(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: GCD4

(1) JBC2FIG (SOUND transformation)

Constructed FIGraph.

(2) Obligation:

FIGraph based on JBC Program:
Graph of 196 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:
Load1169(i100, i98) → Cond_Load1169(i100 > 0 && i98 > 0, i100, i98)
Load1302(i100, i98, i98, i100, i98, i107, i98) → Cond_Load1302(i98 > 0 && i107 >= i98, i100, i98, i98, i100, i98, i107, i98)
Cond_Load1302(TRUE, i100, i98, i98, i100, i98, i107, i98) → Load1302(i100, i98, i98, i100, i98, i107 - i98, i98)
Load1302(i100, i98, i98, i100, i98, i107, i98) → Cond_Load13021(i107 < i98, i100, i98, i98, i100, i98, i107, i98)
The set Q consists of the following terms:
Load1302(x0, x1, x1, x0, x1, x2, x1)
Cond_Load1302(TRUE, x0, x1, x1, x0, x1, x2, x1)
Cond_Load13021(TRUE, x0, x1, x1, x0, x1, x2, x1)

(5) DuplicateArgsRemoverProof (EQUIVALENT transformation)

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

Cond_Load13021(x1, x2, x3, x4, x5, x6, x7, x8) → Cond_Load13021(x1, x5, x7, x8)
Cond_Load1302(x1, x2, x3, x4, x5, x6, x7, x8) → Cond_Load1302(x1, x5, x7, x8)

(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:
Load1169(i100, i98) → Cond_Load1169(i100 > 0 && i98 > 0, i100, i98)
Load1302(i100, i107, i98) → Cond_Load1302(i98 > 0 && i107 >= i98, i100, i107, i98)
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:

Boolean, Integer

The ITRS R consists of the following rules:
Load1169(i100, i98) → Cond_Load1169(i100 > 0 && i98 > 0, i100, i98)
Load1302(i100, i107, i98) → Cond_Load1302(i98 > 0 && i107 >= i98, i100, i107, i98)

The integer pair graph contains the following rules and edges:
(0): LOAD1169(i100[0], i98[0]) → COND_LOAD1169(i100[0] > 0 && i98[0] > 0, i100[0], i98[0])
(2): LOAD1302(i100[2], i107[2], i98[2]) → COND_LOAD1302(i98[2] > 0 && i107[2] >= i98[2], i100[2], i107[2], i98[2])

(0) -> (1), if ((i100[0]* i100[1])∧(i98[0]* i98[1])∧(i100[0] > 0 && i98[0] > 0* TRUE))

(1) -> (2), if ((i100[1]* i107[2])∧(i100[1]* i100[2])∧(i98[1]* i98[2]))

(1) -> (4), if ((i98[1]* i98[4])∧(i100[1]* i107[4])∧(i100[1]* i100[4]))

(2) -> (3), if ((i98[2]* i98[3])∧(i100[2]* i100[3])∧(i107[2]* i107[3])∧(i98[2] > 0 && i107[2] >= i98[2]* TRUE))

(3) -> (2), if ((i107[3] - i98[3]* i107[2])∧(i100[3]* i100[2])∧(i98[3]* i98[2]))

(3) -> (4), if ((i98[3]* i98[4])∧(i100[3]* i100[4])∧(i107[3] - i98[3]* i107[4]))

(4) -> (5), if ((i100[4]* i100[5])∧(i98[4]* i98[5])∧(i107[4]* i107[5])∧(i107[4] < i98[4]* TRUE))

(5) -> (0), if ((i107[5]* i98[0])∧(i98[5]* i100[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:

Boolean, Integer

R is empty.

The integer pair graph contains the following rules and edges:
(0): LOAD1169(i100[0], i98[0]) → COND_LOAD1169(i100[0] > 0 && i98[0] > 0, i100[0], i98[0])
(2): LOAD1302(i100[2], i107[2], i98[2]) → COND_LOAD1302(i98[2] > 0 && i107[2] >= i98[2], i100[2], i107[2], i98[2])

(0) -> (1), if ((i100[0]* i100[1])∧(i98[0]* i98[1])∧(i100[0] > 0 && i98[0] > 0* TRUE))

(1) -> (2), if ((i100[1]* i107[2])∧(i100[1]* i100[2])∧(i98[1]* i98[2]))

(1) -> (4), if ((i98[1]* i98[4])∧(i100[1]* i107[4])∧(i100[1]* i100[4]))

(2) -> (3), if ((i98[2]* i98[3])∧(i100[2]* i100[3])∧(i107[2]* i107[3])∧(i98[2] > 0 && i107[2] >= i98[2]* TRUE))

(3) -> (2), if ((i107[3] - i98[3]* i107[2])∧(i100[3]* i100[2])∧(i98[3]* i98[2]))

(3) -> (4), if ((i98[3]* i98[4])∧(i100[3]* i100[4])∧(i107[3] - i98[3]* i107[4]))

(4) -> (5), if ((i100[4]* i100[5])∧(i98[4]* i98[5])∧(i107[4]* i107[5])∧(i107[4] < i98[4]* TRUE))

(5) -> (0), if ((i107[5]* i98[0])∧(i98[5]* i100[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 LOAD1169(i100, i98) → COND_LOAD1169(&&(>(i100, 0), >(i98, 0)), i100, i98) the following chains were created:
• We consider the chain LOAD1169(i100[0], i98[0]) → COND_LOAD1169(&&(>(i100[0], 0), >(i98[0], 0)), i100[0], i98[0]), COND_LOAD1169(TRUE, i100[1], i98[1]) → LOAD1302(i100[1], i100[1], i98[1]) which results in the following constraint:

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

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

(3)    (i100[0] + [-1] ≥ 0∧i98[0] + [-1] ≥ 0 ⇒ (UIncreasing(COND_LOAD1169(&&(>(i100[0], 0), >(i98[0], 0)), i100[0], i98[0])), ≥)∧[(-1)bni_28 + (-1)Bound*bni_28] + [bni_28]i98[0] ≥ 0∧[(-1)bso_29] ≥ 0)

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

(4)    (i100[0] + [-1] ≥ 0∧i98[0] + [-1] ≥ 0 ⇒ (UIncreasing(COND_LOAD1169(&&(>(i100[0], 0), >(i98[0], 0)), i100[0], i98[0])), ≥)∧[(-1)bni_28 + (-1)Bound*bni_28] + [bni_28]i98[0] ≥ 0∧[(-1)bso_29] ≥ 0)

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

(5)    (i100[0] + [-1] ≥ 0∧i98[0] + [-1] ≥ 0 ⇒ (UIncreasing(COND_LOAD1169(&&(>(i100[0], 0), >(i98[0], 0)), i100[0], i98[0])), ≥)∧[(-1)bni_28 + (-1)Bound*bni_28] + [bni_28]i98[0] ≥ 0∧[(-1)bso_29] ≥ 0)

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

(6)    (i100[0] ≥ 0∧i98[0] + [-1] ≥ 0 ⇒ (UIncreasing(COND_LOAD1169(&&(>(i100[0], 0), >(i98[0], 0)), i100[0], i98[0])), ≥)∧[(-1)bni_28 + (-1)Bound*bni_28] + [bni_28]i98[0] ≥ 0∧[(-1)bso_29] ≥ 0)

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

(7)    (i100[0] ≥ 0∧i98[0] ≥ 0 ⇒ (UIncreasing(COND_LOAD1169(&&(>(i100[0], 0), >(i98[0], 0)), i100[0], i98[0])), ≥)∧[(-1)Bound*bni_28] + [bni_28]i98[0] ≥ 0∧[(-1)bso_29] ≥ 0)

For Pair COND_LOAD1169(TRUE, i100, i98) → LOAD1302(i100, i100, i98) the following chains were created:
• We consider the chain COND_LOAD1169(TRUE, i100[1], i98[1]) → LOAD1302(i100[1], i100[1], i98[1]), LOAD1302(i100[2], i107[2], i98[2]) → COND_LOAD1302(&&(>(i98[2], 0), >=(i107[2], i98[2])), i100[2], i107[2], i98[2]) 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)    ((UIncreasing(LOAD1302(i100[1], i100[1], i98[1])), ≥)∧[(-1)bso_31] ≥ 0)

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

(11)    ((UIncreasing(LOAD1302(i100[1], i100[1], i98[1])), ≥)∧[(-1)bso_31] ≥ 0)

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

(12)    ((UIncreasing(LOAD1302(i100[1], i100[1], i98[1])), ≥)∧[(-1)bso_31] ≥ 0)

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

(13)    ((UIncreasing(LOAD1302(i100[1], i100[1], i98[1])), ≥)∧0 = 0∧0 = 0∧[(-1)bso_31] ≥ 0)

• We consider the chain COND_LOAD1169(TRUE, i100[1], i98[1]) → LOAD1302(i100[1], i100[1], i98[1]), LOAD1302(i100[4], i107[4], i98[4]) → COND_LOAD13021(<(i107[4], i98[4]), i100[4], i107[4], i98[4]) which results in the following constraint:

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

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

(16)    ((UIncreasing(LOAD1302(i100[1], i100[1], i98[1])), ≥)∧[(-1)bso_31] ≥ 0)

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

(17)    ((UIncreasing(LOAD1302(i100[1], i100[1], i98[1])), ≥)∧[(-1)bso_31] ≥ 0)

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

(18)    ((UIncreasing(LOAD1302(i100[1], i100[1], i98[1])), ≥)∧[(-1)bso_31] ≥ 0)

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

(19)    ((UIncreasing(LOAD1302(i100[1], i100[1], i98[1])), ≥)∧0 = 0∧0 = 0∧[(-1)bso_31] ≥ 0)

For Pair LOAD1302(i100, i107, i98) → COND_LOAD1302(&&(>(i98, 0), >=(i107, i98)), i100, i107, i98) the following chains were created:
• We consider the chain LOAD1302(i100[2], i107[2], i98[2]) → COND_LOAD1302(&&(>(i98[2], 0), >=(i107[2], i98[2])), i100[2], i107[2], i98[2]), COND_LOAD1302(TRUE, i100[3], i107[3], i98[3]) → LOAD1302(i100[3], -(i107[3], i98[3]), i98[3]) which results in the following constraint:

We simplified constraint (20) using rules (IV), (IDP_BOOLEAN) which results in the following new constraint:

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

(22)    (i98[2] + [-1] ≥ 0∧i107[2] + [-1]i98[2] ≥ 0 ⇒ (UIncreasing(COND_LOAD1302(&&(>(i98[2], 0), >=(i107[2], i98[2])), i100[2], i107[2], i98[2])), ≥)∧[(-1)bni_32 + (-1)Bound*bni_32] + [bni_32]i98[2] ≥ 0∧[(-1)bso_33] ≥ 0)

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

(23)    (i98[2] + [-1] ≥ 0∧i107[2] + [-1]i98[2] ≥ 0 ⇒ (UIncreasing(COND_LOAD1302(&&(>(i98[2], 0), >=(i107[2], i98[2])), i100[2], i107[2], i98[2])), ≥)∧[(-1)bni_32 + (-1)Bound*bni_32] + [bni_32]i98[2] ≥ 0∧[(-1)bso_33] ≥ 0)

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

(24)    (i98[2] + [-1] ≥ 0∧i107[2] + [-1]i98[2] ≥ 0 ⇒ (UIncreasing(COND_LOAD1302(&&(>(i98[2], 0), >=(i107[2], i98[2])), i100[2], i107[2], i98[2])), ≥)∧[(-1)bni_32 + (-1)Bound*bni_32] + [bni_32]i98[2] ≥ 0∧[(-1)bso_33] ≥ 0)

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

(25)    (i98[2] + [-1] ≥ 0∧i107[2] + [-1]i98[2] ≥ 0 ⇒ (UIncreasing(COND_LOAD1302(&&(>(i98[2], 0), >=(i107[2], i98[2])), i100[2], i107[2], i98[2])), ≥)∧0 = 0∧[(-1)bni_32 + (-1)Bound*bni_32] + [bni_32]i98[2] ≥ 0∧0 = 0∧[(-1)bso_33] ≥ 0)

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

(26)    (i98[2] ≥ 0∧i107[2] + [-1] + [-1]i98[2] ≥ 0 ⇒ (UIncreasing(COND_LOAD1302(&&(>(i98[2], 0), >=(i107[2], i98[2])), i100[2], i107[2], i98[2])), ≥)∧0 = 0∧[(-1)Bound*bni_32] + [bni_32]i98[2] ≥ 0∧0 = 0∧[(-1)bso_33] ≥ 0)

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

(27)    (i98[2] ≥ 0∧i107[2] ≥ 0 ⇒ (UIncreasing(COND_LOAD1302(&&(>(i98[2], 0), >=(i107[2], i98[2])), i100[2], i107[2], i98[2])), ≥)∧0 = 0∧[(-1)Bound*bni_32] + [bni_32]i98[2] ≥ 0∧0 = 0∧[(-1)bso_33] ≥ 0)

For Pair COND_LOAD1302(TRUE, i100, i107, i98) → LOAD1302(i100, -(i107, i98), i98) the following chains were created:
• We consider the chain LOAD1302(i100[2], i107[2], i98[2]) → COND_LOAD1302(&&(>(i98[2], 0), >=(i107[2], i98[2])), i100[2], i107[2], i98[2]), COND_LOAD1302(TRUE, i100[3], i107[3], i98[3]) → LOAD1302(i100[3], -(i107[3], i98[3]), i98[3]), LOAD1302(i100[2], i107[2], i98[2]) → COND_LOAD1302(&&(>(i98[2], 0), >=(i107[2], i98[2])), i100[2], i107[2], i98[2]) which results in the following constraint:

We simplified constraint (28) using rules (III), (IV), (IDP_BOOLEAN) which results in the following new constraint:

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

(30)    (i98[2] + [-1] ≥ 0∧i107[2] + [-1]i98[2] ≥ 0 ⇒ (UIncreasing(LOAD1302(i100[3], -(i107[3], i98[3]), i98[3])), ≥)∧[(-1)bni_34 + (-1)Bound*bni_34] + [bni_34]i98[2] ≥ 0∧[(-1)bso_35] ≥ 0)

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

(31)    (i98[2] + [-1] ≥ 0∧i107[2] + [-1]i98[2] ≥ 0 ⇒ (UIncreasing(LOAD1302(i100[3], -(i107[3], i98[3]), i98[3])), ≥)∧[(-1)bni_34 + (-1)Bound*bni_34] + [bni_34]i98[2] ≥ 0∧[(-1)bso_35] ≥ 0)

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

(32)    (i98[2] + [-1] ≥ 0∧i107[2] + [-1]i98[2] ≥ 0 ⇒ (UIncreasing(LOAD1302(i100[3], -(i107[3], i98[3]), i98[3])), ≥)∧[(-1)bni_34 + (-1)Bound*bni_34] + [bni_34]i98[2] ≥ 0∧[(-1)bso_35] ≥ 0)

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

(33)    (i98[2] + [-1] ≥ 0∧i107[2] + [-1]i98[2] ≥ 0 ⇒ (UIncreasing(LOAD1302(i100[3], -(i107[3], i98[3]), i98[3])), ≥)∧0 = 0∧[(-1)bni_34 + (-1)Bound*bni_34] + [bni_34]i98[2] ≥ 0∧0 = 0∧[(-1)bso_35] ≥ 0)

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

(34)    (i98[2] ≥ 0∧i107[2] + [-1] + [-1]i98[2] ≥ 0 ⇒ (UIncreasing(LOAD1302(i100[3], -(i107[3], i98[3]), i98[3])), ≥)∧0 = 0∧[(-1)Bound*bni_34] + [bni_34]i98[2] ≥ 0∧0 = 0∧[(-1)bso_35] ≥ 0)

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

(35)    (i98[2] ≥ 0∧i107[2] ≥ 0 ⇒ (UIncreasing(LOAD1302(i100[3], -(i107[3], i98[3]), i98[3])), ≥)∧0 = 0∧[(-1)Bound*bni_34] + [bni_34]i98[2] ≥ 0∧0 = 0∧[(-1)bso_35] ≥ 0)

• We consider the chain LOAD1302(i100[2], i107[2], i98[2]) → COND_LOAD1302(&&(>(i98[2], 0), >=(i107[2], i98[2])), i100[2], i107[2], i98[2]), COND_LOAD1302(TRUE, i100[3], i107[3], i98[3]) → LOAD1302(i100[3], -(i107[3], i98[3]), i98[3]), LOAD1302(i100[4], i107[4], i98[4]) → COND_LOAD13021(<(i107[4], i98[4]), i100[4], i107[4], i98[4]) which results in the following constraint:

We simplified constraint (36) using rules (III), (IV), (IDP_BOOLEAN) which results in the following new constraint:

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

(38)    (i98[2] + [-1] ≥ 0∧i107[2] + [-1]i98[2] ≥ 0 ⇒ (UIncreasing(LOAD1302(i100[3], -(i107[3], i98[3]), i98[3])), ≥)∧[(-1)bni_34 + (-1)Bound*bni_34] + [bni_34]i98[2] ≥ 0∧[(-1)bso_35] ≥ 0)

We simplified constraint (38) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:

(39)    (i98[2] + [-1] ≥ 0∧i107[2] + [-1]i98[2] ≥ 0 ⇒ (UIncreasing(LOAD1302(i100[3], -(i107[3], i98[3]), i98[3])), ≥)∧[(-1)bni_34 + (-1)Bound*bni_34] + [bni_34]i98[2] ≥ 0∧[(-1)bso_35] ≥ 0)

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

(40)    (i98[2] + [-1] ≥ 0∧i107[2] + [-1]i98[2] ≥ 0 ⇒ (UIncreasing(LOAD1302(i100[3], -(i107[3], i98[3]), i98[3])), ≥)∧[(-1)bni_34 + (-1)Bound*bni_34] + [bni_34]i98[2] ≥ 0∧[(-1)bso_35] ≥ 0)

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

(41)    (i98[2] + [-1] ≥ 0∧i107[2] + [-1]i98[2] ≥ 0 ⇒ (UIncreasing(LOAD1302(i100[3], -(i107[3], i98[3]), i98[3])), ≥)∧0 = 0∧[(-1)bni_34 + (-1)Bound*bni_34] + [bni_34]i98[2] ≥ 0∧0 = 0∧[(-1)bso_35] ≥ 0)

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

(42)    (i98[2] ≥ 0∧i107[2] + [-1] + [-1]i98[2] ≥ 0 ⇒ (UIncreasing(LOAD1302(i100[3], -(i107[3], i98[3]), i98[3])), ≥)∧0 = 0∧[(-1)Bound*bni_34] + [bni_34]i98[2] ≥ 0∧0 = 0∧[(-1)bso_35] ≥ 0)

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

(43)    (i98[2] ≥ 0∧i107[2] ≥ 0 ⇒ (UIncreasing(LOAD1302(i100[3], -(i107[3], i98[3]), i98[3])), ≥)∧0 = 0∧[(-1)Bound*bni_34] + [bni_34]i98[2] ≥ 0∧0 = 0∧[(-1)bso_35] ≥ 0)

For Pair LOAD1302(i100, i107, i98) → COND_LOAD13021(<(i107, i98), i100, i107, i98) the following chains were created:
• We consider the chain LOAD1302(i100[4], i107[4], i98[4]) → COND_LOAD13021(<(i107[4], i98[4]), i100[4], i107[4], i98[4]), COND_LOAD13021(TRUE, i100[5], i107[5], i98[5]) → LOAD1169(i98[5], i107[5]) which results in the following constraint:

We simplified constraint (44) using rule (IV) which results in the following new constraint:

We simplified constraint (45) using rule (POLY_CONSTRAINTS) which results in the following new constraint:

(46)    (i98[4] + [-1] + [-1]i107[4] ≥ 0 ⇒ (UIncreasing(COND_LOAD13021(<(i107[4], i98[4]), i100[4], i107[4], i98[4])), ≥)∧[(-1)bni_36 + (-1)Bound*bni_36] + [bni_36]i98[4] ≥ 0∧[-1 + (-1)bso_37] + i98[4] + [-1]i107[4] ≥ 0)

We simplified constraint (46) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:

(47)    (i98[4] + [-1] + [-1]i107[4] ≥ 0 ⇒ (UIncreasing(COND_LOAD13021(<(i107[4], i98[4]), i100[4], i107[4], i98[4])), ≥)∧[(-1)bni_36 + (-1)Bound*bni_36] + [bni_36]i98[4] ≥ 0∧[-1 + (-1)bso_37] + i98[4] + [-1]i107[4] ≥ 0)

We simplified constraint (47) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:

(48)    (i98[4] + [-1] + [-1]i107[4] ≥ 0 ⇒ (UIncreasing(COND_LOAD13021(<(i107[4], i98[4]), i100[4], i107[4], i98[4])), ≥)∧[(-1)bni_36 + (-1)Bound*bni_36] + [bni_36]i98[4] ≥ 0∧[-1 + (-1)bso_37] + i98[4] + [-1]i107[4] ≥ 0)

We simplified constraint (48) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint:

(49)    (i98[4] + [-1] + [-1]i107[4] ≥ 0 ⇒ (UIncreasing(COND_LOAD13021(<(i107[4], i98[4]), i100[4], i107[4], i98[4])), ≥)∧0 = 0∧[(-1)bni_36 + (-1)Bound*bni_36] + [bni_36]i98[4] ≥ 0∧0 = 0∧[-1 + (-1)bso_37] + i98[4] + [-1]i107[4] ≥ 0)

We simplified constraint (49) using rule (IDP_SMT_SPLIT) which results in the following new constraint:

(50)    (i98[4] ≥ 0 ⇒ (UIncreasing(COND_LOAD13021(<(i107[4], i98[4]), i100[4], i107[4], i98[4])), ≥)∧0 = 0∧[(-1)Bound*bni_36] + [bni_36]i107[4] + [bni_36]i98[4] ≥ 0∧0 = 0∧[(-1)bso_37] + i98[4] ≥ 0)

We simplified constraint (50) using rule (IDP_SMT_SPLIT) which results in the following new constraints:

(51)    (i98[4] ≥ 0∧i107[4] ≥ 0 ⇒ (UIncreasing(COND_LOAD13021(<(i107[4], i98[4]), i100[4], i107[4], i98[4])), ≥)∧0 = 0∧[(-1)Bound*bni_36] + [(-1)bni_36]i107[4] + [bni_36]i98[4] ≥ 0∧0 = 0∧[(-1)bso_37] + i98[4] ≥ 0)

(52)    (i98[4] ≥ 0∧i107[4] ≥ 0 ⇒ (UIncreasing(COND_LOAD13021(<(i107[4], i98[4]), i100[4], i107[4], i98[4])), ≥)∧0 = 0∧[(-1)Bound*bni_36] + [bni_36]i107[4] + [bni_36]i98[4] ≥ 0∧0 = 0∧[(-1)bso_37] + i98[4] ≥ 0)

For Pair COND_LOAD13021(TRUE, i100, i107, i98) → LOAD1169(i98, i107) the following chains were created:
• We consider the chain COND_LOAD13021(TRUE, i100[5], i107[5], i98[5]) → LOAD1169(i98[5], i107[5]), LOAD1169(i100[0], i98[0]) → COND_LOAD1169(&&(>(i100[0], 0), >(i98[0], 0)), i100[0], i98[0]) which results in the following constraint:

We simplified constraint (53) using rule (IV) which results in the following new constraint:

We simplified constraint (54) using rule (POLY_CONSTRAINTS) which results in the following new constraint:

(55)    ((UIncreasing(LOAD1169(i98[5], i107[5])), ≥)∧[1 + (-1)bso_39] ≥ 0)

We simplified constraint (55) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:

(56)    ((UIncreasing(LOAD1169(i98[5], i107[5])), ≥)∧[1 + (-1)bso_39] ≥ 0)

We simplified constraint (56) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:

(57)    ((UIncreasing(LOAD1169(i98[5], i107[5])), ≥)∧[1 + (-1)bso_39] ≥ 0)

We simplified constraint (57) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint:

(58)    ((UIncreasing(LOAD1169(i98[5], i107[5])), ≥)∧0 = 0∧0 = 0∧0 = 0∧[1 + (-1)bso_39] ≥ 0)

To summarize, we get the following constraints P for the following pairs.
• (i100[0] ≥ 0∧i98[0] ≥ 0 ⇒ (UIncreasing(COND_LOAD1169(&&(>(i100[0], 0), >(i98[0], 0)), i100[0], i98[0])), ≥)∧[(-1)Bound*bni_28] + [bni_28]i98[0] ≥ 0∧[(-1)bso_29] ≥ 0)

• ((UIncreasing(LOAD1302(i100[1], i100[1], i98[1])), ≥)∧0 = 0∧0 = 0∧[(-1)bso_31] ≥ 0)
• ((UIncreasing(LOAD1302(i100[1], i100[1], i98[1])), ≥)∧0 = 0∧0 = 0∧[(-1)bso_31] ≥ 0)

• LOAD1302(i100, i107, i98) → COND_LOAD1302(&&(>(i98, 0), >=(i107, i98)), i100, i107, i98)
• (i98[2] ≥ 0∧i107[2] ≥ 0 ⇒ (UIncreasing(COND_LOAD1302(&&(>(i98[2], 0), >=(i107[2], i98[2])), i100[2], i107[2], i98[2])), ≥)∧0 = 0∧[(-1)Bound*bni_32] + [bni_32]i98[2] ≥ 0∧0 = 0∧[(-1)bso_33] ≥ 0)

• (i98[2] ≥ 0∧i107[2] ≥ 0 ⇒ (UIncreasing(LOAD1302(i100[3], -(i107[3], i98[3]), i98[3])), ≥)∧0 = 0∧[(-1)Bound*bni_34] + [bni_34]i98[2] ≥ 0∧0 = 0∧[(-1)bso_35] ≥ 0)
• (i98[2] ≥ 0∧i107[2] ≥ 0 ⇒ (UIncreasing(LOAD1302(i100[3], -(i107[3], i98[3]), i98[3])), ≥)∧0 = 0∧[(-1)Bound*bni_34] + [bni_34]i98[2] ≥ 0∧0 = 0∧[(-1)bso_35] ≥ 0)

• (i98[4] ≥ 0∧i107[4] ≥ 0 ⇒ (UIncreasing(COND_LOAD13021(<(i107[4], i98[4]), i100[4], i107[4], i98[4])), ≥)∧0 = 0∧[(-1)Bound*bni_36] + [(-1)bni_36]i107[4] + [bni_36]i98[4] ≥ 0∧0 = 0∧[(-1)bso_37] + i98[4] ≥ 0)
• (i98[4] ≥ 0∧i107[4] ≥ 0 ⇒ (UIncreasing(COND_LOAD13021(<(i107[4], i98[4]), i100[4], i107[4], i98[4])), ≥)∧0 = 0∧[(-1)Bound*bni_36] + [bni_36]i107[4] + [bni_36]i98[4] ≥ 0∧0 = 0∧[(-1)bso_37] + i98[4] ≥ 0)

• ((UIncreasing(LOAD1169(i98[5], i107[5])), ≥)∧0 = 0∧0 = 0∧0 = 0∧[1 + (-1)bso_39] ≥ 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(LOAD1169(x1, x2)) = [-1] + x2
POL(COND_LOAD1169(x1, x2, x3)) = [-1] + x3
POL(&&(x1, x2)) = [-1]
POL(>(x1, x2)) = [-1]
POL(0) = 0
POL(LOAD1302(x1, x2, x3)) = [-1] + x3
POL(COND_LOAD1302(x1, x2, x3, x4)) = [-1] + x4
POL(>=(x1, x2)) = [-1]
POL(-(x1, x2)) = x1 + [-1]x2
POL(COND_LOAD13021(x1, x2, x3, x4)) = x3
POL(<(x1, x2)) = [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:

FALSE1&&(FALSE, FALSE)1

(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:

Boolean, Integer

R is empty.

The integer pair graph contains the following rules and edges:
(0): LOAD1169(i100[0], i98[0]) → COND_LOAD1169(i100[0] > 0 && i98[0] > 0, i100[0], i98[0])
(2): LOAD1302(i100[2], i107[2], i98[2]) → COND_LOAD1302(i98[2] > 0 && i107[2] >= i98[2], i100[2], i107[2], i98[2])

(0) -> (1), if ((i100[0]* i100[1])∧(i98[0]* i98[1])∧(i100[0] > 0 && i98[0] > 0* TRUE))

(1) -> (2), if ((i100[1]* i107[2])∧(i100[1]* i100[2])∧(i98[1]* i98[2]))

(3) -> (2), if ((i107[3] - i98[3]* i107[2])∧(i100[3]* i100[2])∧(i98[3]* i98[2]))

(2) -> (3), if ((i98[2]* i98[3])∧(i100[2]* i100[3])∧(i107[2]* i107[3])∧(i98[2] > 0 && i107[2] >= i98[2]* TRUE))

(1) -> (4), if ((i98[1]* i98[4])∧(i100[1]* i107[4])∧(i100[1]* i100[4]))

(3) -> (4), if ((i98[3]* i98[4])∧(i100[3]* i100[4])∧(i107[3] - i98[3]* i107[4]))

The set Q consists of the following terms:

(14) IDependencyGraphProof (EQUIVALENT transformation)

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

(15) 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): LOAD1302(i100[2], i107[2], i98[2]) → COND_LOAD1302(i98[2] > 0 && i107[2] >= i98[2], i100[2], i107[2], i98[2])

(3) -> (2), if ((i107[3] - i98[3]* i107[2])∧(i100[3]* i100[2])∧(i98[3]* i98[2]))

(2) -> (3), if ((i98[2]* i98[3])∧(i100[2]* i100[3])∧(i107[2]* i107[3])∧(i98[2] > 0 && i107[2] >= i98[2]* TRUE))

The set Q consists of the following terms:

(16) 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_LOAD1302(TRUE, i100[3], i107[3], i98[3]) → LOAD1302(i100[3], -(i107[3], i98[3]), i98[3]) the following chains were created:
• We consider the chain LOAD1302(i100[2], i107[2], i98[2]) → COND_LOAD1302(&&(>(i98[2], 0), >=(i107[2], i98[2])), i100[2], i107[2], i98[2]), COND_LOAD1302(TRUE, i100[3], i107[3], i98[3]) → LOAD1302(i100[3], -(i107[3], i98[3]), i98[3]), LOAD1302(i100[2], i107[2], i98[2]) → COND_LOAD1302(&&(>(i98[2], 0), >=(i107[2], i98[2])), i100[2], i107[2], i98[2]) which results in the following constraint:

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

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

(3)    (i98[2] + [-1] ≥ 0∧i107[2] + [-1]i98[2] ≥ 0 ⇒ (UIncreasing(LOAD1302(i100[3], -(i107[3], i98[3]), i98[3])), ≥)∧[(-2)bni_15 + (-1)Bound*bni_15] + [(-1)bni_15]i98[2] + [(2)bni_15]i107[2] ≥ 0∧[-1 + (-1)bso_16] + [2]i98[2] ≥ 0)

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

(4)    (i98[2] + [-1] ≥ 0∧i107[2] + [-1]i98[2] ≥ 0 ⇒ (UIncreasing(LOAD1302(i100[3], -(i107[3], i98[3]), i98[3])), ≥)∧[(-2)bni_15 + (-1)Bound*bni_15] + [(-1)bni_15]i98[2] + [(2)bni_15]i107[2] ≥ 0∧[-1 + (-1)bso_16] + [2]i98[2] ≥ 0)

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

(5)    (i98[2] + [-1] ≥ 0∧i107[2] + [-1]i98[2] ≥ 0 ⇒ (UIncreasing(LOAD1302(i100[3], -(i107[3], i98[3]), i98[3])), ≥)∧[(-2)bni_15 + (-1)Bound*bni_15] + [(-1)bni_15]i98[2] + [(2)bni_15]i107[2] ≥ 0∧[-1 + (-1)bso_16] + [2]i98[2] ≥ 0)

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

(6)    (i98[2] + [-1] ≥ 0∧i107[2] + [-1]i98[2] ≥ 0 ⇒ (UIncreasing(LOAD1302(i100[3], -(i107[3], i98[3]), i98[3])), ≥)∧0 = 0∧[(-2)bni_15 + (-1)Bound*bni_15] + [(-1)bni_15]i98[2] + [(2)bni_15]i107[2] ≥ 0∧0 = 0∧[-1 + (-1)bso_16] + [2]i98[2] ≥ 0)

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

(7)    (i98[2] ≥ 0∧i107[2] + [-1] + [-1]i98[2] ≥ 0 ⇒ (UIncreasing(LOAD1302(i100[3], -(i107[3], i98[3]), i98[3])), ≥)∧0 = 0∧[(-3)bni_15 + (-1)Bound*bni_15] + [(-1)bni_15]i98[2] + [(2)bni_15]i107[2] ≥ 0∧0 = 0∧[1 + (-1)bso_16] + [2]i98[2] ≥ 0)

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

(8)    (i98[2] ≥ 0∧i107[2] ≥ 0 ⇒ (UIncreasing(LOAD1302(i100[3], -(i107[3], i98[3]), i98[3])), ≥)∧0 = 0∧[(-1)bni_15 + (-1)Bound*bni_15] + [bni_15]i98[2] + [(2)bni_15]i107[2] ≥ 0∧0 = 0∧[1 + (-1)bso_16] + [2]i98[2] ≥ 0)

For Pair LOAD1302(i100[2], i107[2], i98[2]) → COND_LOAD1302(&&(>(i98[2], 0), >=(i107[2], i98[2])), i100[2], i107[2], i98[2]) the following chains were created:
• We consider the chain LOAD1302(i100[2], i107[2], i98[2]) → COND_LOAD1302(&&(>(i98[2], 0), >=(i107[2], i98[2])), i100[2], i107[2], i98[2]), COND_LOAD1302(TRUE, i100[3], i107[3], i98[3]) → LOAD1302(i100[3], -(i107[3], i98[3]), i98[3]) which results in the following constraint:

We simplified constraint (9) using rules (IV), (IDP_BOOLEAN) which results in the following new constraint:

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

(11)    (i98[2] + [-1] ≥ 0∧i107[2] + [-1]i98[2] ≥ 0 ⇒ (UIncreasing(COND_LOAD1302(&&(>(i98[2], 0), >=(i107[2], i98[2])), i100[2], i107[2], i98[2])), ≥)∧[(-1)bni_17 + (-1)Bound*bni_17] + [(-1)bni_17]i98[2] + [(2)bni_17]i107[2] ≥ 0∧[(-1)bso_18] ≥ 0)

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

(12)    (i98[2] + [-1] ≥ 0∧i107[2] + [-1]i98[2] ≥ 0 ⇒ (UIncreasing(COND_LOAD1302(&&(>(i98[2], 0), >=(i107[2], i98[2])), i100[2], i107[2], i98[2])), ≥)∧[(-1)bni_17 + (-1)Bound*bni_17] + [(-1)bni_17]i98[2] + [(2)bni_17]i107[2] ≥ 0∧[(-1)bso_18] ≥ 0)

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

(13)    (i98[2] + [-1] ≥ 0∧i107[2] + [-1]i98[2] ≥ 0 ⇒ (UIncreasing(COND_LOAD1302(&&(>(i98[2], 0), >=(i107[2], i98[2])), i100[2], i107[2], i98[2])), ≥)∧[(-1)bni_17 + (-1)Bound*bni_17] + [(-1)bni_17]i98[2] + [(2)bni_17]i107[2] ≥ 0∧[(-1)bso_18] ≥ 0)

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

(14)    (i98[2] + [-1] ≥ 0∧i107[2] + [-1]i98[2] ≥ 0 ⇒ (UIncreasing(COND_LOAD1302(&&(>(i98[2], 0), >=(i107[2], i98[2])), i100[2], i107[2], i98[2])), ≥)∧0 = 0∧[(-1)bni_17 + (-1)Bound*bni_17] + [(-1)bni_17]i98[2] + [(2)bni_17]i107[2] ≥ 0∧0 = 0∧[(-1)bso_18] ≥ 0)

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

(15)    (i98[2] ≥ 0∧i107[2] + [-1] + [-1]i98[2] ≥ 0 ⇒ (UIncreasing(COND_LOAD1302(&&(>(i98[2], 0), >=(i107[2], i98[2])), i100[2], i107[2], i98[2])), ≥)∧0 = 0∧[(-2)bni_17 + (-1)Bound*bni_17] + [(-1)bni_17]i98[2] + [(2)bni_17]i107[2] ≥ 0∧0 = 0∧[(-1)bso_18] ≥ 0)

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

(16)    (i98[2] ≥ 0∧i107[2] ≥ 0 ⇒ (UIncreasing(COND_LOAD1302(&&(>(i98[2], 0), >=(i107[2], i98[2])), i100[2], i107[2], i98[2])), ≥)∧0 = 0∧[(-1)Bound*bni_17] + [bni_17]i98[2] + [(2)bni_17]i107[2] ≥ 0∧0 = 0∧[(-1)bso_18] ≥ 0)

To summarize, we get the following constraints P for the following pairs.
• (i98[2] ≥ 0∧i107[2] ≥ 0 ⇒ (UIncreasing(LOAD1302(i100[3], -(i107[3], i98[3]), i98[3])), ≥)∧0 = 0∧[(-1)bni_15 + (-1)Bound*bni_15] + [bni_15]i98[2] + [(2)bni_15]i107[2] ≥ 0∧0 = 0∧[1 + (-1)bso_16] + [2]i98[2] ≥ 0)

• LOAD1302(i100[2], i107[2], i98[2]) → COND_LOAD1302(&&(>(i98[2], 0), >=(i107[2], i98[2])), i100[2], i107[2], i98[2])
• (i98[2] ≥ 0∧i107[2] ≥ 0 ⇒ (UIncreasing(COND_LOAD1302(&&(>(i98[2], 0), >=(i107[2], i98[2])), i100[2], i107[2], i98[2])), ≥)∧0 = 0∧[(-1)Bound*bni_17] + [bni_17]i98[2] + [(2)bni_17]i107[2] ≥ 0∧0 = 0∧[(-1)bso_18] ≥ 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) = [1]
POL(FALSE) = [2]
POL(COND_LOAD1302(x1, x2, x3, x4)) = [-1] + [-1]x4 + [2]x3 + [-1]x1
POL(LOAD1302(x1, x2, x3)) = [-1] + [-1]x3 + [2]x2
POL(-(x1, x2)) = x1 + [-1]x2
POL(&&(x1, x2)) = 0
POL(>(x1, x2)) = [-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:

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

TRUE1&&(TRUE, TRUE)1
FALSE1&&(TRUE, FALSE)1
FALSE1&&(FALSE, TRUE)1
FALSE1&&(FALSE, FALSE)1

(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:

Boolean, Integer

R is empty.

The integer pair graph contains the following rules and edges:
(2): LOAD1302(i100[2], i107[2], i98[2]) → COND_LOAD1302(i98[2] > 0 && i107[2] >= i98[2], i100[2], i107[2], i98[2])

The set Q consists of the following terms:

(19) IDependencyGraphProof (EQUIVALENT transformation)

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

(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:
none

R is empty.

The integer pair graph is empty.

The set Q consists of the following terms:

(22) IDependencyGraphProof (EQUIVALENT transformation)

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

(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:

Integer

R is empty.

The integer pair graph contains the following rules and edges:

(1) -> (4), if ((i98[1]* i98[4])∧(i100[1]* i107[4])∧(i100[1]* i100[4]))

(4) -> (5), if ((i100[4]* i100[5])∧(i98[4]* i98[5])∧(i107[4]* i107[5])∧(i107[4] < i98[4]* TRUE))

The set Q consists of the following terms: