### (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: PastaA10

### (1) JBC2FIG (SOUND transformation)

Constructed FIGraph.

### (2) Obligation:

FIGraph based on JBC Program:
Graph of 169 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:
Load524(i12, i44) → Cond_Load524(i12 >= 0 && i12 < i44 && i12 + 1 > 0, i12, i44)
Load524(i12, i44) → Cond_Load5241(i44 >= 0 && i12 > i44, i12, i44)
The set Q consists of the following terms:

### (6) Obligation:

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

The following domains are used:

Boolean, Integer

The ITRS R consists of the following rules:
Load524(i12, i44) → Cond_Load524(i12 >= 0 && i12 < i44 && i12 + 1 > 0, i12, i44)
Load524(i12, i44) → Cond_Load5241(i44 >= 0 && i12 > i44, i12, i44)

The integer pair graph contains the following rules and edges:
(0): LOAD524(i12[0], i44[0]) → COND_LOAD524(i12[0] >= 0 && i12[0] < i44[0] && i12[0] + 1 > 0, i12[0], i44[0])
(2): LOAD524(i12[2], i44[2]) → COND_LOAD5241(i44[2] >= 0 && i12[2] > i44[2], i12[2], i44[2])

(0) -> (1), if ((i44[0]* i44[1])∧(i12[0]* i12[1])∧(i12[0] >= 0 && i12[0] < i44[0] && i12[0] + 1 > 0* TRUE))

(1) -> (0), if ((i44[1]* i44[0])∧(i12[1] + 1* i12[0]))

(1) -> (2), if ((i12[1] + 1* i12[2])∧(i44[1]* i44[2]))

(2) -> (3), if ((i44[2]* i44[3])∧(i12[2]* i12[3])∧(i44[2] >= 0 && i12[2] > i44[2]* TRUE))

(3) -> (0), if ((i44[3] + 1* i44[0])∧(i12[3]* i12[0]))

(3) -> (2), if ((i44[3] + 1* i44[2])∧(i12[3]* i12[2]))

The set Q consists of the following terms:

### (7) UsableRulesProof (EQUIVALENT transformation)

As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [LPAR04] we can delete all non-usable rules [FROCOS05] from R.

### (8) Obligation:

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

The following domains are used:

Boolean, Integer

R is empty.

The integer pair graph contains the following rules and edges:
(0): LOAD524(i12[0], i44[0]) → COND_LOAD524(i12[0] >= 0 && i12[0] < i44[0] && i12[0] + 1 > 0, i12[0], i44[0])
(2): LOAD524(i12[2], i44[2]) → COND_LOAD5241(i44[2] >= 0 && i12[2] > i44[2], i12[2], i44[2])

(0) -> (1), if ((i44[0]* i44[1])∧(i12[0]* i12[1])∧(i12[0] >= 0 && i12[0] < i44[0] && i12[0] + 1 > 0* TRUE))

(1) -> (0), if ((i44[1]* i44[0])∧(i12[1] + 1* i12[0]))

(1) -> (2), if ((i12[1] + 1* i12[2])∧(i44[1]* i44[2]))

(2) -> (3), if ((i44[2]* i44[3])∧(i12[2]* i12[3])∧(i44[2] >= 0 && i12[2] > i44[2]* TRUE))

(3) -> (0), if ((i44[3] + 1* i44[0])∧(i12[3]* i12[0]))

(3) -> (2), if ((i44[3] + 1* i44[2])∧(i12[3]* i12[2]))

The set Q consists of the following terms:

### (9) IDPNonInfProof (SOUND transformation)

The constraints were generated the following way:
The DP Problem is simplified using the Induction Calculus [NONINF] with the following steps:
Note that final constraints are written in bold face.

For Pair LOAD524(i12, i44) → COND_LOAD524(&&(&&(>=(i12, 0), <(i12, i44)), >(+(i12, 1), 0)), i12, i44) the following chains were created:
• We consider the chain LOAD524(i12[0], i44[0]) → COND_LOAD524(&&(&&(>=(i12[0], 0), <(i12[0], i44[0])), >(+(i12[0], 1), 0)), i12[0], i44[0]), COND_LOAD524(TRUE, i12[1], i44[1]) → LOAD524(+(i12[1], 1), i44[1]) which results in the following constraint:

(1)    (i44[0]=i44[1]i12[0]=i12[1]&&(&&(>=(i12[0], 0), <(i12[0], i44[0])), >(+(i12[0], 1), 0))=TRUELOAD524(i12[0], i44[0])≥NonInfC∧LOAD524(i12[0], i44[0])≥COND_LOAD524(&&(&&(>=(i12[0], 0), <(i12[0], i44[0])), >(+(i12[0], 1), 0)), i12[0], i44[0])∧(UIncreasing(COND_LOAD524(&&(&&(>=(i12[0], 0), <(i12[0], i44[0])), >(+(i12[0], 1), 0)), i12[0], i44[0])), ≥))

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

(2)    (>(+(i12[0], 1), 0)=TRUE>=(i12[0], 0)=TRUE<(i12[0], i44[0])=TRUELOAD524(i12[0], i44[0])≥NonInfC∧LOAD524(i12[0], i44[0])≥COND_LOAD524(&&(&&(>=(i12[0], 0), <(i12[0], i44[0])), >(+(i12[0], 1), 0)), i12[0], i44[0])∧(UIncreasing(COND_LOAD524(&&(&&(>=(i12[0], 0), <(i12[0], i44[0])), >(+(i12[0], 1), 0)), i12[0], i44[0])), ≥))

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

(3)    (i12[0] ≥ 0∧i12[0] ≥ 0∧i44[0] + [-1] + [-1]i12[0] ≥ 0 ⇒ (UIncreasing(COND_LOAD524(&&(&&(>=(i12[0], 0), <(i12[0], i44[0])), >(+(i12[0], 1), 0)), i12[0], i44[0])), ≥)∧[(-1)bni_15 + (-1)Bound*bni_15] + [bni_15]max{i12[0] + [-1]i44[0], [-1]i12[0] + i44[0]} ≥ 0∧[(-1)bso_16] + max{i12[0] + [-1]i44[0], [-1]i12[0] + i44[0]} + [-1]max{i12[0] + [-1]i44[0], [-1]i12[0] + i44[0]} ≥ 0)

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

(4)    (i12[0] ≥ 0∧i12[0] ≥ 0∧i44[0] + [-1] + [-1]i12[0] ≥ 0 ⇒ (UIncreasing(COND_LOAD524(&&(&&(>=(i12[0], 0), <(i12[0], i44[0])), >(+(i12[0], 1), 0)), i12[0], i44[0])), ≥)∧[(-1)bni_15 + (-1)Bound*bni_15] + [bni_15]max{i12[0] + [-1]i44[0], [-1]i12[0] + i44[0]} ≥ 0∧[(-1)bso_16] + max{i12[0] + [-1]i44[0], [-1]i12[0] + i44[0]} + [-1]max{i12[0] + [-1]i44[0], [-1]i12[0] + i44[0]} ≥ 0)

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

(5)    (i12[0] ≥ 0∧i12[0] ≥ 0∧i44[0] + [-1] + [-1]i12[0] ≥ 0∧[-1] + [-2]i12[0] + [2]i44[0] ≥ 0 ⇒ (UIncreasing(COND_LOAD524(&&(&&(>=(i12[0], 0), <(i12[0], i44[0])), >(+(i12[0], 1), 0)), i12[0], i44[0])), ≥)∧[(-1)bni_15 + (-1)Bound*bni_15] + [(-1)bni_15]i12[0] + [bni_15]i44[0] ≥ 0∧[(-1)bso_16] ≥ 0)

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

(6)    (i12[0] ≥ 0∧i12[0] ≥ 0∧i44[0] ≥ 0∧[1] + [2]i44[0] ≥ 0 ⇒ (UIncreasing(COND_LOAD524(&&(&&(>=(i12[0], 0), <(i12[0], i44[0])), >(+(i12[0], 1), 0)), i12[0], i44[0])), ≥)∧[(-1)Bound*bni_15] + [bni_15]i44[0] ≥ 0∧[(-1)bso_16] ≥ 0)

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

(7)    (i12[0] ≥ 0∧i12[0] ≥ 0∧i44[0] ≥ 0∧i44[0] ≥ 0 ⇒ (UIncreasing(COND_LOAD524(&&(&&(>=(i12[0], 0), <(i12[0], i44[0])), >(+(i12[0], 1), 0)), i12[0], i44[0])), ≥)∧[(-1)Bound*bni_15] + [bni_15]i44[0] ≥ 0∧[(-1)bso_16] ≥ 0)

For Pair COND_LOAD524(TRUE, i12, i44) → LOAD524(+(i12, 1), i44) the following chains were created:
• We consider the chain COND_LOAD524(TRUE, i12[1], i44[1]) → LOAD524(+(i12[1], 1), i44[1]), LOAD524(i12[0], i44[0]) → COND_LOAD524(&&(&&(>=(i12[0], 0), <(i12[0], i44[0])), >(+(i12[0], 1), 0)), i12[0], i44[0]), COND_LOAD524(TRUE, i12[1], i44[1]) → LOAD524(+(i12[1], 1), i44[1]) which results in the following constraint:

We simplified constraint (8) using rules (III), (IDP_BOOLEAN) which results in the following new constraint:

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

(10)    (i12[1] + [1] ≥ 0∧i12[1] + [1] ≥ 0∧i44[0] + [-2] + [-1]i12[1] ≥ 0 ⇒ (UIncreasing(LOAD524(+(i12[1]1, 1), i44[1]1)), ≥)∧[(-1)bni_17 + (-1)Bound*bni_17] + [bni_17]max{[1] + i12[1] + [-1]i44[0], [-1] + [-1]i12[1] + i44[0]} ≥ 0∧[(-1)bso_18] + max{[1] + i12[1] + [-1]i44[0], [-1] + [-1]i12[1] + i44[0]} + [-1]max{[2] + i12[1] + [-1]i44[0], [-2] + [-1]i12[1] + i44[0]} ≥ 0)

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

(11)    (i12[1] + [1] ≥ 0∧i12[1] + [1] ≥ 0∧i44[0] + [-2] + [-1]i12[1] ≥ 0 ⇒ (UIncreasing(LOAD524(+(i12[1]1, 1), i44[1]1)), ≥)∧[(-1)bni_17 + (-1)Bound*bni_17] + [bni_17]max{[1] + i12[1] + [-1]i44[0], [-1] + [-1]i12[1] + i44[0]} ≥ 0∧[(-1)bso_18] + max{[1] + i12[1] + [-1]i44[0], [-1] + [-1]i12[1] + i44[0]} + [-1]max{[2] + i12[1] + [-1]i44[0], [-2] + [-1]i12[1] + i44[0]} ≥ 0)

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

(12)    (i12[1] + [1] ≥ 0∧i12[1] + [1] ≥ 0∧i44[0] + [-2] + [-1]i12[1] ≥ 0∧[-3] + [-2]i12[1] + [2]i44[0] ≥ 0∧[4] + [2]i12[1] + [-2]i44[0] ≥ 0 ⇒ (UIncreasing(LOAD524(+(i12[1]1, 1), i44[1]1)), ≥)∧[(-2)bni_17 + (-1)Bound*bni_17] + [(-1)bni_17]i12[1] + [bni_17]i44[0] ≥ 0∧[-3 + (-1)bso_18] + [-2]i12[1] + [2]i44[0] ≥ 0)

(13)    (i12[1] + [1] ≥ 0∧i12[1] + [1] ≥ 0∧i44[0] + [-2] + [-1]i12[1] ≥ 0∧[-3] + [-2]i12[1] + [2]i44[0] ≥ 0∧[-5] + [-2]i12[1] + [2]i44[0] ≥ 0 ⇒ (UIncreasing(LOAD524(+(i12[1]1, 1), i44[1]1)), ≥)∧[(-2)bni_17 + (-1)Bound*bni_17] + [(-1)bni_17]i12[1] + [bni_17]i44[0] ≥ 0∧[1 + (-1)bso_18] ≥ 0)

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

(14)    (i44[0] + [-1] + [-1]i12[1] ≥ 0∧i44[0] + [-1] + [-1]i12[1] ≥ 0∧i12[1] ≥ 0∧[1] + [2]i12[1] ≥ 0∧[-2]i12[1] ≥ 0 ⇒ (UIncreasing(LOAD524(+(i12[1]1, 1), i44[1]1)), ≥)∧[(-1)Bound*bni_17] + [bni_17]i12[1] ≥ 0∧[1 + (-1)bso_18] + [2]i12[1] ≥ 0)

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

(15)    (i44[0] + [-1] + [-1]i12[1] ≥ 0∧i44[0] + [-1] + [-1]i12[1] ≥ 0∧i12[1] ≥ 0∧[1] + [2]i12[1] ≥ 0∧[-1] + [2]i12[1] ≥ 0 ⇒ (UIncreasing(LOAD524(+(i12[1]1, 1), i44[1]1)), ≥)∧[(-1)Bound*bni_17] + [bni_17]i12[1] ≥ 0∧[1 + (-1)bso_18] ≥ 0)

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

(16)    (i44[0] + [-1]i12[1] ≥ 0∧i44[0] + [-1]i12[1] ≥ 0∧i12[1] ≥ 0∧[1] + [2]i12[1] ≥ 0∧[-2]i12[1] ≥ 0 ⇒ (UIncreasing(LOAD524(+(i12[1]1, 1), i44[1]1)), ≥)∧[(-1)Bound*bni_17] + [bni_17]i12[1] ≥ 0∧[1 + (-1)bso_18] + [2]i12[1] ≥ 0)

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

(17)    (i44[0] ≥ 0∧i44[0] ≥ 0∧0 ≥ 0∧[1] ≥ 0∧0 ≥ 0 ⇒ (UIncreasing(LOAD524(+(i12[1]1, 1), i44[1]1)), ≥)∧[(-1)Bound*bni_17] ≥ 0∧[1 + (-1)bso_18] ≥ 0)

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

(18)    (i44[0] ≥ 0∧i44[0] ≥ 0∧i12[1] ≥ 0∧[1] + [2]i12[1] ≥ 0∧[-1] + [2]i12[1] ≥ 0 ⇒ (UIncreasing(LOAD524(+(i12[1]1, 1), i44[1]1)), ≥)∧[(-1)Bound*bni_17] + [bni_17]i12[1] ≥ 0∧[1 + (-1)bso_18] ≥ 0)

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

(19)    (i44[0] ≥ 0∧i44[0] ≥ 0∧i12[1] ≥ 0∧i12[1] ≥ 0∧i12[1] ≥ 0 ⇒ (UIncreasing(LOAD524(+(i12[1]1, 1), i44[1]1)), ≥)∧[(-1)Bound*bni_17] + [bni_17]i12[1] ≥ 0∧[1 + (-1)bso_18] ≥ 0)

• We consider the chain COND_LOAD5241(TRUE, i12[3], i44[3]) → LOAD524(i12[3], +(i44[3], 1)), LOAD524(i12[0], i44[0]) → COND_LOAD524(&&(&&(>=(i12[0], 0), <(i12[0], i44[0])), >(+(i12[0], 1), 0)), i12[0], i44[0]), COND_LOAD524(TRUE, i12[1], i44[1]) → LOAD524(+(i12[1], 1), i44[1]) which results in the following constraint:

We simplified constraint (20) using rules (III), (IDP_BOOLEAN) which results in the following new constraint:

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

(22)    (i12[0] ≥ 0∧i12[0] ≥ 0∧i44[3] + [-1]i12[0] ≥ 0 ⇒ (UIncreasing(LOAD524(+(i12[1], 1), i44[1])), ≥)∧[(-1)bni_17 + (-1)Bound*bni_17] + [bni_17]max{[-1] + i12[0] + [-1]i44[3], [1] + [-1]i12[0] + i44[3]} ≥ 0∧[(-1)bso_18] + max{[-1] + i12[0] + [-1]i44[3], [1] + [-1]i12[0] + i44[3]} + [-1]max{i12[0] + [-1]i44[3], [-1]i12[0] + i44[3]} ≥ 0)

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

(23)    (i12[0] ≥ 0∧i12[0] ≥ 0∧i44[3] + [-1]i12[0] ≥ 0 ⇒ (UIncreasing(LOAD524(+(i12[1], 1), i44[1])), ≥)∧[(-1)bni_17 + (-1)Bound*bni_17] + [bni_17]max{[-1] + i12[0] + [-1]i44[3], [1] + [-1]i12[0] + i44[3]} ≥ 0∧[(-1)bso_18] + max{[-1] + i12[0] + [-1]i44[3], [1] + [-1]i12[0] + i44[3]} + [-1]max{i12[0] + [-1]i44[3], [-1]i12[0] + i44[3]} ≥ 0)

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

(24)    (i12[0] ≥ 0∧i12[0] ≥ 0∧i44[3] + [-1]i12[0] ≥ 0∧[1] + [-2]i12[0] + [2]i44[3] ≥ 0∧[2]i12[0] + [-2]i44[3] ≥ 0 ⇒ (UIncreasing(LOAD524(+(i12[1], 1), i44[1])), ≥)∧[(-1)Bound*bni_17] + [(-1)bni_17]i12[0] + [bni_17]i44[3] ≥ 0∧[1 + (-1)bso_18] + [-2]i12[0] + [2]i44[3] ≥ 0)

(25)    (i12[0] ≥ 0∧i12[0] ≥ 0∧i44[3] + [-1]i12[0] ≥ 0∧[1] + [-2]i12[0] + [2]i44[3] ≥ 0∧[-1] + [-2]i12[0] + [2]i44[3] ≥ 0 ⇒ (UIncreasing(LOAD524(+(i12[1], 1), i44[1])), ≥)∧[(-1)Bound*bni_17] + [(-1)bni_17]i12[0] + [bni_17]i44[3] ≥ 0∧[1 + (-1)bso_18] ≥ 0)

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

(26)    (i44[3] + i12[0] ≥ 0∧i44[3] + i12[0] ≥ 0∧[-1]i12[0] ≥ 0∧[1] + [-2]i12[0] ≥ 0∧[2]i12[0] ≥ 0 ⇒ (UIncreasing(LOAD524(+(i12[1], 1), i44[1])), ≥)∧[(-1)Bound*bni_17] + [(-1)bni_17]i12[0] ≥ 0∧[1 + (-1)bso_18] + [-2]i12[0] ≥ 0)

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

(27)    (i12[0] ≥ 0∧i12[0] ≥ 0∧i44[3] ≥ 0∧[1] + [2]i44[3] ≥ 0∧[-1] + [2]i44[3] ≥ 0 ⇒ (UIncreasing(LOAD524(+(i12[1], 1), i44[1])), ≥)∧[(-1)Bound*bni_17] + [bni_17]i44[3] ≥ 0∧[1 + (-1)bso_18] ≥ 0)

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

(28)    (i44[3] ≥ 0∧i44[3] ≥ 0∧0 ≥ 0∧[1] ≥ 0∧0 ≥ 0 ⇒ (UIncreasing(LOAD524(+(i12[1], 1), i44[1])), ≥)∧[(-1)Bound*bni_17] ≥ 0∧[1 + (-1)bso_18] ≥ 0)

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

(29)    (i12[0] ≥ 0∧i12[0] ≥ 0∧i44[3] ≥ 0∧i44[3] ≥ 0∧i44[3] ≥ 0 ⇒ (UIncreasing(LOAD524(+(i12[1], 1), i44[1])), ≥)∧[(-1)Bound*bni_17] + [bni_17]i44[3] ≥ 0∧[1 + (-1)bso_18] ≥ 0)

For Pair LOAD524(i12, i44) → COND_LOAD5241(&&(>=(i44, 0), >(i12, i44)), i12, i44) the following chains were created:
• We consider the chain LOAD524(i12[2], i44[2]) → COND_LOAD5241(&&(>=(i44[2], 0), >(i12[2], i44[2])), i12[2], i44[2]), COND_LOAD5241(TRUE, i12[3], i44[3]) → LOAD524(i12[3], +(i44[3], 1)) which results in the following constraint:

We simplified constraint (30) using rules (IV), (IDP_BOOLEAN) which results in the following new constraint:

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

(32)    (i44[2] ≥ 0∧i12[2] + [-1] + [-1]i44[2] ≥ 0 ⇒ (UIncreasing(COND_LOAD5241(&&(>=(i44[2], 0), >(i12[2], i44[2])), i12[2], i44[2])), ≥)∧[(-1)bni_19 + (-1)Bound*bni_19] + [bni_19]max{i12[2] + [-1]i44[2], [-1]i12[2] + i44[2]} ≥ 0∧[(-1)bso_20] + max{i12[2] + [-1]i44[2], [-1]i12[2] + i44[2]} + [-1]max{i12[2] + [-1]i44[2], [-1]i12[2] + i44[2]} ≥ 0)

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

(33)    (i44[2] ≥ 0∧i12[2] + [-1] + [-1]i44[2] ≥ 0 ⇒ (UIncreasing(COND_LOAD5241(&&(>=(i44[2], 0), >(i12[2], i44[2])), i12[2], i44[2])), ≥)∧[(-1)bni_19 + (-1)Bound*bni_19] + [bni_19]max{i12[2] + [-1]i44[2], [-1]i12[2] + i44[2]} ≥ 0∧[(-1)bso_20] + max{i12[2] + [-1]i44[2], [-1]i12[2] + i44[2]} + [-1]max{i12[2] + [-1]i44[2], [-1]i12[2] + i44[2]} ≥ 0)

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

(34)    (i44[2] ≥ 0∧i12[2] + [-1] + [-1]i44[2] ≥ 0∧[2]i12[2] + [-2]i44[2] ≥ 0 ⇒ (UIncreasing(COND_LOAD5241(&&(>=(i44[2], 0), >(i12[2], i44[2])), i12[2], i44[2])), ≥)∧[(-1)bni_19 + (-1)Bound*bni_19] + [bni_19]i12[2] + [(-1)bni_19]i44[2] ≥ 0∧[(-1)bso_20] ≥ 0)

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

(35)    (i44[2] ≥ 0∧i12[2] ≥ 0∧[2] + [2]i12[2] ≥ 0 ⇒ (UIncreasing(COND_LOAD5241(&&(>=(i44[2], 0), >(i12[2], i44[2])), i12[2], i44[2])), ≥)∧[(-1)Bound*bni_19] + [bni_19]i12[2] ≥ 0∧[(-1)bso_20] ≥ 0)

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

(36)    (i44[2] ≥ 0∧i12[2] ≥ 0∧[1] + i12[2] ≥ 0 ⇒ (UIncreasing(COND_LOAD5241(&&(>=(i44[2], 0), >(i12[2], i44[2])), i12[2], i44[2])), ≥)∧[(-1)Bound*bni_19] + [bni_19]i12[2] ≥ 0∧[(-1)bso_20] ≥ 0)

For Pair COND_LOAD5241(TRUE, i12, i44) → LOAD524(i12, +(i44, 1)) the following chains were created:
• We consider the chain COND_LOAD524(TRUE, i12[1], i44[1]) → LOAD524(+(i12[1], 1), i44[1]), LOAD524(i12[2], i44[2]) → COND_LOAD5241(&&(>=(i44[2], 0), >(i12[2], i44[2])), i12[2], i44[2]), COND_LOAD5241(TRUE, i12[3], i44[3]) → LOAD524(i12[3], +(i44[3], 1)) which results in the following constraint:

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

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

(39)    (i44[2] ≥ 0∧i12[1] + [-1]i44[2] ≥ 0 ⇒ (UIncreasing(LOAD524(i12[3], +(i44[3], 1))), ≥)∧[(-1)bni_21 + (-1)Bound*bni_21] + [bni_21]max{[1] + i12[1] + [-1]i44[2], [-1] + [-1]i12[1] + i44[2]} ≥ 0∧[(-1)bso_22] + max{[1] + i12[1] + [-1]i44[2], [-1] + [-1]i12[1] + i44[2]} + [-1]max{i12[1] + [-1]i44[2], [-1]i12[1] + i44[2]} ≥ 0)

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

(40)    (i44[2] ≥ 0∧i12[1] + [-1]i44[2] ≥ 0 ⇒ (UIncreasing(LOAD524(i12[3], +(i44[3], 1))), ≥)∧[(-1)bni_21 + (-1)Bound*bni_21] + [bni_21]max{[1] + i12[1] + [-1]i44[2], [-1] + [-1]i12[1] + i44[2]} ≥ 0∧[(-1)bso_22] + max{[1] + i12[1] + [-1]i44[2], [-1] + [-1]i12[1] + i44[2]} + [-1]max{i12[1] + [-1]i44[2], [-1]i12[1] + i44[2]} ≥ 0)

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

(41)    (i44[2] ≥ 0∧i12[1] + [-1]i44[2] ≥ 0∧[2] + [2]i12[1] + [-2]i44[2] ≥ 0∧[2]i12[1] + [-2]i44[2] ≥ 0 ⇒ (UIncreasing(LOAD524(i12[3], +(i44[3], 1))), ≥)∧[(-1)Bound*bni_21] + [bni_21]i12[1] + [(-1)bni_21]i44[2] ≥ 0∧[1 + (-1)bso_22] ≥ 0)

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

(42)    (i44[2] ≥ 0∧i12[1] ≥ 0∧[2] + [2]i12[1] ≥ 0∧[2]i12[1] ≥ 0 ⇒ (UIncreasing(LOAD524(i12[3], +(i44[3], 1))), ≥)∧[(-1)Bound*bni_21] + [bni_21]i12[1] ≥ 0∧[1 + (-1)bso_22] ≥ 0)

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

(43)    (i44[2] ≥ 0∧i12[1] ≥ 0∧[1] + i12[1] ≥ 0∧i12[1] ≥ 0 ⇒ (UIncreasing(LOAD524(i12[3], +(i44[3], 1))), ≥)∧[(-1)Bound*bni_21] + [bni_21]i12[1] ≥ 0∧[1 + (-1)bso_22] ≥ 0)

• We consider the chain COND_LOAD5241(TRUE, i12[3], i44[3]) → LOAD524(i12[3], +(i44[3], 1)), LOAD524(i12[2], i44[2]) → COND_LOAD5241(&&(>=(i44[2], 0), >(i12[2], i44[2])), i12[2], i44[2]), COND_LOAD5241(TRUE, i12[3], i44[3]) → LOAD524(i12[3], +(i44[3], 1)) which results in the following constraint:

We simplified constraint (44) using rules (III), (IDP_BOOLEAN) which results in the following new constraint:

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

(46)    (i44[3] + [1] ≥ 0∧i12[2] + [-2] + [-1]i44[3] ≥ 0 ⇒ (UIncreasing(LOAD524(i12[3]1, +(i44[3]1, 1))), ≥)∧[(-1)bni_21 + (-1)Bound*bni_21] + [bni_21]max{[-1] + i12[2] + [-1]i44[3], [1] + [-1]i12[2] + i44[3]} ≥ 0∧[(-1)bso_22] + max{[-1] + i12[2] + [-1]i44[3], [1] + [-1]i12[2] + i44[3]} + [-1]max{[-2] + i12[2] + [-1]i44[3], [2] + [-1]i12[2] + i44[3]} ≥ 0)

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

(47)    (i44[3] + [1] ≥ 0∧i12[2] + [-2] + [-1]i44[3] ≥ 0 ⇒ (UIncreasing(LOAD524(i12[3]1, +(i44[3]1, 1))), ≥)∧[(-1)bni_21 + (-1)Bound*bni_21] + [bni_21]max{[-1] + i12[2] + [-1]i44[3], [1] + [-1]i12[2] + i44[3]} ≥ 0∧[(-1)bso_22] + max{[-1] + i12[2] + [-1]i44[3], [1] + [-1]i12[2] + i44[3]} + [-1]max{[-2] + i12[2] + [-1]i44[3], [2] + [-1]i12[2] + i44[3]} ≥ 0)

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

(48)    (i44[3] + [1] ≥ 0∧i12[2] + [-2] + [-1]i44[3] ≥ 0∧[-2] + [2]i12[2] + [-2]i44[3] ≥ 0∧[-4] + [2]i12[2] + [-2]i44[3] ≥ 0 ⇒ (UIncreasing(LOAD524(i12[3]1, +(i44[3]1, 1))), ≥)∧[(-2)bni_21 + (-1)Bound*bni_21] + [bni_21]i12[2] + [(-1)bni_21]i44[3] ≥ 0∧[1 + (-1)bso_22] ≥ 0)

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

(49)    (i12[2] + [-1] + [-1]i44[3] ≥ 0∧i44[3] ≥ 0∧[2] + [2]i44[3] ≥ 0∧[2]i44[3] ≥ 0 ⇒ (UIncreasing(LOAD524(i12[3]1, +(i44[3]1, 1))), ≥)∧[(-1)Bound*bni_21] + [bni_21]i44[3] ≥ 0∧[1 + (-1)bso_22] ≥ 0)

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

(50)    (i12[2] ≥ 0∧i44[3] ≥ 0∧[2] + [2]i44[3] ≥ 0∧[2]i44[3] ≥ 0 ⇒ (UIncreasing(LOAD524(i12[3]1, +(i44[3]1, 1))), ≥)∧[(-1)Bound*bni_21] + [bni_21]i44[3] ≥ 0∧[1 + (-1)bso_22] ≥ 0)

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

(51)    (i12[2] ≥ 0∧i44[3] ≥ 0∧[1] + i44[3] ≥ 0∧i44[3] ≥ 0 ⇒ (UIncreasing(LOAD524(i12[3]1, +(i44[3]1, 1))), ≥)∧[(-1)Bound*bni_21] + [bni_21]i44[3] ≥ 0∧[1 + (-1)bso_22] ≥ 0)

To summarize, we get the following constraints P for the following pairs.
• LOAD524(i12, i44) → COND_LOAD524(&&(&&(>=(i12, 0), <(i12, i44)), >(+(i12, 1), 0)), i12, i44)
• (i12[0] ≥ 0∧i12[0] ≥ 0∧i44[0] ≥ 0∧i44[0] ≥ 0 ⇒ (UIncreasing(COND_LOAD524(&&(&&(>=(i12[0], 0), <(i12[0], i44[0])), >(+(i12[0], 1), 0)), i12[0], i44[0])), ≥)∧[(-1)Bound*bni_15] + [bni_15]i44[0] ≥ 0∧[(-1)bso_16] ≥ 0)

• (i44[0] ≥ 0∧i44[0] ≥ 0∧0 ≥ 0∧[1] ≥ 0∧0 ≥ 0 ⇒ (UIncreasing(LOAD524(+(i12[1]1, 1), i44[1]1)), ≥)∧[(-1)Bound*bni_17] ≥ 0∧[1 + (-1)bso_18] ≥ 0)
• (i44[0] ≥ 0∧i44[0] ≥ 0∧i12[1] ≥ 0∧i12[1] ≥ 0∧i12[1] ≥ 0 ⇒ (UIncreasing(LOAD524(+(i12[1]1, 1), i44[1]1)), ≥)∧[(-1)Bound*bni_17] + [bni_17]i12[1] ≥ 0∧[1 + (-1)bso_18] ≥ 0)
• (i44[3] ≥ 0∧i44[3] ≥ 0∧0 ≥ 0∧[1] ≥ 0∧0 ≥ 0 ⇒ (UIncreasing(LOAD524(+(i12[1], 1), i44[1])), ≥)∧[(-1)Bound*bni_17] ≥ 0∧[1 + (-1)bso_18] ≥ 0)
• (i12[0] ≥ 0∧i12[0] ≥ 0∧i44[3] ≥ 0∧i44[3] ≥ 0∧i44[3] ≥ 0 ⇒ (UIncreasing(LOAD524(+(i12[1], 1), i44[1])), ≥)∧[(-1)Bound*bni_17] + [bni_17]i44[3] ≥ 0∧[1 + (-1)bso_18] ≥ 0)

• (i44[2] ≥ 0∧i12[2] ≥ 0∧[1] + i12[2] ≥ 0 ⇒ (UIncreasing(COND_LOAD5241(&&(>=(i44[2], 0), >(i12[2], i44[2])), i12[2], i44[2])), ≥)∧[(-1)Bound*bni_19] + [bni_19]i12[2] ≥ 0∧[(-1)bso_20] ≥ 0)

• (i44[2] ≥ 0∧i12[1] ≥ 0∧[1] + i12[1] ≥ 0∧i12[1] ≥ 0 ⇒ (UIncreasing(LOAD524(i12[3], +(i44[3], 1))), ≥)∧[(-1)Bound*bni_21] + [bni_21]i12[1] ≥ 0∧[1 + (-1)bso_22] ≥ 0)
• (i12[2] ≥ 0∧i44[3] ≥ 0∧[1] + i44[3] ≥ 0∧i44[3] ≥ 0 ⇒ (UIncreasing(LOAD524(i12[3]1, +(i44[3]1, 1))), ≥)∧[(-1)Bound*bni_21] + [bni_21]i44[3] ≥ 0∧[1 + (-1)bso_22] ≥ 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(LOAD524(x1, x2)) = [-1] + max{x1 + [-1]x2, [-1]x1 + x2}
POL(COND_LOAD524(x1, x2, x3)) = [-1] + max{x2 + [-1]x3, [-1]x2 + x3}
POL(&&(x1, x2)) = [-1]
POL(>=(x1, x2)) = [-1]
POL(0) = 0
POL(<(x1, x2)) = [-1]
POL(>(x1, x2)) = [-1]
POL(+(x1, x2)) = x1 + x2
POL(1) = [1]
POL(COND_LOAD5241(x1, x2, x3)) = [-1] + max{x2 + [-1]x3, [-1]x2 + x3}

The following pairs are in P>:

The following pairs are in Pbound:

LOAD524(i12[0], i44[0]) → COND_LOAD524(&&(&&(>=(i12[0], 0), <(i12[0], i44[0])), >(+(i12[0], 1), 0)), i12[0], i44[0])

The following pairs are in P:

LOAD524(i12[0], i44[0]) → COND_LOAD524(&&(&&(>=(i12[0], 0), <(i12[0], i44[0])), >(+(i12[0], 1), 0)), i12[0], i44[0])

There are no usable rules.

### (11) Obligation:

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

The following domains are used:

Boolean, Integer

R is empty.

The integer pair graph contains the following rules and edges:
(0): LOAD524(i12[0], i44[0]) → COND_LOAD524(i12[0] >= 0 && i12[0] < i44[0] && i12[0] + 1 > 0, i12[0], i44[0])
(2): LOAD524(i12[2], i44[2]) → COND_LOAD5241(i44[2] >= 0 && i12[2] > i44[2], i12[2], i44[2])

The set Q consists of the following terms:

### (12) IDependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs 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:
none

R is empty.

The integer pair graph is empty.

The set Q consists of the following terms: