### (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: PastaA5

### (1) JBC2FIG (SOUND transformation)

Constructed FIGraph.

### (2) Obligation:

FIGraph based on JBC Program:
Graph of 163 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:
Load426(i12, i43) → Cond_Load426(i43 + 1 > 0 && i12 >= i43 + 1, i12, i43)
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:
Load426(i12, i43) → Cond_Load426(i43 + 1 > 0 && i12 >= i43 + 1, i12, i43)

The integer pair graph contains the following rules and edges:
(0): LOAD426(i12[0], i43[0]) → COND_LOAD426(i43[0] + 1 > 0 && i12[0] >= i43[0] + 1, i12[0], i43[0])

(0) -> (1), if ((i43[0]* i43[1])∧(i43[0] + 1 > 0 && i12[0] >= i43[0] + 1* TRUE)∧(i12[0]* i12[1]))

(1) -> (0), if ((i43[1] + 1* i43[0])∧(i12[1]* i12[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:

Boolean, Integer

R is empty.

The integer pair graph contains the following rules and edges:
(0): LOAD426(i12[0], i43[0]) → COND_LOAD426(i43[0] + 1 > 0 && i12[0] >= i43[0] + 1, i12[0], i43[0])

(0) -> (1), if ((i43[0]* i43[1])∧(i43[0] + 1 > 0 && i12[0] >= i43[0] + 1* TRUE)∧(i12[0]* i12[1]))

(1) -> (0), if ((i43[1] + 1* i43[0])∧(i12[1]* i12[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 LOAD426(i12, i43) → COND_LOAD426(&&(>(+(i43, 1), 0), >=(i12, +(i43, 1))), i12, i43) the following chains were created:
• We consider the chain LOAD426(i12[0], i43[0]) → COND_LOAD426(&&(>(+(i43[0], 1), 0), >=(i12[0], +(i43[0], 1))), i12[0], i43[0]), COND_LOAD426(TRUE, i12[1], i43[1]) → LOAD426(i12[1], +(i43[1], 1)) which results in the following constraint:

(1)    (i43[0]=i43[1]&&(>(+(i43[0], 1), 0), >=(i12[0], +(i43[0], 1)))=TRUEi12[0]=i12[1]LOAD426(i12[0], i43[0])≥NonInfC∧LOAD426(i12[0], i43[0])≥COND_LOAD426(&&(>(+(i43[0], 1), 0), >=(i12[0], +(i43[0], 1))), i12[0], i43[0])∧(UIncreasing(COND_LOAD426(&&(>(+(i43[0], 1), 0), >=(i12[0], +(i43[0], 1))), i12[0], i43[0])), ≥))

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

(2)    (>(+(i43[0], 1), 0)=TRUE>=(i12[0], +(i43[0], 1))=TRUELOAD426(i12[0], i43[0])≥NonInfC∧LOAD426(i12[0], i43[0])≥COND_LOAD426(&&(>(+(i43[0], 1), 0), >=(i12[0], +(i43[0], 1))), i12[0], i43[0])∧(UIncreasing(COND_LOAD426(&&(>(+(i43[0], 1), 0), >=(i12[0], +(i43[0], 1))), i12[0], i43[0])), ≥))

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

(3)    (i43[0] ≥ 0∧i12[0] + [-1] + [-1]i43[0] ≥ 0 ⇒ (UIncreasing(COND_LOAD426(&&(>(+(i43[0], 1), 0), >=(i12[0], +(i43[0], 1))), i12[0], i43[0])), ≥)∧[(-1)bni_10 + (-1)Bound*bni_10] + [(-1)bni_10]i43[0] + [(2)bni_10]i12[0] ≥ 0∧[(-1)bso_11] ≥ 0)

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

(4)    (i43[0] ≥ 0∧i12[0] + [-1] + [-1]i43[0] ≥ 0 ⇒ (UIncreasing(COND_LOAD426(&&(>(+(i43[0], 1), 0), >=(i12[0], +(i43[0], 1))), i12[0], i43[0])), ≥)∧[(-1)bni_10 + (-1)Bound*bni_10] + [(-1)bni_10]i43[0] + [(2)bni_10]i12[0] ≥ 0∧[(-1)bso_11] ≥ 0)

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

(5)    (i43[0] ≥ 0∧i12[0] + [-1] + [-1]i43[0] ≥ 0 ⇒ (UIncreasing(COND_LOAD426(&&(>(+(i43[0], 1), 0), >=(i12[0], +(i43[0], 1))), i12[0], i43[0])), ≥)∧[(-1)bni_10 + (-1)Bound*bni_10] + [(-1)bni_10]i43[0] + [(2)bni_10]i12[0] ≥ 0∧[(-1)bso_11] ≥ 0)

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

(6)    (i43[0] ≥ 0∧i12[0] ≥ 0 ⇒ (UIncreasing(COND_LOAD426(&&(>(+(i43[0], 1), 0), >=(i12[0], +(i43[0], 1))), i12[0], i43[0])), ≥)∧[bni_10 + (-1)Bound*bni_10] + [bni_10]i43[0] + [(2)bni_10]i12[0] ≥ 0∧[(-1)bso_11] ≥ 0)

For Pair COND_LOAD426(TRUE, i12, i43) → LOAD426(i12, +(i43, 1)) the following chains were created:
• We consider the chain COND_LOAD426(TRUE, i12[1], i43[1]) → LOAD426(i12[1], +(i43[1], 1)) which results in the following constraint:

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

(8)    ((UIncreasing(LOAD426(i12[1], +(i43[1], 1))), ≥)∧[1 + (-1)bso_13] ≥ 0)

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

(9)    ((UIncreasing(LOAD426(i12[1], +(i43[1], 1))), ≥)∧[1 + (-1)bso_13] ≥ 0)

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

(10)    ((UIncreasing(LOAD426(i12[1], +(i43[1], 1))), ≥)∧[1 + (-1)bso_13] ≥ 0)

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

(11)    ((UIncreasing(LOAD426(i12[1], +(i43[1], 1))), ≥)∧0 = 0∧0 = 0∧[1 + (-1)bso_13] ≥ 0)

To summarize, we get the following constraints P for the following pairs.
• LOAD426(i12, i43) → COND_LOAD426(&&(>(+(i43, 1), 0), >=(i12, +(i43, 1))), i12, i43)
• (i43[0] ≥ 0∧i12[0] ≥ 0 ⇒ (UIncreasing(COND_LOAD426(&&(>(+(i43[0], 1), 0), >=(i12[0], +(i43[0], 1))), i12[0], i43[0])), ≥)∧[bni_10 + (-1)Bound*bni_10] + [bni_10]i43[0] + [(2)bni_10]i12[0] ≥ 0∧[(-1)bso_11] ≥ 0)

• ((UIncreasing(LOAD426(i12[1], +(i43[1], 1))), ≥)∧0 = 0∧0 = 0∧[1 + (-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(LOAD426(x1, x2)) = [-1] + [-1]x2 + [2]x1
POL(COND_LOAD426(x1, x2, x3)) = [-1] + [-1]x3 + [2]x2
POL(&&(x1, x2)) = [-1]
POL(>(x1, x2)) = [-1]
POL(+(x1, x2)) = x1 + x2
POL(1) = [1]
POL(0) = 0
POL(>=(x1, x2)) = [-1]

The following pairs are in P>:

The following pairs are in Pbound:

The following pairs are in P:

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): LOAD426(i12[0], i43[0]) → COND_LOAD426(i43[0] + 1 > 0 && i12[0] >= i43[0] + 1, i12[0], i43[0])

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 1 less node.

### (14) Obligation:

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

The following domains are used:

Integer

R is empty.

The integer pair graph contains the following rules and edges: