### (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: PastaA1

### (1) JBC2FIG (SOUND transformation)

Constructed FIGraph.

### (2) Obligation:

FIGraph based on JBC Program:
Graph of 110 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:
Load274(i15, i21) → Cond_Load274(i21 >= 0 && i21 < i15 && i21 + 1 > 0, i15, i21)
Load274(i15, i21) → Cond_Load2741(i15 > 0 && i21 >= i15, i15, i21)
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:

Integer, Boolean

The ITRS R consists of the following rules:
Load274(i15, i21) → Cond_Load274(i21 >= 0 && i21 < i15 && i21 + 1 > 0, i15, i21)
Load274(i15, i21) → Cond_Load2741(i15 > 0 && i21 >= i15, i15, i21)

The integer pair graph contains the following rules and edges:
(2): LOAD274(i15[2], i21[2]) → COND_LOAD274(i21[2] >= 0 && i21[2] < i15[2] && i21[2] + 1 > 0, i15[2], i21[2])
(4): LOAD274(i15[4], i21[4]) → COND_LOAD2741(i15[4] > 0 && i21[4] >= i15[4], i15[4], i21[4])

(0) -> (1), if ((i15[0]* i15[1])∧(i15[0] > 0* TRUE))

(1) -> (2), if ((0* i21[2])∧(i15[1]* i15[2]))

(1) -> (4), if ((0* i21[4])∧(i15[1]* i15[4]))

(2) -> (3), if ((i15[2]* i15[3])∧(i21[2]* i21[3])∧(i21[2] >= 0 && i21[2] < i15[2] && i21[2] + 1 > 0* TRUE))

(3) -> (2), if ((i21[3] + 1* i21[2])∧(i15[3]* i15[2]))

(3) -> (4), if ((i21[3] + 1* i21[4])∧(i15[3]* i15[4]))

(4) -> (5), if ((i15[4] > 0 && i21[4] >= i15[4]* TRUE)∧(i15[4]* i15[5])∧(i21[4]* i21[5]))

(5) -> (0), if ((i15[5] + -1* i15[0]))

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:

Integer, Boolean

R is empty.

The integer pair graph contains the following rules and edges:
(2): LOAD274(i15[2], i21[2]) → COND_LOAD274(i21[2] >= 0 && i21[2] < i15[2] && i21[2] + 1 > 0, i15[2], i21[2])
(4): LOAD274(i15[4], i21[4]) → COND_LOAD2741(i15[4] > 0 && i21[4] >= i15[4], i15[4], i21[4])

(0) -> (1), if ((i15[0]* i15[1])∧(i15[0] > 0* TRUE))

(1) -> (2), if ((0* i21[2])∧(i15[1]* i15[2]))

(1) -> (4), if ((0* i21[4])∧(i15[1]* i15[4]))

(2) -> (3), if ((i15[2]* i15[3])∧(i21[2]* i21[3])∧(i21[2] >= 0 && i21[2] < i15[2] && i21[2] + 1 > 0* TRUE))

(3) -> (2), if ((i21[3] + 1* i21[2])∧(i15[3]* i15[2]))

(3) -> (4), if ((i21[3] + 1* i21[4])∧(i15[3]* i15[4]))

(4) -> (5), if ((i15[4] > 0 && i21[4] >= i15[4]* TRUE)∧(i15[4]* i15[5])∧(i21[4]* i21[5]))

(5) -> (0), if ((i15[5] + -1* i15[0]))

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 LOAD139(i15) → COND_LOAD139(>(i15, 0), i15) the following chains were created:

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

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

(3)    (i15[0] + [-1] ≥ 0 ⇒ (UIncreasing(COND_LOAD139(>(i15[0], 0), i15[0])), ≥)∧[(-1)bni_16 + (-1)Bound*bni_16] + [bni_16]i15[0] ≥ 0∧[(-1)bso_17] ≥ 0)

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

(4)    (i15[0] + [-1] ≥ 0 ⇒ (UIncreasing(COND_LOAD139(>(i15[0], 0), i15[0])), ≥)∧[(-1)bni_16 + (-1)Bound*bni_16] + [bni_16]i15[0] ≥ 0∧[(-1)bso_17] ≥ 0)

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

(5)    (i15[0] + [-1] ≥ 0 ⇒ (UIncreasing(COND_LOAD139(>(i15[0], 0), i15[0])), ≥)∧[(-1)bni_16 + (-1)Bound*bni_16] + [bni_16]i15[0] ≥ 0∧[(-1)bso_17] ≥ 0)

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

(6)    (i15[0] ≥ 0 ⇒ (UIncreasing(COND_LOAD139(>(i15[0], 0), i15[0])), ≥)∧[(-1)Bound*bni_16] + [bni_16]i15[0] ≥ 0∧[(-1)bso_17] ≥ 0)

For Pair COND_LOAD139(TRUE, i15) → LOAD274(i15, 0) the following chains were created:
• We consider the chain COND_LOAD139(TRUE, i15[1]) → LOAD274(i15[1], 0) which results in the following constraint:

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

(8)    ((UIncreasing(LOAD274(i15[1], 0)), ≥)∧[(-1)bso_19] ≥ 0)

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

(9)    ((UIncreasing(LOAD274(i15[1], 0)), ≥)∧[(-1)bso_19] ≥ 0)

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

(10)    ((UIncreasing(LOAD274(i15[1], 0)), ≥)∧[(-1)bso_19] ≥ 0)

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

(11)    ((UIncreasing(LOAD274(i15[1], 0)), ≥)∧0 = 0∧[(-1)bso_19] ≥ 0)

For Pair LOAD274(i15, i21) → COND_LOAD274(&&(&&(>=(i21, 0), <(i21, i15)), >(+(i21, 1), 0)), i15, i21) the following chains were created:
• We consider the chain LOAD274(i15[2], i21[2]) → COND_LOAD274(&&(&&(>=(i21[2], 0), <(i21[2], i15[2])), >(+(i21[2], 1), 0)), i15[2], i21[2]), COND_LOAD274(TRUE, i15[3], i21[3]) → LOAD274(i15[3], +(i21[3], 1)) which results in the following constraint:

(12)    (i15[2]=i15[3]i21[2]=i21[3]&&(&&(>=(i21[2], 0), <(i21[2], i15[2])), >(+(i21[2], 1), 0))=TRUELOAD274(i15[2], i21[2])≥NonInfC∧LOAD274(i15[2], i21[2])≥COND_LOAD274(&&(&&(>=(i21[2], 0), <(i21[2], i15[2])), >(+(i21[2], 1), 0)), i15[2], i21[2])∧(UIncreasing(COND_LOAD274(&&(&&(>=(i21[2], 0), <(i21[2], i15[2])), >(+(i21[2], 1), 0)), i15[2], i21[2])), ≥))

We simplified constraint (12) using rules (IV), (IDP_BOOLEAN) which results in the following new constraint:

(13)    (>(+(i21[2], 1), 0)=TRUE>=(i21[2], 0)=TRUE<(i21[2], i15[2])=TRUELOAD274(i15[2], i21[2])≥NonInfC∧LOAD274(i15[2], i21[2])≥COND_LOAD274(&&(&&(>=(i21[2], 0), <(i21[2], i15[2])), >(+(i21[2], 1), 0)), i15[2], i21[2])∧(UIncreasing(COND_LOAD274(&&(&&(>=(i21[2], 0), <(i21[2], i15[2])), >(+(i21[2], 1), 0)), i15[2], i21[2])), ≥))

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

(14)    (i21[2] ≥ 0∧i21[2] ≥ 0∧i15[2] + [-1] + [-1]i21[2] ≥ 0 ⇒ (UIncreasing(COND_LOAD274(&&(&&(>=(i21[2], 0), <(i21[2], i15[2])), >(+(i21[2], 1), 0)), i15[2], i21[2])), ≥)∧[(-1)bni_20 + (-1)Bound*bni_20] + [bni_20]i15[2] ≥ 0∧[(-1)bso_21] ≥ 0)

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

(15)    (i21[2] ≥ 0∧i21[2] ≥ 0∧i15[2] + [-1] + [-1]i21[2] ≥ 0 ⇒ (UIncreasing(COND_LOAD274(&&(&&(>=(i21[2], 0), <(i21[2], i15[2])), >(+(i21[2], 1), 0)), i15[2], i21[2])), ≥)∧[(-1)bni_20 + (-1)Bound*bni_20] + [bni_20]i15[2] ≥ 0∧[(-1)bso_21] ≥ 0)

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

(16)    (i21[2] ≥ 0∧i21[2] ≥ 0∧i15[2] + [-1] + [-1]i21[2] ≥ 0 ⇒ (UIncreasing(COND_LOAD274(&&(&&(>=(i21[2], 0), <(i21[2], i15[2])), >(+(i21[2], 1), 0)), i15[2], i21[2])), ≥)∧[(-1)bni_20 + (-1)Bound*bni_20] + [bni_20]i15[2] ≥ 0∧[(-1)bso_21] ≥ 0)

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

(17)    (i21[2] ≥ 0∧i21[2] ≥ 0∧i15[2] ≥ 0 ⇒ (UIncreasing(COND_LOAD274(&&(&&(>=(i21[2], 0), <(i21[2], i15[2])), >(+(i21[2], 1), 0)), i15[2], i21[2])), ≥)∧[(-1)Bound*bni_20] + [bni_20]i21[2] + [bni_20]i15[2] ≥ 0∧[(-1)bso_21] ≥ 0)

For Pair COND_LOAD274(TRUE, i15, i21) → LOAD274(i15, +(i21, 1)) the following chains were created:
• We consider the chain COND_LOAD274(TRUE, i15[3], i21[3]) → LOAD274(i15[3], +(i21[3], 1)) which results in the following constraint:

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

(19)    ((UIncreasing(LOAD274(i15[3], +(i21[3], 1))), ≥)∧[(-1)bso_23] ≥ 0)

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

(20)    ((UIncreasing(LOAD274(i15[3], +(i21[3], 1))), ≥)∧[(-1)bso_23] ≥ 0)

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

(21)    ((UIncreasing(LOAD274(i15[3], +(i21[3], 1))), ≥)∧[(-1)bso_23] ≥ 0)

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

(22)    ((UIncreasing(LOAD274(i15[3], +(i21[3], 1))), ≥)∧0 = 0∧0 = 0∧[(-1)bso_23] ≥ 0)

For Pair LOAD274(i15, i21) → COND_LOAD2741(&&(>(i15, 0), >=(i21, i15)), i15, i21) the following chains were created:
• We consider the chain LOAD274(i15[4], i21[4]) → COND_LOAD2741(&&(>(i15[4], 0), >=(i21[4], i15[4])), i15[4], i21[4]), COND_LOAD2741(TRUE, i15[5], i21[5]) → LOAD139(+(i15[5], -1)) which results in the following constraint:

We simplified constraint (23) using rules (IV), (IDP_BOOLEAN) which results in the following new constraint:

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

(25)    (i15[4] + [-1] ≥ 0∧i21[4] + [-1]i15[4] ≥ 0 ⇒ (UIncreasing(COND_LOAD2741(&&(>(i15[4], 0), >=(i21[4], i15[4])), i15[4], i21[4])), ≥)∧[(-1)bni_24 + (-1)Bound*bni_24] + [bni_24]i15[4] ≥ 0∧[(-1)bso_25] ≥ 0)

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

(26)    (i15[4] + [-1] ≥ 0∧i21[4] + [-1]i15[4] ≥ 0 ⇒ (UIncreasing(COND_LOAD2741(&&(>(i15[4], 0), >=(i21[4], i15[4])), i15[4], i21[4])), ≥)∧[(-1)bni_24 + (-1)Bound*bni_24] + [bni_24]i15[4] ≥ 0∧[(-1)bso_25] ≥ 0)

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

(27)    (i15[4] + [-1] ≥ 0∧i21[4] + [-1]i15[4] ≥ 0 ⇒ (UIncreasing(COND_LOAD2741(&&(>(i15[4], 0), >=(i21[4], i15[4])), i15[4], i21[4])), ≥)∧[(-1)bni_24 + (-1)Bound*bni_24] + [bni_24]i15[4] ≥ 0∧[(-1)bso_25] ≥ 0)

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

(28)    (i15[4] ≥ 0∧i21[4] + [-1] + [-1]i15[4] ≥ 0 ⇒ (UIncreasing(COND_LOAD2741(&&(>(i15[4], 0), >=(i21[4], i15[4])), i15[4], i21[4])), ≥)∧[(-1)Bound*bni_24] + [bni_24]i15[4] ≥ 0∧[(-1)bso_25] ≥ 0)

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

(29)    (i15[4] ≥ 0∧i21[4] ≥ 0 ⇒ (UIncreasing(COND_LOAD2741(&&(>(i15[4], 0), >=(i21[4], i15[4])), i15[4], i21[4])), ≥)∧[(-1)Bound*bni_24] + [bni_24]i15[4] ≥ 0∧[(-1)bso_25] ≥ 0)

For Pair COND_LOAD2741(TRUE, i15, i21) → LOAD139(+(i15, -1)) the following chains were created:
• We consider the chain COND_LOAD2741(TRUE, i15[5], i21[5]) → LOAD139(+(i15[5], -1)) which results in the following constraint:

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

(31)    ((UIncreasing(LOAD139(+(i15[5], -1))), ≥)∧[1 + (-1)bso_27] ≥ 0)

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

(32)    ((UIncreasing(LOAD139(+(i15[5], -1))), ≥)∧[1 + (-1)bso_27] ≥ 0)

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

(33)    ((UIncreasing(LOAD139(+(i15[5], -1))), ≥)∧[1 + (-1)bso_27] ≥ 0)

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

(34)    ((UIncreasing(LOAD139(+(i15[5], -1))), ≥)∧0 = 0∧0 = 0∧[1 + (-1)bso_27] ≥ 0)

To summarize, we get the following constraints P for the following pairs.
• (i15[0] ≥ 0 ⇒ (UIncreasing(COND_LOAD139(>(i15[0], 0), i15[0])), ≥)∧[(-1)Bound*bni_16] + [bni_16]i15[0] ≥ 0∧[(-1)bso_17] ≥ 0)

• ((UIncreasing(LOAD274(i15[1], 0)), ≥)∧0 = 0∧[(-1)bso_19] ≥ 0)

• LOAD274(i15, i21) → COND_LOAD274(&&(&&(>=(i21, 0), <(i21, i15)), >(+(i21, 1), 0)), i15, i21)
• (i21[2] ≥ 0∧i21[2] ≥ 0∧i15[2] ≥ 0 ⇒ (UIncreasing(COND_LOAD274(&&(&&(>=(i21[2], 0), <(i21[2], i15[2])), >(+(i21[2], 1), 0)), i15[2], i21[2])), ≥)∧[(-1)Bound*bni_20] + [bni_20]i21[2] + [bni_20]i15[2] ≥ 0∧[(-1)bso_21] ≥ 0)

• ((UIncreasing(LOAD274(i15[3], +(i21[3], 1))), ≥)∧0 = 0∧0 = 0∧[(-1)bso_23] ≥ 0)

• (i15[4] ≥ 0∧i21[4] ≥ 0 ⇒ (UIncreasing(COND_LOAD2741(&&(>(i15[4], 0), >=(i21[4], i15[4])), i15[4], i21[4])), ≥)∧[(-1)Bound*bni_24] + [bni_24]i15[4] ≥ 0∧[(-1)bso_25] ≥ 0)

• ((UIncreasing(LOAD139(+(i15[5], -1))), ≥)∧0 = 0∧0 = 0∧[1 + (-1)bso_27] ≥ 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_LOAD139(x1, x2)) = [-1] + x2
POL(>(x1, x2)) = [-1]
POL(0) = 0
POL(LOAD274(x1, x2)) = [-1] + x1
POL(COND_LOAD274(x1, x2, x3)) = [-1] + x2
POL(&&(x1, x2)) = [-1]
POL(>=(x1, x2)) = [-1]
POL(<(x1, x2)) = [-1]
POL(+(x1, x2)) = x1 + x2
POL(1) = [1]
POL(COND_LOAD2741(x1, x2, x3)) = [-1] + x2
POL(-1) = [-1]

The following pairs are in P>:

The following pairs are in Pbound:

LOAD274(i15[2], i21[2]) → COND_LOAD274(&&(&&(>=(i21[2], 0), <(i21[2], i15[2])), >(+(i21[2], 1), 0)), i15[2], i21[2])

The following pairs are in P:

LOAD274(i15[2], i21[2]) → COND_LOAD274(&&(&&(>=(i21[2], 0), <(i21[2], i15[2])), >(+(i21[2], 1), 0)), i15[2], i21[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:

Integer, Boolean

R is empty.

The integer pair graph contains the following rules and edges:
(2): LOAD274(i15[2], i21[2]) → COND_LOAD274(i21[2] >= 0 && i21[2] < i15[2] && i21[2] + 1 > 0, i15[2], i21[2])
(4): LOAD274(i15[4], i21[4]) → COND_LOAD2741(i15[4] > 0 && i21[4] >= i15[4], i15[4], i21[4])

(0) -> (1), if ((i15[0]* i15[1])∧(i15[0] > 0* TRUE))

(1) -> (2), if ((0* i21[2])∧(i15[1]* i15[2]))

(3) -> (2), if ((i21[3] + 1* i21[2])∧(i15[3]* i15[2]))

(2) -> (3), if ((i15[2]* i15[3])∧(i21[2]* i21[3])∧(i21[2] >= 0 && i21[2] < i15[2] && i21[2] + 1 > 0* TRUE))

(1) -> (4), if ((0* i21[4])∧(i15[1]* i15[4]))

(3) -> (4), if ((i21[3] + 1* i21[4])∧(i15[3]* i15[4]))

The set Q consists of the following terms:

### (12) IDependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 3 less nodes.

### (13) Obligation:

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

The following domains are used:

Integer, Boolean

R is empty.

The integer pair graph contains the following rules and edges:
(2): LOAD274(i15[2], i21[2]) → COND_LOAD274(i21[2] >= 0 && i21[2] < i15[2] && i21[2] + 1 > 0, i15[2], i21[2])

(3) -> (2), if ((i21[3] + 1* i21[2])∧(i15[3]* i15[2]))

(2) -> (3), if ((i15[2]* i15[3])∧(i21[2]* i21[3])∧(i21[2] >= 0 && i21[2] < i15[2] && i21[2] + 1 > 0* TRUE))

The set Q consists of the following terms:

### (14) 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_LOAD274(TRUE, i15[3], i21[3]) → LOAD274(i15[3], +(i21[3], 1)) the following chains were created:
• We consider the chain COND_LOAD274(TRUE, i15[3], i21[3]) → LOAD274(i15[3], +(i21[3], 1)) which results in the following constraint:

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

(2)    ((UIncreasing(LOAD274(i15[3], +(i21[3], 1))), ≥)∧[1 + (-1)bso_11] ≥ 0)

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

(3)    ((UIncreasing(LOAD274(i15[3], +(i21[3], 1))), ≥)∧[1 + (-1)bso_11] ≥ 0)

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

(4)    ((UIncreasing(LOAD274(i15[3], +(i21[3], 1))), ≥)∧[1 + (-1)bso_11] ≥ 0)

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

(5)    ((UIncreasing(LOAD274(i15[3], +(i21[3], 1))), ≥)∧0 = 0∧0 = 0∧[1 + (-1)bso_11] ≥ 0)

For Pair LOAD274(i15[2], i21[2]) → COND_LOAD274(&&(&&(>=(i21[2], 0), <(i21[2], i15[2])), >(+(i21[2], 1), 0)), i15[2], i21[2]) the following chains were created:
• We consider the chain LOAD274(i15[2], i21[2]) → COND_LOAD274(&&(&&(>=(i21[2], 0), <(i21[2], i15[2])), >(+(i21[2], 1), 0)), i15[2], i21[2]), COND_LOAD274(TRUE, i15[3], i21[3]) → LOAD274(i15[3], +(i21[3], 1)) which results in the following constraint:

(6)    (i15[2]=i15[3]i21[2]=i21[3]&&(&&(>=(i21[2], 0), <(i21[2], i15[2])), >(+(i21[2], 1), 0))=TRUELOAD274(i15[2], i21[2])≥NonInfC∧LOAD274(i15[2], i21[2])≥COND_LOAD274(&&(&&(>=(i21[2], 0), <(i21[2], i15[2])), >(+(i21[2], 1), 0)), i15[2], i21[2])∧(UIncreasing(COND_LOAD274(&&(&&(>=(i21[2], 0), <(i21[2], i15[2])), >(+(i21[2], 1), 0)), i15[2], i21[2])), ≥))

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

(7)    (>(+(i21[2], 1), 0)=TRUE>=(i21[2], 0)=TRUE<(i21[2], i15[2])=TRUELOAD274(i15[2], i21[2])≥NonInfC∧LOAD274(i15[2], i21[2])≥COND_LOAD274(&&(&&(>=(i21[2], 0), <(i21[2], i15[2])), >(+(i21[2], 1), 0)), i15[2], i21[2])∧(UIncreasing(COND_LOAD274(&&(&&(>=(i21[2], 0), <(i21[2], i15[2])), >(+(i21[2], 1), 0)), i15[2], i21[2])), ≥))

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

(8)    (i21[2] ≥ 0∧i21[2] ≥ 0∧i15[2] + [-1] + [-1]i21[2] ≥ 0 ⇒ (UIncreasing(COND_LOAD274(&&(&&(>=(i21[2], 0), <(i21[2], i15[2])), >(+(i21[2], 1), 0)), i15[2], i21[2])), ≥)∧[(-1)bni_12 + (-1)Bound*bni_12] + [(-1)bni_12]i21[2] + [bni_12]i15[2] ≥ 0∧[(-1)bso_13] ≥ 0)

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

(9)    (i21[2] ≥ 0∧i21[2] ≥ 0∧i15[2] + [-1] + [-1]i21[2] ≥ 0 ⇒ (UIncreasing(COND_LOAD274(&&(&&(>=(i21[2], 0), <(i21[2], i15[2])), >(+(i21[2], 1), 0)), i15[2], i21[2])), ≥)∧[(-1)bni_12 + (-1)Bound*bni_12] + [(-1)bni_12]i21[2] + [bni_12]i15[2] ≥ 0∧[(-1)bso_13] ≥ 0)

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

(10)    (i21[2] ≥ 0∧i21[2] ≥ 0∧i15[2] + [-1] + [-1]i21[2] ≥ 0 ⇒ (UIncreasing(COND_LOAD274(&&(&&(>=(i21[2], 0), <(i21[2], i15[2])), >(+(i21[2], 1), 0)), i15[2], i21[2])), ≥)∧[(-1)bni_12 + (-1)Bound*bni_12] + [(-1)bni_12]i21[2] + [bni_12]i15[2] ≥ 0∧[(-1)bso_13] ≥ 0)

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

(11)    (i21[2] ≥ 0∧i21[2] ≥ 0∧i15[2] ≥ 0 ⇒ (UIncreasing(COND_LOAD274(&&(&&(>=(i21[2], 0), <(i21[2], i15[2])), >(+(i21[2], 1), 0)), i15[2], i21[2])), ≥)∧[(-1)Bound*bni_12] + [bni_12]i15[2] ≥ 0∧[(-1)bso_13] ≥ 0)

To summarize, we get the following constraints P for the following pairs.
• ((UIncreasing(LOAD274(i15[3], +(i21[3], 1))), ≥)∧0 = 0∧0 = 0∧[1 + (-1)bso_11] ≥ 0)

• LOAD274(i15[2], i21[2]) → COND_LOAD274(&&(&&(>=(i21[2], 0), <(i21[2], i15[2])), >(+(i21[2], 1), 0)), i15[2], i21[2])
• (i21[2] ≥ 0∧i21[2] ≥ 0∧i15[2] ≥ 0 ⇒ (UIncreasing(COND_LOAD274(&&(&&(>=(i21[2], 0), <(i21[2], i15[2])), >(+(i21[2], 1), 0)), i15[2], i21[2])), ≥)∧[(-1)Bound*bni_12] + [bni_12]i15[2] ≥ 0∧[(-1)bso_13] ≥ 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_LOAD274(x1, x2, x3)) = [-1] + x2 + [-1]x3
POL(LOAD274(x1, x2)) = [-1] + [-1]x2 + x1
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>:

The following pairs are in Pbound:

LOAD274(i15[2], i21[2]) → COND_LOAD274(&&(&&(>=(i21[2], 0), <(i21[2], i15[2])), >(+(i21[2], 1), 0)), i15[2], i21[2])

The following pairs are in P:

LOAD274(i15[2], i21[2]) → COND_LOAD274(&&(&&(>=(i21[2], 0), <(i21[2], i15[2])), >(+(i21[2], 1), 0)), i15[2], i21[2])

There are no usable rules.

### (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:

Boolean, Integer

R is empty.

The integer pair graph contains the following rules and edges:
(2): LOAD274(i15[2], i21[2]) → COND_LOAD274(i21[2] >= 0 && i21[2] < i15[2] && i21[2] + 1 > 0, i15[2], i21[2])

The set Q consists of the following terms:

### (17) IDependencyGraphProof (EQUIVALENT transformation)

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

### (19) Obligation:

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

The following domains are used:

Integer

R is empty.

The integer pair graph contains the following rules and edges:

The set Q consists of the following terms:

### (20) IDependencyGraphProof (EQUIVALENT transformation)

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

### (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:

Integer

R is empty.

The integer pair graph contains the following rules and edges:

The set Q consists of the following terms: