### (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: Mod

### (1) JBC2FIG (SOUND transformation)

Constructed FIGraph.

### (2) Obligation:

FIGraph based on JBC Program:
Graph of 164 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:
Load720(i90, i84, i90) → Cond_Load720(i90 > 0 && i84 >= i90, i90, i84, i90)
Load793(i90, i90, i90, i94, i97) → Cond_Load793(i97 > 0, i90, i90, i90, i94, i97)
Cond_Load793(TRUE, i90, i90, i90, i94, i97) → Load793(i90, i90, i90, i94 + -1, i97 + -1)
The set Q consists of the following terms:
Cond_Load793(TRUE, x0, x0, x0, x1, x2)

### (5) DuplicateArgsRemoverProof (EQUIVALENT transformation)

Some arguments are removed because they only appear as duplicates.
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:
Load720(i84, i90) → Cond_Load720(i90 > 0 && i84 >= i90, i84, i90)
Cond_Load793(TRUE, i90, i94, i97) → Load793(i90, i94 + -1, i97 + -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:
Load720(i84, i90) → Cond_Load720(i90 > 0 && i84 >= i90, i84, i90)
Cond_Load793(TRUE, i90, i94, i97) → Load793(i90, i94 + -1, i97 + -1)

The integer pair graph contains the following rules and edges:
(0): LOAD720(i84[0], i90[0]) → COND_LOAD720(i90[0] > 0 && i84[0] >= i90[0], i84[0], i90[0])
(3): COND_LOAD793(TRUE, i90[3], i94[3], i97[3]) → LOAD793(i90[3], i94[3] + -1, i97[3] + -1)

(0) -> (1), if ((i90[0]* i90[1])∧(i84[0]* i84[1])∧(i90[0] > 0 && i84[0] >= i90[0]* TRUE))

(1) -> (2), if ((i90[1]* i97[2])∧(i90[1]* i90[2])∧(i84[1]* i94[2]))

(1) -> (4), if ((i84[1]* i94[4])∧(i90[1]* 0)∧(i90[1]* i90[4]))

(2) -> (3), if ((i90[2]* i90[3])∧(i94[2]* i94[3])∧(i97[2] > 0* TRUE)∧(i97[2]* i97[3]))

(3) -> (2), if ((i90[3]* i90[2])∧(i97[3] + -1* i97[2])∧(i94[3] + -1* i94[2]))

(3) -> (4), if ((i97[3] + -1* 0)∧(i90[3]* i90[4])∧(i94[3] + -1* i94[4]))

(4) -> (0), if ((i94[4]* i84[0])∧(i90[4]* 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): LOAD720(i84[0], i90[0]) → COND_LOAD720(i90[0] > 0 && i84[0] >= i90[0], i84[0], i90[0])
(3): COND_LOAD793(TRUE, i90[3], i94[3], i97[3]) → LOAD793(i90[3], i94[3] + -1, i97[3] + -1)

(0) -> (1), if ((i90[0]* i90[1])∧(i84[0]* i84[1])∧(i90[0] > 0 && i84[0] >= i90[0]* TRUE))

(1) -> (2), if ((i90[1]* i97[2])∧(i90[1]* i90[2])∧(i84[1]* i94[2]))

(1) -> (4), if ((i84[1]* i94[4])∧(i90[1]* 0)∧(i90[1]* i90[4]))

(2) -> (3), if ((i90[2]* i90[3])∧(i94[2]* i94[3])∧(i97[2] > 0* TRUE)∧(i97[2]* i97[3]))

(3) -> (2), if ((i90[3]* i90[2])∧(i97[3] + -1* i97[2])∧(i94[3] + -1* i94[2]))

(3) -> (4), if ((i97[3] + -1* 0)∧(i90[3]* i90[4])∧(i94[3] + -1* i94[4]))

(4) -> (0), if ((i94[4]* i84[0])∧(i90[4]* 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 LOAD720(i84, i90) → COND_LOAD720(&&(>(i90, 0), >=(i84, i90)), i84, i90) the following chains were created:
• We consider the chain LOAD720(i84[0], i90[0]) → COND_LOAD720(&&(>(i90[0], 0), >=(i84[0], i90[0])), i84[0], i90[0]), COND_LOAD720(TRUE, i84[1], i90[1]) → LOAD793(i90[1], i84[1], i90[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)    (i90[0] + [-1] ≥ 0∧i84[0] + [-1]i90[0] ≥ 0 ⇒ (UIncreasing(COND_LOAD720(&&(>(i90[0], 0), >=(i84[0], i90[0])), i84[0], i90[0])), ≥)∧[(-1)bni_22 + (-1)Bound*bni_22] + [bni_22]i84[0] ≥ 0∧[(-1)bso_23] + i90[0] ≥ 0)

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

(4)    (i90[0] + [-1] ≥ 0∧i84[0] + [-1]i90[0] ≥ 0 ⇒ (UIncreasing(COND_LOAD720(&&(>(i90[0], 0), >=(i84[0], i90[0])), i84[0], i90[0])), ≥)∧[(-1)bni_22 + (-1)Bound*bni_22] + [bni_22]i84[0] ≥ 0∧[(-1)bso_23] + i90[0] ≥ 0)

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

(5)    (i90[0] + [-1] ≥ 0∧i84[0] + [-1]i90[0] ≥ 0 ⇒ (UIncreasing(COND_LOAD720(&&(>(i90[0], 0), >=(i84[0], i90[0])), i84[0], i90[0])), ≥)∧[(-1)bni_22 + (-1)Bound*bni_22] + [bni_22]i84[0] ≥ 0∧[(-1)bso_23] + i90[0] ≥ 0)

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

(6)    (i90[0] ≥ 0∧i84[0] + [-1] + [-1]i90[0] ≥ 0 ⇒ (UIncreasing(COND_LOAD720(&&(>(i90[0], 0), >=(i84[0], i90[0])), i84[0], i90[0])), ≥)∧[(-1)bni_22 + (-1)Bound*bni_22] + [bni_22]i84[0] ≥ 0∧[1 + (-1)bso_23] + i90[0] ≥ 0)

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

(7)    (i90[0] ≥ 0∧i84[0] ≥ 0 ⇒ (UIncreasing(COND_LOAD720(&&(>(i90[0], 0), >=(i84[0], i90[0])), i84[0], i90[0])), ≥)∧[(-1)Bound*bni_22] + [bni_22]i90[0] + [bni_22]i84[0] ≥ 0∧[1 + (-1)bso_23] + i90[0] ≥ 0)

For Pair COND_LOAD720(TRUE, i84, i90) → LOAD793(i90, i84, i90) the following chains were created:
• We consider the chain COND_LOAD720(TRUE, i84[1], i90[1]) → LOAD793(i90[1], i84[1], i90[1]), LOAD793(i90[2], i94[2], i97[2]) → COND_LOAD793(>(i97[2], 0), i90[2], i94[2], i97[2]) 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)    ((UIncreasing(LOAD793(i90[1], i84[1], i90[1])), ≥)∧[(-1)bso_25] ≥ 0)

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

(11)    ((UIncreasing(LOAD793(i90[1], i84[1], i90[1])), ≥)∧[(-1)bso_25] ≥ 0)

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

(12)    ((UIncreasing(LOAD793(i90[1], i84[1], i90[1])), ≥)∧[(-1)bso_25] ≥ 0)

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

(13)    ((UIncreasing(LOAD793(i90[1], i84[1], i90[1])), ≥)∧0 = 0∧0 = 0∧[(-1)bso_25] ≥ 0)

• We consider the chain COND_LOAD720(TRUE, i84[1], i90[1]) → LOAD793(i90[1], i84[1], i90[1]), LOAD793(i90[4], i94[4], 0) → LOAD720(i94[4], i90[4]) which results in the following constraint:

We simplified constraint (14) using rules (III), (IV) which results in the following new constraint:

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

(16)    ((UIncreasing(LOAD793(i90[1], i84[1], i90[1])), ≥)∧[(-1)bso_25] ≥ 0)

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

(17)    ((UIncreasing(LOAD793(i90[1], i84[1], i90[1])), ≥)∧[(-1)bso_25] ≥ 0)

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

(18)    ((UIncreasing(LOAD793(i90[1], i84[1], i90[1])), ≥)∧[(-1)bso_25] ≥ 0)

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

(19)    ((UIncreasing(LOAD793(i90[1], i84[1], i90[1])), ≥)∧0 = 0∧[(-1)bso_25] ≥ 0)

For Pair LOAD793(i90, i94, i97) → COND_LOAD793(>(i97, 0), i90, i94, i97) the following chains were created:
• We consider the chain LOAD793(i90[2], i94[2], i97[2]) → COND_LOAD793(>(i97[2], 0), i90[2], i94[2], i97[2]), COND_LOAD793(TRUE, i90[3], i94[3], i97[3]) → LOAD793(i90[3], +(i94[3], -1), +(i97[3], -1)) which results in the following constraint:

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

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

(22)    (i97[2] + [-1] ≥ 0 ⇒ (UIncreasing(COND_LOAD793(>(i97[2], 0), i90[2], i94[2], i97[2])), ≥)∧[(-1)bni_26 + (-1)Bound*bni_26] + [(-1)bni_26]i97[2] + [bni_26]i94[2] ≥ 0∧[(-1)bso_27] ≥ 0)

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

(23)    (i97[2] + [-1] ≥ 0 ⇒ (UIncreasing(COND_LOAD793(>(i97[2], 0), i90[2], i94[2], i97[2])), ≥)∧[(-1)bni_26 + (-1)Bound*bni_26] + [(-1)bni_26]i97[2] + [bni_26]i94[2] ≥ 0∧[(-1)bso_27] ≥ 0)

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

(24)    (i97[2] + [-1] ≥ 0 ⇒ (UIncreasing(COND_LOAD793(>(i97[2], 0), i90[2], i94[2], i97[2])), ≥)∧[(-1)bni_26 + (-1)Bound*bni_26] + [(-1)bni_26]i97[2] + [bni_26]i94[2] ≥ 0∧[(-1)bso_27] ≥ 0)

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

(25)    (i97[2] + [-1] ≥ 0 ⇒ (UIncreasing(COND_LOAD793(>(i97[2], 0), i90[2], i94[2], i97[2])), ≥)∧[bni_26] = 0∧0 = 0∧[(-1)bni_26 + (-1)Bound*bni_26] + [(-1)bni_26]i97[2] ≥ 0∧0 = 0∧0 = 0∧[(-1)bso_27] ≥ 0)

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

(26)    (i97[2] ≥ 0 ⇒ (UIncreasing(COND_LOAD793(>(i97[2], 0), i90[2], i94[2], i97[2])), ≥)∧[bni_26] = 0∧0 = 0∧[(-2)bni_26 + (-1)Bound*bni_26] + [(-1)bni_26]i97[2] ≥ 0∧0 = 0∧0 = 0∧[(-1)bso_27] ≥ 0)

For Pair COND_LOAD793(TRUE, i90, i94, i97) → LOAD793(i90, +(i94, -1), +(i97, -1)) the following chains were created:
• We consider the chain LOAD793(i90[2], i94[2], i97[2]) → COND_LOAD793(>(i97[2], 0), i90[2], i94[2], i97[2]), COND_LOAD793(TRUE, i90[3], i94[3], i97[3]) → LOAD793(i90[3], +(i94[3], -1), +(i97[3], -1)), LOAD793(i90[2], i94[2], i97[2]) → COND_LOAD793(>(i97[2], 0), i90[2], i94[2], i97[2]) which results in the following constraint:

We simplified constraint (27) using rules (III), (IV) which results in the following new constraint:

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

(29)    (i97[2] + [-1] ≥ 0 ⇒ (UIncreasing(LOAD793(i90[3], +(i94[3], -1), +(i97[3], -1))), ≥)∧[(-1)bni_28 + (-1)Bound*bni_28] + [(-1)bni_28]i97[2] + [bni_28]i94[2] ≥ 0∧[(-1)bso_29] ≥ 0)

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

(30)    (i97[2] + [-1] ≥ 0 ⇒ (UIncreasing(LOAD793(i90[3], +(i94[3], -1), +(i97[3], -1))), ≥)∧[(-1)bni_28 + (-1)Bound*bni_28] + [(-1)bni_28]i97[2] + [bni_28]i94[2] ≥ 0∧[(-1)bso_29] ≥ 0)

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

(31)    (i97[2] + [-1] ≥ 0 ⇒ (UIncreasing(LOAD793(i90[3], +(i94[3], -1), +(i97[3], -1))), ≥)∧[(-1)bni_28 + (-1)Bound*bni_28] + [(-1)bni_28]i97[2] + [bni_28]i94[2] ≥ 0∧[(-1)bso_29] ≥ 0)

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

(32)    (i97[2] + [-1] ≥ 0 ⇒ (UIncreasing(LOAD793(i90[3], +(i94[3], -1), +(i97[3], -1))), ≥)∧[bni_28] = 0∧0 = 0∧[(-1)bni_28 + (-1)Bound*bni_28] + [(-1)bni_28]i97[2] ≥ 0∧0 = 0∧0 = 0∧[(-1)bso_29] ≥ 0)

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

(33)    (i97[2] ≥ 0 ⇒ (UIncreasing(LOAD793(i90[3], +(i94[3], -1), +(i97[3], -1))), ≥)∧[bni_28] = 0∧0 = 0∧[(-2)bni_28 + (-1)Bound*bni_28] + [(-1)bni_28]i97[2] ≥ 0∧0 = 0∧0 = 0∧[(-1)bso_29] ≥ 0)

• We consider the chain LOAD793(i90[2], i94[2], i97[2]) → COND_LOAD793(>(i97[2], 0), i90[2], i94[2], i97[2]), COND_LOAD793(TRUE, i90[3], i94[3], i97[3]) → LOAD793(i90[3], +(i94[3], -1), +(i97[3], -1)), LOAD793(i90[4], i94[4], 0) → LOAD720(i94[4], i90[4]) which results in the following constraint:

We simplified constraint (34) using rules (III), (IV) which results in the following new constraint:

We simplified constraint (35) using rule (POLY_CONSTRAINTS) which results in the following new constraint:

(36)    (i97[2] + [-1] ≥ 0∧i97[2] + [-1] ≥ 0 ⇒ (UIncreasing(LOAD793(i90[3], +(i94[3], -1), +(i97[3], -1))), ≥)∧[(-1)bni_28 + (-1)Bound*bni_28] + [(-1)bni_28]i97[2] + [bni_28]i94[2] ≥ 0∧[(-1)bso_29] ≥ 0)

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

(37)    (i97[2] + [-1] ≥ 0∧i97[2] + [-1] ≥ 0 ⇒ (UIncreasing(LOAD793(i90[3], +(i94[3], -1), +(i97[3], -1))), ≥)∧[(-1)bni_28 + (-1)Bound*bni_28] + [(-1)bni_28]i97[2] + [bni_28]i94[2] ≥ 0∧[(-1)bso_29] ≥ 0)

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

(38)    (i97[2] + [-1] ≥ 0∧i97[2] + [-1] ≥ 0 ⇒ (UIncreasing(LOAD793(i90[3], +(i94[3], -1), +(i97[3], -1))), ≥)∧[(-1)bni_28 + (-1)Bound*bni_28] + [(-1)bni_28]i97[2] + [bni_28]i94[2] ≥ 0∧[(-1)bso_29] ≥ 0)

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

(39)    (i97[2] + [-1] ≥ 0∧i97[2] + [-1] ≥ 0 ⇒ (UIncreasing(LOAD793(i90[3], +(i94[3], -1), +(i97[3], -1))), ≥)∧[bni_28] = 0∧0 = 0∧[(-1)bni_28 + (-1)Bound*bni_28] + [(-1)bni_28]i97[2] ≥ 0∧0 = 0∧0 = 0∧[(-1)bso_29] ≥ 0)

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

(40)    (i97[2] ≥ 0∧i97[2] ≥ 0 ⇒ (UIncreasing(LOAD793(i90[3], +(i94[3], -1), +(i97[3], -1))), ≥)∧[bni_28] = 0∧0 = 0∧[(-2)bni_28 + (-1)Bound*bni_28] + [(-1)bni_28]i97[2] ≥ 0∧0 = 0∧0 = 0∧[(-1)bso_29] ≥ 0)

For Pair LOAD793(i90, i94, 0) → LOAD720(i94, i90) the following chains were created:
• We consider the chain LOAD793(i90[4], i94[4], 0) → LOAD720(i94[4], i90[4]), LOAD720(i84[0], i90[0]) → COND_LOAD720(&&(>(i90[0], 0), >=(i84[0], i90[0])), i84[0], i90[0]) which results in the following constraint:

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

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

(43)    ((UIncreasing(LOAD720(i94[4], i90[4])), ≥)∧[(-1)bso_31] ≥ 0)

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

(44)    ((UIncreasing(LOAD720(i94[4], i90[4])), ≥)∧[(-1)bso_31] ≥ 0)

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

(45)    ((UIncreasing(LOAD720(i94[4], i90[4])), ≥)∧[(-1)bso_31] ≥ 0)

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

(46)    ((UIncreasing(LOAD720(i94[4], i90[4])), ≥)∧0 = 0∧0 = 0∧[(-1)bso_31] ≥ 0)

To summarize, we get the following constraints P for the following pairs.
• (i90[0] ≥ 0∧i84[0] ≥ 0 ⇒ (UIncreasing(COND_LOAD720(&&(>(i90[0], 0), >=(i84[0], i90[0])), i84[0], i90[0])), ≥)∧[(-1)Bound*bni_22] + [bni_22]i90[0] + [bni_22]i84[0] ≥ 0∧[1 + (-1)bso_23] + i90[0] ≥ 0)

• ((UIncreasing(LOAD793(i90[1], i84[1], i90[1])), ≥)∧0 = 0∧0 = 0∧[(-1)bso_25] ≥ 0)
• ((UIncreasing(LOAD793(i90[1], i84[1], i90[1])), ≥)∧0 = 0∧[(-1)bso_25] ≥ 0)

• (i97[2] ≥ 0 ⇒ (UIncreasing(COND_LOAD793(>(i97[2], 0), i90[2], i94[2], i97[2])), ≥)∧[bni_26] = 0∧0 = 0∧[(-2)bni_26 + (-1)Bound*bni_26] + [(-1)bni_26]i97[2] ≥ 0∧0 = 0∧0 = 0∧[(-1)bso_27] ≥ 0)

• (i97[2] ≥ 0 ⇒ (UIncreasing(LOAD793(i90[3], +(i94[3], -1), +(i97[3], -1))), ≥)∧[bni_28] = 0∧0 = 0∧[(-2)bni_28 + (-1)Bound*bni_28] + [(-1)bni_28]i97[2] ≥ 0∧0 = 0∧0 = 0∧[(-1)bso_29] ≥ 0)
• (i97[2] ≥ 0∧i97[2] ≥ 0 ⇒ (UIncreasing(LOAD793(i90[3], +(i94[3], -1), +(i97[3], -1))), ≥)∧[bni_28] = 0∧0 = 0∧[(-2)bni_28 + (-1)Bound*bni_28] + [(-1)bni_28]i97[2] ≥ 0∧0 = 0∧0 = 0∧[(-1)bso_29] ≥ 0)

• ((UIncreasing(LOAD720(i94[4], i90[4])), ≥)∧0 = 0∧0 = 0∧[(-1)bso_31] ≥ 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(LOAD720(x1, x2)) = [-1] + x1
POL(COND_LOAD720(x1, x2, x3)) = [-1] + [-1]x3 + x2 + [-1]x1
POL(&&(x1, x2)) = 0
POL(>(x1, x2)) = [-1]
POL(0) = 0
POL(>=(x1, x2)) = [-1]
POL(LOAD793(x1, x2, x3)) = [-1] + [-1]x3 + x2
POL(COND_LOAD793(x1, x2, x3, x4)) = [-1] + [-1]x4 + x3
POL(+(x1, x2)) = x1 + x2
POL(-1) = [-1]

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:

&&(TRUE, TRUE)1TRUE1
&&(TRUE, FALSE)1FALSE1
&&(FALSE, TRUE)1FALSE1
&&(FALSE, FALSE)1FALSE1

### (12) Obligation:

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

The following domains are used:

Integer

R is empty.

The integer pair graph contains the following rules and edges:
(3): COND_LOAD793(TRUE, i90[3], i94[3], i97[3]) → LOAD793(i90[3], i94[3] + -1, i97[3] + -1)

(1) -> (2), if ((i90[1]* i97[2])∧(i90[1]* i90[2])∧(i84[1]* i94[2]))

(3) -> (2), if ((i90[3]* i90[2])∧(i97[3] + -1* i97[2])∧(i94[3] + -1* i94[2]))

(2) -> (3), if ((i90[2]* i90[3])∧(i94[2]* i94[3])∧(i97[2] > 0* TRUE)∧(i97[2]* i97[3]))

(1) -> (4), if ((i84[1]* i94[4])∧(i90[1]* 0)∧(i90[1]* i90[4]))

(3) -> (4), if ((i97[3] + -1* 0)∧(i90[3]* i90[4])∧(i94[3] + -1* i94[4]))

The set Q consists of the following terms:

### (13) IDependencyGraphProof (EQUIVALENT transformation)

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

### (14) Obligation:

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

The following domains are used:

Integer

R is empty.

The integer pair graph contains the following rules and edges:
(3): COND_LOAD793(TRUE, i90[3], i94[3], i97[3]) → LOAD793(i90[3], i94[3] + -1, i97[3] + -1)

(3) -> (2), if ((i90[3]* i90[2])∧(i97[3] + -1* i97[2])∧(i94[3] + -1* i94[2]))

(2) -> (3), if ((i90[2]* i90[3])∧(i94[2]* i94[3])∧(i97[2] > 0* TRUE)∧(i97[2]* i97[3]))

The set Q consists of the following terms:

### (15) 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_LOAD793(TRUE, i90[3], i94[3], i97[3]) → LOAD793(i90[3], +(i94[3], -1), +(i97[3], -1)) the following chains were created:
• We consider the chain LOAD793(i90[2], i94[2], i97[2]) → COND_LOAD793(>(i97[2], 0), i90[2], i94[2], i97[2]), COND_LOAD793(TRUE, i90[3], i94[3], i97[3]) → LOAD793(i90[3], +(i94[3], -1), +(i97[3], -1)), LOAD793(i90[2], i94[2], i97[2]) → COND_LOAD793(>(i97[2], 0), i90[2], i94[2], i97[2]) 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)    (i97[2] + [-1] ≥ 0 ⇒ (UIncreasing(LOAD793(i90[3], +(i94[3], -1), +(i97[3], -1))), ≥)∧[(-1)bni_13 + (-1)Bound*bni_13] + [bni_13]i97[2] ≥ 0∧[1 + (-1)bso_14] ≥ 0)

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

(4)    (i97[2] + [-1] ≥ 0 ⇒ (UIncreasing(LOAD793(i90[3], +(i94[3], -1), +(i97[3], -1))), ≥)∧[(-1)bni_13 + (-1)Bound*bni_13] + [bni_13]i97[2] ≥ 0∧[1 + (-1)bso_14] ≥ 0)

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

(5)    (i97[2] + [-1] ≥ 0 ⇒ (UIncreasing(LOAD793(i90[3], +(i94[3], -1), +(i97[3], -1))), ≥)∧[(-1)bni_13 + (-1)Bound*bni_13] + [bni_13]i97[2] ≥ 0∧[1 + (-1)bso_14] ≥ 0)

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

(6)    (i97[2] + [-1] ≥ 0 ⇒ (UIncreasing(LOAD793(i90[3], +(i94[3], -1), +(i97[3], -1))), ≥)∧0 = 0∧0 = 0∧[(-1)bni_13 + (-1)Bound*bni_13] + [bni_13]i97[2] ≥ 0∧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)    (i97[2] ≥ 0 ⇒ (UIncreasing(LOAD793(i90[3], +(i94[3], -1), +(i97[3], -1))), ≥)∧0 = 0∧0 = 0∧[(-1)Bound*bni_13] + [bni_13]i97[2] ≥ 0∧0 = 0∧0 = 0∧[1 + (-1)bso_14] ≥ 0)

For Pair LOAD793(i90[2], i94[2], i97[2]) → COND_LOAD793(>(i97[2], 0), i90[2], i94[2], i97[2]) the following chains were created:
• We consider the chain LOAD793(i90[2], i94[2], i97[2]) → COND_LOAD793(>(i97[2], 0), i90[2], i94[2], i97[2]), COND_LOAD793(TRUE, i90[3], i94[3], i97[3]) → LOAD793(i90[3], +(i94[3], -1), +(i97[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)    (i97[2] + [-1] ≥ 0 ⇒ (UIncreasing(COND_LOAD793(>(i97[2], 0), i90[2], i94[2], i97[2])), ≥)∧[(-1)bni_15 + (-1)Bound*bni_15] + [bni_15]i97[2] ≥ 0∧[(-1)bso_16] ≥ 0)

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

(11)    (i97[2] + [-1] ≥ 0 ⇒ (UIncreasing(COND_LOAD793(>(i97[2], 0), i90[2], i94[2], i97[2])), ≥)∧[(-1)bni_15 + (-1)Bound*bni_15] + [bni_15]i97[2] ≥ 0∧[(-1)bso_16] ≥ 0)

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

(12)    (i97[2] + [-1] ≥ 0 ⇒ (UIncreasing(COND_LOAD793(>(i97[2], 0), i90[2], i94[2], i97[2])), ≥)∧[(-1)bni_15 + (-1)Bound*bni_15] + [bni_15]i97[2] ≥ 0∧[(-1)bso_16] ≥ 0)

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

(13)    (i97[2] + [-1] ≥ 0 ⇒ (UIncreasing(COND_LOAD793(>(i97[2], 0), i90[2], i94[2], i97[2])), ≥)∧0 = 0∧0 = 0∧[(-1)bni_15 + (-1)Bound*bni_15] + [bni_15]i97[2] ≥ 0∧0 = 0∧0 = 0∧[(-1)bso_16] ≥ 0)

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

(14)    (i97[2] ≥ 0 ⇒ (UIncreasing(COND_LOAD793(>(i97[2], 0), i90[2], i94[2], i97[2])), ≥)∧0 = 0∧0 = 0∧[(-1)Bound*bni_15] + [bni_15]i97[2] ≥ 0∧0 = 0∧0 = 0∧[(-1)bso_16] ≥ 0)

To summarize, we get the following constraints P for the following pairs.
• (i97[2] ≥ 0 ⇒ (UIncreasing(LOAD793(i90[3], +(i94[3], -1), +(i97[3], -1))), ≥)∧0 = 0∧0 = 0∧[(-1)Bound*bni_13] + [bni_13]i97[2] ≥ 0∧0 = 0∧0 = 0∧[1 + (-1)bso_14] ≥ 0)

• (i97[2] ≥ 0 ⇒ (UIncreasing(COND_LOAD793(>(i97[2], 0), i90[2], i94[2], i97[2])), ≥)∧0 = 0∧0 = 0∧[(-1)Bound*bni_15] + [bni_15]i97[2] ≥ 0∧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_LOAD793(x1, x2, x3, x4)) = [-1] + x4
POL(LOAD793(x1, x2, x3)) = [-1] + x3
POL(+(x1, x2)) = x1 + x2
POL(-1) = [-1]
POL(>(x1, x2)) = [-1]
POL(0) = 0

The following pairs are in P>:

The following pairs are in Pbound:

The following pairs are in P:

There are no usable rules.

### (17) 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:

### (18) IDependencyGraphProof (EQUIVALENT transformation)

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

### (20) 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: