### (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: example_2/Test

### (1) JBC2FIG (SOUND transformation)

Constructed FIGraph.

### (2) Obligation:

FIGraph based on JBC Program:
Graph of 84 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:
Cond_Load219(TRUE, i28, i24) → Load219(i28 / 2, i24 + i28 / 2)
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

The ITRS R consists of the following rules:
Cond_Load219(TRUE, i28, i24) → Load219(i28 / 2, i24 + i28 / 2)

The integer pair graph contains the following rules and edges:
(1): COND_LOAD219(TRUE, i28[1], i24[1]) → LOAD219(i28[1] / 2, i24[1] + i28[1] / 2)

(0) -> (1), if ((i28[0] > 0* TRUE)∧(i28[0]* i28[1])∧(i24[0]* i24[1]))

(1) -> (0), if ((i24[1] + i28[1] / 2* i24[0])∧(i28[1] / 2* i28[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

R is empty.

The integer pair graph contains the following rules and edges:
(1): COND_LOAD219(TRUE, i28[1], i24[1]) → LOAD219(i28[1] / 2, i24[1] + i28[1] / 2)

(0) -> (1), if ((i28[0] > 0* TRUE)∧(i28[0]* i28[1])∧(i24[0]* i24[1]))

(1) -> (0), if ((i24[1] + i28[1] / 2* i24[0])∧(i28[1] / 2* i28[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 LOAD219(i28, i24) → COND_LOAD219(>(i28, 0), i28, i24) the following chains were created:
• We consider the chain LOAD219(i28[0], i24[0]) → COND_LOAD219(>(i28[0], 0), i28[0], i24[0]), COND_LOAD219(TRUE, i28[1], i24[1]) → LOAD219(/(i28[1], 2), +(i24[1], /(i28[1], 2))) which results in the following constraint:

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)    (i28[0] + [-1] ≥ 0 ⇒ (UIncreasing(COND_LOAD219(>(i28[0], 0), i28[0], i24[0])), ≥)∧[(2)bni_11 + (-1)Bound*bni_11] + [bni_11]i28[0] ≥ 0∧[(-1)bso_12] ≥ 0)

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

(4)    (i28[0] + [-1] ≥ 0 ⇒ (UIncreasing(COND_LOAD219(>(i28[0], 0), i28[0], i24[0])), ≥)∧[(2)bni_11 + (-1)Bound*bni_11] + [bni_11]i28[0] ≥ 0∧[(-1)bso_12] ≥ 0)

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

(5)    (i28[0] + [-1] ≥ 0 ⇒ (UIncreasing(COND_LOAD219(>(i28[0], 0), i28[0], i24[0])), ≥)∧[(2)bni_11 + (-1)Bound*bni_11] + [bni_11]i28[0] ≥ 0∧[(-1)bso_12] ≥ 0)

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

(6)    (i28[0] + [-1] ≥ 0 ⇒ (UIncreasing(COND_LOAD219(>(i28[0], 0), i28[0], i24[0])), ≥)∧0 = 0∧[(2)bni_11 + (-1)Bound*bni_11] + [bni_11]i28[0] ≥ 0∧0 = 0∧[(-1)bso_12] ≥ 0)

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

(7)    (i28[0] ≥ 0 ⇒ (UIncreasing(COND_LOAD219(>(i28[0], 0), i28[0], i24[0])), ≥)∧0 = 0∧[(3)bni_11 + (-1)Bound*bni_11] + [bni_11]i28[0] ≥ 0∧0 = 0∧[(-1)bso_12] ≥ 0)

For Pair COND_LOAD219(TRUE, i28, i24) → LOAD219(/(i28, 2), +(i24, /(i28, 2))) the following chains were created:
• We consider the chain LOAD219(i28[0], i24[0]) → COND_LOAD219(>(i28[0], 0), i28[0], i24[0]), COND_LOAD219(TRUE, i28[1], i24[1]) → LOAD219(/(i28[1], 2), +(i24[1], /(i28[1], 2))), LOAD219(i28[0], i24[0]) → COND_LOAD219(>(i28[0], 0), i28[0], i24[0]) which results in the following constraint:

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

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

(10)    (i28[0] + [-1] ≥ 0 ⇒ (UIncreasing(LOAD219(/(i28[1], 2), +(i24[1], /(i28[1], 2)))), ≥)∧[(2)bni_13 + (-1)Bound*bni_13] + [bni_13]i28[0] ≥ 0∧[1 + (-1)bso_20] + i28[0] + [-1]max{i28[0], [-1]i28[0]} ≥ 0)

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

(11)    (i28[0] + [-1] ≥ 0 ⇒ (UIncreasing(LOAD219(/(i28[1], 2), +(i24[1], /(i28[1], 2)))), ≥)∧[(2)bni_13 + (-1)Bound*bni_13] + [bni_13]i28[0] ≥ 0∧[1 + (-1)bso_20] + i28[0] + [-1]max{i28[0], [-1]i28[0]} ≥ 0)

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

(12)    (i28[0] + [-1] ≥ 0∧[2]i28[0] ≥ 0 ⇒ (UIncreasing(LOAD219(/(i28[1], 2), +(i24[1], /(i28[1], 2)))), ≥)∧[(2)bni_13 + (-1)Bound*bni_13] + [bni_13]i28[0] ≥ 0∧[1 + (-1)bso_20] ≥ 0)

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

(13)    (i28[0] + [-1] ≥ 0∧[2]i28[0] ≥ 0 ⇒ (UIncreasing(LOAD219(/(i28[1], 2), +(i24[1], /(i28[1], 2)))), ≥)∧0 = 0∧[(2)bni_13 + (-1)Bound*bni_13] + [bni_13]i28[0] ≥ 0∧0 = 0∧[1 + (-1)bso_20] ≥ 0)

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

(14)    (i28[0] ≥ 0∧[2] + [2]i28[0] ≥ 0 ⇒ (UIncreasing(LOAD219(/(i28[1], 2), +(i24[1], /(i28[1], 2)))), ≥)∧0 = 0∧[(3)bni_13 + (-1)Bound*bni_13] + [bni_13]i28[0] ≥ 0∧0 = 0∧[1 + (-1)bso_20] ≥ 0)

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

(15)    (i28[0] ≥ 0∧[1] + i28[0] ≥ 0 ⇒ (UIncreasing(LOAD219(/(i28[1], 2), +(i24[1], /(i28[1], 2)))), ≥)∧0 = 0∧[(3)bni_13 + (-1)Bound*bni_13] + [bni_13]i28[0] ≥ 0∧0 = 0∧[1 + (-1)bso_20] ≥ 0)

To summarize, we get the following constraints P for the following pairs.
• (i28[0] ≥ 0 ⇒ (UIncreasing(COND_LOAD219(>(i28[0], 0), i28[0], i24[0])), ≥)∧0 = 0∧[(3)bni_11 + (-1)Bound*bni_11] + [bni_11]i28[0] ≥ 0∧0 = 0∧[(-1)bso_12] ≥ 0)

• (i28[0] ≥ 0∧[1] + i28[0] ≥ 0 ⇒ (UIncreasing(LOAD219(/(i28[1], 2), +(i24[1], /(i28[1], 2)))), ≥)∧0 = 0∧[(3)bni_13 + (-1)Bound*bni_13] + [bni_13]i28[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) = [3]
POL(FALSE) = 0
POL(LOAD219(x1, x2)) = [2] + x1
POL(COND_LOAD219(x1, x2, x3)) = [2] + x2
POL(>(x1, x2)) = [-1]
POL(0) = 0
POL(2) = [2]
POL(+(x1, x2)) = x1 + x2

Polynomial Interpretations with Context Sensitive Arithemetic Replacement
POL(TermCSAR-Mode @ Context)

POL(/(x1, 2)1 @ {LOAD219_2/0}) = max{x1, [-1]x1} + [-1]
POL(/(x1, 2)1 @ {LOAD219_2/1, +_2/1}) = [-1]max{x1, [-1]x1} + [1]

The following pairs are in P>:

The following pairs are in Pbound:

The following pairs are in P:

At least the following rules have been oriented under context sensitive arithmetic replacement:

/1

### (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

R is empty.

The integer pair graph contains the following rules and edges:

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:
none

R is empty.

The integer pair graph is empty.

The set Q consists of the following terms: