### (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: PastaC3

### (1) JBC2FIG (SOUND transformation)

Constructed FIGraph.

### (2) Obligation:

FIGraph based on JBC Program:
Graph of 230 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:
Load875(i12, i35, i76) → Cond_Load875(i76 >= 0 && i12 >= i76 && i12 < i35 && i76 + 1 > 0, i12, i35, i76)
Load875(i12, i35, i76) → Cond_Load8751(i12 >= 0 && i12 < i76 && i12 < i35 && i12 + 1 > 0, i12, i35, i76)
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:
Load875(i12, i35, i76) → Cond_Load875(i76 >= 0 && i12 >= i76 && i12 < i35 && i76 + 1 > 0, i12, i35, i76)
Load875(i12, i35, i76) → Cond_Load8751(i12 >= 0 && i12 < i76 && i12 < i35 && i12 + 1 > 0, i12, i35, i76)

The integer pair graph contains the following rules and edges:
(0): LOAD875(i12[0], i35[0], i76[0]) → COND_LOAD875(i76[0] >= 0 && i12[0] >= i76[0] && i12[0] < i35[0] && i76[0] + 1 > 0, i12[0], i35[0], i76[0])
(2): LOAD875(i12[2], i35[2], i76[2]) → COND_LOAD8751(i12[2] >= 0 && i12[2] < i76[2] && i12[2] < i35[2] && i12[2] + 1 > 0, i12[2], i35[2], i76[2])

(0) -> (1), if ((i76[0] >= 0 && i12[0] >= i76[0] && i12[0] < i35[0] && i76[0] + 1 > 0* TRUE)∧(i76[0]* i76[1])∧(i12[0]* i12[1])∧(i35[0]* i35[1]))

(1) -> (0), if ((i35[1]* i35[0])∧(i12[1]* i12[0])∧(i76[1] + 1* i76[0]))

(1) -> (2), if ((i12[1]* i12[2])∧(i35[1]* i35[2])∧(i76[1] + 1* i76[2]))

(2) -> (3), if ((i35[2]* i35[3])∧(i12[2] >= 0 && i12[2] < i76[2] && i12[2] < i35[2] && i12[2] + 1 > 0* TRUE)∧(i76[2]* i76[3])∧(i12[2]* i12[3]))

(3) -> (0), if ((i35[3]* i35[0])∧(i76[3]* i76[0])∧(i12[3] + 1* i12[0]))

(3) -> (2), if ((i35[3]* i35[2])∧(i76[3]* i76[2])∧(i12[3] + 1* 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): LOAD875(i12[0], i35[0], i76[0]) → COND_LOAD875(i76[0] >= 0 && i12[0] >= i76[0] && i12[0] < i35[0] && i76[0] + 1 > 0, i12[0], i35[0], i76[0])
(2): LOAD875(i12[2], i35[2], i76[2]) → COND_LOAD8751(i12[2] >= 0 && i12[2] < i76[2] && i12[2] < i35[2] && i12[2] + 1 > 0, i12[2], i35[2], i76[2])

(0) -> (1), if ((i76[0] >= 0 && i12[0] >= i76[0] && i12[0] < i35[0] && i76[0] + 1 > 0* TRUE)∧(i76[0]* i76[1])∧(i12[0]* i12[1])∧(i35[0]* i35[1]))

(1) -> (0), if ((i35[1]* i35[0])∧(i12[1]* i12[0])∧(i76[1] + 1* i76[0]))

(1) -> (2), if ((i12[1]* i12[2])∧(i35[1]* i35[2])∧(i76[1] + 1* i76[2]))

(2) -> (3), if ((i35[2]* i35[3])∧(i12[2] >= 0 && i12[2] < i76[2] && i12[2] < i35[2] && i12[2] + 1 > 0* TRUE)∧(i76[2]* i76[3])∧(i12[2]* i12[3]))

(3) -> (0), if ((i35[3]* i35[0])∧(i76[3]* i76[0])∧(i12[3] + 1* i12[0]))

(3) -> (2), if ((i35[3]* i35[2])∧(i76[3]* i76[2])∧(i12[3] + 1* 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 LOAD875(i12, i35, i76) → COND_LOAD875(&&(&&(&&(>=(i76, 0), >=(i12, i76)), <(i12, i35)), >(+(i76, 1), 0)), i12, i35, i76) the following chains were created:
• We consider the chain LOAD875(i12[0], i35[0], i76[0]) → COND_LOAD875(&&(&&(&&(>=(i76[0], 0), >=(i12[0], i76[0])), <(i12[0], i35[0])), >(+(i76[0], 1), 0)), i12[0], i35[0], i76[0]), COND_LOAD875(TRUE, i12[1], i35[1], i76[1]) → LOAD875(i12[1], i35[1], +(i76[1], 1)) which results in the following constraint:

(1)    (&&(&&(&&(>=(i76[0], 0), >=(i12[0], i76[0])), <(i12[0], i35[0])), >(+(i76[0], 1), 0))=TRUEi76[0]=i76[1]i12[0]=i12[1]i35[0]=i35[1]LOAD875(i12[0], i35[0], i76[0])≥NonInfC∧LOAD875(i12[0], i35[0], i76[0])≥COND_LOAD875(&&(&&(&&(>=(i76[0], 0), >=(i12[0], i76[0])), <(i12[0], i35[0])), >(+(i76[0], 1), 0)), i12[0], i35[0], i76[0])∧(UIncreasing(COND_LOAD875(&&(&&(&&(>=(i76[0], 0), >=(i12[0], i76[0])), <(i12[0], i35[0])), >(+(i76[0], 1), 0)), i12[0], i35[0], i76[0])), ≥))

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

(2)    (>(+(i76[0], 1), 0)=TRUE<(i12[0], i35[0])=TRUE>=(i76[0], 0)=TRUE>=(i12[0], i76[0])=TRUELOAD875(i12[0], i35[0], i76[0])≥NonInfC∧LOAD875(i12[0], i35[0], i76[0])≥COND_LOAD875(&&(&&(&&(>=(i76[0], 0), >=(i12[0], i76[0])), <(i12[0], i35[0])), >(+(i76[0], 1), 0)), i12[0], i35[0], i76[0])∧(UIncreasing(COND_LOAD875(&&(&&(&&(>=(i76[0], 0), >=(i12[0], i76[0])), <(i12[0], i35[0])), >(+(i76[0], 1), 0)), i12[0], i35[0], i76[0])), ≥))

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

(3)    (i76[0] ≥ 0∧i35[0] + [-1] + [-1]i12[0] ≥ 0∧i76[0] ≥ 0∧i12[0] + [-1]i76[0] ≥ 0 ⇒ (UIncreasing(COND_LOAD875(&&(&&(&&(>=(i76[0], 0), >=(i12[0], i76[0])), <(i12[0], i35[0])), >(+(i76[0], 1), 0)), i12[0], i35[0], i76[0])), ≥)∧[(2)bni_13 + (-1)Bound*bni_13] + [(-1)bni_13]i76[0] + [(2)bni_13]i35[0] + [(-1)bni_13]i12[0] ≥ 0∧[(-1)bso_14] ≥ 0)

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

(4)    (i76[0] ≥ 0∧i35[0] + [-1] + [-1]i12[0] ≥ 0∧i76[0] ≥ 0∧i12[0] + [-1]i76[0] ≥ 0 ⇒ (UIncreasing(COND_LOAD875(&&(&&(&&(>=(i76[0], 0), >=(i12[0], i76[0])), <(i12[0], i35[0])), >(+(i76[0], 1), 0)), i12[0], i35[0], i76[0])), ≥)∧[(2)bni_13 + (-1)Bound*bni_13] + [(-1)bni_13]i76[0] + [(2)bni_13]i35[0] + [(-1)bni_13]i12[0] ≥ 0∧[(-1)bso_14] ≥ 0)

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

(5)    (i76[0] ≥ 0∧i35[0] + [-1] + [-1]i12[0] ≥ 0∧i76[0] ≥ 0∧i12[0] + [-1]i76[0] ≥ 0 ⇒ (UIncreasing(COND_LOAD875(&&(&&(&&(>=(i76[0], 0), >=(i12[0], i76[0])), <(i12[0], i35[0])), >(+(i76[0], 1), 0)), i12[0], i35[0], i76[0])), ≥)∧[(2)bni_13 + (-1)Bound*bni_13] + [(-1)bni_13]i76[0] + [(2)bni_13]i35[0] + [(-1)bni_13]i12[0] ≥ 0∧[(-1)bso_14] ≥ 0)

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

(6)    (i76[0] ≥ 0∧i35[0] ≥ 0∧i76[0] ≥ 0∧i12[0] + [-1]i76[0] ≥ 0 ⇒ (UIncreasing(COND_LOAD875(&&(&&(&&(>=(i76[0], 0), >=(i12[0], i76[0])), <(i12[0], i35[0])), >(+(i76[0], 1), 0)), i12[0], i35[0], i76[0])), ≥)∧[(4)bni_13 + (-1)Bound*bni_13] + [(-1)bni_13]i76[0] + [bni_13]i12[0] + [(2)bni_13]i35[0] ≥ 0∧[(-1)bso_14] ≥ 0)

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

(7)    (i76[0] ≥ 0∧i35[0] ≥ 0∧i76[0] ≥ 0∧i12[0] ≥ 0 ⇒ (UIncreasing(COND_LOAD875(&&(&&(&&(>=(i76[0], 0), >=(i12[0], i76[0])), <(i12[0], i35[0])), >(+(i76[0], 1), 0)), i12[0], i35[0], i76[0])), ≥)∧[(4)bni_13 + (-1)Bound*bni_13] + [bni_13]i12[0] + [(2)bni_13]i35[0] ≥ 0∧[(-1)bso_14] ≥ 0)

For Pair COND_LOAD875(TRUE, i12, i35, i76) → LOAD875(i12, i35, +(i76, 1)) the following chains were created:
• We consider the chain COND_LOAD875(TRUE, i12[1], i35[1], i76[1]) → LOAD875(i12[1], i35[1], +(i76[1], 1)) which results in the following constraint:

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

(9)    ((UIncreasing(LOAD875(i12[1], i35[1], +(i76[1], 1))), ≥)∧[1 + (-1)bso_16] ≥ 0)

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

(10)    ((UIncreasing(LOAD875(i12[1], i35[1], +(i76[1], 1))), ≥)∧[1 + (-1)bso_16] ≥ 0)

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

(11)    ((UIncreasing(LOAD875(i12[1], i35[1], +(i76[1], 1))), ≥)∧[1 + (-1)bso_16] ≥ 0)

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

(12)    ((UIncreasing(LOAD875(i12[1], i35[1], +(i76[1], 1))), ≥)∧0 = 0∧0 = 0∧0 = 0∧[1 + (-1)bso_16] ≥ 0)

For Pair LOAD875(i12, i35, i76) → COND_LOAD8751(&&(&&(&&(>=(i12, 0), <(i12, i76)), <(i12, i35)), >(+(i12, 1), 0)), i12, i35, i76) the following chains were created:
• We consider the chain LOAD875(i12[2], i35[2], i76[2]) → COND_LOAD8751(&&(&&(&&(>=(i12[2], 0), <(i12[2], i76[2])), <(i12[2], i35[2])), >(+(i12[2], 1), 0)), i12[2], i35[2], i76[2]), COND_LOAD8751(TRUE, i12[3], i35[3], i76[3]) → LOAD875(+(i12[3], 1), i35[3], i76[3]) which results in the following constraint:

(13)    (i35[2]=i35[3]&&(&&(&&(>=(i12[2], 0), <(i12[2], i76[2])), <(i12[2], i35[2])), >(+(i12[2], 1), 0))=TRUEi76[2]=i76[3]i12[2]=i12[3]LOAD875(i12[2], i35[2], i76[2])≥NonInfC∧LOAD875(i12[2], i35[2], i76[2])≥COND_LOAD8751(&&(&&(&&(>=(i12[2], 0), <(i12[2], i76[2])), <(i12[2], i35[2])), >(+(i12[2], 1), 0)), i12[2], i35[2], i76[2])∧(UIncreasing(COND_LOAD8751(&&(&&(&&(>=(i12[2], 0), <(i12[2], i76[2])), <(i12[2], i35[2])), >(+(i12[2], 1), 0)), i12[2], i35[2], i76[2])), ≥))

We simplified constraint (13) using rules (IV), (IDP_BOOLEAN) which results in the following new constraint:

(14)    (>(+(i12[2], 1), 0)=TRUE<(i12[2], i35[2])=TRUE>=(i12[2], 0)=TRUE<(i12[2], i76[2])=TRUELOAD875(i12[2], i35[2], i76[2])≥NonInfC∧LOAD875(i12[2], i35[2], i76[2])≥COND_LOAD8751(&&(&&(&&(>=(i12[2], 0), <(i12[2], i76[2])), <(i12[2], i35[2])), >(+(i12[2], 1), 0)), i12[2], i35[2], i76[2])∧(UIncreasing(COND_LOAD8751(&&(&&(&&(>=(i12[2], 0), <(i12[2], i76[2])), <(i12[2], i35[2])), >(+(i12[2], 1), 0)), i12[2], i35[2], i76[2])), ≥))

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

(15)    (i12[2] ≥ 0∧i35[2] + [-1] + [-1]i12[2] ≥ 0∧i12[2] ≥ 0∧i76[2] + [-1] + [-1]i12[2] ≥ 0 ⇒ (UIncreasing(COND_LOAD8751(&&(&&(&&(>=(i12[2], 0), <(i12[2], i76[2])), <(i12[2], i35[2])), >(+(i12[2], 1), 0)), i12[2], i35[2], i76[2])), ≥)∧[(2)bni_17 + (-1)Bound*bni_17] + [(-1)bni_17]i76[2] + [(2)bni_17]i35[2] + [(-1)bni_17]i12[2] ≥ 0∧[(-1)bso_18] ≥ 0)

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

(16)    (i12[2] ≥ 0∧i35[2] + [-1] + [-1]i12[2] ≥ 0∧i12[2] ≥ 0∧i76[2] + [-1] + [-1]i12[2] ≥ 0 ⇒ (UIncreasing(COND_LOAD8751(&&(&&(&&(>=(i12[2], 0), <(i12[2], i76[2])), <(i12[2], i35[2])), >(+(i12[2], 1), 0)), i12[2], i35[2], i76[2])), ≥)∧[(2)bni_17 + (-1)Bound*bni_17] + [(-1)bni_17]i76[2] + [(2)bni_17]i35[2] + [(-1)bni_17]i12[2] ≥ 0∧[(-1)bso_18] ≥ 0)

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

(17)    (i12[2] ≥ 0∧i35[2] + [-1] + [-1]i12[2] ≥ 0∧i12[2] ≥ 0∧i76[2] + [-1] + [-1]i12[2] ≥ 0 ⇒ (UIncreasing(COND_LOAD8751(&&(&&(&&(>=(i12[2], 0), <(i12[2], i76[2])), <(i12[2], i35[2])), >(+(i12[2], 1), 0)), i12[2], i35[2], i76[2])), ≥)∧[(2)bni_17 + (-1)Bound*bni_17] + [(-1)bni_17]i76[2] + [(2)bni_17]i35[2] + [(-1)bni_17]i12[2] ≥ 0∧[(-1)bso_18] ≥ 0)

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

(18)    (i12[2] ≥ 0∧i35[2] ≥ 0∧i12[2] ≥ 0∧i76[2] + [-1] + [-1]i12[2] ≥ 0 ⇒ (UIncreasing(COND_LOAD8751(&&(&&(&&(>=(i12[2], 0), <(i12[2], i76[2])), <(i12[2], i35[2])), >(+(i12[2], 1), 0)), i12[2], i35[2], i76[2])), ≥)∧[(4)bni_17 + (-1)Bound*bni_17] + [(-1)bni_17]i76[2] + [bni_17]i12[2] + [(2)bni_17]i35[2] ≥ 0∧[(-1)bso_18] ≥ 0)

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

(19)    (i12[2] ≥ 0∧i35[2] ≥ 0∧i12[2] ≥ 0∧i76[2] ≥ 0 ⇒ (UIncreasing(COND_LOAD8751(&&(&&(&&(>=(i12[2], 0), <(i12[2], i76[2])), <(i12[2], i35[2])), >(+(i12[2], 1), 0)), i12[2], i35[2], i76[2])), ≥)∧[(3)bni_17 + (-1)Bound*bni_17] + [(-1)bni_17]i76[2] + [(2)bni_17]i35[2] ≥ 0∧[(-1)bso_18] ≥ 0)

For Pair COND_LOAD8751(TRUE, i12, i35, i76) → LOAD875(+(i12, 1), i35, i76) the following chains were created:
• We consider the chain COND_LOAD8751(TRUE, i12[3], i35[3], i76[3]) → LOAD875(+(i12[3], 1), i35[3], i76[3]) which results in the following constraint:

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

(21)    ((UIncreasing(LOAD875(+(i12[3], 1), i35[3], i76[3])), ≥)∧[1 + (-1)bso_20] ≥ 0)

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

(22)    ((UIncreasing(LOAD875(+(i12[3], 1), i35[3], i76[3])), ≥)∧[1 + (-1)bso_20] ≥ 0)

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

(23)    ((UIncreasing(LOAD875(+(i12[3], 1), i35[3], i76[3])), ≥)∧[1 + (-1)bso_20] ≥ 0)

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

(24)    ((UIncreasing(LOAD875(+(i12[3], 1), i35[3], i76[3])), ≥)∧0 = 0∧0 = 0∧0 = 0∧[1 + (-1)bso_20] ≥ 0)

To summarize, we get the following constraints P for the following pairs.
• LOAD875(i12, i35, i76) → COND_LOAD875(&&(&&(&&(>=(i76, 0), >=(i12, i76)), <(i12, i35)), >(+(i76, 1), 0)), i12, i35, i76)
• (i76[0] ≥ 0∧i35[0] ≥ 0∧i76[0] ≥ 0∧i12[0] ≥ 0 ⇒ (UIncreasing(COND_LOAD875(&&(&&(&&(>=(i76[0], 0), >=(i12[0], i76[0])), <(i12[0], i35[0])), >(+(i76[0], 1), 0)), i12[0], i35[0], i76[0])), ≥)∧[(4)bni_13 + (-1)Bound*bni_13] + [bni_13]i12[0] + [(2)bni_13]i35[0] ≥ 0∧[(-1)bso_14] ≥ 0)

• ((UIncreasing(LOAD875(i12[1], i35[1], +(i76[1], 1))), ≥)∧0 = 0∧0 = 0∧0 = 0∧[1 + (-1)bso_16] ≥ 0)

• LOAD875(i12, i35, i76) → COND_LOAD8751(&&(&&(&&(>=(i12, 0), <(i12, i76)), <(i12, i35)), >(+(i12, 1), 0)), i12, i35, i76)
• (i12[2] ≥ 0∧i35[2] ≥ 0∧i12[2] ≥ 0∧i76[2] ≥ 0 ⇒ (UIncreasing(COND_LOAD8751(&&(&&(&&(>=(i12[2], 0), <(i12[2], i76[2])), <(i12[2], i35[2])), >(+(i12[2], 1), 0)), i12[2], i35[2], i76[2])), ≥)∧[(3)bni_17 + (-1)Bound*bni_17] + [(-1)bni_17]i76[2] + [(2)bni_17]i35[2] ≥ 0∧[(-1)bso_18] ≥ 0)

• ((UIncreasing(LOAD875(+(i12[3], 1), i35[3], i76[3])), ≥)∧0 = 0∧0 = 0∧0 = 0∧[1 + (-1)bso_20] ≥ 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(LOAD875(x1, x2, x3)) = [2] + [-1]x3 + [2]x2 + [-1]x1
POL(COND_LOAD875(x1, x2, x3, x4)) = [2] + [-1]x4 + [2]x3 + [-1]x2
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_LOAD8751(x1, x2, x3, x4)) = [2] + [-1]x4 + [2]x3 + [-1]x2

The following pairs are in P>:

The following pairs are in Pbound:

LOAD875(i12[0], i35[0], i76[0]) → COND_LOAD875(&&(&&(&&(>=(i76[0], 0), >=(i12[0], i76[0])), <(i12[0], i35[0])), >(+(i76[0], 1), 0)), i12[0], i35[0], i76[0])

The following pairs are in P:

LOAD875(i12[0], i35[0], i76[0]) → COND_LOAD875(&&(&&(&&(>=(i76[0], 0), >=(i12[0], i76[0])), <(i12[0], i35[0])), >(+(i76[0], 1), 0)), i12[0], i35[0], i76[0])
LOAD875(i12[2], i35[2], i76[2]) → COND_LOAD8751(&&(&&(&&(>=(i12[2], 0), <(i12[2], i76[2])), <(i12[2], i35[2])), >(+(i12[2], 1), 0)), i12[2], i35[2], i76[2])

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): LOAD875(i12[0], i35[0], i76[0]) → COND_LOAD875(i76[0] >= 0 && i12[0] >= i76[0] && i12[0] < i35[0] && i76[0] + 1 > 0, i12[0], i35[0], i76[0])
(2): LOAD875(i12[2], i35[2], i76[2]) → COND_LOAD8751(i12[2] >= 0 && i12[2] < i76[2] && i12[2] < i35[2] && i12[2] + 1 > 0, i12[2], i35[2], i76[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:

Integer, Boolean

R is empty.

The integer pair graph contains the following rules and edges:
(2): LOAD875(i12[2], i35[2], i76[2]) → COND_LOAD8751(i12[2] >= 0 && i12[2] < i76[2] && i12[2] < i35[2] && i12[2] + 1 > 0, i12[2], i35[2], i76[2])

(1) -> (2), if ((i12[1]* i12[2])∧(i35[1]* i35[2])∧(i76[1] + 1* i76[2]))

(3) -> (2), if ((i35[3]* i35[2])∧(i76[3]* i76[2])∧(i12[3] + 1* i12[2]))

(2) -> (3), if ((i35[2]* i35[3])∧(i12[2] >= 0 && i12[2] < i76[2] && i12[2] < i35[2] && i12[2] + 1 > 0* TRUE)∧(i76[2]* i76[3])∧(i12[2]* i12[3]))

The set Q consists of the following terms:

### (15) IDependencyGraphProof (EQUIVALENT transformation)

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

### (16) Obligation:

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

The following domains are used:

Integer, Boolean

R is empty.

The integer pair graph contains the following rules and edges:
(2): LOAD875(i12[2], i35[2], i76[2]) → COND_LOAD8751(i12[2] >= 0 && i12[2] < i76[2] && i12[2] < i35[2] && i12[2] + 1 > 0, i12[2], i35[2], i76[2])

(3) -> (2), if ((i35[3]* i35[2])∧(i76[3]* i76[2])∧(i12[3] + 1* i12[2]))

(2) -> (3), if ((i35[2]* i35[3])∧(i12[2] >= 0 && i12[2] < i76[2] && i12[2] < i35[2] && i12[2] + 1 > 0* TRUE)∧(i76[2]* i76[3])∧(i12[2]* i12[3]))

The set Q consists of the following terms:

### (17) 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_LOAD8751(TRUE, i12[3], i35[3], i76[3]) → LOAD875(+(i12[3], 1), i35[3], i76[3]) the following chains were created:
• We consider the chain COND_LOAD8751(TRUE, i12[3], i35[3], i76[3]) → LOAD875(+(i12[3], 1), i35[3], i76[3]) which results in the following constraint:

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

(2)    ((UIncreasing(LOAD875(+(i12[3], 1), i35[3], i76[3])), ≥)∧[(-1)bso_13] ≥ 0)

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

(3)    ((UIncreasing(LOAD875(+(i12[3], 1), i35[3], i76[3])), ≥)∧[(-1)bso_13] ≥ 0)

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

(4)    ((UIncreasing(LOAD875(+(i12[3], 1), i35[3], i76[3])), ≥)∧[(-1)bso_13] ≥ 0)

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

(5)    ((UIncreasing(LOAD875(+(i12[3], 1), i35[3], i76[3])), ≥)∧0 = 0∧0 = 0∧0 = 0∧[(-1)bso_13] ≥ 0)

For Pair LOAD875(i12[2], i35[2], i76[2]) → COND_LOAD8751(&&(&&(&&(>=(i12[2], 0), <(i12[2], i76[2])), <(i12[2], i35[2])), >(+(i12[2], 1), 0)), i12[2], i35[2], i76[2]) the following chains were created:
• We consider the chain LOAD875(i12[2], i35[2], i76[2]) → COND_LOAD8751(&&(&&(&&(>=(i12[2], 0), <(i12[2], i76[2])), <(i12[2], i35[2])), >(+(i12[2], 1), 0)), i12[2], i35[2], i76[2]), COND_LOAD8751(TRUE, i12[3], i35[3], i76[3]) → LOAD875(+(i12[3], 1), i35[3], i76[3]) which results in the following constraint:

(6)    (i35[2]=i35[3]&&(&&(&&(>=(i12[2], 0), <(i12[2], i76[2])), <(i12[2], i35[2])), >(+(i12[2], 1), 0))=TRUEi76[2]=i76[3]i12[2]=i12[3]LOAD875(i12[2], i35[2], i76[2])≥NonInfC∧LOAD875(i12[2], i35[2], i76[2])≥COND_LOAD8751(&&(&&(&&(>=(i12[2], 0), <(i12[2], i76[2])), <(i12[2], i35[2])), >(+(i12[2], 1), 0)), i12[2], i35[2], i76[2])∧(UIncreasing(COND_LOAD8751(&&(&&(&&(>=(i12[2], 0), <(i12[2], i76[2])), <(i12[2], i35[2])), >(+(i12[2], 1), 0)), i12[2], i35[2], i76[2])), ≥))

We simplified constraint (6) using rules (IV), (IDP_BOOLEAN) which results in the following new constraint:

(7)    (>(+(i12[2], 1), 0)=TRUE<(i12[2], i35[2])=TRUE>=(i12[2], 0)=TRUE<(i12[2], i76[2])=TRUELOAD875(i12[2], i35[2], i76[2])≥NonInfC∧LOAD875(i12[2], i35[2], i76[2])≥COND_LOAD8751(&&(&&(&&(>=(i12[2], 0), <(i12[2], i76[2])), <(i12[2], i35[2])), >(+(i12[2], 1), 0)), i12[2], i35[2], i76[2])∧(UIncreasing(COND_LOAD8751(&&(&&(&&(>=(i12[2], 0), <(i12[2], i76[2])), <(i12[2], i35[2])), >(+(i12[2], 1), 0)), i12[2], i35[2], i76[2])), ≥))

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

(8)    (i12[2] ≥ 0∧i35[2] + [-1] + [-1]i12[2] ≥ 0∧i12[2] ≥ 0∧i76[2] + [-1] + [-1]i12[2] ≥ 0 ⇒ (UIncreasing(COND_LOAD8751(&&(&&(&&(>=(i12[2], 0), <(i12[2], i76[2])), <(i12[2], i35[2])), >(+(i12[2], 1), 0)), i12[2], i35[2], i76[2])), ≥)∧[(-1)Bound*bni_14] + [(-1)bni_14]i12[2] + [bni_14]i76[2] + [(2)bni_14]i35[2] ≥ 0∧[1 + (-1)bso_15] ≥ 0)

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

(9)    (i12[2] ≥ 0∧i35[2] + [-1] + [-1]i12[2] ≥ 0∧i12[2] ≥ 0∧i76[2] + [-1] + [-1]i12[2] ≥ 0 ⇒ (UIncreasing(COND_LOAD8751(&&(&&(&&(>=(i12[2], 0), <(i12[2], i76[2])), <(i12[2], i35[2])), >(+(i12[2], 1), 0)), i12[2], i35[2], i76[2])), ≥)∧[(-1)Bound*bni_14] + [(-1)bni_14]i12[2] + [bni_14]i76[2] + [(2)bni_14]i35[2] ≥ 0∧[1 + (-1)bso_15] ≥ 0)

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

(10)    (i12[2] ≥ 0∧i35[2] + [-1] + [-1]i12[2] ≥ 0∧i12[2] ≥ 0∧i76[2] + [-1] + [-1]i12[2] ≥ 0 ⇒ (UIncreasing(COND_LOAD8751(&&(&&(&&(>=(i12[2], 0), <(i12[2], i76[2])), <(i12[2], i35[2])), >(+(i12[2], 1), 0)), i12[2], i35[2], i76[2])), ≥)∧[(-1)Bound*bni_14] + [(-1)bni_14]i12[2] + [bni_14]i76[2] + [(2)bni_14]i35[2] ≥ 0∧[1 + (-1)bso_15] ≥ 0)

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

(11)    (i12[2] ≥ 0∧i35[2] ≥ 0∧i12[2] ≥ 0∧i76[2] + [-1] + [-1]i12[2] ≥ 0 ⇒ (UIncreasing(COND_LOAD8751(&&(&&(&&(>=(i12[2], 0), <(i12[2], i76[2])), <(i12[2], i35[2])), >(+(i12[2], 1), 0)), i12[2], i35[2], i76[2])), ≥)∧[(-1)Bound*bni_14 + (2)bni_14] + [bni_14]i12[2] + [bni_14]i76[2] + [(2)bni_14]i35[2] ≥ 0∧[1 + (-1)bso_15] ≥ 0)

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

(12)    (i12[2] ≥ 0∧i35[2] ≥ 0∧i12[2] ≥ 0∧i76[2] ≥ 0 ⇒ (UIncreasing(COND_LOAD8751(&&(&&(&&(>=(i12[2], 0), <(i12[2], i76[2])), <(i12[2], i35[2])), >(+(i12[2], 1), 0)), i12[2], i35[2], i76[2])), ≥)∧[(-1)Bound*bni_14 + (3)bni_14] + [(2)bni_14]i12[2] + [bni_14]i76[2] + [(2)bni_14]i35[2] ≥ 0∧[1 + (-1)bso_15] ≥ 0)

To summarize, we get the following constraints P for the following pairs.
• ((UIncreasing(LOAD875(+(i12[3], 1), i35[3], i76[3])), ≥)∧0 = 0∧0 = 0∧0 = 0∧[(-1)bso_13] ≥ 0)

• LOAD875(i12[2], i35[2], i76[2]) → COND_LOAD8751(&&(&&(&&(>=(i12[2], 0), <(i12[2], i76[2])), <(i12[2], i35[2])), >(+(i12[2], 1), 0)), i12[2], i35[2], i76[2])
• (i12[2] ≥ 0∧i35[2] ≥ 0∧i12[2] ≥ 0∧i76[2] ≥ 0 ⇒ (UIncreasing(COND_LOAD8751(&&(&&(&&(>=(i12[2], 0), <(i12[2], i76[2])), <(i12[2], i35[2])), >(+(i12[2], 1), 0)), i12[2], i35[2], i76[2])), ≥)∧[(-1)Bound*bni_14 + (3)bni_14] + [(2)bni_14]i12[2] + [bni_14]i76[2] + [(2)bni_14]i35[2] ≥ 0∧[1 + (-1)bso_15] ≥ 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_LOAD8751(x1, x2, x3, x4)) = [-1] + x4 + [2]x3 + [-1]x2
POL(LOAD875(x1, x2, x3)) = [-1]x1 + x3 + [2]x2
POL(+(x1, x2)) = x1 + x2
POL(1) = [1]
POL(&&(x1, x2)) = [-1]
POL(>=(x1, x2)) = [-1]
POL(0) = 0
POL(<(x1, x2)) = [-1]
POL(>(x1, x2)) = [-1]

The following pairs are in P>:

LOAD875(i12[2], i35[2], i76[2]) → COND_LOAD8751(&&(&&(&&(>=(i12[2], 0), <(i12[2], i76[2])), <(i12[2], i35[2])), >(+(i12[2], 1), 0)), i12[2], i35[2], i76[2])

The following pairs are in Pbound:

LOAD875(i12[2], i35[2], i76[2]) → COND_LOAD8751(&&(&&(&&(>=(i12[2], 0), <(i12[2], i76[2])), <(i12[2], i35[2])), >(+(i12[2], 1), 0)), i12[2], i35[2], i76[2])

The following pairs are in P:

There are no usable rules.

### (18) Obligation:

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

The following domains are used:

Integer

R is empty.

The integer pair graph contains the following rules and edges: