(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: PastaB14

(1) JBC2FIG (SOUND transformation)

Constructed FIGraph.

(2) Obligation:

FIGraph based on JBC Program:
Graph of 175 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:
The set Q consists of the following terms:

(5) DuplicateArgsRemoverProof (EQUIVALENT transformation)

Some arguments are removed because they only appear as duplicates.
We removed arguments according to the following replacements:

(6) 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:
The set Q consists of the following terms:

(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

The ITRS R consists of the following rules:

The integer pair graph contains the following rules and edges:

(0) -> (1), if ((i104[0] > 0* TRUE)∧(i104[0]* i104[1]))

(1) -> (2), if ((i104[1]* i123[2])∧(i104[1]* i120[2]))

(1) -> (4), if ((i104[1]* i120[4])∧(i104[1]* 0))

(2) -> (3), if ((i123[2]* i123[3])∧(i123[2] > 0* TRUE)∧(i120[2]* i120[3]))

(3) -> (2), if ((i123[3] + -1* i123[2])∧(i120[3] + -1* i120[2]))

(3) -> (4), if ((i120[3] + -1* i120[4])∧(i123[3] + -1* 0))

(4) -> (0), if ((i120[4]* i104[0])∧(0* i104[0]))

The set Q consists of the following terms:

(9) 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.

(10) 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:

(0) -> (1), if ((i104[0] > 0* TRUE)∧(i104[0]* i104[1]))

(1) -> (2), if ((i104[1]* i123[2])∧(i104[1]* i120[2]))

(1) -> (4), if ((i104[1]* i120[4])∧(i104[1]* 0))

(2) -> (3), if ((i123[2]* i123[3])∧(i123[2] > 0* TRUE)∧(i120[2]* i120[3]))

(3) -> (2), if ((i123[3] + -1* i123[2])∧(i120[3] + -1* i120[2]))

(3) -> (4), if ((i120[3] + -1* i120[4])∧(i123[3] + -1* 0))

(4) -> (0), if ((i120[4]* i104[0])∧(0* i104[0]))

The set Q consists of the following terms:

(11) 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 LOAD661(i104, i104) → COND_LOAD661(>(i104, 0), i104) 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)    (i104[0] + [-1] ≥ 0 ⇒ (UIncreasing(COND_LOAD661(>(i104[0], 0), i104[0])), ≥)∧[(2)bni_17 + (-1)Bound*bni_17] + [(2)bni_17]i104[0] ≥ 0∧[(-1)bso_18] + i104[0] ≥ 0)

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

(4)    (i104[0] + [-1] ≥ 0 ⇒ (UIncreasing(COND_LOAD661(>(i104[0], 0), i104[0])), ≥)∧[(2)bni_17 + (-1)Bound*bni_17] + [(2)bni_17]i104[0] ≥ 0∧[(-1)bso_18] + i104[0] ≥ 0)

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

(5)    (i104[0] + [-1] ≥ 0 ⇒ (UIncreasing(COND_LOAD661(>(i104[0], 0), i104[0])), ≥)∧[(2)bni_17 + (-1)Bound*bni_17] + [(2)bni_17]i104[0] ≥ 0∧[(-1)bso_18] + i104[0] ≥ 0)

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

(6)    (i104[0] ≥ 0 ⇒ (UIncreasing(COND_LOAD661(>(i104[0], 0), i104[0])), ≥)∧[(4)bni_17 + (-1)Bound*bni_17] + [(2)bni_17]i104[0] ≥ 0∧[1 + (-1)bso_18] + i104[0] ≥ 0)

For Pair COND_LOAD661(TRUE, i104) → LOAD759(i104, i104) the following chains were created:
• We consider the chain COND_LOAD661(TRUE, i104[1]) → LOAD759(i104[1], i104[1]), LOAD759(i120[2], i123[2]) → COND_LOAD759(>(i123[2], 0), i120[2], i123[2]) which results in the following constraint:

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

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

(9)    ((UIncreasing(LOAD759(i104[1], i104[1])), ≥)∧[(-1)bso_20] ≥ 0)

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

(10)    ((UIncreasing(LOAD759(i104[1], i104[1])), ≥)∧[(-1)bso_20] ≥ 0)

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

(11)    ((UIncreasing(LOAD759(i104[1], i104[1])), ≥)∧[(-1)bso_20] ≥ 0)

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

(12)    ((UIncreasing(LOAD759(i104[1], i104[1])), ≥)∧0 = 0∧[(-1)bso_20] ≥ 0)

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

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

(15)    ((UIncreasing(LOAD759(i104[1], i104[1])), ≥)∧[(-1)bso_20] ≥ 0)

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

(16)    ((UIncreasing(LOAD759(i104[1], i104[1])), ≥)∧[(-1)bso_20] ≥ 0)

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

(17)    ((UIncreasing(LOAD759(i104[1], i104[1])), ≥)∧[(-1)bso_20] ≥ 0)

For Pair LOAD759(i120, i123) → COND_LOAD759(>(i123, 0), i120, i123) the following chains were created:
• We consider the chain LOAD759(i120[2], i123[2]) → COND_LOAD759(>(i123[2], 0), i120[2], i123[2]), COND_LOAD759(TRUE, i120[3], i123[3]) → LOAD759(+(i120[3], -1), +(i123[3], -1)) which results in the following constraint:

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

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

(20)    (i123[2] + [-1] ≥ 0 ⇒ (UIncreasing(COND_LOAD759(>(i123[2], 0), i120[2], i123[2])), ≥)∧[(2)bni_21 + (-1)Bound*bni_21] + [(-1)bni_21]i123[2] + [(2)bni_21]i120[2] ≥ 0∧[(-1)bso_22] ≥ 0)

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

(21)    (i123[2] + [-1] ≥ 0 ⇒ (UIncreasing(COND_LOAD759(>(i123[2], 0), i120[2], i123[2])), ≥)∧[(2)bni_21 + (-1)Bound*bni_21] + [(-1)bni_21]i123[2] + [(2)bni_21]i120[2] ≥ 0∧[(-1)bso_22] ≥ 0)

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

(22)    (i123[2] + [-1] ≥ 0 ⇒ (UIncreasing(COND_LOAD759(>(i123[2], 0), i120[2], i123[2])), ≥)∧[(2)bni_21 + (-1)Bound*bni_21] + [(-1)bni_21]i123[2] + [(2)bni_21]i120[2] ≥ 0∧[(-1)bso_22] ≥ 0)

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

(23)    (i123[2] + [-1] ≥ 0 ⇒ (UIncreasing(COND_LOAD759(>(i123[2], 0), i120[2], i123[2])), ≥)∧[(2)bni_21] = 0∧[(2)bni_21 + (-1)Bound*bni_21] + [(-1)bni_21]i123[2] ≥ 0∧0 = 0∧[(-1)bso_22] ≥ 0)

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

(24)    (i123[2] ≥ 0 ⇒ (UIncreasing(COND_LOAD759(>(i123[2], 0), i120[2], i123[2])), ≥)∧[(2)bni_21] = 0∧[bni_21 + (-1)Bound*bni_21] + [(-1)bni_21]i123[2] ≥ 0∧0 = 0∧[(-1)bso_22] ≥ 0)

For Pair COND_LOAD759(TRUE, i120, i123) → LOAD759(+(i120, -1), +(i123, -1)) the following chains were created:

We simplified constraint (25) using rules (III), (IV) which results in the following new constraint:

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

(27)    (i123[2] + [-1] ≥ 0 ⇒ (UIncreasing(LOAD759(+(i120[3], -1), +(i123[3], -1))), ≥)∧[(2)bni_23 + (-1)Bound*bni_23] + [(-1)bni_23]i123[2] + [(2)bni_23]i120[2] ≥ 0∧[1 + (-1)bso_24] ≥ 0)

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

(28)    (i123[2] + [-1] ≥ 0 ⇒ (UIncreasing(LOAD759(+(i120[3], -1), +(i123[3], -1))), ≥)∧[(2)bni_23 + (-1)Bound*bni_23] + [(-1)bni_23]i123[2] + [(2)bni_23]i120[2] ≥ 0∧[1 + (-1)bso_24] ≥ 0)

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

(29)    (i123[2] + [-1] ≥ 0 ⇒ (UIncreasing(LOAD759(+(i120[3], -1), +(i123[3], -1))), ≥)∧[(2)bni_23 + (-1)Bound*bni_23] + [(-1)bni_23]i123[2] + [(2)bni_23]i120[2] ≥ 0∧[1 + (-1)bso_24] ≥ 0)

We simplified constraint (29) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint:

(30)    (i123[2] + [-1] ≥ 0 ⇒ (UIncreasing(LOAD759(+(i120[3], -1), +(i123[3], -1))), ≥)∧[(2)bni_23] = 0∧[(2)bni_23 + (-1)Bound*bni_23] + [(-1)bni_23]i123[2] ≥ 0∧0 = 0∧[1 + (-1)bso_24] ≥ 0)

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

(31)    (i123[2] ≥ 0 ⇒ (UIncreasing(LOAD759(+(i120[3], -1), +(i123[3], -1))), ≥)∧[(2)bni_23] = 0∧[bni_23 + (-1)Bound*bni_23] + [(-1)bni_23]i123[2] ≥ 0∧0 = 0∧[1 + (-1)bso_24] ≥ 0)

We simplified constraint (32) using rules (III), (IV) which results in the following new constraint:

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

(34)    (i123[2] + [-1] ≥ 0∧i123[2] + [-1] ≥ 0 ⇒ (UIncreasing(LOAD759(+(i120[3], -1), +(i123[3], -1))), ≥)∧[(2)bni_23 + (-1)Bound*bni_23] + [(-1)bni_23]i123[2] + [(2)bni_23]i120[2] ≥ 0∧[1 + (-1)bso_24] ≥ 0)

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

(35)    (i123[2] + [-1] ≥ 0∧i123[2] + [-1] ≥ 0 ⇒ (UIncreasing(LOAD759(+(i120[3], -1), +(i123[3], -1))), ≥)∧[(2)bni_23 + (-1)Bound*bni_23] + [(-1)bni_23]i123[2] + [(2)bni_23]i120[2] ≥ 0∧[1 + (-1)bso_24] ≥ 0)

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

(36)    (i123[2] + [-1] ≥ 0∧i123[2] + [-1] ≥ 0 ⇒ (UIncreasing(LOAD759(+(i120[3], -1), +(i123[3], -1))), ≥)∧[(2)bni_23 + (-1)Bound*bni_23] + [(-1)bni_23]i123[2] + [(2)bni_23]i120[2] ≥ 0∧[1 + (-1)bso_24] ≥ 0)

We simplified constraint (36) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint:

(37)    (i123[2] + [-1] ≥ 0∧i123[2] + [-1] ≥ 0 ⇒ (UIncreasing(LOAD759(+(i120[3], -1), +(i123[3], -1))), ≥)∧[(2)bni_23] = 0∧[(2)bni_23 + (-1)Bound*bni_23] + [(-1)bni_23]i123[2] ≥ 0∧0 = 0∧[1 + (-1)bso_24] ≥ 0)

We simplified constraint (37) using rule (IDP_SMT_SPLIT) which results in the following new constraint:

(38)    (i123[2] ≥ 0∧i123[2] ≥ 0 ⇒ (UIncreasing(LOAD759(+(i120[3], -1), +(i123[3], -1))), ≥)∧[(2)bni_23] = 0∧[bni_23 + (-1)Bound*bni_23] + [(-1)bni_23]i123[2] ≥ 0∧0 = 0∧[1 + (-1)bso_24] ≥ 0)

For Pair LOAD759(i120, 0) → LOAD661(i120, 0) the following chains were created:

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

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

(41)    ((UIncreasing(LOAD661(i120[4], 0)), ≥)∧[(-1)bso_26] ≥ 0)

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

(42)    ((UIncreasing(LOAD661(i120[4], 0)), ≥)∧[(-1)bso_26] ≥ 0)

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

(43)    ((UIncreasing(LOAD661(i120[4], 0)), ≥)∧[(-1)bso_26] ≥ 0)

To summarize, we get the following constraints P for the following pairs.
• (i104[0] ≥ 0 ⇒ (UIncreasing(COND_LOAD661(>(i104[0], 0), i104[0])), ≥)∧[(4)bni_17 + (-1)Bound*bni_17] + [(2)bni_17]i104[0] ≥ 0∧[1 + (-1)bso_18] + i104[0] ≥ 0)

• ((UIncreasing(LOAD759(i104[1], i104[1])), ≥)∧0 = 0∧[(-1)bso_20] ≥ 0)
• ((UIncreasing(LOAD759(i104[1], i104[1])), ≥)∧[(-1)bso_20] ≥ 0)

• (i123[2] ≥ 0 ⇒ (UIncreasing(COND_LOAD759(>(i123[2], 0), i120[2], i123[2])), ≥)∧[(2)bni_21] = 0∧[bni_21 + (-1)Bound*bni_21] + [(-1)bni_21]i123[2] ≥ 0∧0 = 0∧[(-1)bso_22] ≥ 0)

• (i123[2] ≥ 0 ⇒ (UIncreasing(LOAD759(+(i120[3], -1), +(i123[3], -1))), ≥)∧[(2)bni_23] = 0∧[bni_23 + (-1)Bound*bni_23] + [(-1)bni_23]i123[2] ≥ 0∧0 = 0∧[1 + (-1)bso_24] ≥ 0)
• (i123[2] ≥ 0∧i123[2] ≥ 0 ⇒ (UIncreasing(LOAD759(+(i120[3], -1), +(i123[3], -1))), ≥)∧[(2)bni_23] = 0∧[bni_23 + (-1)Bound*bni_23] + [(-1)bni_23]i123[2] ≥ 0∧0 = 0∧[1 + (-1)bso_24] ≥ 0)

• ((UIncreasing(LOAD661(i120[4], 0)), ≥)∧[(-1)bso_26] ≥ 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(LOAD661(x1, x2)) = [2] + x2 + x1
POL(COND_LOAD661(x1, x2)) = [2] + x2
POL(>(x1, x2)) = [-1]
POL(0) = 0
POL(LOAD759(x1, x2)) = [2] + [-1]x2 + [2]x1
POL(COND_LOAD759(x1, x2, x3)) = [2] + [-1]x3 + [2]x2
POL(+(x1, x2)) = x1 + x2
POL(-1) = [-1]

The following pairs are in P>:

The following pairs are in Pbound:

The following pairs are in P:

There are no usable rules.

(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

R is empty.

The integer pair graph contains the following rules and edges:

(1) -> (2), if ((i104[1]* i123[2])∧(i104[1]* i120[2]))

(1) -> (4), if ((i104[1]* i120[4])∧(i104[1]* 0))

The set Q consists of the following terms:

(14) IDependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 3 less nodes.

(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

R is empty.

The integer pair graph contains the following rules and edges:

(1) -> (2), if ((i104[1]* i123[2])∧(i104[1]* i120[2]))

(3) -> (2), if ((i123[3] + -1* i123[2])∧(i120[3] + -1* i120[2]))

(2) -> (3), if ((i123[2]* i123[3])∧(i123[2] > 0* TRUE)∧(i120[2]* i120[3]))

(1) -> (4), if ((i104[1]* i120[4])∧(i104[1]* 0))

(3) -> (4), if ((i120[3] + -1* i120[4])∧(i123[3] + -1* 0))

The set Q consists of the following terms:

(17) IDependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 2 less nodes.

(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:

(3) -> (2), if ((i123[3] + -1* i123[2])∧(i120[3] + -1* i120[2]))

(2) -> (3), if ((i123[2]* i123[3])∧(i123[2] > 0* TRUE)∧(i120[2]* i120[3]))

The set Q consists of the following terms:

(19) 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_LOAD759(TRUE, i120[3], i123[3]) → LOAD759(+(i120[3], -1), +(i123[3], -1)) the following chains were created:

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

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

(3)    (i123[2] + [-1] ≥ 0 ⇒ (UIncreasing(LOAD759(+(i120[3], -1), +(i123[3], -1))), ≥)∧[(-1)bni_11 + (-1)Bound*bni_11] + [bni_11]i123[2] ≥ 0∧[1 + (-1)bso_12] ≥ 0)

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

(4)    (i123[2] + [-1] ≥ 0 ⇒ (UIncreasing(LOAD759(+(i120[3], -1), +(i123[3], -1))), ≥)∧[(-1)bni_11 + (-1)Bound*bni_11] + [bni_11]i123[2] ≥ 0∧[1 + (-1)bso_12] ≥ 0)

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

(5)    (i123[2] + [-1] ≥ 0 ⇒ (UIncreasing(LOAD759(+(i120[3], -1), +(i123[3], -1))), ≥)∧[(-1)bni_11 + (-1)Bound*bni_11] + [bni_11]i123[2] ≥ 0∧[1 + (-1)bso_12] ≥ 0)

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

(6)    (i123[2] + [-1] ≥ 0 ⇒ (UIncreasing(LOAD759(+(i120[3], -1), +(i123[3], -1))), ≥)∧0 = 0∧[(-1)bni_11 + (-1)Bound*bni_11] + [bni_11]i123[2] ≥ 0∧0 = 0∧[1 + (-1)bso_12] ≥ 0)

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

(7)    (i123[2] ≥ 0 ⇒ (UIncreasing(LOAD759(+(i120[3], -1), +(i123[3], -1))), ≥)∧0 = 0∧[(-1)Bound*bni_11] + [bni_11]i123[2] ≥ 0∧0 = 0∧[1 + (-1)bso_12] ≥ 0)

For Pair LOAD759(i120[2], i123[2]) → COND_LOAD759(>(i123[2], 0), i120[2], i123[2]) the following chains were created:
• We consider the chain LOAD759(i120[2], i123[2]) → COND_LOAD759(>(i123[2], 0), i120[2], i123[2]), COND_LOAD759(TRUE, i120[3], i123[3]) → LOAD759(+(i120[3], -1), +(i123[3], -1)) which results in the following constraint:

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

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

(10)    (i123[2] + [-1] ≥ 0 ⇒ (UIncreasing(COND_LOAD759(>(i123[2], 0), i120[2], i123[2])), ≥)∧[(-1)bni_13 + (-1)Bound*bni_13] + [bni_13]i123[2] ≥ 0∧[(-1)bso_14] ≥ 0)

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

(11)    (i123[2] + [-1] ≥ 0 ⇒ (UIncreasing(COND_LOAD759(>(i123[2], 0), i120[2], i123[2])), ≥)∧[(-1)bni_13 + (-1)Bound*bni_13] + [bni_13]i123[2] ≥ 0∧[(-1)bso_14] ≥ 0)

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

(12)    (i123[2] + [-1] ≥ 0 ⇒ (UIncreasing(COND_LOAD759(>(i123[2], 0), i120[2], i123[2])), ≥)∧[(-1)bni_13 + (-1)Bound*bni_13] + [bni_13]i123[2] ≥ 0∧[(-1)bso_14] ≥ 0)

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

(13)    (i123[2] + [-1] ≥ 0 ⇒ (UIncreasing(COND_LOAD759(>(i123[2], 0), i120[2], i123[2])), ≥)∧0 = 0∧[(-1)bni_13 + (-1)Bound*bni_13] + [bni_13]i123[2] ≥ 0∧0 = 0∧[(-1)bso_14] ≥ 0)

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

(14)    (i123[2] ≥ 0 ⇒ (UIncreasing(COND_LOAD759(>(i123[2], 0), i120[2], i123[2])), ≥)∧0 = 0∧[(-1)Bound*bni_13] + [bni_13]i123[2] ≥ 0∧0 = 0∧[(-1)bso_14] ≥ 0)

To summarize, we get the following constraints P for the following pairs.
• (i123[2] ≥ 0 ⇒ (UIncreasing(LOAD759(+(i120[3], -1), +(i123[3], -1))), ≥)∧0 = 0∧[(-1)Bound*bni_11] + [bni_11]i123[2] ≥ 0∧0 = 0∧[1 + (-1)bso_12] ≥ 0)

• (i123[2] ≥ 0 ⇒ (UIncreasing(COND_LOAD759(>(i123[2], 0), i120[2], i123[2])), ≥)∧0 = 0∧[(-1)Bound*bni_13] + [bni_13]i123[2] ≥ 0∧0 = 0∧[(-1)bso_14] ≥ 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_LOAD759(x1, x2, x3)) = [-1] + x3
POL(LOAD759(x1, x2)) = [-1] + x2
POL(+(x1, x2)) = x1 + x2
POL(-1) = [-1]
POL(>(x1, x2)) = [2]
POL(0) = 0

The following pairs are in P>:

The following pairs are in Pbound:

The following pairs are in P:

There are no usable rules.

(21) 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:

(22) IDependencyGraphProof (EQUIVALENT transformation)

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

(24) 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:

(25) IDependencyGraphProof (EQUIVALENT transformation)

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