(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: PastaB17

(1) JBC2FIG (SOUND transformation)

Constructed FIGraph.

(2) Obligation:

FIGraph based on JBC Program:
Graph of 232 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:
Load1129(i128, i129, i88) → Cond_Load1129(i88 >= 0 && i129 > i88, i128, i129, i88)
Load1115(i128, i129, i88) → Cond_Load1115(i88 >= 0 && i129 > i88 && i128 > i88, i128, i129, i88)
Load1115(i128, i129, i88) → Cond_Load11151(i129 <= i88 && i88 >= 0 && i128 > i88, i128, i129, i88)
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:
Load1129(i128, i129, i88) → Cond_Load1129(i88 >= 0 && i129 > i88, i128, i129, i88)
Load1115(i128, i129, i88) → Cond_Load1115(i88 >= 0 && i129 > i88 && i128 > i88, i128, i129, i88)
Load1115(i128, i129, i88) → Cond_Load11151(i129 <= i88 && i88 >= 0 && i128 > i88, i128, i129, i88)

The integer pair graph contains the following rules and edges:
(0): LOAD1129(i128[0], i129[0], i88[0]) → COND_LOAD1129(i88[0] >= 0 && i129[0] > i88[0], i128[0], i129[0], i88[0])
(2): LOAD1115(i128[2], i129[2], i88[2]) → COND_LOAD1115(i88[2] >= 0 && i129[2] > i88[2] && i128[2] > i88[2], i128[2], i129[2], i88[2])
(6): LOAD1115(i128[6], i129[6], i88[6]) → COND_LOAD11151(i129[6] <= i88[6] && i88[6] >= 0 && i128[6] > i88[6], i128[6], i129[6], i88[6])

(0) -> (1), if ((i88[0]* i88[1])∧(i88[0] >= 0 && i129[0] > i88[0]* TRUE)∧(i129[0]* i129[1])∧(i128[0]* i128[1]))

(1) -> (0), if ((i128[1]* i128[0])∧(i129[1] + -1* i129[0])∧(i88[1]* i88[0]))

(1) -> (4), if ((i128[1]* i128[4])∧(i129[1] + -1* i129[4])∧(i88[1]* i88[4]))

(2) -> (3), if ((i128[2]* i128[3])∧(i88[2]* i88[3])∧(i129[2]* i129[3])∧(i88[2] >= 0 && i129[2] > i88[2] && i128[2] > i88[2]* TRUE))

(3) -> (0), if ((i88[3]* i88[0])∧(i129[3] + -1* i129[0])∧(i128[3]* i128[0]))

(3) -> (4), if ((i129[3] + -1* i129[4])∧(i128[3]* i128[4])∧(i88[3]* i88[4]))

(4) -> (5), if ((i88[4]* i88[5])∧(i129[4]* i129[5])∧(i129[4] <= i88[4]* TRUE)∧(i128[4]* i128[5]))

(5) -> (2), if ((i88[5]* i88[2])∧(i128[5] + -1* i128[2])∧(i129[5]* i129[2]))

(5) -> (6), if ((i88[5]* i88[6])∧(i128[5] + -1* i128[6])∧(i129[5]* i129[6]))

(6) -> (7), if ((i129[6] <= i88[6] && i88[6] >= 0 && i128[6] > i88[6]* TRUE)∧(i128[6]* i128[7])∧(i88[6]* i88[7])∧(i129[6]* i129[7]))

(7) -> (2), if ((i128[7] + -1* i128[2])∧(i129[7]* i129[2])∧(i88[7]* i88[2]))

(7) -> (6), if ((i129[7]* i129[6])∧(i128[7] + -1* i128[6])∧(i88[7]* i88[6]))

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): LOAD1129(i128[0], i129[0], i88[0]) → COND_LOAD1129(i88[0] >= 0 && i129[0] > i88[0], i128[0], i129[0], i88[0])
(2): LOAD1115(i128[2], i129[2], i88[2]) → COND_LOAD1115(i88[2] >= 0 && i129[2] > i88[2] && i128[2] > i88[2], i128[2], i129[2], i88[2])
(6): LOAD1115(i128[6], i129[6], i88[6]) → COND_LOAD11151(i129[6] <= i88[6] && i88[6] >= 0 && i128[6] > i88[6], i128[6], i129[6], i88[6])

(0) -> (1), if ((i88[0]* i88[1])∧(i88[0] >= 0 && i129[0] > i88[0]* TRUE)∧(i129[0]* i129[1])∧(i128[0]* i128[1]))

(1) -> (0), if ((i128[1]* i128[0])∧(i129[1] + -1* i129[0])∧(i88[1]* i88[0]))

(1) -> (4), if ((i128[1]* i128[4])∧(i129[1] + -1* i129[4])∧(i88[1]* i88[4]))

(2) -> (3), if ((i128[2]* i128[3])∧(i88[2]* i88[3])∧(i129[2]* i129[3])∧(i88[2] >= 0 && i129[2] > i88[2] && i128[2] > i88[2]* TRUE))

(3) -> (0), if ((i88[3]* i88[0])∧(i129[3] + -1* i129[0])∧(i128[3]* i128[0]))

(3) -> (4), if ((i129[3] + -1* i129[4])∧(i128[3]* i128[4])∧(i88[3]* i88[4]))

(4) -> (5), if ((i88[4]* i88[5])∧(i129[4]* i129[5])∧(i129[4] <= i88[4]* TRUE)∧(i128[4]* i128[5]))

(5) -> (2), if ((i88[5]* i88[2])∧(i128[5] + -1* i128[2])∧(i129[5]* i129[2]))

(5) -> (6), if ((i88[5]* i88[6])∧(i128[5] + -1* i128[6])∧(i129[5]* i129[6]))

(6) -> (7), if ((i129[6] <= i88[6] && i88[6] >= 0 && i128[6] > i88[6]* TRUE)∧(i128[6]* i128[7])∧(i88[6]* i88[7])∧(i129[6]* i129[7]))

(7) -> (2), if ((i128[7] + -1* i128[2])∧(i129[7]* i129[2])∧(i88[7]* i88[2]))

(7) -> (6), if ((i129[7]* i129[6])∧(i128[7] + -1* i128[6])∧(i88[7]* i88[6]))

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 LOAD1129(i128, i129, i88) → COND_LOAD1129(&&(>=(i88, 0), >(i129, i88)), i128, i129, i88) the following chains were created:
• We consider the chain LOAD1129(i128[0], i129[0], i88[0]) → COND_LOAD1129(&&(>=(i88[0], 0), >(i129[0], i88[0])), i128[0], i129[0], i88[0]), COND_LOAD1129(TRUE, i128[1], i129[1], i88[1]) → LOAD1129(i128[1], +(i129[1], -1), i88[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)    (i88[0] ≥ 0∧i129[0] + [-1] + [-1]i88[0] ≥ 0 ⇒ (UIncreasing(COND_LOAD1129(&&(>=(i88[0], 0), >(i129[0], i88[0])), i128[0], i129[0], i88[0])), ≥)∧[(-1)bni_16 + (-1)Bound*bni_16] + [(-1)bni_16]i88[0] + [bni_16]i128[0] ≥ 0∧[(-1)bso_17] ≥ 0)

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

(4)    (i88[0] ≥ 0∧i129[0] + [-1] + [-1]i88[0] ≥ 0 ⇒ (UIncreasing(COND_LOAD1129(&&(>=(i88[0], 0), >(i129[0], i88[0])), i128[0], i129[0], i88[0])), ≥)∧[(-1)bni_16 + (-1)Bound*bni_16] + [(-1)bni_16]i88[0] + [bni_16]i128[0] ≥ 0∧[(-1)bso_17] ≥ 0)

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

(5)    (i88[0] ≥ 0∧i129[0] + [-1] + [-1]i88[0] ≥ 0 ⇒ (UIncreasing(COND_LOAD1129(&&(>=(i88[0], 0), >(i129[0], i88[0])), i128[0], i129[0], i88[0])), ≥)∧[(-1)bni_16 + (-1)Bound*bni_16] + [(-1)bni_16]i88[0] + [bni_16]i128[0] ≥ 0∧[(-1)bso_17] ≥ 0)

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

(6)    (i88[0] ≥ 0∧i129[0] + [-1] + [-1]i88[0] ≥ 0 ⇒ (UIncreasing(COND_LOAD1129(&&(>=(i88[0], 0), >(i129[0], i88[0])), i128[0], i129[0], i88[0])), ≥)∧[bni_16] = 0∧[(-1)bni_16 + (-1)Bound*bni_16] + [(-1)bni_16]i88[0] ≥ 0∧0 = 0∧[(-1)bso_17] ≥ 0)

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

(7)    (i88[0] ≥ 0∧i129[0] ≥ 0 ⇒ (UIncreasing(COND_LOAD1129(&&(>=(i88[0], 0), >(i129[0], i88[0])), i128[0], i129[0], i88[0])), ≥)∧[bni_16] = 0∧[(-1)bni_16 + (-1)Bound*bni_16] + [(-1)bni_16]i88[0] ≥ 0∧0 = 0∧[(-1)bso_17] ≥ 0)

For Pair COND_LOAD1129(TRUE, i128, i129, i88) → LOAD1129(i128, +(i129, -1), i88) the following chains were created:
• We consider the chain COND_LOAD1129(TRUE, i128[1], i129[1], i88[1]) → LOAD1129(i128[1], +(i129[1], -1), i88[1]) which results in the following constraint:

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

(9)    ((UIncreasing(LOAD1129(i128[1], +(i129[1], -1), i88[1])), ≥)∧[(-1)bso_19] ≥ 0)

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

(10)    ((UIncreasing(LOAD1129(i128[1], +(i129[1], -1), i88[1])), ≥)∧[(-1)bso_19] ≥ 0)

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

(11)    ((UIncreasing(LOAD1129(i128[1], +(i129[1], -1), i88[1])), ≥)∧[(-1)bso_19] ≥ 0)

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

(12)    ((UIncreasing(LOAD1129(i128[1], +(i129[1], -1), i88[1])), ≥)∧0 = 0∧0 = 0∧0 = 0∧[(-1)bso_19] ≥ 0)

For Pair LOAD1115(i128, i129, i88) → COND_LOAD1115(&&(&&(>=(i88, 0), >(i129, i88)), >(i128, i88)), i128, i129, i88) the following chains were created:
• We consider the chain LOAD1115(i128[2], i129[2], i88[2]) → COND_LOAD1115(&&(&&(>=(i88[2], 0), >(i129[2], i88[2])), >(i128[2], i88[2])), i128[2], i129[2], i88[2]), COND_LOAD1115(TRUE, i128[3], i129[3], i88[3]) → LOAD1129(i128[3], +(i129[3], -1), i88[3]) which results in the following constraint:

(13)    (i128[2]=i128[3]i88[2]=i88[3]i129[2]=i129[3]&&(&&(>=(i88[2], 0), >(i129[2], i88[2])), >(i128[2], i88[2]))=TRUELOAD1115(i128[2], i129[2], i88[2])≥NonInfC∧LOAD1115(i128[2], i129[2], i88[2])≥COND_LOAD1115(&&(&&(>=(i88[2], 0), >(i129[2], i88[2])), >(i128[2], i88[2])), i128[2], i129[2], i88[2])∧(UIncreasing(COND_LOAD1115(&&(&&(>=(i88[2], 0), >(i129[2], i88[2])), >(i128[2], i88[2])), i128[2], i129[2], i88[2])), ≥))

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

(14)    (>(i128[2], i88[2])=TRUE>=(i88[2], 0)=TRUE>(i129[2], i88[2])=TRUELOAD1115(i128[2], i129[2], i88[2])≥NonInfC∧LOAD1115(i128[2], i129[2], i88[2])≥COND_LOAD1115(&&(&&(>=(i88[2], 0), >(i129[2], i88[2])), >(i128[2], i88[2])), i128[2], i129[2], i88[2])∧(UIncreasing(COND_LOAD1115(&&(&&(>=(i88[2], 0), >(i129[2], i88[2])), >(i128[2], i88[2])), i128[2], i129[2], i88[2])), ≥))

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

(15)    (i128[2] + [-1] + [-1]i88[2] ≥ 0∧i88[2] ≥ 0∧i129[2] + [-1] + [-1]i88[2] ≥ 0 ⇒ (UIncreasing(COND_LOAD1115(&&(&&(>=(i88[2], 0), >(i129[2], i88[2])), >(i128[2], i88[2])), i128[2], i129[2], i88[2])), ≥)∧[(-1)bni_20 + (-1)Bound*bni_20] + [(-1)bni_20]i88[2] + [bni_20]i128[2] ≥ 0∧[(-1)bso_21] ≥ 0)

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

(16)    (i128[2] + [-1] + [-1]i88[2] ≥ 0∧i88[2] ≥ 0∧i129[2] + [-1] + [-1]i88[2] ≥ 0 ⇒ (UIncreasing(COND_LOAD1115(&&(&&(>=(i88[2], 0), >(i129[2], i88[2])), >(i128[2], i88[2])), i128[2], i129[2], i88[2])), ≥)∧[(-1)bni_20 + (-1)Bound*bni_20] + [(-1)bni_20]i88[2] + [bni_20]i128[2] ≥ 0∧[(-1)bso_21] ≥ 0)

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

(17)    (i128[2] + [-1] + [-1]i88[2] ≥ 0∧i88[2] ≥ 0∧i129[2] + [-1] + [-1]i88[2] ≥ 0 ⇒ (UIncreasing(COND_LOAD1115(&&(&&(>=(i88[2], 0), >(i129[2], i88[2])), >(i128[2], i88[2])), i128[2], i129[2], i88[2])), ≥)∧[(-1)bni_20 + (-1)Bound*bni_20] + [(-1)bni_20]i88[2] + [bni_20]i128[2] ≥ 0∧[(-1)bso_21] ≥ 0)

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

(18)    (i128[2] ≥ 0∧i88[2] ≥ 0∧i129[2] + [-1] + [-1]i88[2] ≥ 0 ⇒ (UIncreasing(COND_LOAD1115(&&(&&(>=(i88[2], 0), >(i129[2], i88[2])), >(i128[2], i88[2])), i128[2], i129[2], i88[2])), ≥)∧[(-1)Bound*bni_20] + [bni_20]i128[2] ≥ 0∧[(-1)bso_21] ≥ 0)

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

(19)    (i128[2] ≥ 0∧i88[2] ≥ 0∧i129[2] ≥ 0 ⇒ (UIncreasing(COND_LOAD1115(&&(&&(>=(i88[2], 0), >(i129[2], i88[2])), >(i128[2], i88[2])), i128[2], i129[2], i88[2])), ≥)∧[(-1)Bound*bni_20] + [bni_20]i128[2] ≥ 0∧[(-1)bso_21] ≥ 0)

For Pair COND_LOAD1115(TRUE, i128, i129, i88) → LOAD1129(i128, +(i129, -1), i88) the following chains were created:
• We consider the chain COND_LOAD1115(TRUE, i128[3], i129[3], i88[3]) → LOAD1129(i128[3], +(i129[3], -1), i88[3]) which results in the following constraint:

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

(21)    ((UIncreasing(LOAD1129(i128[3], +(i129[3], -1), i88[3])), ≥)∧[(-1)bso_23] ≥ 0)

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

(22)    ((UIncreasing(LOAD1129(i128[3], +(i129[3], -1), i88[3])), ≥)∧[(-1)bso_23] ≥ 0)

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

(23)    ((UIncreasing(LOAD1129(i128[3], +(i129[3], -1), i88[3])), ≥)∧[(-1)bso_23] ≥ 0)

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

(24)    ((UIncreasing(LOAD1129(i128[3], +(i129[3], -1), i88[3])), ≥)∧0 = 0∧0 = 0∧0 = 0∧[(-1)bso_23] ≥ 0)

For Pair LOAD1129(i128, i129, i88) → COND_LOAD11291(<=(i129, i88), i128, i129, i88) the following chains were created:
• We consider the chain LOAD1129(i128[4], i129[4], i88[4]) → COND_LOAD11291(<=(i129[4], i88[4]), i128[4], i129[4], i88[4]), COND_LOAD11291(TRUE, i128[5], i129[5], i88[5]) → LOAD1115(+(i128[5], -1), i129[5], i88[5]) which results in the following constraint:

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

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

(27)    (i88[4] + [-1]i129[4] ≥ 0 ⇒ (UIncreasing(COND_LOAD11291(<=(i129[4], i88[4]), i128[4], i129[4], i88[4])), ≥)∧[(-1)bni_24 + (-1)Bound*bni_24] + [(-1)bni_24]i88[4] + [bni_24]i128[4] ≥ 0∧[(-1)bso_25] ≥ 0)

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

(28)    (i88[4] + [-1]i129[4] ≥ 0 ⇒ (UIncreasing(COND_LOAD11291(<=(i129[4], i88[4]), i128[4], i129[4], i88[4])), ≥)∧[(-1)bni_24 + (-1)Bound*bni_24] + [(-1)bni_24]i88[4] + [bni_24]i128[4] ≥ 0∧[(-1)bso_25] ≥ 0)

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

(29)    (i88[4] + [-1]i129[4] ≥ 0 ⇒ (UIncreasing(COND_LOAD11291(<=(i129[4], i88[4]), i128[4], i129[4], i88[4])), ≥)∧[(-1)bni_24 + (-1)Bound*bni_24] + [(-1)bni_24]i88[4] + [bni_24]i128[4] ≥ 0∧[(-1)bso_25] ≥ 0)

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

(30)    (i88[4] + [-1]i129[4] ≥ 0 ⇒ (UIncreasing(COND_LOAD11291(<=(i129[4], i88[4]), i128[4], i129[4], i88[4])), ≥)∧[bni_24] = 0∧[(-1)bni_24 + (-1)Bound*bni_24] + [(-1)bni_24]i88[4] ≥ 0∧0 = 0∧[(-1)bso_25] ≥ 0)

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

(31)    (i88[4] ≥ 0 ⇒ (UIncreasing(COND_LOAD11291(<=(i129[4], i88[4]), i128[4], i129[4], i88[4])), ≥)∧[bni_24] = 0∧[(-1)bni_24 + (-1)Bound*bni_24] + [(-1)bni_24]i129[4] + [(-1)bni_24]i88[4] ≥ 0∧0 = 0∧[(-1)bso_25] ≥ 0)

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

(32)    (i88[4] ≥ 0∧i129[4] ≥ 0 ⇒ (UIncreasing(COND_LOAD11291(<=(i129[4], i88[4]), i128[4], i129[4], i88[4])), ≥)∧[bni_24] = 0∧[(-1)bni_24 + (-1)Bound*bni_24] + [bni_24]i129[4] + [(-1)bni_24]i88[4] ≥ 0∧0 = 0∧[(-1)bso_25] ≥ 0)

(33)    (i88[4] ≥ 0∧i129[4] ≥ 0 ⇒ (UIncreasing(COND_LOAD11291(<=(i129[4], i88[4]), i128[4], i129[4], i88[4])), ≥)∧[bni_24] = 0∧[(-1)bni_24 + (-1)Bound*bni_24] + [(-1)bni_24]i129[4] + [(-1)bni_24]i88[4] ≥ 0∧0 = 0∧[(-1)bso_25] ≥ 0)

For Pair COND_LOAD11291(TRUE, i128, i129, i88) → LOAD1115(+(i128, -1), i129, i88) the following chains were created:
• We consider the chain COND_LOAD11291(TRUE, i128[5], i129[5], i88[5]) → LOAD1115(+(i128[5], -1), i129[5], i88[5]) which results in the following constraint:

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

(35)    ((UIncreasing(LOAD1115(+(i128[5], -1), i129[5], i88[5])), ≥)∧[1 + (-1)bso_27] ≥ 0)

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

(36)    ((UIncreasing(LOAD1115(+(i128[5], -1), i129[5], i88[5])), ≥)∧[1 + (-1)bso_27] ≥ 0)

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

(37)    ((UIncreasing(LOAD1115(+(i128[5], -1), i129[5], i88[5])), ≥)∧[1 + (-1)bso_27] ≥ 0)

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

(38)    ((UIncreasing(LOAD1115(+(i128[5], -1), i129[5], i88[5])), ≥)∧0 = 0∧0 = 0∧0 = 0∧[1 + (-1)bso_27] ≥ 0)

For Pair LOAD1115(i128, i129, i88) → COND_LOAD11151(&&(&&(<=(i129, i88), >=(i88, 0)), >(i128, i88)), i128, i129, i88) the following chains were created:
• We consider the chain LOAD1115(i128[6], i129[6], i88[6]) → COND_LOAD11151(&&(&&(<=(i129[6], i88[6]), >=(i88[6], 0)), >(i128[6], i88[6])), i128[6], i129[6], i88[6]), COND_LOAD11151(TRUE, i128[7], i129[7], i88[7]) → LOAD1115(+(i128[7], -1), i129[7], i88[7]) which results in the following constraint:

(39)    (&&(&&(<=(i129[6], i88[6]), >=(i88[6], 0)), >(i128[6], i88[6]))=TRUEi128[6]=i128[7]i88[6]=i88[7]i129[6]=i129[7]LOAD1115(i128[6], i129[6], i88[6])≥NonInfC∧LOAD1115(i128[6], i129[6], i88[6])≥COND_LOAD11151(&&(&&(<=(i129[6], i88[6]), >=(i88[6], 0)), >(i128[6], i88[6])), i128[6], i129[6], i88[6])∧(UIncreasing(COND_LOAD11151(&&(&&(<=(i129[6], i88[6]), >=(i88[6], 0)), >(i128[6], i88[6])), i128[6], i129[6], i88[6])), ≥))

We simplified constraint (39) using rules (IV), (IDP_BOOLEAN) which results in the following new constraint:

(40)    (>(i128[6], i88[6])=TRUE<=(i129[6], i88[6])=TRUE>=(i88[6], 0)=TRUELOAD1115(i128[6], i129[6], i88[6])≥NonInfC∧LOAD1115(i128[6], i129[6], i88[6])≥COND_LOAD11151(&&(&&(<=(i129[6], i88[6]), >=(i88[6], 0)), >(i128[6], i88[6])), i128[6], i129[6], i88[6])∧(UIncreasing(COND_LOAD11151(&&(&&(<=(i129[6], i88[6]), >=(i88[6], 0)), >(i128[6], i88[6])), i128[6], i129[6], i88[6])), ≥))

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

(41)    (i128[6] + [-1] + [-1]i88[6] ≥ 0∧i88[6] + [-1]i129[6] ≥ 0∧i88[6] ≥ 0 ⇒ (UIncreasing(COND_LOAD11151(&&(&&(<=(i129[6], i88[6]), >=(i88[6], 0)), >(i128[6], i88[6])), i128[6], i129[6], i88[6])), ≥)∧[(-1)bni_28 + (-1)Bound*bni_28] + [(-1)bni_28]i88[6] + [bni_28]i128[6] ≥ 0∧[(-1)bso_29] ≥ 0)

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

(42)    (i128[6] + [-1] + [-1]i88[6] ≥ 0∧i88[6] + [-1]i129[6] ≥ 0∧i88[6] ≥ 0 ⇒ (UIncreasing(COND_LOAD11151(&&(&&(<=(i129[6], i88[6]), >=(i88[6], 0)), >(i128[6], i88[6])), i128[6], i129[6], i88[6])), ≥)∧[(-1)bni_28 + (-1)Bound*bni_28] + [(-1)bni_28]i88[6] + [bni_28]i128[6] ≥ 0∧[(-1)bso_29] ≥ 0)

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

(43)    (i128[6] + [-1] + [-1]i88[6] ≥ 0∧i88[6] + [-1]i129[6] ≥ 0∧i88[6] ≥ 0 ⇒ (UIncreasing(COND_LOAD11151(&&(&&(<=(i129[6], i88[6]), >=(i88[6], 0)), >(i128[6], i88[6])), i128[6], i129[6], i88[6])), ≥)∧[(-1)bni_28 + (-1)Bound*bni_28] + [(-1)bni_28]i88[6] + [bni_28]i128[6] ≥ 0∧[(-1)bso_29] ≥ 0)

We simplified constraint (43) using rule (IDP_SMT_SPLIT) which results in the following new constraint:

(44)    (i128[6] ≥ 0∧i88[6] + [-1]i129[6] ≥ 0∧i88[6] ≥ 0 ⇒ (UIncreasing(COND_LOAD11151(&&(&&(<=(i129[6], i88[6]), >=(i88[6], 0)), >(i128[6], i88[6])), i128[6], i129[6], i88[6])), ≥)∧[(-1)Bound*bni_28] + [bni_28]i128[6] ≥ 0∧[(-1)bso_29] ≥ 0)

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

(45)    (i128[6] ≥ 0∧i88[6] + [-1]i129[6] ≥ 0∧i88[6] ≥ 0∧i129[6] ≥ 0 ⇒ (UIncreasing(COND_LOAD11151(&&(&&(<=(i129[6], i88[6]), >=(i88[6], 0)), >(i128[6], i88[6])), i128[6], i129[6], i88[6])), ≥)∧[(-1)Bound*bni_28] + [bni_28]i128[6] ≥ 0∧[(-1)bso_29] ≥ 0)

(46)    (i128[6] ≥ 0∧i88[6] + i129[6] ≥ 0∧i88[6] ≥ 0∧i129[6] ≥ 0 ⇒ (UIncreasing(COND_LOAD11151(&&(&&(<=(i129[6], i88[6]), >=(i88[6], 0)), >(i128[6], i88[6])), i128[6], i129[6], i88[6])), ≥)∧[(-1)Bound*bni_28] + [bni_28]i128[6] ≥ 0∧[(-1)bso_29] ≥ 0)

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

(47)    (i128[6] ≥ 0∧i88[6] ≥ 0∧i129[6] + i88[6] ≥ 0∧i129[6] ≥ 0 ⇒ (UIncreasing(COND_LOAD11151(&&(&&(<=(i129[6], i88[6]), >=(i88[6], 0)), >(i128[6], i88[6])), i128[6], i129[6], i88[6])), ≥)∧[(-1)Bound*bni_28] + [bni_28]i128[6] ≥ 0∧[(-1)bso_29] ≥ 0)

For Pair COND_LOAD11151(TRUE, i128, i129, i88) → LOAD1115(+(i128, -1), i129, i88) the following chains were created:
• We consider the chain COND_LOAD11151(TRUE, i128[7], i129[7], i88[7]) → LOAD1115(+(i128[7], -1), i129[7], i88[7]) which results in the following constraint:

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

(49)    ((UIncreasing(LOAD1115(+(i128[7], -1), i129[7], i88[7])), ≥)∧[1 + (-1)bso_31] ≥ 0)

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

(50)    ((UIncreasing(LOAD1115(+(i128[7], -1), i129[7], i88[7])), ≥)∧[1 + (-1)bso_31] ≥ 0)

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

(51)    ((UIncreasing(LOAD1115(+(i128[7], -1), i129[7], i88[7])), ≥)∧[1 + (-1)bso_31] ≥ 0)

We simplified constraint (51) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint:

(52)    ((UIncreasing(LOAD1115(+(i128[7], -1), i129[7], i88[7])), ≥)∧0 = 0∧0 = 0∧0 = 0∧[1 + (-1)bso_31] ≥ 0)

To summarize, we get the following constraints P for the following pairs.
• LOAD1129(i128, i129, i88) → COND_LOAD1129(&&(>=(i88, 0), >(i129, i88)), i128, i129, i88)
• (i88[0] ≥ 0∧i129[0] ≥ 0 ⇒ (UIncreasing(COND_LOAD1129(&&(>=(i88[0], 0), >(i129[0], i88[0])), i128[0], i129[0], i88[0])), ≥)∧[bni_16] = 0∧[(-1)bni_16 + (-1)Bound*bni_16] + [(-1)bni_16]i88[0] ≥ 0∧0 = 0∧[(-1)bso_17] ≥ 0)

• ((UIncreasing(LOAD1129(i128[1], +(i129[1], -1), i88[1])), ≥)∧0 = 0∧0 = 0∧0 = 0∧[(-1)bso_19] ≥ 0)

• LOAD1115(i128, i129, i88) → COND_LOAD1115(&&(&&(>=(i88, 0), >(i129, i88)), >(i128, i88)), i128, i129, i88)
• (i128[2] ≥ 0∧i88[2] ≥ 0∧i129[2] ≥ 0 ⇒ (UIncreasing(COND_LOAD1115(&&(&&(>=(i88[2], 0), >(i129[2], i88[2])), >(i128[2], i88[2])), i128[2], i129[2], i88[2])), ≥)∧[(-1)Bound*bni_20] + [bni_20]i128[2] ≥ 0∧[(-1)bso_21] ≥ 0)

• ((UIncreasing(LOAD1129(i128[3], +(i129[3], -1), i88[3])), ≥)∧0 = 0∧0 = 0∧0 = 0∧[(-1)bso_23] ≥ 0)

• (i88[4] ≥ 0∧i129[4] ≥ 0 ⇒ (UIncreasing(COND_LOAD11291(<=(i129[4], i88[4]), i128[4], i129[4], i88[4])), ≥)∧[bni_24] = 0∧[(-1)bni_24 + (-1)Bound*bni_24] + [bni_24]i129[4] + [(-1)bni_24]i88[4] ≥ 0∧0 = 0∧[(-1)bso_25] ≥ 0)
• (i88[4] ≥ 0∧i129[4] ≥ 0 ⇒ (UIncreasing(COND_LOAD11291(<=(i129[4], i88[4]), i128[4], i129[4], i88[4])), ≥)∧[bni_24] = 0∧[(-1)bni_24 + (-1)Bound*bni_24] + [(-1)bni_24]i129[4] + [(-1)bni_24]i88[4] ≥ 0∧0 = 0∧[(-1)bso_25] ≥ 0)

• ((UIncreasing(LOAD1115(+(i128[5], -1), i129[5], i88[5])), ≥)∧0 = 0∧0 = 0∧0 = 0∧[1 + (-1)bso_27] ≥ 0)

• LOAD1115(i128, i129, i88) → COND_LOAD11151(&&(&&(<=(i129, i88), >=(i88, 0)), >(i128, i88)), i128, i129, i88)
• (i128[6] ≥ 0∧i88[6] + i129[6] ≥ 0∧i88[6] ≥ 0∧i129[6] ≥ 0 ⇒ (UIncreasing(COND_LOAD11151(&&(&&(<=(i129[6], i88[6]), >=(i88[6], 0)), >(i128[6], i88[6])), i128[6], i129[6], i88[6])), ≥)∧[(-1)Bound*bni_28] + [bni_28]i128[6] ≥ 0∧[(-1)bso_29] ≥ 0)
• (i128[6] ≥ 0∧i88[6] ≥ 0∧i129[6] + i88[6] ≥ 0∧i129[6] ≥ 0 ⇒ (UIncreasing(COND_LOAD11151(&&(&&(<=(i129[6], i88[6]), >=(i88[6], 0)), >(i128[6], i88[6])), i128[6], i129[6], i88[6])), ≥)∧[(-1)Bound*bni_28] + [bni_28]i128[6] ≥ 0∧[(-1)bso_29] ≥ 0)

• ((UIncreasing(LOAD1115(+(i128[7], -1), i129[7], i88[7])), ≥)∧0 = 0∧0 = 0∧0 = 0∧[1 + (-1)bso_31] ≥ 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(LOAD1129(x1, x2, x3)) = [-1] + [-1]x3 + x1
POL(COND_LOAD1129(x1, x2, x3, x4)) = [-1] + [-1]x4 + x2
POL(&&(x1, x2)) = [-1]
POL(>=(x1, x2)) = [-1]
POL(0) = 0
POL(>(x1, x2)) = [-1]
POL(+(x1, x2)) = x1 + x2
POL(-1) = [-1]
POL(LOAD1115(x1, x2, x3)) = [-1] + [-1]x3 + x1
POL(COND_LOAD1115(x1, x2, x3, x4)) = [-1] + [-1]x4 + x2
POL(COND_LOAD11291(x1, x2, x3, x4)) = [-1] + [-1]x4 + x2
POL(<=(x1, x2)) = [-1]
POL(COND_LOAD11151(x1, x2, x3, x4)) = [-1] + [-1]x4 + x2

The following pairs are in P>:

The following pairs are in Pbound:

LOAD1115(i128[2], i129[2], i88[2]) → COND_LOAD1115(&&(&&(>=(i88[2], 0), >(i129[2], i88[2])), >(i128[2], i88[2])), i128[2], i129[2], i88[2])
LOAD1115(i128[6], i129[6], i88[6]) → COND_LOAD11151(&&(&&(<=(i129[6], i88[6]), >=(i88[6], 0)), >(i128[6], i88[6])), i128[6], i129[6], i88[6])

The following pairs are in P:

LOAD1115(i128[2], i129[2], i88[2]) → COND_LOAD1115(&&(&&(>=(i88[2], 0), >(i129[2], i88[2])), >(i128[2], i88[2])), i128[2], i129[2], i88[2])
LOAD1115(i128[6], i129[6], i88[6]) → COND_LOAD11151(&&(&&(<=(i129[6], i88[6]), >=(i88[6], 0)), >(i128[6], i88[6])), i128[6], i129[6], i88[6])

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): LOAD1129(i128[0], i129[0], i88[0]) → COND_LOAD1129(i88[0] >= 0 && i129[0] > i88[0], i128[0], i129[0], i88[0])
(2): LOAD1115(i128[2], i129[2], i88[2]) → COND_LOAD1115(i88[2] >= 0 && i129[2] > i88[2] && i128[2] > i88[2], i128[2], i129[2], i88[2])
(6): LOAD1115(i128[6], i129[6], i88[6]) → COND_LOAD11151(i129[6] <= i88[6] && i88[6] >= 0 && i128[6] > i88[6], i128[6], i129[6], i88[6])

(1) -> (0), if ((i128[1]* i128[0])∧(i129[1] + -1* i129[0])∧(i88[1]* i88[0]))

(3) -> (0), if ((i88[3]* i88[0])∧(i129[3] + -1* i129[0])∧(i128[3]* i128[0]))

(0) -> (1), if ((i88[0]* i88[1])∧(i88[0] >= 0 && i129[0] > i88[0]* TRUE)∧(i129[0]* i129[1])∧(i128[0]* i128[1]))

(2) -> (3), if ((i128[2]* i128[3])∧(i88[2]* i88[3])∧(i129[2]* i129[3])∧(i88[2] >= 0 && i129[2] > i88[2] && i128[2] > i88[2]* TRUE))

(1) -> (4), if ((i128[1]* i128[4])∧(i129[1] + -1* i129[4])∧(i88[1]* i88[4]))

(3) -> (4), if ((i129[3] + -1* i129[4])∧(i128[3]* i128[4])∧(i88[3]* i88[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 4 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:
(0): LOAD1129(i128[0], i129[0], i88[0]) → COND_LOAD1129(i88[0] >= 0 && i129[0] > i88[0], i128[0], i129[0], i88[0])

(1) -> (0), if ((i128[1]* i128[0])∧(i129[1] + -1* i129[0])∧(i88[1]* i88[0]))

(0) -> (1), if ((i88[0]* i88[1])∧(i88[0] >= 0 && i129[0] > i88[0]* TRUE)∧(i129[0]* i129[1])∧(i128[0]* i128[1]))

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_LOAD1129(TRUE, i128[1], i129[1], i88[1]) → LOAD1129(i128[1], +(i129[1], -1), i88[1]) the following chains were created:
• We consider the chain COND_LOAD1129(TRUE, i128[1], i129[1], i88[1]) → LOAD1129(i128[1], +(i129[1], -1), i88[1]) which results in the following constraint:

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

(2)    ((UIncreasing(LOAD1129(i128[1], +(i129[1], -1), i88[1])), ≥)∧[1 + (-1)bso_9] ≥ 0)

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

(3)    ((UIncreasing(LOAD1129(i128[1], +(i129[1], -1), i88[1])), ≥)∧[1 + (-1)bso_9] ≥ 0)

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

(4)    ((UIncreasing(LOAD1129(i128[1], +(i129[1], -1), i88[1])), ≥)∧[1 + (-1)bso_9] ≥ 0)

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

(5)    ((UIncreasing(LOAD1129(i128[1], +(i129[1], -1), i88[1])), ≥)∧0 = 0∧0 = 0∧[1 + (-1)bso_9] ≥ 0)

For Pair LOAD1129(i128[0], i129[0], i88[0]) → COND_LOAD1129(&&(>=(i88[0], 0), >(i129[0], i88[0])), i128[0], i129[0], i88[0]) the following chains were created:
• We consider the chain LOAD1129(i128[0], i129[0], i88[0]) → COND_LOAD1129(&&(>=(i88[0], 0), >(i129[0], i88[0])), i128[0], i129[0], i88[0]), COND_LOAD1129(TRUE, i128[1], i129[1], i88[1]) → LOAD1129(i128[1], +(i129[1], -1), i88[1]) which results in the following constraint:

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

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

(8)    (i88[0] ≥ 0∧i129[0] + [-1] + [-1]i88[0] ≥ 0 ⇒ (UIncreasing(COND_LOAD1129(&&(>=(i88[0], 0), >(i129[0], i88[0])), i128[0], i129[0], i88[0])), ≥)∧[bni_10 + (-1)Bound*bni_10] + [bni_10]i129[0] + [bni_10]i88[0] ≥ 0∧[(-1)bso_11] ≥ 0)

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

(9)    (i88[0] ≥ 0∧i129[0] + [-1] + [-1]i88[0] ≥ 0 ⇒ (UIncreasing(COND_LOAD1129(&&(>=(i88[0], 0), >(i129[0], i88[0])), i128[0], i129[0], i88[0])), ≥)∧[bni_10 + (-1)Bound*bni_10] + [bni_10]i129[0] + [bni_10]i88[0] ≥ 0∧[(-1)bso_11] ≥ 0)

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

(10)    (i88[0] ≥ 0∧i129[0] + [-1] + [-1]i88[0] ≥ 0 ⇒ (UIncreasing(COND_LOAD1129(&&(>=(i88[0], 0), >(i129[0], i88[0])), i128[0], i129[0], i88[0])), ≥)∧[bni_10 + (-1)Bound*bni_10] + [bni_10]i129[0] + [bni_10]i88[0] ≥ 0∧[(-1)bso_11] ≥ 0)

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

(11)    (i88[0] ≥ 0∧i129[0] ≥ 0 ⇒ (UIncreasing(COND_LOAD1129(&&(>=(i88[0], 0), >(i129[0], i88[0])), i128[0], i129[0], i88[0])), ≥)∧[(2)bni_10 + (-1)Bound*bni_10] + [(2)bni_10]i88[0] + [bni_10]i129[0] ≥ 0∧[(-1)bso_11] ≥ 0)

To summarize, we get the following constraints P for the following pairs.
• ((UIncreasing(LOAD1129(i128[1], +(i129[1], -1), i88[1])), ≥)∧0 = 0∧0 = 0∧[1 + (-1)bso_9] ≥ 0)

• LOAD1129(i128[0], i129[0], i88[0]) → COND_LOAD1129(&&(>=(i88[0], 0), >(i129[0], i88[0])), i128[0], i129[0], i88[0])
• (i88[0] ≥ 0∧i129[0] ≥ 0 ⇒ (UIncreasing(COND_LOAD1129(&&(>=(i88[0], 0), >(i129[0], i88[0])), i128[0], i129[0], i88[0])), ≥)∧[(2)bni_10 + (-1)Bound*bni_10] + [(2)bni_10]i88[0] + [bni_10]i129[0] ≥ 0∧[(-1)bso_11] ≥ 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_LOAD1129(x1, x2, x3, x4)) = [1] + x4 + x3
POL(LOAD1129(x1, x2, x3)) = [1] + x2 + x3
POL(+(x1, x2)) = x1 + x2
POL(-1) = [-1]
POL(&&(x1, x2)) = [1]
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:

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:
(0): LOAD1129(i128[0], i129[0], i88[0]) → COND_LOAD1129(i88[0] >= 0 && i129[0] > i88[0], i128[0], i129[0], i88[0])

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:

Boolean, Integer

R is empty.

The integer pair graph contains the following rules and edges:
(0): LOAD1129(i128[0], i129[0], i88[0]) → COND_LOAD1129(i88[0] >= 0 && i129[0] > i88[0], i128[0], i129[0], i88[0])

(1) -> (0), if ((i128[1]* i128[0])∧(i129[1] + -1* i129[0])∧(i88[1]* i88[0]))

(3) -> (0), if ((i88[3]* i88[0])∧(i129[3] + -1* i129[0])∧(i128[3]* i128[0]))

(0) -> (1), if ((i88[0]* i88[1])∧(i88[0] >= 0 && i129[0] > i88[0]* TRUE)∧(i129[0]* i129[1])∧(i128[0]* i128[1]))

(1) -> (4), if ((i128[1]* i128[4])∧(i129[1] + -1* i129[4])∧(i88[1]* i88[4]))

(3) -> (4), if ((i129[3] + -1* i129[4])∧(i128[3]* i128[4])∧(i88[3]* i88[4]))

(4) -> (5), if ((i88[4]* i88[5])∧(i129[4]* i129[5])∧(i129[4] <= i88[4]* TRUE)∧(i128[4]* i128[5]))

The set Q consists of the following terms:

(23) IDependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 4 less nodes.

(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, Boolean

R is empty.

The integer pair graph contains the following rules and edges:
(0): LOAD1129(i128[0], i129[0], i88[0]) → COND_LOAD1129(i88[0] >= 0 && i129[0] > i88[0], i128[0], i129[0], i88[0])

(1) -> (0), if ((i128[1]* i128[0])∧(i129[1] + -1* i129[0])∧(i88[1]* i88[0]))

(0) -> (1), if ((i88[0]* i88[1])∧(i88[0] >= 0 && i129[0] > i88[0]* TRUE)∧(i129[0]* i129[1])∧(i128[0]* i128[1]))

The set Q consists of the following terms:

(25) 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_LOAD1129(TRUE, i128[1], i129[1], i88[1]) → LOAD1129(i128[1], +(i129[1], -1), i88[1]) the following chains were created:
• We consider the chain COND_LOAD1129(TRUE, i128[1], i129[1], i88[1]) → LOAD1129(i128[1], +(i129[1], -1), i88[1]) which results in the following constraint:

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

(2)    ((UIncreasing(LOAD1129(i128[1], +(i129[1], -1), i88[1])), ≥)∧[1 + (-1)bso_11] ≥ 0)

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

(3)    ((UIncreasing(LOAD1129(i128[1], +(i129[1], -1), i88[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(LOAD1129(i128[1], +(i129[1], -1), i88[1])), ≥)∧[1 + (-1)bso_11] ≥ 0)

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

(5)    ((UIncreasing(LOAD1129(i128[1], +(i129[1], -1), i88[1])), ≥)∧0 = 0∧0 = 0∧[1 + (-1)bso_11] ≥ 0)

For Pair LOAD1129(i128[0], i129[0], i88[0]) → COND_LOAD1129(&&(>=(i88[0], 0), >(i129[0], i88[0])), i128[0], i129[0], i88[0]) the following chains were created:
• We consider the chain LOAD1129(i128[0], i129[0], i88[0]) → COND_LOAD1129(&&(>=(i88[0], 0), >(i129[0], i88[0])), i128[0], i129[0], i88[0]), COND_LOAD1129(TRUE, i128[1], i129[1], i88[1]) → LOAD1129(i128[1], +(i129[1], -1), i88[1]) which results in the following constraint:

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

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

(8)    (i88[0] ≥ 0∧i129[0] + [-1] + [-1]i88[0] ≥ 0 ⇒ (UIncreasing(COND_LOAD1129(&&(>=(i88[0], 0), >(i129[0], i88[0])), i128[0], i129[0], i88[0])), ≥)∧[(-1)bni_12 + (-1)Bound*bni_12] + [bni_12]i129[0] + [bni_12]i88[0] ≥ 0∧[(-1)bso_13] ≥ 0)

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

(9)    (i88[0] ≥ 0∧i129[0] + [-1] + [-1]i88[0] ≥ 0 ⇒ (UIncreasing(COND_LOAD1129(&&(>=(i88[0], 0), >(i129[0], i88[0])), i128[0], i129[0], i88[0])), ≥)∧[(-1)bni_12 + (-1)Bound*bni_12] + [bni_12]i129[0] + [bni_12]i88[0] ≥ 0∧[(-1)bso_13] ≥ 0)

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

(10)    (i88[0] ≥ 0∧i129[0] + [-1] + [-1]i88[0] ≥ 0 ⇒ (UIncreasing(COND_LOAD1129(&&(>=(i88[0], 0), >(i129[0], i88[0])), i128[0], i129[0], i88[0])), ≥)∧[(-1)bni_12 + (-1)Bound*bni_12] + [bni_12]i129[0] + [bni_12]i88[0] ≥ 0∧[(-1)bso_13] ≥ 0)

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

(11)    (i88[0] ≥ 0∧i129[0] ≥ 0 ⇒ (UIncreasing(COND_LOAD1129(&&(>=(i88[0], 0), >(i129[0], i88[0])), i128[0], i129[0], i88[0])), ≥)∧[(-1)Bound*bni_12] + [(2)bni_12]i88[0] + [bni_12]i129[0] ≥ 0∧[(-1)bso_13] ≥ 0)

To summarize, we get the following constraints P for the following pairs.
• ((UIncreasing(LOAD1129(i128[1], +(i129[1], -1), i88[1])), ≥)∧0 = 0∧0 = 0∧[1 + (-1)bso_11] ≥ 0)

• LOAD1129(i128[0], i129[0], i88[0]) → COND_LOAD1129(&&(>=(i88[0], 0), >(i129[0], i88[0])), i128[0], i129[0], i88[0])
• (i88[0] ≥ 0∧i129[0] ≥ 0 ⇒ (UIncreasing(COND_LOAD1129(&&(>=(i88[0], 0), >(i129[0], i88[0])), i128[0], i129[0], i88[0])), ≥)∧[(-1)Bound*bni_12] + [(2)bni_12]i88[0] + [bni_12]i129[0] ≥ 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_LOAD1129(x1, x2, x3, x4)) = [-1] + x4 + x3
POL(LOAD1129(x1, x2, x3)) = [-1] + x2 + x3
POL(+(x1, x2)) = x1 + x2
POL(-1) = [-1]
POL(&&(x1, x2)) = [-1]
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:

There are no usable rules.

(27) 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): LOAD1129(i128[0], i129[0], i88[0]) → COND_LOAD1129(i88[0] >= 0 && i129[0] > i88[0], i128[0], i129[0], i88[0])

The set Q consists of the following terms:

(28) IDependencyGraphProof (EQUIVALENT transformation)

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

(30) 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:

(31) IDependencyGraphProof (EQUIVALENT transformation)

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