### (0) Obligation:

JBC Problem based on JBC Program:

Manifest-Version: 1.0 Created-By: 1.6.0_16 (Sun Microsystems Inc.) Main-Class: PastaC1

### (1) JBC2FIG (SOUND transformation)

Constructed FIGraph.

### (2) Obligation:

FIGraph based on JBC Program:
Graph of 114 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:
Load439(1, i36, i37) → Cond_Load439(i37 > 0 && i36 > i37, 1, i36, i37)
Load439(1, i36, i37) → Cond_Load4391(i36 >= 0 && i37 > 0 && i36 <= i37, 1, i36, i37)
The set Q consists of the following terms:

### (5) GroundTermsRemoverProof (EQUIVALENT transformation)

Some arguments are removed because they always contain the same ground term.
We removed the following ground terms:
• 1

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:
Load439(i36, i37) → Cond_Load439(i37 > 0 && i36 > i37, i36, i37)
Load439(i36, i37) → Cond_Load4391(i36 >= 0 && i37 > 0 && i36 <= i37, i36, i37)
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, Boolean

The ITRS R consists of the following rules:
Load439(i36, i37) → Cond_Load439(i37 > 0 && i36 > i37, i36, i37)
Load439(i36, i37) → Cond_Load4391(i36 >= 0 && i37 > 0 && i36 <= i37, i36, i37)

The integer pair graph contains the following rules and edges:
(2): LOAD439(i36[2], i37[2]) → COND_LOAD439(i37[2] > 0 && i36[2] > i37[2], i36[2], i37[2])
(4): LOAD439(i36[4], i37[4]) → COND_LOAD4391(i36[4] >= 0 && i37[4] > 0 && i36[4] <= i37[4], i36[4], i37[4])

(0) -> (1), if ((i26[0]* i26[1])∧(i26[0] >= 0* TRUE))

(1) -> (2), if ((1* i37[2])∧(i26[1]* i36[2]))

(1) -> (4), if ((1* i37[4])∧(i26[1]* i36[4]))

(2) -> (3), if ((i37[2]* i37[3])∧(i36[2]* i36[3])∧(i37[2] > 0 && i36[2] > i37[2]* TRUE))

(3) -> (2), if ((2 * i37[3]* i37[2])∧(i36[3]* i36[2]))

(3) -> (4), if ((2 * i37[3]* i37[4])∧(i36[3]* i36[4]))

(4) -> (5), if ((i36[4] >= 0 && i37[4] > 0 && i36[4] <= i37[4]* TRUE)∧(i36[4]* i36[5])∧(i37[4]* i37[5]))

(5) -> (0), if ((i36[5] + -1* i26[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, Boolean

R is empty.

The integer pair graph contains the following rules and edges:
(2): LOAD439(i36[2], i37[2]) → COND_LOAD439(i37[2] > 0 && i36[2] > i37[2], i36[2], i37[2])
(4): LOAD439(i36[4], i37[4]) → COND_LOAD4391(i36[4] >= 0 && i37[4] > 0 && i36[4] <= i37[4], i36[4], i37[4])

(0) -> (1), if ((i26[0]* i26[1])∧(i26[0] >= 0* TRUE))

(1) -> (2), if ((1* i37[2])∧(i26[1]* i36[2]))

(1) -> (4), if ((1* i37[4])∧(i26[1]* i36[4]))

(2) -> (3), if ((i37[2]* i37[3])∧(i36[2]* i36[3])∧(i37[2] > 0 && i36[2] > i37[2]* TRUE))

(3) -> (2), if ((2 * i37[3]* i37[2])∧(i36[3]* i36[2]))

(3) -> (4), if ((2 * i37[3]* i37[4])∧(i36[3]* i36[4]))

(4) -> (5), if ((i36[4] >= 0 && i37[4] > 0 && i36[4] <= i37[4]* TRUE)∧(i36[4]* i36[5])∧(i37[4]* i37[5]))

(5) -> (0), if ((i36[5] + -1* i26[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 LOAD299(i26) → COND_LOAD299(>=(i26, 0), i26) 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)    (i26[0] ≥ 0 ⇒ (UIncreasing(COND_LOAD299(>=(i26[0], 0), i26[0])), ≥)∧[(-1)Bound*bni_23] + [bni_23]i26[0] ≥ 0∧[(-1)bso_24] ≥ 0)

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

(4)    (i26[0] ≥ 0 ⇒ (UIncreasing(COND_LOAD299(>=(i26[0], 0), i26[0])), ≥)∧[(-1)Bound*bni_23] + [bni_23]i26[0] ≥ 0∧[(-1)bso_24] ≥ 0)

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

(5)    (i26[0] ≥ 0 ⇒ (UIncreasing(COND_LOAD299(>=(i26[0], 0), i26[0])), ≥)∧[(-1)Bound*bni_23] + [bni_23]i26[0] ≥ 0∧[(-1)bso_24] ≥ 0)

For Pair COND_LOAD299(TRUE, i26) → LOAD439(i26, 1) the following chains were created:

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

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

(8)    (i26[0] ≥ 0 ⇒ (UIncreasing(LOAD439(i26[1], 1)), ≥)∧[(-1)Bound*bni_25] + [bni_25]i26[0] ≥ 0∧[1 + (-1)bso_26] ≥ 0)

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

(9)    (i26[0] ≥ 0 ⇒ (UIncreasing(LOAD439(i26[1], 1)), ≥)∧[(-1)Bound*bni_25] + [bni_25]i26[0] ≥ 0∧[1 + (-1)bso_26] ≥ 0)

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

(10)    (i26[0] ≥ 0 ⇒ (UIncreasing(LOAD439(i26[1], 1)), ≥)∧[(-1)Bound*bni_25] + [bni_25]i26[0] ≥ 0∧[1 + (-1)bso_26] ≥ 0)

We simplified constraint (11) using rules (III), (IV) which results in the following new constraint:

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

(13)    (i26[0] ≥ 0 ⇒ (UIncreasing(LOAD439(i26[1], 1)), ≥)∧[(-1)Bound*bni_25] + [bni_25]i26[0] ≥ 0∧[1 + (-1)bso_26] ≥ 0)

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

(14)    (i26[0] ≥ 0 ⇒ (UIncreasing(LOAD439(i26[1], 1)), ≥)∧[(-1)Bound*bni_25] + [bni_25]i26[0] ≥ 0∧[1 + (-1)bso_26] ≥ 0)

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

(15)    (i26[0] ≥ 0 ⇒ (UIncreasing(LOAD439(i26[1], 1)), ≥)∧[(-1)Bound*bni_25] + [bni_25]i26[0] ≥ 0∧[1 + (-1)bso_26] ≥ 0)

For Pair LOAD439(i36, i37) → COND_LOAD439(&&(>(i37, 0), >(i36, i37)), i36, i37) the following chains were created:
• We consider the chain LOAD439(i36[2], i37[2]) → COND_LOAD439(&&(>(i37[2], 0), >(i36[2], i37[2])), i36[2], i37[2]), COND_LOAD439(TRUE, i36[3], i37[3]) → LOAD439(i36[3], *(2, i37[3])) which results in the following constraint:

We simplified constraint (16) using rules (IV), (IDP_BOOLEAN) which results in the following new constraint:

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

(18)    (i37[2] + [-1] ≥ 0∧i36[2] + [-1] + [-1]i37[2] ≥ 0 ⇒ (UIncreasing(COND_LOAD439(&&(>(i37[2], 0), >(i36[2], i37[2])), i36[2], i37[2])), ≥)∧[(-1)bni_27 + (-1)Bound*bni_27] + [bni_27]i36[2] ≥ 0∧[(-1)bso_28] ≥ 0)

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

(19)    (i37[2] + [-1] ≥ 0∧i36[2] + [-1] + [-1]i37[2] ≥ 0 ⇒ (UIncreasing(COND_LOAD439(&&(>(i37[2], 0), >(i36[2], i37[2])), i36[2], i37[2])), ≥)∧[(-1)bni_27 + (-1)Bound*bni_27] + [bni_27]i36[2] ≥ 0∧[(-1)bso_28] ≥ 0)

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

(20)    (i37[2] + [-1] ≥ 0∧i36[2] + [-1] + [-1]i37[2] ≥ 0 ⇒ (UIncreasing(COND_LOAD439(&&(>(i37[2], 0), >(i36[2], i37[2])), i36[2], i37[2])), ≥)∧[(-1)bni_27 + (-1)Bound*bni_27] + [bni_27]i36[2] ≥ 0∧[(-1)bso_28] ≥ 0)

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

(21)    (i37[2] ≥ 0∧i36[2] + [-2] + [-1]i37[2] ≥ 0 ⇒ (UIncreasing(COND_LOAD439(&&(>(i37[2], 0), >(i36[2], i37[2])), i36[2], i37[2])), ≥)∧[(-1)bni_27 + (-1)Bound*bni_27] + [bni_27]i36[2] ≥ 0∧[(-1)bso_28] ≥ 0)

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

(22)    (i37[2] ≥ 0∧i36[2] ≥ 0 ⇒ (UIncreasing(COND_LOAD439(&&(>(i37[2], 0), >(i36[2], i37[2])), i36[2], i37[2])), ≥)∧[bni_27 + (-1)Bound*bni_27] + [bni_27]i37[2] + [bni_27]i36[2] ≥ 0∧[(-1)bso_28] ≥ 0)

For Pair COND_LOAD439(TRUE, i36, i37) → LOAD439(i36, *(2, i37)) the following chains were created:
• We consider the chain LOAD439(i36[2], i37[2]) → COND_LOAD439(&&(>(i37[2], 0), >(i36[2], i37[2])), i36[2], i37[2]), COND_LOAD439(TRUE, i36[3], i37[3]) → LOAD439(i36[3], *(2, i37[3])), LOAD439(i36[2], i37[2]) → COND_LOAD439(&&(>(i37[2], 0), >(i36[2], i37[2])), i36[2], i37[2]) which results in the following constraint:

We simplified constraint (23) using rules (III), (IV), (IDP_BOOLEAN) which results in the following new constraint:

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

(25)    (i37[2] + [-1] ≥ 0∧i36[2] + [-1] + [-1]i37[2] ≥ 0 ⇒ (UIncreasing(LOAD439(i36[3], *(2, i37[3]))), ≥)∧[(-1)bni_29 + (-1)Bound*bni_29] + [bni_29]i36[2] ≥ 0∧[(-1)bso_30] ≥ 0)

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

(26)    (i37[2] + [-1] ≥ 0∧i36[2] + [-1] + [-1]i37[2] ≥ 0 ⇒ (UIncreasing(LOAD439(i36[3], *(2, i37[3]))), ≥)∧[(-1)bni_29 + (-1)Bound*bni_29] + [bni_29]i36[2] ≥ 0∧[(-1)bso_30] ≥ 0)

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

(27)    (i37[2] + [-1] ≥ 0∧i36[2] + [-1] + [-1]i37[2] ≥ 0 ⇒ (UIncreasing(LOAD439(i36[3], *(2, i37[3]))), ≥)∧[(-1)bni_29 + (-1)Bound*bni_29] + [bni_29]i36[2] ≥ 0∧[(-1)bso_30] ≥ 0)

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

(28)    (i37[2] ≥ 0∧i36[2] + [-2] + [-1]i37[2] ≥ 0 ⇒ (UIncreasing(LOAD439(i36[3], *(2, i37[3]))), ≥)∧[(-1)bni_29 + (-1)Bound*bni_29] + [bni_29]i36[2] ≥ 0∧[(-1)bso_30] ≥ 0)

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

(29)    (i37[2] ≥ 0∧i36[2] ≥ 0 ⇒ (UIncreasing(LOAD439(i36[3], *(2, i37[3]))), ≥)∧[bni_29 + (-1)Bound*bni_29] + [bni_29]i37[2] + [bni_29]i36[2] ≥ 0∧[(-1)bso_30] ≥ 0)

• We consider the chain LOAD439(i36[2], i37[2]) → COND_LOAD439(&&(>(i37[2], 0), >(i36[2], i37[2])), i36[2], i37[2]), COND_LOAD439(TRUE, i36[3], i37[3]) → LOAD439(i36[3], *(2, i37[3])), LOAD439(i36[4], i37[4]) → COND_LOAD4391(&&(&&(>=(i36[4], 0), >(i37[4], 0)), <=(i36[4], i37[4])), i36[4], i37[4]) which results in the following constraint:

We simplified constraint (30) using rules (III), (IV), (IDP_BOOLEAN) which results in the following new constraint:

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

(32)    (i37[2] + [-1] ≥ 0∧i36[2] + [-1] + [-1]i37[2] ≥ 0 ⇒ (UIncreasing(LOAD439(i36[3], *(2, i37[3]))), ≥)∧[(-1)bni_29 + (-1)Bound*bni_29] + [bni_29]i36[2] ≥ 0∧[(-1)bso_30] ≥ 0)

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

(33)    (i37[2] + [-1] ≥ 0∧i36[2] + [-1] + [-1]i37[2] ≥ 0 ⇒ (UIncreasing(LOAD439(i36[3], *(2, i37[3]))), ≥)∧[(-1)bni_29 + (-1)Bound*bni_29] + [bni_29]i36[2] ≥ 0∧[(-1)bso_30] ≥ 0)

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

(34)    (i37[2] + [-1] ≥ 0∧i36[2] + [-1] + [-1]i37[2] ≥ 0 ⇒ (UIncreasing(LOAD439(i36[3], *(2, i37[3]))), ≥)∧[(-1)bni_29 + (-1)Bound*bni_29] + [bni_29]i36[2] ≥ 0∧[(-1)bso_30] ≥ 0)

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

(35)    (i37[2] ≥ 0∧i36[2] + [-2] + [-1]i37[2] ≥ 0 ⇒ (UIncreasing(LOAD439(i36[3], *(2, i37[3]))), ≥)∧[(-1)bni_29 + (-1)Bound*bni_29] + [bni_29]i36[2] ≥ 0∧[(-1)bso_30] ≥ 0)

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

(36)    (i37[2] ≥ 0∧i36[2] ≥ 0 ⇒ (UIncreasing(LOAD439(i36[3], *(2, i37[3]))), ≥)∧[bni_29 + (-1)Bound*bni_29] + [bni_29]i37[2] + [bni_29]i36[2] ≥ 0∧[(-1)bso_30] ≥ 0)

For Pair LOAD439(i36, i37) → COND_LOAD4391(&&(&&(>=(i36, 0), >(i37, 0)), <=(i36, i37)), i36, i37) the following chains were created:
• We consider the chain LOAD439(i36[4], i37[4]) → COND_LOAD4391(&&(&&(>=(i36[4], 0), >(i37[4], 0)), <=(i36[4], i37[4])), i36[4], i37[4]), COND_LOAD4391(TRUE, i36[5], i37[5]) → LOAD299(+(i36[5], -1)) which results in the following constraint:

(37)    (&&(&&(>=(i36[4], 0), >(i37[4], 0)), <=(i36[4], i37[4]))=TRUEi36[4]=i36[5]i37[4]=i37[5]LOAD439(i36[4], i37[4])≥NonInfC∧LOAD439(i36[4], i37[4])≥COND_LOAD4391(&&(&&(>=(i36[4], 0), >(i37[4], 0)), <=(i36[4], i37[4])), i36[4], i37[4])∧(UIncreasing(COND_LOAD4391(&&(&&(>=(i36[4], 0), >(i37[4], 0)), <=(i36[4], i37[4])), i36[4], i37[4])), ≥))

We simplified constraint (37) using rules (IV), (IDP_BOOLEAN) which results in the following new constraint:

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

(39)    (i37[4] + [-1]i36[4] ≥ 0∧i36[4] ≥ 0∧i37[4] + [-1] ≥ 0 ⇒ (UIncreasing(COND_LOAD4391(&&(&&(>=(i36[4], 0), >(i37[4], 0)), <=(i36[4], i37[4])), i36[4], i37[4])), ≥)∧[(-1)bni_31 + (-1)Bound*bni_31] + [bni_31]i36[4] ≥ 0∧[(-1)bso_32] ≥ 0)

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

(40)    (i37[4] + [-1]i36[4] ≥ 0∧i36[4] ≥ 0∧i37[4] + [-1] ≥ 0 ⇒ (UIncreasing(COND_LOAD4391(&&(&&(>=(i36[4], 0), >(i37[4], 0)), <=(i36[4], i37[4])), i36[4], i37[4])), ≥)∧[(-1)bni_31 + (-1)Bound*bni_31] + [bni_31]i36[4] ≥ 0∧[(-1)bso_32] ≥ 0)

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

(41)    (i37[4] + [-1]i36[4] ≥ 0∧i36[4] ≥ 0∧i37[4] + [-1] ≥ 0 ⇒ (UIncreasing(COND_LOAD4391(&&(&&(>=(i36[4], 0), >(i37[4], 0)), <=(i36[4], i37[4])), i36[4], i37[4])), ≥)∧[(-1)bni_31 + (-1)Bound*bni_31] + [bni_31]i36[4] ≥ 0∧[(-1)bso_32] ≥ 0)

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

(42)    (i37[4] ≥ 0∧i36[4] ≥ 0∧i36[4] + [-1] + i37[4] ≥ 0 ⇒ (UIncreasing(COND_LOAD4391(&&(&&(>=(i36[4], 0), >(i37[4], 0)), <=(i36[4], i37[4])), i36[4], i37[4])), ≥)∧[(-1)bni_31 + (-1)Bound*bni_31] + [bni_31]i36[4] ≥ 0∧[(-1)bso_32] ≥ 0)

For Pair COND_LOAD4391(TRUE, i36, i37) → LOAD299(+(i36, -1)) the following chains were created:

We simplified constraint (43) using rules (III), (IV), (IDP_BOOLEAN) which results in the following new constraint:

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

(45)    (i37[4] + [-1]i36[4] ≥ 0∧i36[4] ≥ 0∧i37[4] + [-1] ≥ 0 ⇒ (UIncreasing(LOAD299(+(i36[5], -1))), ≥)∧[(-1)bni_33 + (-1)Bound*bni_33] + [bni_33]i36[4] ≥ 0∧[(-1)bso_34] ≥ 0)

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

(46)    (i37[4] + [-1]i36[4] ≥ 0∧i36[4] ≥ 0∧i37[4] + [-1] ≥ 0 ⇒ (UIncreasing(LOAD299(+(i36[5], -1))), ≥)∧[(-1)bni_33 + (-1)Bound*bni_33] + [bni_33]i36[4] ≥ 0∧[(-1)bso_34] ≥ 0)

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

(47)    (i37[4] + [-1]i36[4] ≥ 0∧i36[4] ≥ 0∧i37[4] + [-1] ≥ 0 ⇒ (UIncreasing(LOAD299(+(i36[5], -1))), ≥)∧[(-1)bni_33 + (-1)Bound*bni_33] + [bni_33]i36[4] ≥ 0∧[(-1)bso_34] ≥ 0)

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

(48)    (i37[4] ≥ 0∧i36[4] ≥ 0∧i36[4] + [-1] + i37[4] ≥ 0 ⇒ (UIncreasing(LOAD299(+(i36[5], -1))), ≥)∧[(-1)bni_33 + (-1)Bound*bni_33] + [bni_33]i36[4] ≥ 0∧[(-1)bso_34] ≥ 0)

To summarize, we get the following constraints P for the following pairs.
• (i26[0] ≥ 0 ⇒ (UIncreasing(COND_LOAD299(>=(i26[0], 0), i26[0])), ≥)∧[(-1)Bound*bni_23] + [bni_23]i26[0] ≥ 0∧[(-1)bso_24] ≥ 0)

• (i26[0] ≥ 0 ⇒ (UIncreasing(LOAD439(i26[1], 1)), ≥)∧[(-1)Bound*bni_25] + [bni_25]i26[0] ≥ 0∧[1 + (-1)bso_26] ≥ 0)
• (i26[0] ≥ 0 ⇒ (UIncreasing(LOAD439(i26[1], 1)), ≥)∧[(-1)Bound*bni_25] + [bni_25]i26[0] ≥ 0∧[1 + (-1)bso_26] ≥ 0)

• (i37[2] ≥ 0∧i36[2] ≥ 0 ⇒ (UIncreasing(COND_LOAD439(&&(>(i37[2], 0), >(i36[2], i37[2])), i36[2], i37[2])), ≥)∧[bni_27 + (-1)Bound*bni_27] + [bni_27]i37[2] + [bni_27]i36[2] ≥ 0∧[(-1)bso_28] ≥ 0)

• (i37[2] ≥ 0∧i36[2] ≥ 0 ⇒ (UIncreasing(LOAD439(i36[3], *(2, i37[3]))), ≥)∧[bni_29 + (-1)Bound*bni_29] + [bni_29]i37[2] + [bni_29]i36[2] ≥ 0∧[(-1)bso_30] ≥ 0)
• (i37[2] ≥ 0∧i36[2] ≥ 0 ⇒ (UIncreasing(LOAD439(i36[3], *(2, i37[3]))), ≥)∧[bni_29 + (-1)Bound*bni_29] + [bni_29]i37[2] + [bni_29]i36[2] ≥ 0∧[(-1)bso_30] ≥ 0)

• LOAD439(i36, i37) → COND_LOAD4391(&&(&&(>=(i36, 0), >(i37, 0)), <=(i36, i37)), i36, i37)
• (i37[4] ≥ 0∧i36[4] ≥ 0∧i36[4] + [-1] + i37[4] ≥ 0 ⇒ (UIncreasing(COND_LOAD4391(&&(&&(>=(i36[4], 0), >(i37[4], 0)), <=(i36[4], i37[4])), i36[4], i37[4])), ≥)∧[(-1)bni_31 + (-1)Bound*bni_31] + [bni_31]i36[4] ≥ 0∧[(-1)bso_32] ≥ 0)

• (i37[4] ≥ 0∧i36[4] ≥ 0∧i36[4] + [-1] + i37[4] ≥ 0 ⇒ (UIncreasing(LOAD299(+(i36[5], -1))), ≥)∧[(-1)bni_33 + (-1)Bound*bni_33] + [bni_33]i36[4] ≥ 0∧[(-1)bso_34] ≥ 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) = [1]
POL(>=(x1, x2)) = [-1]
POL(0) = 0
POL(LOAD439(x1, x2)) = [-1] + x1
POL(1) = [1]
POL(COND_LOAD439(x1, x2, x3)) = [-1] + x2
POL(&&(x1, x2)) = [-1]
POL(>(x1, x2)) = [-1]
POL(*(x1, x2)) = x1·x2
POL(2) = [2]
POL(COND_LOAD4391(x1, x2, x3)) = [-1] + x2
POL(<=(x1, x2)) = [-1]
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:

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

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

Integer, Boolean

R is empty.

The integer pair graph contains the following rules and edges:
(2): LOAD439(i36[2], i37[2]) → COND_LOAD439(i37[2] > 0 && i36[2] > i37[2], i36[2], i37[2])
(4): LOAD439(i36[4], i37[4]) → COND_LOAD4391(i36[4] >= 0 && i37[4] > 0 && i36[4] <= i37[4], i36[4], i37[4])

(5) -> (0), if ((i36[5] + -1* i26[0]))

(3) -> (2), if ((2 * i37[3]* i37[2])∧(i36[3]* i36[2]))

(2) -> (3), if ((i37[2]* i37[3])∧(i36[2]* i36[3])∧(i37[2] > 0 && i36[2] > i37[2]* TRUE))

(3) -> (4), if ((2 * i37[3]* i37[4])∧(i36[3]* i36[4]))

(4) -> (5), if ((i36[4] >= 0 && i37[4] > 0 && i36[4] <= i37[4]* TRUE)∧(i36[4]* i36[5])∧(i37[4]* i37[5]))

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): LOAD439(i36[2], i37[2]) → COND_LOAD439(i37[2] > 0 && i36[2] > i37[2], i36[2], i37[2])

(3) -> (2), if ((2 * i37[3]* i37[2])∧(i36[3]* i36[2]))

(2) -> (3), if ((i37[2]* i37[3])∧(i36[2]* i36[3])∧(i37[2] > 0 && i36[2] > i37[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_LOAD439(TRUE, i36[3], i37[3]) → LOAD439(i36[3], *(2, i37[3])) the following chains were created:
• We consider the chain LOAD439(i36[2], i37[2]) → COND_LOAD439(&&(>(i37[2], 0), >(i36[2], i37[2])), i36[2], i37[2]), COND_LOAD439(TRUE, i36[3], i37[3]) → LOAD439(i36[3], *(2, i37[3])), LOAD439(i36[2], i37[2]) → COND_LOAD439(&&(>(i37[2], 0), >(i36[2], i37[2])), i36[2], i37[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)    (i37[2] + [-1] ≥ 0∧i36[2] + [-1] + [-1]i37[2] ≥ 0 ⇒ (UIncreasing(LOAD439(i36[3], *(2, i37[3]))), ≥)∧[(-1)Bound*bni_12] + [(-1)bni_12]i37[2] + [bni_12]i36[2] ≥ 0∧[(-1)bso_13] + i37[2] ≥ 0)

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

(4)    (i37[2] + [-1] ≥ 0∧i36[2] + [-1] + [-1]i37[2] ≥ 0 ⇒ (UIncreasing(LOAD439(i36[3], *(2, i37[3]))), ≥)∧[(-1)Bound*bni_12] + [(-1)bni_12]i37[2] + [bni_12]i36[2] ≥ 0∧[(-1)bso_13] + i37[2] ≥ 0)

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

(5)    (i37[2] + [-1] ≥ 0∧i36[2] + [-1] + [-1]i37[2] ≥ 0 ⇒ (UIncreasing(LOAD439(i36[3], *(2, i37[3]))), ≥)∧[(-1)Bound*bni_12] + [(-1)bni_12]i37[2] + [bni_12]i36[2] ≥ 0∧[(-1)bso_13] + i37[2] ≥ 0)

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

(6)    (i37[2] ≥ 0∧i36[2] + [-2] + [-1]i37[2] ≥ 0 ⇒ (UIncreasing(LOAD439(i36[3], *(2, i37[3]))), ≥)∧[(-1)Bound*bni_12 + (-1)bni_12] + [(-1)bni_12]i37[2] + [bni_12]i36[2] ≥ 0∧[1 + (-1)bso_13] + i37[2] ≥ 0)

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

(7)    (i37[2] ≥ 0∧i36[2] ≥ 0 ⇒ (UIncreasing(LOAD439(i36[3], *(2, i37[3]))), ≥)∧[(-1)Bound*bni_12 + bni_12] + [bni_12]i36[2] ≥ 0∧[1 + (-1)bso_13] + i37[2] ≥ 0)

For Pair LOAD439(i36[2], i37[2]) → COND_LOAD439(&&(>(i37[2], 0), >(i36[2], i37[2])), i36[2], i37[2]) the following chains were created:
• We consider the chain LOAD439(i36[2], i37[2]) → COND_LOAD439(&&(>(i37[2], 0), >(i36[2], i37[2])), i36[2], i37[2]), COND_LOAD439(TRUE, i36[3], i37[3]) → LOAD439(i36[3], *(2, i37[3])) which results in the following constraint:

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

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

(10)    (i37[2] + [-1] ≥ 0∧i36[2] + [-1] + [-1]i37[2] ≥ 0 ⇒ (UIncreasing(COND_LOAD439(&&(>(i37[2], 0), >(i36[2], i37[2])), i36[2], i37[2])), ≥)∧[(-1)Bound*bni_14] + [(-1)bni_14]i37[2] + [bni_14]i36[2] ≥ 0∧[(-1)bso_15] ≥ 0)

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

(11)    (i37[2] + [-1] ≥ 0∧i36[2] + [-1] + [-1]i37[2] ≥ 0 ⇒ (UIncreasing(COND_LOAD439(&&(>(i37[2], 0), >(i36[2], i37[2])), i36[2], i37[2])), ≥)∧[(-1)Bound*bni_14] + [(-1)bni_14]i37[2] + [bni_14]i36[2] ≥ 0∧[(-1)bso_15] ≥ 0)

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

(12)    (i37[2] + [-1] ≥ 0∧i36[2] + [-1] + [-1]i37[2] ≥ 0 ⇒ (UIncreasing(COND_LOAD439(&&(>(i37[2], 0), >(i36[2], i37[2])), i36[2], i37[2])), ≥)∧[(-1)Bound*bni_14] + [(-1)bni_14]i37[2] + [bni_14]i36[2] ≥ 0∧[(-1)bso_15] ≥ 0)

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

(13)    (i37[2] ≥ 0∧i36[2] + [-2] + [-1]i37[2] ≥ 0 ⇒ (UIncreasing(COND_LOAD439(&&(>(i37[2], 0), >(i36[2], i37[2])), i36[2], i37[2])), ≥)∧[(-1)Bound*bni_14 + (-1)bni_14] + [(-1)bni_14]i37[2] + [bni_14]i36[2] ≥ 0∧[(-1)bso_15] ≥ 0)

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

(14)    (i37[2] ≥ 0∧i36[2] ≥ 0 ⇒ (UIncreasing(COND_LOAD439(&&(>(i37[2], 0), >(i36[2], i37[2])), i36[2], i37[2])), ≥)∧[(-1)Bound*bni_14 + bni_14] + [bni_14]i36[2] ≥ 0∧[(-1)bso_15] ≥ 0)

To summarize, we get the following constraints P for the following pairs.
• (i37[2] ≥ 0∧i36[2] ≥ 0 ⇒ (UIncreasing(LOAD439(i36[3], *(2, i37[3]))), ≥)∧[(-1)Bound*bni_12 + bni_12] + [bni_12]i36[2] ≥ 0∧[1 + (-1)bso_13] + i37[2] ≥ 0)

• (i37[2] ≥ 0∧i36[2] ≥ 0 ⇒ (UIncreasing(COND_LOAD439(&&(>(i37[2], 0), >(i36[2], i37[2])), i36[2], i37[2])), ≥)∧[(-1)Bound*bni_14 + bni_14] + [bni_14]i36[2] ≥ 0∧[(-1)bso_15] ≥ 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) = [2]
POL(FALSE) = [2]
POL(COND_LOAD439(x1, x2, x3)) = [2] + [-1]x3 + x2 + [-1]x1
POL(LOAD439(x1, x2)) = [-1]x2 + x1
POL(*(x1, x2)) = x1·x2
POL(2) = [2]
POL(&&(x1, x2)) = [2]
POL(>(x1, x2)) = [-1]
POL(0) = 0

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:

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

### (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): LOAD439(i36[2], i37[2]) → COND_LOAD439(i37[2] > 0 && i36[2] > i37[2], i36[2], i37[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:
none

R is empty.

The integer pair graph is empty.

The set Q consists of the following terms: