33] ≥ 0)

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

(31)    (i103[2] + [-1] ≥ 0 ⇒ (UIncreasing(LOAD866(i90[3], i95[3], +(i100[3], -1), +(i103[3], -1))), ≥)∧[(-1)bni_32 + (-1)Bound*bni_32] + [(-1)bni_32]i103[2] + [(2)bni_32]i100[2] ≥ 0∧[1 + (-1)bso_33] ≥ 0)

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

(32)    (i103[2] + [-1] ≥ 0 ⇒ (UIncreasing(LOAD866(i90[3], i95[3], +(i100[3], -1), +(i103[3], -1))), ≥)∧[(-1)bni_32 + (-1)Bound*bni_32] + [(-1)bni_32]i103[2] + [(2)bni_32]i100[2] ≥ 0∧[1 + (-1)bso_33] ≥ 0)

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

(33)    (i103[2] + [-1] ≥ 0 ⇒ (UIncreasing(LOAD866(i90[3], i95[3], +(i100[3], -1), +(i103[3], -1))), ≥)∧[(2)bni_32] = 0∧0 = 0∧0 = 0∧[(-1)bni_32 + (-1)Bound*bni_32] + [(-1)bni_32]i103[2] ≥ 0∧0 = 0∧0 = 0∧0 = 0∧[1 + (-1)bso_33] ≥ 0)

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

(34)    (i103[2] ≥ 0 ⇒ (UIncreasing(LOAD866(i90[3], i95[3], +(i100[3], -1), +(i103[3], -1))), ≥)∧[(2)bni_32] = 0∧0 = 0∧0 = 0∧[(-2)bni_32 + (-1)Bound*bni_32] + [(-1)bni_32]i103[2] ≥ 0∧0 = 0∧0 = 0∧0 = 0∧[1 + (-1)bso_33] ≥ 0)

• We consider the chain LOAD866(i90[2], i95[2], i100[2], i103[2]) → COND_LOAD866(>(i103[2], 0), i90[2], i95[2], i100[2], i103[2]), COND_LOAD866(TRUE, i90[3], i95[3], i100[3], i103[3]) → LOAD866(i90[3], i95[3], +(i100[3], -1), +(i103[3], -1)), LOAD866(i90[4], i95[4], i100[4], 0) → LOAD783(i100[4], i95[4], +(i90[4], 1)) which results in the following constraint:

(35)    (i100[2]=i100[3]i90[2]=i90[3]i103[2]=i103[3]i95[2]=i95[3]>(i103[2], 0)=TRUE+(i100[3], -1)=i100[4]i95[3]=i95[4]+(i103[3], -1)=0i90[3]=i90[4]COND_LOAD866(TRUE, i90[3], i95[3], i100[3], i103[3])≥NonInfC∧COND_LOAD866(TRUE, i90[3], i95[3], i100[3], i103[3])≥LOAD866(i90[3], i95[3], +(i100[3], -1), +(i103[3], -1))∧(UIncreasing(LOAD866(i90[3], i95[3], +(i100[3], -1), +(i103[3], -1))), ≥))

We simplified constraint (35) using rules (III), (IV) which results in the following new constraint:

(36)    (>(i103[2], 0)=TRUE+(i103[2], -1)=0COND_LOAD866(TRUE, i90[2], i95[2], i100[2], i103[2])≥NonInfC∧COND_LOAD866(TRUE, i90[2], i95[2], i100[2], i103[2])≥LOAD866(i90[2], i95[2], +(i100[2], -1), +(i103[2], -1))∧(UIncreasing(LOAD866(i90[3], i95[3], +(i100[3], -1), +(i103[3], -1))), ≥))

We simplified constraint (36) using rule (POLY_CONSTRAINTS) which results in the following new constraint:

(37)    (i103[2] + [-1] ≥ 0∧i103[2] + [-1] ≥ 0 ⇒ (UIncreasing(LOAD866(i90[3], i95[3], +(i100[3], -1), +(i103[3], -1))), ≥)∧[(-1)bni_32 + (-1)Bound*bni_32] + [(-1)bni_32]i103[2] + [(2)bni_32]i100[2] ≥ 0∧[1 + (-1)bso_33] ≥ 0)

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

(38)    (i103[2] + [-1] ≥ 0∧i103[2] + [-1] ≥ 0 ⇒ (UIncreasing(LOAD866(i90[3], i95[3], +(i100[3], -1), +(i103[3], -1))), ≥)∧[(-1)bni_32 + (-1)Bound*bni_32] + [(-1)bni_32]i103[2] + [(2)bni_32]i100[2] ≥ 0∧[1 + (-1)bso_33] ≥ 0)

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

(39)    (i103[2] + [-1] ≥ 0∧i103[2] + [-1] ≥ 0 ⇒ (UIncreasing(LOAD866(i90[3], i95[3], +(i100[3], -1), +(i103[3], -1))), ≥)∧[(-1)bni_32 + (-1)Bound*bni_32] + [(-1)bni_32]i103[2] + [(2)bni_32]i100[2] ≥ 0∧[1 + (-1)bso_33] ≥ 0)

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

(40)    (i103[2] + [-1] ≥ 0∧i103[2] + [-1] ≥ 0 ⇒ (UIncreasing(LOAD866(i90[3], i95[3], +(i100[3], -1), +(i103[3], -1))), ≥)∧[(2)bni_32] = 0∧0 = 0∧0 = 0∧[(-1)bni_32 + (-1)Bound*bni_32] + [(-1)bni_32]i103[2] ≥ 0∧0 = 0∧0 = 0∧0 = 0∧[1 + (-1)bso_33] ≥ 0)

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

(41)    (i103[2] ≥ 0∧i103[2] ≥ 0 ⇒ (UIncreasing(LOAD866(i90[3], i95[3], +(i100[3], -1), +(i103[3], -1))), ≥)∧[(2)bni_32] = 0∧0 = 0∧0 = 0∧[(-2)bni_32 + (-1)Bound*bni_32] + [(-1)bni_32]i103[2] ≥ 0∧0 = 0∧0 = 0∧0 = 0∧[1 + (-1)bso_33] ≥ 0)

For Pair LOAD866(i90, i95, i100, 0) → LOAD783(i100, i95, +(i90, 1)) the following chains were created:
• We consider the chain COND_LOAD783(TRUE, i88[1], i95[1], i90[1]) → LOAD866(i90[1], i95[1], i88[1], i95[1]), LOAD866(i90[4], i95[4], i100[4], 0) → LOAD783(i100[4], i95[4], +(i90[4], 1)), LOAD783(i88[0], i95[0], i90[0]) → COND_LOAD783(&&(>(i95[0], 0), >=(i88[0], i95[0])), i88[0], i95[0], i90[0]) which results in the following constraint:

We simplified constraint (42) using rules (III), (IV) which results in the following new constraint:

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

(44)    ((UIncreasing(LOAD783(i100[4], i95[4], +(i90[4], 1))), ≥)∧[(-1)bso_35] ≥ 0)

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

(45)    ((UIncreasing(LOAD783(i100[4], i95[4], +(i90[4], 1))), ≥)∧[(-1)bso_35] ≥ 0)

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

(46)    ((UIncreasing(LOAD783(i100[4], i95[4], +(i90[4], 1))), ≥)∧[(-1)bso_35] ≥ 0)

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

(47)    ((UIncreasing(LOAD783(i100[4], i95[4], +(i90[4], 1))), ≥)∧0 = 0∧0 = 0∧[(-1)bso_35] ≥ 0)

• We consider the chain COND_LOAD866(TRUE, i90[3], i95[3], i100[3], i103[3]) → LOAD866(i90[3], i95[3], +(i100[3], -1), +(i103[3], -1)), LOAD866(i90[4], i95[4], i100[4], 0) → LOAD783(i100[4], i95[4], +(i90[4], 1)), LOAD783(i88[0], i95[0], i90[0]) → COND_LOAD783(&&(>(i95[0], 0), >=(i88[0], i95[0])), i88[0], i95[0], i90[0]) which results in the following constraint:

We simplified constraint (48) using rules (III), (IV) which results in the following new constraint:

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

(50)    (i103[3] + [-1] ≥ 0 ⇒ (UIncreasing(LOAD783(i100[4], i95[4], +(i90[4], 1))), ≥)∧0 ≥ 0∧[(-1)bso_35] ≥ 0)

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

(51)    (i103[3] + [-1] ≥ 0 ⇒ (UIncreasing(LOAD783(i100[4], i95[4], +(i90[4], 1))), ≥)∧0 ≥ 0∧[(-1)bso_35] ≥ 0)

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

(52)    (i103[3] + [-1] ≥ 0 ⇒ (UIncreasing(LOAD783(i100[4], i95[4], +(i90[4], 1))), ≥)∧0 ≥ 0∧[(-1)bso_35] ≥ 0)

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

(53)    (i103[3] + [-1] ≥ 0 ⇒ (UIncreasing(LOAD783(i100[4], i95[4], +(i90[4], 1))), ≥)∧0 = 0∧0 = 0∧0 = 0∧0 ≥ 0∧0 = 0∧0 = 0∧0 = 0∧[(-1)bso_35] ≥ 0)

To summarize, we get the following constraints P for the following pairs.
• LOAD783(i88, i95, i90) → COND_LOAD783(&&(>(i95, 0), >=(i88, i95)), i88, i95, i90)
• (i95[0] ≥ 0∧i88[0] ≥ 0 ⇒ (UIncreasing(COND_LOAD783(&&(>(i95[0], 0), >=(i88[0], i95[0])), i88[0], i95[0], i90[0])), ≥)∧0 = 0∧[bni_26 + (-1)Bound*bni_26] + [(2)bni_26]i95[0] + [(2)bni_26]i88[0] ≥ 0∧0 = 0∧[1 + (-1)bso_27] + i95[0] ≥ 0)

• ((UIncreasing(LOAD866(i90[1], i95[1], i88[1], i95[1])), ≥)∧0 = 0∧0 = 0∧0 = 0∧[(-1)bso_29] ≥ 0)
• ((UIncreasing(LOAD866(i90[1], i95[1], i88[1], i95[1])), ≥)∧0 = 0∧0 = 0∧[(-1)bso_29] ≥ 0)

• LOAD866(i90, i95, i100, i103) → COND_LOAD866(>(i103, 0), i90, i95, i100, i103)
• (i103[2] ≥ 0 ⇒ (UIncreasing(COND_LOAD866(>(i103[2], 0), i90[2], i95[2], i100[2], i103[2])), ≥)∧[(2)bni_30] = 0∧0 = 0∧0 = 0∧[(-2)bni_30 + (-1)Bound*bni_30] + [(-1)bni_30]i103[2] ≥ 0∧0 = 0∧0 = 0∧0 = 0∧[(-1)bso_31] ≥ 0)

• COND_LOAD866(TRUE, i90, i95, i100, i103) → LOAD866(i90, i95, +(i100, -1), +(i103, -1))
• (i103[2] ≥ 0 ⇒ (UIncreasing(LOAD866(i90[3], i95[3], +(i100[3], -1), +(i103[3], -1))), ≥)∧[(2)bni_32] = 0∧0 = 0∧0 = 0∧[(-2)bni_32 + (-1)Bound*bni_32] + [(-1)bni_32]i103[2] ≥ 0∧0 = 0∧0 = 0∧0 = 0∧[1 + (-1)bso_33] ≥ 0)
• (i103[2] ≥ 0∧i103[2] ≥ 0 ⇒ (UIncreasing(LOAD866(i90[3], i95[3], +(i100[3], -1), +(i103[3], -1))), ≥)∧[(2)bni_32] = 0∧0 = 0∧0 = 0∧[(-2)bni_32 + (-1)Bound*bni_32] + [(-1)bni_32]i103[2] ≥ 0∧0 = 0∧0 = 0∧0 = 0∧[1 + (-1)bso_33] ≥ 0)

• ((UIncreasing(LOAD783(i100[4], i95[4], +(i90[4], 1))), ≥)∧0 = 0∧0 = 0∧[(-1)bso_35] ≥ 0)
• (i103[3] + [-1] ≥ 0 ⇒ (UIncreasing(LOAD783(i100[4], i95[4], +(i90[4], 1))), ≥)∧0 = 0∧0 = 0∧0 = 0∧0 ≥ 0∧0 = 0∧0 = 0∧0 = 0∧[(-1)bso_35] ≥ 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(LOAD783(x1, x2, x3)) = [-1] + [2]x1
POL(COND_LOAD783(x1, x2, x3, x4)) = [-1] + [-1]x3 + [2]x2
POL(&&(x1, x2)) = 0
POL(>(x1, x2)) = [-1]
POL(0) = 0
POL(>=(x1, x2)) = [-1]
POL(LOAD866(x1, x2, x3, x4)) = [-1] + [-1]x4 + [2]x3
POL(COND_LOAD866(x1, x2, x3, x4, x5)) = [-1] + [-1]x5 + [2]x4
POL(+(x1, x2)) = x1 + x2
POL(-1) = [-1]
POL(1) = [1]

The following pairs are in P>:

COND_LOAD866(TRUE, i90[3], i95[3], i100[3], i103[3]) → LOAD866(i90[3], i95[3], +(i100[3], -1), +(i103[3], -1))

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:

FALSE1&&(TRUE, FALSE)1

### (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:
(2): LOAD866(i90[2], i95[2], i100[2], i103[2]) → COND_LOAD866(i103[2] > 0, i90[2], i95[2], i100[2], i103[2])

(1) -> (2), if ((i95[1]* i95[2])∧(i90[1]* i90[2])∧(i95[1]* i103[2])∧(i88[1]* i100[2]))

(1) -> (4), if ((i95[1]* 0)∧(i88[1]* i100[4])∧(i90[1]* i90[4])∧(i95[1]* i95[4]))

The set Q consists of the following terms:

### (14) IDependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 3 less nodes.

### (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:
(2): LOAD866(i90[2], i95[2], i100[2], i103[2]) → COND_LOAD866(i103[2] > 0, i90[2], i95[2], i100[2], i103[2])
(3): COND_LOAD866(TRUE, i90[3], i95[3], i100[3], i103[3]) → LOAD866(i90[3], i95[3], i100[3] + -1, i103[3] + -1)

(1) -> (2), if ((i95[1]* i95[2])∧(i90[1]* i90[2])∧(i95[1]* i103[2])∧(i88[1]* i100[2]))

(3) -> (2), if ((i103[3] + -1* i103[2])∧(i95[3]* i95[2])∧(i90[3]* i90[2])∧(i100[3] + -1* i100[2]))

(2) -> (3), if ((i100[2]* i100[3])∧(i90[2]* i90[3])∧(i103[2]* i103[3])∧(i95[2]* i95[3])∧(i103[2] > 0* TRUE))

(1) -> (4), if ((i95[1]* 0)∧(i88[1]* i100[4])∧(i90[1]* i90[4])∧(i95[1]* i95[4]))

(3) -> (4), if ((i100[3] + -1* i100[4])∧(i95[3]* i95[4])∧(i103[3] + -1* 0)∧(i90[3]* i90[4]))

The set Q consists of the following terms:

### (17) IDependencyGraphProof (EQUIVALENT transformation)

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

### (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:
(3): COND_LOAD866(TRUE, i90[3], i95[3], i100[3], i103[3]) → LOAD866(i90[3], i95[3], i100[3] + -1, i103[3] + -1)
(2): LOAD866(i90[2], i95[2], i100[2], i103[2]) → COND_LOAD866(i103[2] > 0, i90[2], i95[2], i100[2], i103[2])

(3) -> (2), if ((i103[3] + -1* i103[2])∧(i95[3]* i95[2])∧(i90[3]* i90[2])∧(i100[3] + -1* i100[2]))

(2) -> (3), if ((i100[2]* i100[3])∧(i90[2]* i90[3])∧(i103[2]* i103[3])∧(i95[2]* i95[3])∧(i103[2] > 0* TRUE))

The set Q consists of the following terms:

### (19) 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_LOAD866(TRUE, i90[3], i95[3], i100[3], i103[3]) → LOAD866(i90[3], i95[3], +(i100[3], -1), +(i103[3], -1)) the following chains were created:
• We consider the chain LOAD866(i90[2], i95[2], i100[2], i103[2]) → COND_LOAD866(>(i103[2], 0), i90[2], i95[2], i100[2], i103[2]), COND_LOAD866(TRUE, i90[3], i95[3], i100[3], i103[3]) → LOAD866(i90[3], i95[3], +(i100[3], -1), +(i103[3], -1)), LOAD866(i90[2], i95[2], i100[2], i103[2]) → COND_LOAD866(>(i103[2], 0), i90[2], i95[2], i100[2], i103[2]) which results in the following constraint:

(1)    (i100[2]=i100[3]i90[2]=i90[3]i103[2]=i103[3]i95[2]=i95[3]>(i103[2], 0)=TRUE+(i103[3], -1)=i103[2]1i95[3]=i95[2]1i90[3]=i90[2]1+(i100[3], -1)=i100[2]1COND_LOAD866(TRUE, i90[3], i95[3], i100[3], i103[3])≥NonInfC∧COND_LOAD866(TRUE, i90[3], i95[3], i100[3], i103[3])≥LOAD866(i90[3], i95[3], +(i100[3], -1), +(i103[3], -1))∧(UIncreasing(LOAD866(i90[3], i95[3], +(i100[3], -1), +(i103[3], -1))), ≥))

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)    (i103[2] + [-1] ≥ 0 ⇒ (UIncreasing(LOAD866(i90[3], i95[3], +(i100[3], -1), +(i103[3], -1))), ≥)∧[(2)bni_15 + (-1)Bound*bni_15] + [bni_15]i103[2] ≥ 0∧[1 + (-1)bso_16] ≥ 0)

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

(4)    (i103[2] + [-1] ≥ 0 ⇒ (UIncreasing(LOAD866(i90[3], i95[3], +(i100[3], -1), +(i103[3], -1))), ≥)∧[(2)bni_15 + (-1)Bound*bni_15] + [bni_15]i103[2] ≥ 0∧[1 + (-1)bso_16] ≥ 0)

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

(5)    (i103[2] + [-1] ≥ 0 ⇒ (UIncreasing(LOAD866(i90[3], i95[3], +(i100[3], -1), +(i103[3], -1))), ≥)∧[(2)bni_15 + (-1)Bound*bni_15] + [bni_15]i103[2] ≥ 0∧[1 + (-1)bso_16] ≥ 0)

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

(6)    (i103[2] + [-1] ≥ 0 ⇒ (UIncreasing(LOAD866(i90[3], i95[3], +(i100[3], -1), +(i103[3], -1))), ≥)∧0 = 0∧0 = 0∧0 = 0∧[(2)bni_15 + (-1)Bound*bni_15] + [bni_15]i103[2] ≥ 0∧0 = 0∧0 = 0∧0 = 0∧[1 + (-1)bso_16] ≥ 0)

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

(7)    (i103[2] ≥ 0 ⇒ (UIncreasing(LOAD866(i90[3], i95[3], +(i100[3], -1), +(i103[3], -1))), ≥)∧0 = 0∧0 = 0∧0 = 0∧[(3)bni_15 + (-1)Bound*bni_15] + [bni_15]i103[2] ≥ 0∧0 = 0∧0 = 0∧0 = 0∧[1 + (-1)bso_16] ≥ 0)

For Pair LOAD866(i90[2], i95[2], i100[2], i103[2]) → COND_LOAD866(>(i103[2], 0), i90[2], i95[2], i100[2], i103[2]) the following chains were created:
• We consider the chain LOAD866(i90[2], i95[2], i100[2], i103[2]) → COND_LOAD866(>(i103[2], 0), i90[2], i95[2], i100[2], i103[2]), COND_LOAD866(TRUE, i90[3], i95[3], i100[3], i103[3]) → LOAD866(i90[3], i95[3], +(i100[3], -1), +(i103[3], -1)) which results in the following constraint:

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

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

(10)    (i103[2] + [-1] ≥ 0 ⇒ (UIncreasing(COND_LOAD866(>(i103[2], 0), i90[2], i95[2], i100[2], i103[2])), ≥)∧[(2)bni_17 + (-1)Bound*bni_17] + [bni_17]i103[2] ≥ 0∧[(-1)bso_18] ≥ 0)

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

(11)    (i103[2] + [-1] ≥ 0 ⇒ (UIncreasing(COND_LOAD866(>(i103[2], 0), i90[2], i95[2], i100[2], i103[2])), ≥)∧[(2)bni_17 + (-1)Bound*bni_17] + [bni_17]i103[2] ≥ 0∧[(-1)bso_18] ≥ 0)

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

(12)    (i103[2] + [-1] ≥ 0 ⇒ (UIncreasing(COND_LOAD866(>(i103[2], 0), i90[2], i95[2], i100[2], i103[2])), ≥)∧[(2)bni_17 + (-1)Bound*bni_17] + [bni_17]i103[2] ≥ 0∧[(-1)bso_18] ≥ 0)

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

(13)    (i103[2] + [-1] ≥ 0 ⇒ (UIncreasing(COND_LOAD866(>(i103[2], 0), i90[2], i95[2], i100[2], i103[2])), ≥)∧0 = 0∧0 = 0∧0 = 0∧[(2)bni_17 + (-1)Bound*bni_17] + [bni_17]i103[2] ≥ 0∧0 = 0∧0 = 0∧0 = 0∧[(-1)bso_18] ≥ 0)

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

(14)    (i103[2] ≥ 0 ⇒ (UIncreasing(COND_LOAD866(>(i103[2], 0), i90[2], i95[2], i100[2], i103[2])), ≥)∧0 = 0∧0 = 0∧0 = 0∧[(3)bni_17 + (-1)Bound*bni_17] + [bni_17]i103[2] ≥ 0∧0 = 0∧0 = 0∧0 = 0∧[(-1)bso_18] ≥ 0)

To summarize, we get the following constraints P for the following pairs.
• COND_LOAD866(TRUE, i90[3], i95[3], i100[3], i103[3]) → LOAD866(i90[3], i95[3], +(i100[3], -1), +(i103[3], -1))
• (i103[2] ≥ 0 ⇒ (UIncreasing(LOAD866(i90[3], i95[3], +(i100[3], -1), +(i103[3], -1))), ≥)∧0 = 0∧0 = 0∧0 = 0∧[(3)bni_15 + (-1)Bound*bni_15] + [bni_15]i103[2] ≥ 0∧0 = 0∧0 = 0∧0 = 0∧[1 + (-1)bso_16] ≥ 0)

• LOAD866(i90[2], i95[2], i100[2], i103[2]) → COND_LOAD866(>(i103[2], 0), i90[2], i95[2], i100[2], i103[2])
• (i103[2] ≥ 0 ⇒ (UIncreasing(COND_LOAD866(>(i103[2], 0), i90[2], i95[2], i100[2], i103[2])), ≥)∧0 = 0∧0 = 0∧0 = 0∧[(3)bni_17 + (-1)Bound*bni_17] + [bni_17]i103[2] ≥ 0∧0 = 0∧0 = 0∧0 = 0∧[(-1)bso_18] ≥ 0)

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

POL(TRUE) = [2]
POL(FALSE) = 0
POL(COND_LOAD866(x1, x2, x3, x4, x5)) = [2] + x5
POL(LOAD866(x1, x2, x3, x4)) = [2] + x4
POL(+(x1, x2)) = x1 + x2
POL(-1) = [-1]
POL(>(x1, x2)) = [-1]
POL(0) = 0

The following pairs are in P>:

COND_LOAD866(TRUE, i90[3], i95[3], i100[3], i103[3]) → LOAD866(i90[3], i95[3], +(i100[3], -1), +(i103[3], -1))

The following pairs are in Pbound:

COND_LOAD866(TRUE, i90[3], i95[3], i100[3], i103[3]) → LOAD866(i90[3], i95[3], +(i100[3], -1), +(i103[3], -1))

The following pairs are in P:

There are no usable rules.

### (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:
(2): LOAD866(i90[2], i95[2], i100[2], i103[2]) → COND_LOAD866(i103[2] > 0, i90[2], i95[2], i100[2], i103[2])

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:
none

R is empty.

The integer pair graph is empty.

The set Q consists of the following terms: