### (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: DivWithoutMinus

### (1) JBC2FIG (SOUND transformation)

Constructed FIGraph.

### (2) Obligation:

FIGraph based on JBC Program:
Graph of 187 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:
Load645(i66, 0, i82, i78) → Cond_Load645(i82 > 0 && i78 + 1 > 0, i66, 0, i82, i78)
Cond_Load645(TRUE, i66, 0, i82, i78) → Load645(i66, i82, i82, i78 + 1)
Load645(i85, i83, i82, i78) → Cond_Load6451(i83 > 0 && i85 > 0 && i82 > 0, i85, i83, i82, i78)
Cond_Load6451(TRUE, i85, i83, i82, i78) → Load645(i85 + -1, i83 + -1, i82, i78)
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:
Load645(i66, 0, i82, i78) → Cond_Load645(i82 > 0 && i78 + 1 > 0, i66, i82, i78)
Load645(i85, i83, i82, i78) → Cond_Load6451(i83 > 0 && i85 > 0 && i82 > 0, i85, i83, i82, i78)
Cond_Load6451(TRUE, i85, i83, i82, i78) → Load645(i85 + -1, i83 + -1, i82, i78)
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:
Load645(i66, 0, i82, i78) → Cond_Load645(i82 > 0 && i78 + 1 > 0, i66, i82, i78)
Load645(i85, i83, i82, i78) → Cond_Load6451(i83 > 0 && i85 > 0 && i82 > 0, i85, i83, i82, i78)
Cond_Load6451(TRUE, i85, i83, i82, i78) → Load645(i85 + -1, i83 + -1, i82, i78)

The integer pair graph contains the following rules and edges:
(0): LOAD645(i66[0], 0, i82[0], i78[0]) → COND_LOAD645(i82[0] > 0 && i78[0] + 1 > 0, i66[0], i82[0], i78[0])
(1): COND_LOAD645(TRUE, i66[1], i82[1], i78[1]) → LOAD645(i66[1], i82[1], i82[1], i78[1] + 1)
(2): LOAD645(i85[2], i83[2], i82[2], i78[2]) → COND_LOAD6451(i83[2] > 0 && i85[2] > 0 && i82[2] > 0, i85[2], i83[2], i82[2], i78[2])
(3): COND_LOAD6451(TRUE, i85[3], i83[3], i82[3], i78[3]) → LOAD645(i85[3] + -1, i83[3] + -1, i82[3], i78[3])

(0) -> (1), if ((i66[0]* i66[1])∧(i82[0] > 0 && i78[0] + 1 > 0* TRUE)∧(i78[0]* i78[1])∧(i82[0]* i82[1]))

(1) -> (0), if ((i78[1] + 1* i78[0])∧(i66[1]* i66[0])∧(i82[1]* 0)∧(i82[1]* i82[0]))

(1) -> (2), if ((i66[1]* i85[2])∧(i82[1]* i83[2])∧(i82[1]* i82[2])∧(i78[1] + 1* i78[2]))

(2) -> (3), if ((i85[2]* i85[3])∧(i78[2]* i78[3])∧(i83[2] > 0 && i85[2] > 0 && i82[2] > 0* TRUE)∧(i82[2]* i82[3])∧(i83[2]* i83[3]))

(3) -> (0), if ((i83[3] + -1* 0)∧(i82[3]* i82[0])∧(i85[3] + -1* i66[0])∧(i78[3]* i78[0]))

(3) -> (2), if ((i78[3]* i78[2])∧(i83[3] + -1* i83[2])∧(i85[3] + -1* i85[2])∧(i82[3]* i82[2]))

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): LOAD645(i66[0], 0, i82[0], i78[0]) → COND_LOAD645(i82[0] > 0 && i78[0] + 1 > 0, i66[0], i82[0], i78[0])
(1): COND_LOAD645(TRUE, i66[1], i82[1], i78[1]) → LOAD645(i66[1], i82[1], i82[1], i78[1] + 1)
(2): LOAD645(i85[2], i83[2], i82[2], i78[2]) → COND_LOAD6451(i83[2] > 0 && i85[2] > 0 && i82[2] > 0, i85[2], i83[2], i82[2], i78[2])
(3): COND_LOAD6451(TRUE, i85[3], i83[3], i82[3], i78[3]) → LOAD645(i85[3] + -1, i83[3] + -1, i82[3], i78[3])

(0) -> (1), if ((i66[0]* i66[1])∧(i82[0] > 0 && i78[0] + 1 > 0* TRUE)∧(i78[0]* i78[1])∧(i82[0]* i82[1]))

(1) -> (0), if ((i78[1] + 1* i78[0])∧(i66[1]* i66[0])∧(i82[1]* 0)∧(i82[1]* i82[0]))

(1) -> (2), if ((i66[1]* i85[2])∧(i82[1]* i83[2])∧(i82[1]* i82[2])∧(i78[1] + 1* i78[2]))

(2) -> (3), if ((i85[2]* i85[3])∧(i78[2]* i78[3])∧(i83[2] > 0 && i85[2] > 0 && i82[2] > 0* TRUE)∧(i82[2]* i82[3])∧(i83[2]* i83[3]))

(3) -> (0), if ((i83[3] + -1* 0)∧(i82[3]* i82[0])∧(i85[3] + -1* i66[0])∧(i78[3]* i78[0]))

(3) -> (2), if ((i78[3]* i78[2])∧(i83[3] + -1* i83[2])∧(i85[3] + -1* i85[2])∧(i82[3]* i82[2]))

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 LOAD645(i66, 0, i82, i78) → COND_LOAD645(&&(>(i82, 0), >(+(i78, 1), 0)), i66, i82, i78) the following chains were created:
• We consider the chain LOAD645(i66[0], 0, i82[0], i78[0]) → COND_LOAD645(&&(>(i82[0], 0), >(+(i78[0], 1), 0)), i66[0], i82[0], i78[0]), COND_LOAD645(TRUE, i66[1], i82[1], i78[1]) → LOAD645(i66[1], i82[1], i82[1], +(i78[1], 1)) which results in the following constraint:

(1)    (i66[0]=i66[1]&&(>(i82[0], 0), >(+(i78[0], 1), 0))=TRUEi78[0]=i78[1]i82[0]=i82[1]LOAD645(i66[0], 0, i82[0], i78[0])≥NonInfC∧LOAD645(i66[0], 0, i82[0], i78[0])≥COND_LOAD645(&&(>(i82[0], 0), >(+(i78[0], 1), 0)), i66[0], i82[0], i78[0])∧(UIncreasing(COND_LOAD645(&&(>(i82[0], 0), >(+(i78[0], 1), 0)), i66[0], i82[0], i78[0])), ≥))

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

(2)    (>(i82[0], 0)=TRUE>(+(i78[0], 1), 0)=TRUELOAD645(i66[0], 0, i82[0], i78[0])≥NonInfC∧LOAD645(i66[0], 0, i82[0], i78[0])≥COND_LOAD645(&&(>(i82[0], 0), >(+(i78[0], 1), 0)), i66[0], i82[0], i78[0])∧(UIncreasing(COND_LOAD645(&&(>(i82[0], 0), >(+(i78[0], 1), 0)), i66[0], i82[0], i78[0])), ≥))

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

(3)    (i82[0] + [-1] ≥ 0∧i78[0] ≥ 0 ⇒ (UIncreasing(COND_LOAD645(&&(>(i82[0], 0), >(+(i78[0], 1), 0)), i66[0], i82[0], i78[0])), ≥)∧[(-1)Bound*bni_21] + [bni_21]i66[0] ≥ 0∧[(-1)bso_22] ≥ 0)

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

(4)    (i82[0] + [-1] ≥ 0∧i78[0] ≥ 0 ⇒ (UIncreasing(COND_LOAD645(&&(>(i82[0], 0), >(+(i78[0], 1), 0)), i66[0], i82[0], i78[0])), ≥)∧[(-1)Bound*bni_21] + [bni_21]i66[0] ≥ 0∧[(-1)bso_22] ≥ 0)

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

(5)    (i82[0] + [-1] ≥ 0∧i78[0] ≥ 0 ⇒ (UIncreasing(COND_LOAD645(&&(>(i82[0], 0), >(+(i78[0], 1), 0)), i66[0], i82[0], i78[0])), ≥)∧[(-1)Bound*bni_21] + [bni_21]i66[0] ≥ 0∧[(-1)bso_22] ≥ 0)

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

(6)    (i82[0] + [-1] ≥ 0∧i78[0] ≥ 0 ⇒ (UIncreasing(COND_LOAD645(&&(>(i82[0], 0), >(+(i78[0], 1), 0)), i66[0], i82[0], i78[0])), ≥)∧[bni_21] = 0∧[(-1)Bound*bni_21] ≥ 0∧0 = 0∧[(-1)bso_22] ≥ 0)

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

(7)    (i82[0] ≥ 0∧i78[0] ≥ 0 ⇒ (UIncreasing(COND_LOAD645(&&(>(i82[0], 0), >(+(i78[0], 1), 0)), i66[0], i82[0], i78[0])), ≥)∧[bni_21] = 0∧[(-1)Bound*bni_21] ≥ 0∧0 = 0∧[(-1)bso_22] ≥ 0)

For Pair COND_LOAD645(TRUE, i66, i82, i78) → LOAD645(i66, i82, i82, +(i78, 1)) the following chains were created:
• We consider the chain LOAD645(i66[0], 0, i82[0], i78[0]) → COND_LOAD645(&&(>(i82[0], 0), >(+(i78[0], 1), 0)), i66[0], i82[0], i78[0]), COND_LOAD645(TRUE, i66[1], i82[1], i78[1]) → LOAD645(i66[1], i82[1], i82[1], +(i78[1], 1)), LOAD645(i66[0], 0, i82[0], i78[0]) → COND_LOAD645(&&(>(i82[0], 0), >(+(i78[0], 1), 0)), i66[0], i82[0], i78[0]) which results in the following constraint:

We solved constraint (8) using rules (I), (II), (III), (IV), (IDP_CONSTANT_FOLD).
• We consider the chain LOAD645(i66[0], 0, i82[0], i78[0]) → COND_LOAD645(&&(>(i82[0], 0), >(+(i78[0], 1), 0)), i66[0], i82[0], i78[0]), COND_LOAD645(TRUE, i66[1], i82[1], i78[1]) → LOAD645(i66[1], i82[1], i82[1], +(i78[1], 1)), LOAD645(i85[2], i83[2], i82[2], i78[2]) → COND_LOAD6451(&&(&&(>(i83[2], 0), >(i85[2], 0)), >(i82[2], 0)), i85[2], i83[2], i82[2], i78[2]) which results in the following constraint:

We simplified constraint (9) using rules (III), (IV), (IDP_BOOLEAN) which results in the following new constraint:

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

(11)    (i82[0] + [-1] ≥ 0∧i78[0] ≥ 0 ⇒ (UIncreasing(LOAD645(i66[1], i82[1], i82[1], +(i78[1], 1))), ≥)∧[(-1)Bound*bni_23] + [bni_23]i66[0] ≥ 0∧[(-1)bso_24] ≥ 0)

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

(12)    (i82[0] + [-1] ≥ 0∧i78[0] ≥ 0 ⇒ (UIncreasing(LOAD645(i66[1], i82[1], i82[1], +(i78[1], 1))), ≥)∧[(-1)Bound*bni_23] + [bni_23]i66[0] ≥ 0∧[(-1)bso_24] ≥ 0)

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

(13)    (i82[0] + [-1] ≥ 0∧i78[0] ≥ 0 ⇒ (UIncreasing(LOAD645(i66[1], i82[1], i82[1], +(i78[1], 1))), ≥)∧[(-1)Bound*bni_23] + [bni_23]i66[0] ≥ 0∧[(-1)bso_24] ≥ 0)

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

(14)    (i82[0] + [-1] ≥ 0∧i78[0] ≥ 0 ⇒ (UIncreasing(LOAD645(i66[1], i82[1], i82[1], +(i78[1], 1))), ≥)∧[bni_23] = 0∧[(-1)Bound*bni_23] ≥ 0∧0 = 0∧[(-1)bso_24] ≥ 0)

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

(15)    (i82[0] ≥ 0∧i78[0] ≥ 0 ⇒ (UIncreasing(LOAD645(i66[1], i82[1], i82[1], +(i78[1], 1))), ≥)∧[bni_23] = 0∧[(-1)Bound*bni_23] ≥ 0∧0 = 0∧[(-1)bso_24] ≥ 0)

For Pair LOAD645(i85, i83, i82, i78) → COND_LOAD6451(&&(&&(>(i83, 0), >(i85, 0)), >(i82, 0)), i85, i83, i82, i78) the following chains were created:
• We consider the chain LOAD645(i85[2], i83[2], i82[2], i78[2]) → COND_LOAD6451(&&(&&(>(i83[2], 0), >(i85[2], 0)), >(i82[2], 0)), i85[2], i83[2], i82[2], i78[2]), COND_LOAD6451(TRUE, i85[3], i83[3], i82[3], i78[3]) → LOAD645(+(i85[3], -1), +(i83[3], -1), i82[3], i78[3]) which results in the following constraint:

(16)    (i85[2]=i85[3]i78[2]=i78[3]&&(&&(>(i83[2], 0), >(i85[2], 0)), >(i82[2], 0))=TRUEi82[2]=i82[3]i83[2]=i83[3]LOAD645(i85[2], i83[2], i82[2], i78[2])≥NonInfC∧LOAD645(i85[2], i83[2], i82[2], i78[2])≥COND_LOAD6451(&&(&&(>(i83[2], 0), >(i85[2], 0)), >(i82[2], 0)), i85[2], i83[2], i82[2], i78[2])∧(UIncreasing(COND_LOAD6451(&&(&&(>(i83[2], 0), >(i85[2], 0)), >(i82[2], 0)), i85[2], i83[2], i82[2], i78[2])), ≥))

We simplified constraint (16) using rules (IV), (IDP_BOOLEAN) which results in the following new constraint:

(17)    (>(i82[2], 0)=TRUE>(i83[2], 0)=TRUE>(i85[2], 0)=TRUELOAD645(i85[2], i83[2], i82[2], i78[2])≥NonInfC∧LOAD645(i85[2], i83[2], i82[2], i78[2])≥COND_LOAD6451(&&(&&(>(i83[2], 0), >(i85[2], 0)), >(i82[2], 0)), i85[2], i83[2], i82[2], i78[2])∧(UIncreasing(COND_LOAD6451(&&(&&(>(i83[2], 0), >(i85[2], 0)), >(i82[2], 0)), i85[2], i83[2], i82[2], i78[2])), ≥))

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

(18)    (i82[2] + [-1] ≥ 0∧i83[2] + [-1] ≥ 0∧i85[2] + [-1] ≥ 0 ⇒ (UIncreasing(COND_LOAD6451(&&(&&(>(i83[2], 0), >(i85[2], 0)), >(i82[2], 0)), i85[2], i83[2], i82[2], i78[2])), ≥)∧[(-1)Bound*bni_25] + [bni_25]i85[2] ≥ 0∧[1 + (-1)bso_26] ≥ 0)

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

(19)    (i82[2] + [-1] ≥ 0∧i83[2] + [-1] ≥ 0∧i85[2] + [-1] ≥ 0 ⇒ (UIncreasing(COND_LOAD6451(&&(&&(>(i83[2], 0), >(i85[2], 0)), >(i82[2], 0)), i85[2], i83[2], i82[2], i78[2])), ≥)∧[(-1)Bound*bni_25] + [bni_25]i85[2] ≥ 0∧[1 + (-1)bso_26] ≥ 0)

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

(20)    (i82[2] + [-1] ≥ 0∧i83[2] + [-1] ≥ 0∧i85[2] + [-1] ≥ 0 ⇒ (UIncreasing(COND_LOAD6451(&&(&&(>(i83[2], 0), >(i85[2], 0)), >(i82[2], 0)), i85[2], i83[2], i82[2], i78[2])), ≥)∧[(-1)Bound*bni_25] + [bni_25]i85[2] ≥ 0∧[1 + (-1)bso_26] ≥ 0)

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

(21)    (i82[2] + [-1] ≥ 0∧i83[2] + [-1] ≥ 0∧i85[2] + [-1] ≥ 0 ⇒ (UIncreasing(COND_LOAD6451(&&(&&(>(i83[2], 0), >(i85[2], 0)), >(i82[2], 0)), i85[2], i83[2], i82[2], i78[2])), ≥)∧0 = 0∧[(-1)Bound*bni_25] + [bni_25]i85[2] ≥ 0∧0 = 0∧[1 + (-1)bso_26] ≥ 0)

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

(22)    (i82[2] ≥ 0∧i83[2] + [-1] ≥ 0∧i85[2] + [-1] ≥ 0 ⇒ (UIncreasing(COND_LOAD6451(&&(&&(>(i83[2], 0), >(i85[2], 0)), >(i82[2], 0)), i85[2], i83[2], i82[2], i78[2])), ≥)∧0 = 0∧[(-1)Bound*bni_25] + [bni_25]i85[2] ≥ 0∧0 = 0∧[1 + (-1)bso_26] ≥ 0)

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

(23)    (i82[2] ≥ 0∧i83[2] ≥ 0∧i85[2] + [-1] ≥ 0 ⇒ (UIncreasing(COND_LOAD6451(&&(&&(>(i83[2], 0), >(i85[2], 0)), >(i82[2], 0)), i85[2], i83[2], i82[2], i78[2])), ≥)∧0 = 0∧[(-1)Bound*bni_25] + [bni_25]i85[2] ≥ 0∧0 = 0∧[1 + (-1)bso_26] ≥ 0)

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

(24)    (i82[2] ≥ 0∧i83[2] ≥ 0∧i85[2] ≥ 0 ⇒ (UIncreasing(COND_LOAD6451(&&(&&(>(i83[2], 0), >(i85[2], 0)), >(i82[2], 0)), i85[2], i83[2], i82[2], i78[2])), ≥)∧0 = 0∧[(-1)Bound*bni_25 + bni_25] + [bni_25]i85[2] ≥ 0∧0 = 0∧[1 + (-1)bso_26] ≥ 0)

For Pair COND_LOAD6451(TRUE, i85, i83, i82, i78) → LOAD645(+(i85, -1), +(i83, -1), i82, i78) the following chains were created:
• We consider the chain LOAD645(i85[2], i83[2], i82[2], i78[2]) → COND_LOAD6451(&&(&&(>(i83[2], 0), >(i85[2], 0)), >(i82[2], 0)), i85[2], i83[2], i82[2], i78[2]), COND_LOAD6451(TRUE, i85[3], i83[3], i82[3], i78[3]) → LOAD645(+(i85[3], -1), +(i83[3], -1), i82[3], i78[3]), LOAD645(i66[0], 0, i82[0], i78[0]) → COND_LOAD645(&&(>(i82[0], 0), >(+(i78[0], 1), 0)), i66[0], i82[0], i78[0]) which results in the following constraint:

(25)    (i85[2]=i85[3]i78[2]=i78[3]&&(&&(>(i83[2], 0), >(i85[2], 0)), >(i82[2], 0))=TRUEi82[2]=i82[3]i83[2]=i83[3]+(i83[3], -1)=0i82[3]=i82[0]+(i85[3], -1)=i66[0]i78[3]=i78[0]COND_LOAD6451(TRUE, i85[3], i83[3], i82[3], i78[3])≥NonInfC∧COND_LOAD6451(TRUE, i85[3], i83[3], i82[3], i78[3])≥LOAD645(+(i85[3], -1), +(i83[3], -1), i82[3], i78[3])∧(UIncreasing(LOAD645(+(i85[3], -1), +(i83[3], -1), i82[3], i78[3])), ≥))

We simplified constraint (25) using rules (III), (IV), (IDP_BOOLEAN) which results in the following new constraint:

(26)    (+(i83[2], -1)=0>(i82[2], 0)=TRUE>(i83[2], 0)=TRUE>(i85[2], 0)=TRUECOND_LOAD6451(TRUE, i85[2], i83[2], i82[2], i78[2])≥NonInfC∧COND_LOAD6451(TRUE, i85[2], i83[2], i82[2], i78[2])≥LOAD645(+(i85[2], -1), +(i83[2], -1), i82[2], i78[2])∧(UIncreasing(LOAD645(+(i85[3], -1), +(i83[3], -1), i82[3], i78[3])), ≥))

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

(27)    (i83[2] + [-1] ≥ 0∧i82[2] + [-1] ≥ 0∧i83[2] + [-1] ≥ 0∧i85[2] + [-1] ≥ 0 ⇒ (UIncreasing(LOAD645(+(i85[3], -1), +(i83[3], -1), i82[3], i78[3])), ≥)∧[(-1)bni_27 + (-1)Bound*bni_27] + [bni_27]i85[2] ≥ 0∧[(-1)bso_28] ≥ 0)

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

(28)    (i83[2] + [-1] ≥ 0∧i82[2] + [-1] ≥ 0∧i83[2] + [-1] ≥ 0∧i85[2] + [-1] ≥ 0 ⇒ (UIncreasing(LOAD645(+(i85[3], -1), +(i83[3], -1), i82[3], i78[3])), ≥)∧[(-1)bni_27 + (-1)Bound*bni_27] + [bni_27]i85[2] ≥ 0∧[(-1)bso_28] ≥ 0)

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

(29)    (i83[2] + [-1] ≥ 0∧i82[2] + [-1] ≥ 0∧i83[2] + [-1] ≥ 0∧i85[2] + [-1] ≥ 0 ⇒ (UIncreasing(LOAD645(+(i85[3], -1), +(i83[3], -1), i82[3], i78[3])), ≥)∧[(-1)bni_27 + (-1)Bound*bni_27] + [bni_27]i85[2] ≥ 0∧[(-1)bso_28] ≥ 0)

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

(30)    (i83[2] + [-1] ≥ 0∧i82[2] + [-1] ≥ 0∧i83[2] + [-1] ≥ 0∧i85[2] + [-1] ≥ 0 ⇒ (UIncreasing(LOAD645(+(i85[3], -1), +(i83[3], -1), i82[3], i78[3])), ≥)∧0 = 0∧[(-1)bni_27 + (-1)Bound*bni_27] + [bni_27]i85[2] ≥ 0∧0 = 0∧[(-1)bso_28] ≥ 0)

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

(31)    (i83[2] ≥ 0∧i82[2] + [-1] ≥ 0∧i83[2] ≥ 0∧i85[2] + [-1] ≥ 0 ⇒ (UIncreasing(LOAD645(+(i85[3], -1), +(i83[3], -1), i82[3], i78[3])), ≥)∧0 = 0∧[(-1)bni_27 + (-1)Bound*bni_27] + [bni_27]i85[2] ≥ 0∧0 = 0∧[(-1)bso_28] ≥ 0)

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

(32)    (i83[2] ≥ 0∧i82[2] ≥ 0∧i83[2] ≥ 0∧i85[2] + [-1] ≥ 0 ⇒ (UIncreasing(LOAD645(+(i85[3], -1), +(i83[3], -1), i82[3], i78[3])), ≥)∧0 = 0∧[(-1)bni_27 + (-1)Bound*bni_27] + [bni_27]i85[2] ≥ 0∧0 = 0∧[(-1)bso_28] ≥ 0)

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

(33)    (i83[2] ≥ 0∧i82[2] ≥ 0∧i83[2] ≥ 0∧i85[2] ≥ 0 ⇒ (UIncreasing(LOAD645(+(i85[3], -1), +(i83[3], -1), i82[3], i78[3])), ≥)∧0 = 0∧[(-1)Bound*bni_27] + [bni_27]i85[2] ≥ 0∧0 = 0∧[(-1)bso_28] ≥ 0)

• We consider the chain LOAD645(i85[2], i83[2], i82[2], i78[2]) → COND_LOAD6451(&&(&&(>(i83[2], 0), >(i85[2], 0)), >(i82[2], 0)), i85[2], i83[2], i82[2], i78[2]), COND_LOAD6451(TRUE, i85[3], i83[3], i82[3], i78[3]) → LOAD645(+(i85[3], -1), +(i83[3], -1), i82[3], i78[3]), LOAD645(i85[2], i83[2], i82[2], i78[2]) → COND_LOAD6451(&&(&&(>(i83[2], 0), >(i85[2], 0)), >(i82[2], 0)), i85[2], i83[2], i82[2], i78[2]) which results in the following constraint:

(34)    (i85[2]=i85[3]i78[2]=i78[3]&&(&&(>(i83[2], 0), >(i85[2], 0)), >(i82[2], 0))=TRUEi82[2]=i82[3]i83[2]=i83[3]i78[3]=i78[2]1+(i83[3], -1)=i83[2]1+(i85[3], -1)=i85[2]1i82[3]=i82[2]1COND_LOAD6451(TRUE, i85[3], i83[3], i82[3], i78[3])≥NonInfC∧COND_LOAD6451(TRUE, i85[3], i83[3], i82[3], i78[3])≥LOAD645(+(i85[3], -1), +(i83[3], -1), i82[3], i78[3])∧(UIncreasing(LOAD645(+(i85[3], -1), +(i83[3], -1), i82[3], i78[3])), ≥))

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

(35)    (>(i82[2], 0)=TRUE>(i83[2], 0)=TRUE>(i85[2], 0)=TRUECOND_LOAD6451(TRUE, i85[2], i83[2], i82[2], i78[2])≥NonInfC∧COND_LOAD6451(TRUE, i85[2], i83[2], i82[2], i78[2])≥LOAD645(+(i85[2], -1), +(i83[2], -1), i82[2], i78[2])∧(UIncreasing(LOAD645(+(i85[3], -1), +(i83[3], -1), i82[3], i78[3])), ≥))

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

(36)    (i82[2] + [-1] ≥ 0∧i83[2] + [-1] ≥ 0∧i85[2] + [-1] ≥ 0 ⇒ (UIncreasing(LOAD645(+(i85[3], -1), +(i83[3], -1), i82[3], i78[3])), ≥)∧[(-1)bni_27 + (-1)Bound*bni_27] + [bni_27]i85[2] ≥ 0∧[(-1)bso_28] ≥ 0)

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

(37)    (i82[2] + [-1] ≥ 0∧i83[2] + [-1] ≥ 0∧i85[2] + [-1] ≥ 0 ⇒ (UIncreasing(LOAD645(+(i85[3], -1), +(i83[3], -1), i82[3], i78[3])), ≥)∧[(-1)bni_27 + (-1)Bound*bni_27] + [bni_27]i85[2] ≥ 0∧[(-1)bso_28] ≥ 0)

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

(38)    (i82[2] + [-1] ≥ 0∧i83[2] + [-1] ≥ 0∧i85[2] + [-1] ≥ 0 ⇒ (UIncreasing(LOAD645(+(i85[3], -1), +(i83[3], -1), i82[3], i78[3])), ≥)∧[(-1)bni_27 + (-1)Bound*bni_27] + [bni_27]i85[2] ≥ 0∧[(-1)bso_28] ≥ 0)

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

(39)    (i82[2] + [-1] ≥ 0∧i83[2] + [-1] ≥ 0∧i85[2] + [-1] ≥ 0 ⇒ (UIncreasing(LOAD645(+(i85[3], -1), +(i83[3], -1), i82[3], i78[3])), ≥)∧0 = 0∧[(-1)bni_27 + (-1)Bound*bni_27] + [bni_27]i85[2] ≥ 0∧0 = 0∧[(-1)bso_28] ≥ 0)

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

(40)    (i82[2] ≥ 0∧i83[2] + [-1] ≥ 0∧i85[2] + [-1] ≥ 0 ⇒ (UIncreasing(LOAD645(+(i85[3], -1), +(i83[3], -1), i82[3], i78[3])), ≥)∧0 = 0∧[(-1)bni_27 + (-1)Bound*bni_27] + [bni_27]i85[2] ≥ 0∧0 = 0∧[(-1)bso_28] ≥ 0)

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

(41)    (i82[2] ≥ 0∧i83[2] ≥ 0∧i85[2] + [-1] ≥ 0 ⇒ (UIncreasing(LOAD645(+(i85[3], -1), +(i83[3], -1), i82[3], i78[3])), ≥)∧0 = 0∧[(-1)bni_27 + (-1)Bound*bni_27] + [bni_27]i85[2] ≥ 0∧0 = 0∧[(-1)bso_28] ≥ 0)

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

(42)    (i82[2] ≥ 0∧i83[2] ≥ 0∧i85[2] ≥ 0 ⇒ (UIncreasing(LOAD645(+(i85[3], -1), +(i83[3], -1), i82[3], i78[3])), ≥)∧0 = 0∧[(-1)Bound*bni_27] + [bni_27]i85[2] ≥ 0∧0 = 0∧[(-1)bso_28] ≥ 0)

To summarize, we get the following constraints P for the following pairs.
• LOAD645(i66, 0, i82, i78) → COND_LOAD645(&&(>(i82, 0), >(+(i78, 1), 0)), i66, i82, i78)
• (i82[0] ≥ 0∧i78[0] ≥ 0 ⇒ (UIncreasing(COND_LOAD645(&&(>(i82[0], 0), >(+(i78[0], 1), 0)), i66[0], i82[0], i78[0])), ≥)∧[bni_21] = 0∧[(-1)Bound*bni_21] ≥ 0∧0 = 0∧[(-1)bso_22] ≥ 0)

• (i82[0] ≥ 0∧i78[0] ≥ 0 ⇒ (UIncreasing(LOAD645(i66[1], i82[1], i82[1], +(i78[1], 1))), ≥)∧[bni_23] = 0∧[(-1)Bound*bni_23] ≥ 0∧0 = 0∧[(-1)bso_24] ≥ 0)

• LOAD645(i85, i83, i82, i78) → COND_LOAD6451(&&(&&(>(i83, 0), >(i85, 0)), >(i82, 0)), i85, i83, i82, i78)
• (i82[2] ≥ 0∧i83[2] ≥ 0∧i85[2] ≥ 0 ⇒ (UIncreasing(COND_LOAD6451(&&(&&(>(i83[2], 0), >(i85[2], 0)), >(i82[2], 0)), i85[2], i83[2], i82[2], i78[2])), ≥)∧0 = 0∧[(-1)Bound*bni_25 + bni_25] + [bni_25]i85[2] ≥ 0∧0 = 0∧[1 + (-1)bso_26] ≥ 0)

• COND_LOAD6451(TRUE, i85, i83, i82, i78) → LOAD645(+(i85, -1), +(i83, -1), i82, i78)
• (i83[2] ≥ 0∧i82[2] ≥ 0∧i83[2] ≥ 0∧i85[2] ≥ 0 ⇒ (UIncreasing(LOAD645(+(i85[3], -1), +(i83[3], -1), i82[3], i78[3])), ≥)∧0 = 0∧[(-1)Bound*bni_27] + [bni_27]i85[2] ≥ 0∧0 = 0∧[(-1)bso_28] ≥ 0)
• (i82[2] ≥ 0∧i83[2] ≥ 0∧i85[2] ≥ 0 ⇒ (UIncreasing(LOAD645(+(i85[3], -1), +(i83[3], -1), i82[3], i78[3])), ≥)∧0 = 0∧[(-1)Bound*bni_27] + [bni_27]i85[2] ≥ 0∧0 = 0∧[(-1)bso_28] ≥ 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) = [1]
POL(FALSE) = [1]
POL(LOAD645(x1, x2, x3, x4)) = x1
POL(0) = 0
POL(COND_LOAD645(x1, x2, x3, x4)) = [1] + x2 + [-1]x1
POL(&&(x1, x2)) = [1]
POL(>(x1, x2)) = [-1]
POL(+(x1, x2)) = x1 + x2
POL(1) = [1]
POL(COND_LOAD6451(x1, x2, x3, x4, x5)) = [-1] + x2
POL(-1) = [-1]

The following pairs are in P>:

LOAD645(i85[2], i83[2], i82[2], i78[2]) → COND_LOAD6451(&&(&&(>(i83[2], 0), >(i85[2], 0)), >(i82[2], 0)), i85[2], i83[2], i82[2], i78[2])

The following pairs are in Pbound:

LOAD645(i85[2], i83[2], i82[2], i78[2]) → COND_LOAD6451(&&(&&(>(i83[2], 0), >(i85[2], 0)), >(i82[2], 0)), i85[2], i83[2], i82[2], i78[2])
COND_LOAD6451(TRUE, i85[3], i83[3], i82[3], i78[3]) → LOAD645(+(i85[3], -1), +(i83[3], -1), i82[3], i78[3])

The following pairs are in P:

LOAD645(i66[0], 0, i82[0], i78[0]) → COND_LOAD645(&&(>(i82[0], 0), >(+(i78[0], 1), 0)), i66[0], i82[0], i78[0])
COND_LOAD6451(TRUE, i85[3], i83[3], i82[3], i78[3]) → LOAD645(+(i85[3], -1), +(i83[3], -1), i82[3], i78[3])

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

### (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:

Boolean, Integer

R is empty.

The integer pair graph contains the following rules and edges:
(0): LOAD645(i66[0], 0, i82[0], i78[0]) → COND_LOAD645(i82[0] > 0 && i78[0] + 1 > 0, i66[0], i82[0], i78[0])
(1): COND_LOAD645(TRUE, i66[1], i82[1], i78[1]) → LOAD645(i66[1], i82[1], i82[1], i78[1] + 1)
(3): COND_LOAD6451(TRUE, i85[3], i83[3], i82[3], i78[3]) → LOAD645(i85[3] + -1, i83[3] + -1, i82[3], i78[3])

(1) -> (0), if ((i78[1] + 1* i78[0])∧(i66[1]* i66[0])∧(i82[1]* 0)∧(i82[1]* i82[0]))

(3) -> (0), if ((i83[3] + -1* 0)∧(i82[3]* i82[0])∧(i85[3] + -1* i66[0])∧(i78[3]* i78[0]))

(0) -> (1), if ((i66[0]* i66[1])∧(i82[0] > 0 && i78[0] + 1 > 0* TRUE)∧(i78[0]* i78[1])∧(i82[0]* i82[1]))

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 1 less node.

### (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, Boolean

R is empty.

The integer pair graph contains the following rules and edges:
(1): COND_LOAD645(TRUE, i66[1], i82[1], i78[1]) → LOAD645(i66[1], i82[1], i82[1], i78[1] + 1)
(0): LOAD645(i66[0], 0, i82[0], i78[0]) → COND_LOAD645(i82[0] > 0 && i78[0] + 1 > 0, i66[0], i82[0], i78[0])

(1) -> (0), if ((i78[1] + 1* i78[0])∧(i66[1]* i66[0])∧(i82[1]* 0)∧(i82[1]* i82[0]))

(0) -> (1), if ((i66[0]* i66[1])∧(i82[0] > 0 && i78[0] + 1 > 0* TRUE)∧(i78[0]* i78[1])∧(i82[0]* i82[1]))

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_LOAD645(TRUE, i66[1], i82[1], i78[1]) → LOAD645(i66[1], i82[1], i82[1], +(i78[1], 1)) the following chains were created:
• We consider the chain LOAD645(i66[0], 0, i82[0], i78[0]) → COND_LOAD645(&&(>(i82[0], 0), >(+(i78[0], 1), 0)), i66[0], i82[0], i78[0]), COND_LOAD645(TRUE, i66[1], i82[1], i78[1]) → LOAD645(i66[1], i82[1], i82[1], +(i78[1], 1)), LOAD645(i66[0], 0, i82[0], i78[0]) → COND_LOAD645(&&(>(i82[0], 0), >(+(i78[0], 1), 0)), i66[0], i82[0], i78[0]) which results in the following constraint:

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

For Pair LOAD645(i66[0], 0, i82[0], i78[0]) → COND_LOAD645(&&(>(i82[0], 0), >(+(i78[0], 1), 0)), i66[0], i82[0], i78[0]) the following chains were created:
• We consider the chain LOAD645(i66[0], 0, i82[0], i78[0]) → COND_LOAD645(&&(>(i82[0], 0), >(+(i78[0], 1), 0)), i66[0], i82[0], i78[0]), COND_LOAD645(TRUE, i66[1], i82[1], i78[1]) → LOAD645(i66[1], i82[1], i82[1], +(i78[1], 1)) which results in the following constraint:

(2)    (i66[0]=i66[1]&&(>(i82[0], 0), >(+(i78[0], 1), 0))=TRUEi78[0]=i78[1]i82[0]=i82[1]LOAD645(i66[0], 0, i82[0], i78[0])≥NonInfC∧LOAD645(i66[0], 0, i82[0], i78[0])≥COND_LOAD645(&&(>(i82[0], 0), >(+(i78[0], 1), 0)), i66[0], i82[0], i78[0])∧(UIncreasing(COND_LOAD645(&&(>(i82[0], 0), >(+(i78[0], 1), 0)), i66[0], i82[0], i78[0])), ≥))

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

(3)    (&&(>(i82[0], 0), >(+(i78[0], 1), 0))=TRUELOAD645(i66[0], 0, i82[0], i78[0])≥NonInfC∧LOAD645(i66[0], 0, i82[0], i78[0])≥COND_LOAD645(&&(>(i82[0], 0), >(+(i78[0], 1), 0)), i66[0], i82[0], i78[0])∧(UIncreasing(COND_LOAD645(&&(>(i82[0], 0), >(+(i78[0], 1), 0)), i66[0], i82[0], i78[0])), ≥))

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

(4)    (0 ≥ 0 ⇒ (UIncreasing(COND_LOAD645(&&(>(i82[0], 0), >(+(i78[0], 1), 0)), i66[0], i82[0], i78[0])), ≥)∧[(2)bni_11 + (-1)Bound*bni_11] + [(2)bni_11]i78[0] + [(2)bni_11]i82[0] + [(2)bni_11]i66[0] ≥ 0∧[2 + (-1)bso_12] + i78[0] + i82[0] + i66[0] ≥ 0)

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

(5)    (0 ≥ 0 ⇒ (UIncreasing(COND_LOAD645(&&(>(i82[0], 0), >(+(i78[0], 1), 0)), i66[0], i82[0], i78[0])), ≥)∧[(2)bni_11 + (-1)Bound*bni_11] + [(2)bni_11]i78[0] + [(2)bni_11]i82[0] + [(2)bni_11]i66[0] ≥ 0∧[2 + (-1)bso_12] + i78[0] + i82[0] + i66[0] ≥ 0)

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

(6)    (0 ≥ 0 ⇒ (UIncreasing(COND_LOAD645(&&(>(i82[0], 0), >(+(i78[0], 1), 0)), i66[0], i82[0], i78[0])), ≥)∧[(2)bni_11 + (-1)Bound*bni_11] + [(2)bni_11]i78[0] + [(2)bni_11]i82[0] + [(2)bni_11]i66[0] ≥ 0∧[2 + (-1)bso_12] + i78[0] + i82[0] + i66[0] ≥ 0)

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

(7)    (0 ≥ 0 ⇒ (UIncreasing(COND_LOAD645(&&(>(i82[0], 0), >(+(i78[0], 1), 0)), i66[0], i82[0], i78[0])), ≥)∧[(2)bni_11] ≥ 0∧[(2)bni_11] ≥ 0∧[(2)bni_11] ≥ 0∧[(2)bni_11 + (-1)Bound*bni_11] ≥ 0∧[1] ≥ 0∧[1] ≥ 0∧[1] ≥ 0∧[2 + (-1)bso_12] ≥ 0)

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

• LOAD645(i66[0], 0, i82[0], i78[0]) → COND_LOAD645(&&(>(i82[0], 0), >(+(i78[0], 1), 0)), i66[0], i82[0], i78[0])
• (0 ≥ 0 ⇒ (UIncreasing(COND_LOAD645(&&(>(i82[0], 0), >(+(i78[0], 1), 0)), i66[0], i82[0], i78[0])), ≥)∧[(2)bni_11] ≥ 0∧[(2)bni_11] ≥ 0∧[(2)bni_11] ≥ 0∧[(2)bni_11 + (-1)Bound*bni_11] ≥ 0∧[1] ≥ 0∧[1] ≥ 0∧[1] ≥ 0∧[2 + (-1)bso_12] ≥ 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 with natural coefficients for all symbols [NONINF][POLO]:

POL(TRUE) = 0
POL(FALSE) = 0
POL(COND_LOAD645(x1, x2, x3, x4)) = x4 + x3 + x2
POL(LOAD645(x1, x2, x3, x4)) = [2] + [2]x4 + [2]x3 + [2]x1
POL(+(x1, x2)) = 0
POL(1) = 0
POL(0) = 0
POL(&&(x1, x2)) = 0
POL(>(x1, x2)) = 0

The following pairs are in P>:

LOAD645(i66[0], 0, i82[0], i78[0]) → COND_LOAD645(&&(>(i82[0], 0), >(+(i78[0], 1), 0)), i66[0], i82[0], i78[0])

The following pairs are in Pbound:

LOAD645(i66[0], 0, i82[0], i78[0]) → COND_LOAD645(&&(>(i82[0], 0), >(+(i78[0], 1), 0)), i66[0], i82[0], i78[0])

The following pairs are in P:
none

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

### (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:
none

R is empty.

The integer pair graph is empty.

The set Q consists of the following terms:

### (18) IDependencyGraphProof (EQUIVALENT transformation)

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

### (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:

Boolean, Integer

R is empty.

The integer pair graph contains the following rules and edges:
(0): LOAD645(i66[0], 0, i82[0], i78[0]) → COND_LOAD645(i82[0] > 0 && i78[0] + 1 > 0, i66[0], i82[0], i78[0])
(1): COND_LOAD645(TRUE, i66[1], i82[1], i78[1]) → LOAD645(i66[1], i82[1], i82[1], i78[1] + 1)

(1) -> (0), if ((i78[1] + 1* i78[0])∧(i66[1]* i66[0])∧(i82[1]* 0)∧(i82[1]* i82[0]))

(0) -> (1), if ((i66[0]* i66[1])∧(i82[0] > 0 && i78[0] + 1 > 0* TRUE)∧(i78[0]* i78[1])∧(i82[0]* i82[1]))

The set Q consists of the following terms:

### (21) 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 LOAD645(i66[0], 0, i82[0], i78[0]) → COND_LOAD645(&&(>(i82[0], 0), >(+(i78[0], 1), 0)), i66[0], i82[0], i78[0]) the following chains were created:
• We consider the chain LOAD645(i66[0], 0, i82[0], i78[0]) → COND_LOAD645(&&(>(i82[0], 0), >(+(i78[0], 1), 0)), i66[0], i82[0], i78[0]), COND_LOAD645(TRUE, i66[1], i82[1], i78[1]) → LOAD645(i66[1], i82[1], i82[1], +(i78[1], 1)) which results in the following constraint:

(1)    (i66[0]=i66[1]&&(>(i82[0], 0), >(+(i78[0], 1), 0))=TRUEi78[0]=i78[1]i82[0]=i82[1]LOAD645(i66[0], 0, i82[0], i78[0])≥NonInfC∧LOAD645(i66[0], 0, i82[0], i78[0])≥COND_LOAD645(&&(>(i82[0], 0), >(+(i78[0], 1), 0)), i66[0], i82[0], i78[0])∧(UIncreasing(COND_LOAD645(&&(>(i82[0], 0), >(+(i78[0], 1), 0)), i66[0], i82[0], i78[0])), ≥))

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

(2)    (&&(>(i82[0], 0), >(+(i78[0], 1), 0))=TRUELOAD645(i66[0], 0, i82[0], i78[0])≥NonInfC∧LOAD645(i66[0], 0, i82[0], i78[0])≥COND_LOAD645(&&(>(i82[0], 0), >(+(i78[0], 1), 0)), i66[0], i82[0], i78[0])∧(UIncreasing(COND_LOAD645(&&(>(i82[0], 0), >(+(i78[0], 1), 0)), i66[0], i82[0], i78[0])), ≥))

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

(3)    (0 ≥ 0 ⇒ (UIncreasing(COND_LOAD645(&&(>(i82[0], 0), >(+(i78[0], 1), 0)), i66[0], i82[0], i78[0])), ≥)∧[(2)bni_11 + (-1)Bound*bni_11] + [(2)bni_11]i78[0] + [(2)bni_11]i82[0] + [(2)bni_11]i66[0] ≥ 0∧[2 + (-1)bso_12] + i78[0] + i82[0] + i66[0] ≥ 0)

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

(4)    (0 ≥ 0 ⇒ (UIncreasing(COND_LOAD645(&&(>(i82[0], 0), >(+(i78[0], 1), 0)), i66[0], i82[0], i78[0])), ≥)∧[(2)bni_11 + (-1)Bound*bni_11] + [(2)bni_11]i78[0] + [(2)bni_11]i82[0] + [(2)bni_11]i66[0] ≥ 0∧[2 + (-1)bso_12] + i78[0] + i82[0] + i66[0] ≥ 0)

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

(5)    (0 ≥ 0 ⇒ (UIncreasing(COND_LOAD645(&&(>(i82[0], 0), >(+(i78[0], 1), 0)), i66[0], i82[0], i78[0])), ≥)∧[(2)bni_11 + (-1)Bound*bni_11] + [(2)bni_11]i78[0] + [(2)bni_11]i82[0] + [(2)bni_11]i66[0] ≥ 0∧[2 + (-1)bso_12] + i78[0] + i82[0] + i66[0] ≥ 0)

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

(6)    (0 ≥ 0 ⇒ (UIncreasing(COND_LOAD645(&&(>(i82[0], 0), >(+(i78[0], 1), 0)), i66[0], i82[0], i78[0])), ≥)∧[(2)bni_11] ≥ 0∧[(2)bni_11] ≥ 0∧[(2)bni_11] ≥ 0∧[(2)bni_11 + (-1)Bound*bni_11] ≥ 0∧[1] ≥ 0∧[1] ≥ 0∧[1] ≥ 0∧[2 + (-1)bso_12] ≥ 0)

For Pair COND_LOAD645(TRUE, i66[1], i82[1], i78[1]) → LOAD645(i66[1], i82[1], i82[1], +(i78[1], 1)) the following chains were created:
• We consider the chain LOAD645(i66[0], 0, i82[0], i78[0]) → COND_LOAD645(&&(>(i82[0], 0), >(+(i78[0], 1), 0)), i66[0], i82[0], i78[0]), COND_LOAD645(TRUE, i66[1], i82[1], i78[1]) → LOAD645(i66[1], i82[1], i82[1], +(i78[1], 1)), LOAD645(i66[0], 0, i82[0], i78[0]) → COND_LOAD645(&&(>(i82[0], 0), >(+(i78[0], 1), 0)), i66[0], i82[0], i78[0]) which results in the following constraint:

We solved constraint (7) using rules (I), (II), (III), (IV), (IDP_CONSTANT_FOLD).

To summarize, we get the following constraints P for the following pairs.
• LOAD645(i66[0], 0, i82[0], i78[0]) → COND_LOAD645(&&(>(i82[0], 0), >(+(i78[0], 1), 0)), i66[0], i82[0], i78[0])
• (0 ≥ 0 ⇒ (UIncreasing(COND_LOAD645(&&(>(i82[0], 0), >(+(i78[0], 1), 0)), i66[0], i82[0], i78[0])), ≥)∧[(2)bni_11] ≥ 0∧[(2)bni_11] ≥ 0∧[(2)bni_11] ≥ 0∧[(2)bni_11 + (-1)Bound*bni_11] ≥ 0∧[1] ≥ 0∧[1] ≥ 0∧[1] ≥ 0∧[2 + (-1)bso_12] ≥ 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 with natural coefficients for all symbols [NONINF][POLO]:

POL(TRUE) = 0
POL(FALSE) = 0
POL(LOAD645(x1, x2, x3, x4)) = [2] + [2]x4 + [2]x3 + [2]x1
POL(0) = 0
POL(COND_LOAD645(x1, x2, x3, x4)) = x4 + x3 + x2
POL(&&(x1, x2)) = 0
POL(>(x1, x2)) = 0
POL(+(x1, x2)) = 0
POL(1) = 0

The following pairs are in P>:

LOAD645(i66[0], 0, i82[0], i78[0]) → COND_LOAD645(&&(>(i82[0], 0), >(+(i78[0], 1), 0)), i66[0], i82[0], i78[0])

The following pairs are in Pbound:

LOAD645(i66[0], 0, i82[0], i78[0]) → COND_LOAD645(&&(>(i82[0], 0), >(+(i78[0], 1), 0)), i66[0], i82[0], i78[0])

The following pairs are in P:
none

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

### (22) Obligation:

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

The following domains are used:
none

R is empty.

The integer pair graph is empty.

The set Q consists of the following terms: