### (0) Obligation:

JBC Problem based on JBC Program:
Manifest-Version: 1.0 Created-By: 1.6.0_16 (Sun Microsystems Inc.) Main-Class: DivWithoutMinus
`public class DivWithoutMinus{  // adaption of the algorithm from [Kolbe 95]  public static void main(String[] args) {    Random.args = args;    int x = Random.random();    int y = Random.random();    int z = y;    int res = 0;    while (z > 0 && (y == 0 || y > 0 && x > 0))	{      if (y == 0) {        res++;        y = z;      }      else {        x--;        y--;      }    }  }}public class Random {  static String[] args;  static int index = 0;  public static int random() {    String string = args[index];    index++;    return string.length();  }}`

### (1) JBC2FIG (SOUND transformation)

Constructed FIGraph.

### (2) Obligation:

FIGraph based on JBC Program:
Graph of 213 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:
Load1200(i175, 0, i237, i227) → Cond_Load1200(i237 > 0 && i227 + 1 > 0, i175, 0, i237, i227)
Cond_Load1200(TRUE, i175, 0, i237, i227) → Load1200(i175, i237, i237, i227 + 1)
Load1200(i268, i255, i237, i227) → Cond_Load12001(i255 > 0 && i268 > 0 && i237 > 0, i268, i255, i237, i227)
Cond_Load12001(TRUE, i268, i255, i237, i227) → Load1200(i268 + -1, i255 + -1, i237, i227)
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:
• 0

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:
Load1200(i175, 0, i237, i227) → Cond_Load1200(i237 > 0 && i227 + 1 > 0, i175, i237, i227)
Load1200(i268, i255, i237, i227) → Cond_Load12001(i255 > 0 && i268 > 0 && i237 > 0, i268, i255, i237, i227)
Cond_Load12001(TRUE, i268, i255, i237, i227) → Load1200(i268 + -1, i255 + -1, i237, i227)
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:
Load1200(i175, 0, i237, i227) → Cond_Load1200(i237 > 0 && i227 + 1 > 0, i175, i237, i227)
Load1200(i268, i255, i237, i227) → Cond_Load12001(i255 > 0 && i268 > 0 && i237 > 0, i268, i255, i237, i227)
Cond_Load12001(TRUE, i268, i255, i237, i227) → Load1200(i268 + -1, i255 + -1, i237, i227)

The integer pair graph contains the following rules and edges:
(0): LOAD1200(i175[0], 0, i237[0], i227[0]) → COND_LOAD1200(i237[0] > 0 && i227[0] + 1 > 0, i175[0], i237[0], i227[0])
(1): COND_LOAD1200(TRUE, i175[1], i237[1], i227[1]) → LOAD1200(i175[1], i237[1], i237[1], i227[1] + 1)
(2): LOAD1200(i268[2], i255[2], i237[2], i227[2]) → COND_LOAD12001(i255[2] > 0 && i268[2] > 0 && i237[2] > 0, i268[2], i255[2], i237[2], i227[2])
(3): COND_LOAD12001(TRUE, i268[3], i255[3], i237[3], i227[3]) → LOAD1200(i268[3] + -1, i255[3] + -1, i237[3], i227[3])

(0) -> (1), if ((i237[0]* i237[1])∧(i175[0]* i175[1])∧(i237[0] > 0 && i227[0] + 1 > 0* TRUE)∧(i227[0]* i227[1]))

(1) -> (0), if ((i237[1]* i237[0])∧(i227[1] + 1* i227[0])∧(i237[1]* 0)∧(i175[1]* i175[0]))

(1) -> (2), if ((i237[1]* i237[2])∧(i175[1]* i268[2])∧(i227[1] + 1* i227[2])∧(i237[1]* i255[2]))

(2) -> (3), if ((i255[2] > 0 && i268[2] > 0 && i237[2] > 0* TRUE)∧(i255[2]* i255[3])∧(i227[2]* i227[3])∧(i237[2]* i237[3])∧(i268[2]* i268[3]))

(3) -> (0), if ((i255[3] + -1* 0)∧(i237[3]* i237[0])∧(i227[3]* i227[0])∧(i268[3] + -1* i175[0]))

(3) -> (2), if ((i255[3] + -1* i255[2])∧(i237[3]* i237[2])∧(i268[3] + -1* i268[2])∧(i227[3]* i227[2]))

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): LOAD1200(i175[0], 0, i237[0], i227[0]) → COND_LOAD1200(i237[0] > 0 && i227[0] + 1 > 0, i175[0], i237[0], i227[0])
(1): COND_LOAD1200(TRUE, i175[1], i237[1], i227[1]) → LOAD1200(i175[1], i237[1], i237[1], i227[1] + 1)
(2): LOAD1200(i268[2], i255[2], i237[2], i227[2]) → COND_LOAD12001(i255[2] > 0 && i268[2] > 0 && i237[2] > 0, i268[2], i255[2], i237[2], i227[2])
(3): COND_LOAD12001(TRUE, i268[3], i255[3], i237[3], i227[3]) → LOAD1200(i268[3] + -1, i255[3] + -1, i237[3], i227[3])

(0) -> (1), if ((i237[0]* i237[1])∧(i175[0]* i175[1])∧(i237[0] > 0 && i227[0] + 1 > 0* TRUE)∧(i227[0]* i227[1]))

(1) -> (0), if ((i237[1]* i237[0])∧(i227[1] + 1* i227[0])∧(i237[1]* 0)∧(i175[1]* i175[0]))

(1) -> (2), if ((i237[1]* i237[2])∧(i175[1]* i268[2])∧(i227[1] + 1* i227[2])∧(i237[1]* i255[2]))

(2) -> (3), if ((i255[2] > 0 && i268[2] > 0 && i237[2] > 0* TRUE)∧(i255[2]* i255[3])∧(i227[2]* i227[3])∧(i237[2]* i237[3])∧(i268[2]* i268[3]))

(3) -> (0), if ((i255[3] + -1* 0)∧(i237[3]* i237[0])∧(i227[3]* i227[0])∧(i268[3] + -1* i175[0]))

(3) -> (2), if ((i255[3] + -1* i255[2])∧(i237[3]* i237[2])∧(i268[3] + -1* i268[2])∧(i227[3]* i227[2]))

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 LOAD1200(i175, 0, i237, i227) → COND_LOAD1200(&&(>(i237, 0), >(+(i227, 1), 0)), i175, i237, i227) the following chains were created:
• We consider the chain LOAD1200(i175[0], 0, i237[0], i227[0]) → COND_LOAD1200(&&(>(i237[0], 0), >(+(i227[0], 1), 0)), i175[0], i237[0], i227[0]), COND_LOAD1200(TRUE, i175[1], i237[1], i227[1]) → LOAD1200(i175[1], i237[1], i237[1], +(i227[1], 1)) which results in the following constraint:

(1)    (i237[0]=i237[1]i175[0]=i175[1]&&(>(i237[0], 0), >(+(i227[0], 1), 0))=TRUEi227[0]=i227[1]LOAD1200(i175[0], 0, i237[0], i227[0])≥NonInfC∧LOAD1200(i175[0], 0, i237[0], i227[0])≥COND_LOAD1200(&&(>(i237[0], 0), >(+(i227[0], 1), 0)), i175[0], i237[0], i227[0])∧(UIncreasing(COND_LOAD1200(&&(>(i237[0], 0), >(+(i227[0], 1), 0)), i175[0], i237[0], i227[0])), ≥))

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

(2)    (>(i237[0], 0)=TRUE>(+(i227[0], 1), 0)=TRUELOAD1200(i175[0], 0, i237[0], i227[0])≥NonInfC∧LOAD1200(i175[0], 0, i237[0], i227[0])≥COND_LOAD1200(&&(>(i237[0], 0), >(+(i227[0], 1), 0)), i175[0], i237[0], i227[0])∧(UIncreasing(COND_LOAD1200(&&(>(i237[0], 0), >(+(i227[0], 1), 0)), i175[0], i237[0], i227[0])), ≥))

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

(3)    (i237[0] + [-1] ≥ 0∧i227[0] ≥ 0 ⇒ (UIncreasing(COND_LOAD1200(&&(>(i237[0], 0), >(+(i227[0], 1), 0)), i175[0], i237[0], i227[0])), ≥)∧[(-1)bni_21 + (-1)Bound*bni_21] + [bni_21]i175[0] ≥ 0∧[(-1)bso_22] ≥ 0)

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

(4)    (i237[0] + [-1] ≥ 0∧i227[0] ≥ 0 ⇒ (UIncreasing(COND_LOAD1200(&&(>(i237[0], 0), >(+(i227[0], 1), 0)), i175[0], i237[0], i227[0])), ≥)∧[(-1)bni_21 + (-1)Bound*bni_21] + [bni_21]i175[0] ≥ 0∧[(-1)bso_22] ≥ 0)

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

(5)    (i237[0] + [-1] ≥ 0∧i227[0] ≥ 0 ⇒ (UIncreasing(COND_LOAD1200(&&(>(i237[0], 0), >(+(i227[0], 1), 0)), i175[0], i237[0], i227[0])), ≥)∧[(-1)bni_21 + (-1)Bound*bni_21] + [bni_21]i175[0] ≥ 0∧[(-1)bso_22] ≥ 0)

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

(6)    (i237[0] + [-1] ≥ 0∧i227[0] ≥ 0 ⇒ (UIncreasing(COND_LOAD1200(&&(>(i237[0], 0), >(+(i227[0], 1), 0)), i175[0], i237[0], i227[0])), ≥)∧[bni_21] = 0∧[(-1)bni_21 + (-1)Bound*bni_21] ≥ 0∧0 = 0∧[(-1)bso_22] ≥ 0)

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

(7)    (i237[0] ≥ 0∧i227[0] ≥ 0 ⇒ (UIncreasing(COND_LOAD1200(&&(>(i237[0], 0), >(+(i227[0], 1), 0)), i175[0], i237[0], i227[0])), ≥)∧[bni_21] = 0∧[(-1)bni_21 + (-1)Bound*bni_21] ≥ 0∧0 = 0∧[(-1)bso_22] ≥ 0)

For Pair COND_LOAD1200(TRUE, i175, i237, i227) → LOAD1200(i175, i237, i237, +(i227, 1)) the following chains were created:
• We consider the chain LOAD1200(i175[0], 0, i237[0], i227[0]) → COND_LOAD1200(&&(>(i237[0], 0), >(+(i227[0], 1), 0)), i175[0], i237[0], i227[0]), COND_LOAD1200(TRUE, i175[1], i237[1], i227[1]) → LOAD1200(i175[1], i237[1], i237[1], +(i227[1], 1)), LOAD1200(i175[0], 0, i237[0], i227[0]) → COND_LOAD1200(&&(>(i237[0], 0), >(+(i227[0], 1), 0)), i175[0], i237[0], i227[0]) which results in the following constraint:

We solved constraint (8) using rules (I), (II), (III), (IV), (IDP_CONSTANT_FOLD).
• We consider the chain LOAD1200(i175[0], 0, i237[0], i227[0]) → COND_LOAD1200(&&(>(i237[0], 0), >(+(i227[0], 1), 0)), i175[0], i237[0], i227[0]), COND_LOAD1200(TRUE, i175[1], i237[1], i227[1]) → LOAD1200(i175[1], i237[1], i237[1], +(i227[1], 1)), LOAD1200(i268[2], i255[2], i237[2], i227[2]) → COND_LOAD12001(&&(&&(>(i255[2], 0), >(i268[2], 0)), >(i237[2], 0)), i268[2], i255[2], i237[2], i227[2]) which results in the following constraint:

We simplified constraint (9) using rules (III), (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)    (i237[0] + [-1] ≥ 0∧i227[0] ≥ 0 ⇒ (UIncreasing(LOAD1200(i175[1], i237[1], i237[1], +(i227[1], 1))), ≥)∧[(-1)bni_23 + (-1)Bound*bni_23] + [bni_23]i175[0] ≥ 0∧[(-1)bso_24] ≥ 0)

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

(12)    (i237[0] + [-1] ≥ 0∧i227[0] ≥ 0 ⇒ (UIncreasing(LOAD1200(i175[1], i237[1], i237[1], +(i227[1], 1))), ≥)∧[(-1)bni_23 + (-1)Bound*bni_23] + [bni_23]i175[0] ≥ 0∧[(-1)bso_24] ≥ 0)

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

(13)    (i237[0] + [-1] ≥ 0∧i227[0] ≥ 0 ⇒ (UIncreasing(LOAD1200(i175[1], i237[1], i237[1], +(i227[1], 1))), ≥)∧[(-1)bni_23 + (-1)Bound*bni_23] + [bni_23]i175[0] ≥ 0∧[(-1)bso_24] ≥ 0)

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

(14)    (i237[0] + [-1] ≥ 0∧i227[0] ≥ 0 ⇒ (UIncreasing(LOAD1200(i175[1], i237[1], i237[1], +(i227[1], 1))), ≥)∧[bni_23] = 0∧[(-1)bni_23 + (-1)Bound*bni_23] ≥ 0∧0 = 0∧[(-1)bso_24] ≥ 0)

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

(15)    (i237[0] ≥ 0∧i227[0] ≥ 0 ⇒ (UIncreasing(LOAD1200(i175[1], i237[1], i237[1], +(i227[1], 1))), ≥)∧[bni_23] = 0∧[(-1)bni_23 + (-1)Bound*bni_23] ≥ 0∧0 = 0∧[(-1)bso_24] ≥ 0)

For Pair LOAD1200(i268, i255, i237, i227) → COND_LOAD12001(&&(&&(>(i255, 0), >(i268, 0)), >(i237, 0)), i268, i255, i237, i227) the following chains were created:
• We consider the chain LOAD1200(i268[2], i255[2], i237[2], i227[2]) → COND_LOAD12001(&&(&&(>(i255[2], 0), >(i268[2], 0)), >(i237[2], 0)), i268[2], i255[2], i237[2], i227[2]), COND_LOAD12001(TRUE, i268[3], i255[3], i237[3], i227[3]) → LOAD1200(+(i268[3], -1), +(i255[3], -1), i237[3], i227[3]) which results in the following constraint:

(16)    (&&(&&(>(i255[2], 0), >(i268[2], 0)), >(i237[2], 0))=TRUEi255[2]=i255[3]i227[2]=i227[3]i237[2]=i237[3]i268[2]=i268[3]LOAD1200(i268[2], i255[2], i237[2], i227[2])≥NonInfC∧LOAD1200(i268[2], i255[2], i237[2], i227[2])≥COND_LOAD12001(&&(&&(>(i255[2], 0), >(i268[2], 0)), >(i237[2], 0)), i268[2], i255[2], i237[2], i227[2])∧(UIncreasing(COND_LOAD12001(&&(&&(>(i255[2], 0), >(i268[2], 0)), >(i237[2], 0)), i268[2], i255[2], i237[2], i227[2])), ≥))

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

(17)    (>(i237[2], 0)=TRUE>(i255[2], 0)=TRUE>(i268[2], 0)=TRUELOAD1200(i268[2], i255[2], i237[2], i227[2])≥NonInfC∧LOAD1200(i268[2], i255[2], i237[2], i227[2])≥COND_LOAD12001(&&(&&(>(i255[2], 0), >(i268[2], 0)), >(i237[2], 0)), i268[2], i255[2], i237[2], i227[2])∧(UIncreasing(COND_LOAD12001(&&(&&(>(i255[2], 0), >(i268[2], 0)), >(i237[2], 0)), i268[2], i255[2], i237[2], i227[2])), ≥))

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

(18)    (i237[2] + [-1] ≥ 0∧i255[2] + [-1] ≥ 0∧i268[2] + [-1] ≥ 0 ⇒ (UIncreasing(COND_LOAD12001(&&(&&(>(i255[2], 0), >(i268[2], 0)), >(i237[2], 0)), i268[2], i255[2], i237[2], i227[2])), ≥)∧[(-1)bni_25 + (-1)Bound*bni_25] + [bni_25]i268[2] ≥ 0∧[(-1)bso_26] ≥ 0)

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

(19)    (i237[2] + [-1] ≥ 0∧i255[2] + [-1] ≥ 0∧i268[2] + [-1] ≥ 0 ⇒ (UIncreasing(COND_LOAD12001(&&(&&(>(i255[2], 0), >(i268[2], 0)), >(i237[2], 0)), i268[2], i255[2], i237[2], i227[2])), ≥)∧[(-1)bni_25 + (-1)Bound*bni_25] + [bni_25]i268[2] ≥ 0∧[(-1)bso_26] ≥ 0)

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

(20)    (i237[2] + [-1] ≥ 0∧i255[2] + [-1] ≥ 0∧i268[2] + [-1] ≥ 0 ⇒ (UIncreasing(COND_LOAD12001(&&(&&(>(i255[2], 0), >(i268[2], 0)), >(i237[2], 0)), i268[2], i255[2], i237[2], i227[2])), ≥)∧[(-1)bni_25 + (-1)Bound*bni_25] + [bni_25]i268[2] ≥ 0∧[(-1)bso_26] ≥ 0)

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

(21)    (i237[2] + [-1] ≥ 0∧i255[2] + [-1] ≥ 0∧i268[2] + [-1] ≥ 0 ⇒ (UIncreasing(COND_LOAD12001(&&(&&(>(i255[2], 0), >(i268[2], 0)), >(i237[2], 0)), i268[2], i255[2], i237[2], i227[2])), ≥)∧0 = 0∧[(-1)bni_25 + (-1)Bound*bni_25] + [bni_25]i268[2] ≥ 0∧0 = 0∧[(-1)bso_26] ≥ 0)

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

(22)    (i237[2] ≥ 0∧i255[2] + [-1] ≥ 0∧i268[2] + [-1] ≥ 0 ⇒ (UIncreasing(COND_LOAD12001(&&(&&(>(i255[2], 0), >(i268[2], 0)), >(i237[2], 0)), i268[2], i255[2], i237[2], i227[2])), ≥)∧0 = 0∧[(-1)bni_25 + (-1)Bound*bni_25] + [bni_25]i268[2] ≥ 0∧0 = 0∧[(-1)bso_26] ≥ 0)

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

(23)    (i237[2] ≥ 0∧i255[2] ≥ 0∧i268[2] + [-1] ≥ 0 ⇒ (UIncreasing(COND_LOAD12001(&&(&&(>(i255[2], 0), >(i268[2], 0)), >(i237[2], 0)), i268[2], i255[2], i237[2], i227[2])), ≥)∧0 = 0∧[(-1)bni_25 + (-1)Bound*bni_25] + [bni_25]i268[2] ≥ 0∧0 = 0∧[(-1)bso_26] ≥ 0)

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

(24)    (i237[2] ≥ 0∧i255[2] ≥ 0∧i268[2] ≥ 0 ⇒ (UIncreasing(COND_LOAD12001(&&(&&(>(i255[2], 0), >(i268[2], 0)), >(i237[2], 0)), i268[2], i255[2], i237[2], i227[2])), ≥)∧0 = 0∧[(-1)Bound*bni_25] + [bni_25]i268[2] ≥ 0∧0 = 0∧[(-1)bso_26] ≥ 0)

For Pair COND_LOAD12001(TRUE, i268, i255, i237, i227) → LOAD1200(+(i268, -1), +(i255, -1), i237, i227) the following chains were created:
• We consider the chain LOAD1200(i268[2], i255[2], i237[2], i227[2]) → COND_LOAD12001(&&(&&(>(i255[2], 0), >(i268[2], 0)), >(i237[2], 0)), i268[2], i255[2], i237[2], i227[2]), COND_LOAD12001(TRUE, i268[3], i255[3], i237[3], i227[3]) → LOAD1200(+(i268[3], -1), +(i255[3], -1), i237[3], i227[3]), LOAD1200(i175[0], 0, i237[0], i227[0]) → COND_LOAD1200(&&(>(i237[0], 0), >(+(i227[0], 1), 0)), i175[0], i237[0], i227[0]) which results in the following constraint:

(25)    (&&(&&(>(i255[2], 0), >(i268[2], 0)), >(i237[2], 0))=TRUEi255[2]=i255[3]i227[2]=i227[3]i237[2]=i237[3]i268[2]=i268[3]+(i255[3], -1)=0i237[3]=i237[0]i227[3]=i227[0]+(i268[3], -1)=i175[0]COND_LOAD12001(TRUE, i268[3], i255[3], i237[3], i227[3])≥NonInfC∧COND_LOAD12001(TRUE, i268[3], i255[3], i237[3], i227[3])≥LOAD1200(+(i268[3], -1), +(i255[3], -1), i237[3], i227[3])∧(UIncreasing(LOAD1200(+(i268[3], -1), +(i255[3], -1), i237[3], i227[3])), ≥))

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

(26)    (+(i255[2], -1)=0>(i237[2], 0)=TRUE>(i255[2], 0)=TRUE>(i268[2], 0)=TRUECOND_LOAD12001(TRUE, i268[2], i255[2], i237[2], i227[2])≥NonInfC∧COND_LOAD12001(TRUE, i268[2], i255[2], i237[2], i227[2])≥LOAD1200(+(i268[2], -1), +(i255[2], -1), i237[2], i227[2])∧(UIncreasing(LOAD1200(+(i268[3], -1), +(i255[3], -1), i237[3], i227[3])), ≥))

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

(27)    (i255[2] + [-1] ≥ 0∧i237[2] + [-1] ≥ 0∧i255[2] + [-1] ≥ 0∧i268[2] + [-1] ≥ 0 ⇒ (UIncreasing(LOAD1200(+(i268[3], -1), +(i255[3], -1), i237[3], i227[3])), ≥)∧[(-1)bni_27 + (-1)Bound*bni_27] + [bni_27]i268[2] ≥ 0∧[1 + (-1)bso_28] ≥ 0)

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

(28)    (i255[2] + [-1] ≥ 0∧i237[2] + [-1] ≥ 0∧i255[2] + [-1] ≥ 0∧i268[2] + [-1] ≥ 0 ⇒ (UIncreasing(LOAD1200(+(i268[3], -1), +(i255[3], -1), i237[3], i227[3])), ≥)∧[(-1)bni_27 + (-1)Bound*bni_27] + [bni_27]i268[2] ≥ 0∧[1 + (-1)bso_28] ≥ 0)

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

(29)    (i255[2] + [-1] ≥ 0∧i237[2] + [-1] ≥ 0∧i255[2] + [-1] ≥ 0∧i268[2] + [-1] ≥ 0 ⇒ (UIncreasing(LOAD1200(+(i268[3], -1), +(i255[3], -1), i237[3], i227[3])), ≥)∧[(-1)bni_27 + (-1)Bound*bni_27] + [bni_27]i268[2] ≥ 0∧[1 + (-1)bso_28] ≥ 0)

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

(30)    (i255[2] + [-1] ≥ 0∧i237[2] + [-1] ≥ 0∧i255[2] + [-1] ≥ 0∧i268[2] + [-1] ≥ 0 ⇒ (UIncreasing(LOAD1200(+(i268[3], -1), +(i255[3], -1), i237[3], i227[3])), ≥)∧0 = 0∧[(-1)bni_27 + (-1)Bound*bni_27] + [bni_27]i268[2] ≥ 0∧0 = 0∧[1 + (-1)bso_28] ≥ 0)

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

(31)    (i255[2] ≥ 0∧i237[2] + [-1] ≥ 0∧i255[2] ≥ 0∧i268[2] + [-1] ≥ 0 ⇒ (UIncreasing(LOAD1200(+(i268[3], -1), +(i255[3], -1), i237[3], i227[3])), ≥)∧0 = 0∧[(-1)bni_27 + (-1)Bound*bni_27] + [bni_27]i268[2] ≥ 0∧0 = 0∧[1 + (-1)bso_28] ≥ 0)

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

(32)    (i255[2] ≥ 0∧i237[2] ≥ 0∧i255[2] ≥ 0∧i268[2] + [-1] ≥ 0 ⇒ (UIncreasing(LOAD1200(+(i268[3], -1), +(i255[3], -1), i237[3], i227[3])), ≥)∧0 = 0∧[(-1)bni_27 + (-1)Bound*bni_27] + [bni_27]i268[2] ≥ 0∧0 = 0∧[1 + (-1)bso_28] ≥ 0)

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

(33)    (i255[2] ≥ 0∧i237[2] ≥ 0∧i255[2] ≥ 0∧i268[2] ≥ 0 ⇒ (UIncreasing(LOAD1200(+(i268[3], -1), +(i255[3], -1), i237[3], i227[3])), ≥)∧0 = 0∧[(-1)Bound*bni_27] + [bni_27]i268[2] ≥ 0∧0 = 0∧[1 + (-1)bso_28] ≥ 0)

• We consider the chain LOAD1200(i268[2], i255[2], i237[2], i227[2]) → COND_LOAD12001(&&(&&(>(i255[2], 0), >(i268[2], 0)), >(i237[2], 0)), i268[2], i255[2], i237[2], i227[2]), COND_LOAD12001(TRUE, i268[3], i255[3], i237[3], i227[3]) → LOAD1200(+(i268[3], -1), +(i255[3], -1), i237[3], i227[3]), LOAD1200(i268[2], i255[2], i237[2], i227[2]) → COND_LOAD12001(&&(&&(>(i255[2], 0), >(i268[2], 0)), >(i237[2], 0)), i268[2], i255[2], i237[2], i227[2]) which results in the following constraint:

(34)    (&&(&&(>(i255[2], 0), >(i268[2], 0)), >(i237[2], 0))=TRUEi255[2]=i255[3]i227[2]=i227[3]i237[2]=i237[3]i268[2]=i268[3]+(i255[3], -1)=i255[2]1i237[3]=i237[2]1+(i268[3], -1)=i268[2]1i227[3]=i227[2]1COND_LOAD12001(TRUE, i268[3], i255[3], i237[3], i227[3])≥NonInfC∧COND_LOAD12001(TRUE, i268[3], i255[3], i237[3], i227[3])≥LOAD1200(+(i268[3], -1), +(i255[3], -1), i237[3], i227[3])∧(UIncreasing(LOAD1200(+(i268[3], -1), +(i255[3], -1), i237[3], i227[3])), ≥))

We simplified constraint (34) using rules (III), (IV), (IDP_BOOLEAN) which results in the following new constraint:

(35)    (>(i237[2], 0)=TRUE>(i255[2], 0)=TRUE>(i268[2], 0)=TRUECOND_LOAD12001(TRUE, i268[2], i255[2], i237[2], i227[2])≥NonInfC∧COND_LOAD12001(TRUE, i268[2], i255[2], i237[2], i227[2])≥LOAD1200(+(i268[2], -1), +(i255[2], -1), i237[2], i227[2])∧(UIncreasing(LOAD1200(+(i268[3], -1), +(i255[3], -1), i237[3], i227[3])), ≥))

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

(36)    (i237[2] + [-1] ≥ 0∧i255[2] + [-1] ≥ 0∧i268[2] + [-1] ≥ 0 ⇒ (UIncreasing(LOAD1200(+(i268[3], -1), +(i255[3], -1), i237[3], i227[3])), ≥)∧[(-1)bni_27 + (-1)Bound*bni_27] + [bni_27]i268[2] ≥ 0∧[1 + (-1)bso_28] ≥ 0)

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

(37)    (i237[2] + [-1] ≥ 0∧i255[2] + [-1] ≥ 0∧i268[2] + [-1] ≥ 0 ⇒ (UIncreasing(LOAD1200(+(i268[3], -1), +(i255[3], -1), i237[3], i227[3])), ≥)∧[(-1)bni_27 + (-1)Bound*bni_27] + [bni_27]i268[2] ≥ 0∧[1 + (-1)bso_28] ≥ 0)

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

(38)    (i237[2] + [-1] ≥ 0∧i255[2] + [-1] ≥ 0∧i268[2] + [-1] ≥ 0 ⇒ (UIncreasing(LOAD1200(+(i268[3], -1), +(i255[3], -1), i237[3], i227[3])), ≥)∧[(-1)bni_27 + (-1)Bound*bni_27] + [bni_27]i268[2] ≥ 0∧[1 + (-1)bso_28] ≥ 0)

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

(39)    (i237[2] + [-1] ≥ 0∧i255[2] + [-1] ≥ 0∧i268[2] + [-1] ≥ 0 ⇒ (UIncreasing(LOAD1200(+(i268[3], -1), +(i255[3], -1), i237[3], i227[3])), ≥)∧0 = 0∧[(-1)bni_27 + (-1)Bound*bni_27] + [bni_27]i268[2] ≥ 0∧0 = 0∧[1 + (-1)bso_28] ≥ 0)

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

(40)    (i237[2] ≥ 0∧i255[2] + [-1] ≥ 0∧i268[2] + [-1] ≥ 0 ⇒ (UIncreasing(LOAD1200(+(i268[3], -1), +(i255[3], -1), i237[3], i227[3])), ≥)∧0 = 0∧[(-1)bni_27 + (-1)Bound*bni_27] + [bni_27]i268[2] ≥ 0∧0 = 0∧[1 + (-1)bso_28] ≥ 0)

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

(41)    (i237[2] ≥ 0∧i255[2] ≥ 0∧i268[2] + [-1] ≥ 0 ⇒ (UIncreasing(LOAD1200(+(i268[3], -1), +(i255[3], -1), i237[3], i227[3])), ≥)∧0 = 0∧[(-1)bni_27 + (-1)Bound*bni_27] + [bni_27]i268[2] ≥ 0∧0 = 0∧[1 + (-1)bso_28] ≥ 0)

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

(42)    (i237[2] ≥ 0∧i255[2] ≥ 0∧i268[2] ≥ 0 ⇒ (UIncreasing(LOAD1200(+(i268[3], -1), +(i255[3], -1), i237[3], i227[3])), ≥)∧0 = 0∧[(-1)Bound*bni_27] + [bni_27]i268[2] ≥ 0∧0 = 0∧[1 + (-1)bso_28] ≥ 0)

To summarize, we get the following constraints P for the following pairs.
• LOAD1200(i175, 0, i237, i227) → COND_LOAD1200(&&(>(i237, 0), >(+(i227, 1), 0)), i175, i237, i227)
• (i237[0] ≥ 0∧i227[0] ≥ 0 ⇒ (UIncreasing(COND_LOAD1200(&&(>(i237[0], 0), >(+(i227[0], 1), 0)), i175[0], i237[0], i227[0])), ≥)∧[bni_21] = 0∧[(-1)bni_21 + (-1)Bound*bni_21] ≥ 0∧0 = 0∧[(-1)bso_22] ≥ 0)

• (i237[0] ≥ 0∧i227[0] ≥ 0 ⇒ (UIncreasing(LOAD1200(i175[1], i237[1], i237[1], +(i227[1], 1))), ≥)∧[bni_23] = 0∧[(-1)bni_23 + (-1)Bound*bni_23] ≥ 0∧0 = 0∧[(-1)bso_24] ≥ 0)

• LOAD1200(i268, i255, i237, i227) → COND_LOAD12001(&&(&&(>(i255, 0), >(i268, 0)), >(i237, 0)), i268, i255, i237, i227)
• (i237[2] ≥ 0∧i255[2] ≥ 0∧i268[2] ≥ 0 ⇒ (UIncreasing(COND_LOAD12001(&&(&&(>(i255[2], 0), >(i268[2], 0)), >(i237[2], 0)), i268[2], i255[2], i237[2], i227[2])), ≥)∧0 = 0∧[(-1)Bound*bni_25] + [bni_25]i268[2] ≥ 0∧0 = 0∧[(-1)bso_26] ≥ 0)

• COND_LOAD12001(TRUE, i268, i255, i237, i227) → LOAD1200(+(i268, -1), +(i255, -1), i237, i227)
• (i255[2] ≥ 0∧i237[2] ≥ 0∧i255[2] ≥ 0∧i268[2] ≥ 0 ⇒ (UIncreasing(LOAD1200(+(i268[3], -1), +(i255[3], -1), i237[3], i227[3])), ≥)∧0 = 0∧[(-1)Bound*bni_27] + [bni_27]i268[2] ≥ 0∧0 = 0∧[1 + (-1)bso_28] ≥ 0)
• (i237[2] ≥ 0∧i255[2] ≥ 0∧i268[2] ≥ 0 ⇒ (UIncreasing(LOAD1200(+(i268[3], -1), +(i255[3], -1), i237[3], i227[3])), ≥)∧0 = 0∧[(-1)Bound*bni_27] + [bni_27]i268[2] ≥ 0∧0 = 0∧[1 + (-1)bso_28] ≥ 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) = [3]
POL(LOAD1200(x1, x2, x3, x4)) = [-1] + x1
POL(0) = 0
POL(COND_LOAD1200(x1, x2, x3, x4)) = [-1] + x2
POL(&&(x1, x2)) = [-1]
POL(>(x1, x2)) = [-1]
POL(+(x1, x2)) = x1 + x2
POL(1) = [1]
POL(COND_LOAD12001(x1, x2, x3, x4, x5)) = [-1] + x2
POL(-1) = [-1]

The following pairs are in P>:

COND_LOAD12001(TRUE, i268[3], i255[3], i237[3], i227[3]) → LOAD1200(+(i268[3], -1), +(i255[3], -1), i237[3], i227[3])

The following pairs are in Pbound:

LOAD1200(i268[2], i255[2], i237[2], i227[2]) → COND_LOAD12001(&&(&&(>(i255[2], 0), >(i268[2], 0)), >(i237[2], 0)), i268[2], i255[2], i237[2], i227[2])
COND_LOAD12001(TRUE, i268[3], i255[3], i237[3], i227[3]) → LOAD1200(+(i268[3], -1), +(i255[3], -1), i237[3], i227[3])

The following pairs are in P:

LOAD1200(i175[0], 0, i237[0], i227[0]) → COND_LOAD1200(&&(>(i237[0], 0), >(+(i227[0], 1), 0)), i175[0], i237[0], i227[0])
LOAD1200(i268[2], i255[2], i237[2], i227[2]) → COND_LOAD12001(&&(&&(>(i255[2], 0), >(i268[2], 0)), >(i237[2], 0)), i268[2], i255[2], i237[2], i227[2])

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

### (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): LOAD1200(i175[0], 0, i237[0], i227[0]) → COND_LOAD1200(i237[0] > 0 && i227[0] + 1 > 0, i175[0], i237[0], i227[0])
(1): COND_LOAD1200(TRUE, i175[1], i237[1], i227[1]) → LOAD1200(i175[1], i237[1], i237[1], i227[1] + 1)
(2): LOAD1200(i268[2], i255[2], i237[2], i227[2]) → COND_LOAD12001(i255[2] > 0 && i268[2] > 0 && i237[2] > 0, i268[2], i255[2], i237[2], i227[2])

(1) -> (0), if ((i237[1]* i237[0])∧(i227[1] + 1* i227[0])∧(i237[1]* 0)∧(i175[1]* i175[0]))

(0) -> (1), if ((i237[0]* i237[1])∧(i175[0]* i175[1])∧(i237[0] > 0 && i227[0] + 1 > 0* TRUE)∧(i227[0]* i227[1]))

(1) -> (2), if ((i237[1]* i237[2])∧(i175[1]* i268[2])∧(i227[1] + 1* i227[2])∧(i237[1]* i255[2]))

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 1 less node.

### (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:
(1): COND_LOAD1200(TRUE, i175[1], i237[1], i227[1]) → LOAD1200(i175[1], i237[1], i237[1], i227[1] + 1)
(0): LOAD1200(i175[0], 0, i237[0], i227[0]) → COND_LOAD1200(i237[0] > 0 && i227[0] + 1 > 0, i175[0], i237[0], i227[0])

(1) -> (0), if ((i237[1]* i237[0])∧(i227[1] + 1* i227[0])∧(i237[1]* 0)∧(i175[1]* i175[0]))

(0) -> (1), if ((i237[0]* i237[1])∧(i175[0]* i175[1])∧(i237[0] > 0 && i227[0] + 1 > 0* TRUE)∧(i227[0]* i227[1]))

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_LOAD1200(TRUE, i175[1], i237[1], i227[1]) → LOAD1200(i175[1], i237[1], i237[1], +(i227[1], 1)) the following chains were created:
• We consider the chain LOAD1200(i175[0], 0, i237[0], i227[0]) → COND_LOAD1200(&&(>(i237[0], 0), >(+(i227[0], 1), 0)), i175[0], i237[0], i227[0]), COND_LOAD1200(TRUE, i175[1], i237[1], i227[1]) → LOAD1200(i175[1], i237[1], i237[1], +(i227[1], 1)), LOAD1200(i175[0], 0, i237[0], i227[0]) → COND_LOAD1200(&&(>(i237[0], 0), >(+(i227[0], 1), 0)), i175[0], i237[0], i227[0]) which results in the following constraint:

We solved constraint (1) using rules (I), (II), (III), (IV), (IDP_CONSTANT_FOLD).

For Pair LOAD1200(i175[0], 0, i237[0], i227[0]) → COND_LOAD1200(&&(>(i237[0], 0), >(+(i227[0], 1), 0)), i175[0], i237[0], i227[0]) the following chains were created:
• We consider the chain LOAD1200(i175[0], 0, i237[0], i227[0]) → COND_LOAD1200(&&(>(i237[0], 0), >(+(i227[0], 1), 0)), i175[0], i237[0], i227[0]), COND_LOAD1200(TRUE, i175[1], i237[1], i227[1]) → LOAD1200(i175[1], i237[1], i237[1], +(i227[1], 1)) which results in the following constraint:

(2)    (i237[0]=i237[1]i175[0]=i175[1]&&(>(i237[0], 0), >(+(i227[0], 1), 0))=TRUEi227[0]=i227[1]LOAD1200(i175[0], 0, i237[0], i227[0])≥NonInfC∧LOAD1200(i175[0], 0, i237[0], i227[0])≥COND_LOAD1200(&&(>(i237[0], 0), >(+(i227[0], 1), 0)), i175[0], i237[0], i227[0])∧(UIncreasing(COND_LOAD1200(&&(>(i237[0], 0), >(+(i227[0], 1), 0)), i175[0], i237[0], i227[0])), ≥))

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

(3)    (&&(>(i237[0], 0), >(+(i227[0], 1), 0))=TRUELOAD1200(i175[0], 0, i237[0], i227[0])≥NonInfC∧LOAD1200(i175[0], 0, i237[0], i227[0])≥COND_LOAD1200(&&(>(i237[0], 0), >(+(i227[0], 1), 0)), i175[0], i237[0], i227[0])∧(UIncreasing(COND_LOAD1200(&&(>(i237[0], 0), >(+(i227[0], 1), 0)), i175[0], i237[0], i227[0])), ≥))

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

(4)    (0 ≥ 0 ⇒ (UIncreasing(COND_LOAD1200(&&(>(i237[0], 0), >(+(i227[0], 1), 0)), i175[0], i237[0], i227[0])), ≥)∧[bni_11 + (-1)Bound*bni_11] + [bni_11]i227[0] + [bni_11]i237[0] + [bni_11]i175[0] ≥ 0∧[2 + (-1)bso_12] + i227[0] + i237[0] + i175[0] ≥ 0)

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

(5)    (0 ≥ 0 ⇒ (UIncreasing(COND_LOAD1200(&&(>(i237[0], 0), >(+(i227[0], 1), 0)), i175[0], i237[0], i227[0])), ≥)∧[bni_11 + (-1)Bound*bni_11] + [bni_11]i227[0] + [bni_11]i237[0] + [bni_11]i175[0] ≥ 0∧[2 + (-1)bso_12] + i227[0] + i237[0] + i175[0] ≥ 0)

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

(6)    (0 ≥ 0 ⇒ (UIncreasing(COND_LOAD1200(&&(>(i237[0], 0), >(+(i227[0], 1), 0)), i175[0], i237[0], i227[0])), ≥)∧[bni_11 + (-1)Bound*bni_11] + [bni_11]i227[0] + [bni_11]i237[0] + [bni_11]i175[0] ≥ 0∧[2 + (-1)bso_12] + i227[0] + i237[0] + i175[0] ≥ 0)

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

(7)    (0 ≥ 0 ⇒ (UIncreasing(COND_LOAD1200(&&(>(i237[0], 0), >(+(i227[0], 1), 0)), i175[0], i237[0], i227[0])), ≥)∧[bni_11] ≥ 0∧[bni_11] ≥ 0∧[bni_11] ≥ 0∧[bni_11 + (-1)Bound*bni_11] ≥ 0∧[1] ≥ 0∧[1] ≥ 0∧[1] ≥ 0∧[2 + (-1)bso_12] ≥ 0)

To summarize, we get the following constraints P for the following pairs.

• LOAD1200(i175[0], 0, i237[0], i227[0]) → COND_LOAD1200(&&(>(i237[0], 0), >(+(i227[0], 1), 0)), i175[0], i237[0], i227[0])
• (0 ≥ 0 ⇒ (UIncreasing(COND_LOAD1200(&&(>(i237[0], 0), >(+(i227[0], 1), 0)), i175[0], i237[0], i227[0])), ≥)∧[bni_11] ≥ 0∧[bni_11] ≥ 0∧[bni_11] ≥ 0∧[bni_11 + (-1)Bound*bni_11] ≥ 0∧[1] ≥ 0∧[1] ≥ 0∧[1] ≥ 0∧[2 + (-1)bso_12] ≥ 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 with natural coefficients for non-tuple symbols [NONINF][POLO]:

POL(TRUE) = 0
POL(FALSE) = 0
POL(COND_LOAD1200(x1, x2, x3, x4)) = [-1] + [2]x1
POL(LOAD1200(x1, x2, x3, x4)) = [1] + x4 + x3 + [-1]x2 + x1
POL(+(x1, x2)) = 0
POL(1) = 0
POL(0) = 0
POL(&&(x1, x2)) = 0
POL(>(x1, x2)) = 0

The following pairs are in P>:

LOAD1200(i175[0], 0, i237[0], i227[0]) → COND_LOAD1200(&&(>(i237[0], 0), >(+(i227[0], 1), 0)), i175[0], i237[0], i227[0])

The following pairs are in Pbound:

LOAD1200(i175[0], 0, i237[0], i227[0]) → COND_LOAD1200(&&(>(i237[0], 0), >(+(i227[0], 1), 0)), i175[0], i237[0], i227[0])

The following pairs are in P:
none

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

### (17) 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:

### (18) IDependencyGraphProof (EQUIVALENT transformation)

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

### (20) 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): LOAD1200(i175[0], 0, i237[0], i227[0]) → COND_LOAD1200(i237[0] > 0 && i227[0] + 1 > 0, i175[0], i237[0], i227[0])
(1): COND_LOAD1200(TRUE, i175[1], i237[1], i227[1]) → LOAD1200(i175[1], i237[1], i237[1], i227[1] + 1)

(1) -> (0), if ((i237[1]* i237[0])∧(i227[1] + 1* i227[0])∧(i237[1]* 0)∧(i175[1]* i175[0]))

(0) -> (1), if ((i237[0]* i237[1])∧(i175[0]* i175[1])∧(i237[0] > 0 && i227[0] + 1 > 0* TRUE)∧(i227[0]* i227[1]))

The set Q consists of the following terms:

### (21) 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 LOAD1200(i175[0], 0, i237[0], i227[0]) → COND_LOAD1200(&&(>(i237[0], 0), >(+(i227[0], 1), 0)), i175[0], i237[0], i227[0]) the following chains were created:
• We consider the chain LOAD1200(i175[0], 0, i237[0], i227[0]) → COND_LOAD1200(&&(>(i237[0], 0), >(+(i227[0], 1), 0)), i175[0], i237[0], i227[0]), COND_LOAD1200(TRUE, i175[1], i237[1], i227[1]) → LOAD1200(i175[1], i237[1], i237[1], +(i227[1], 1)) which results in the following constraint:

(1)    (i237[0]=i237[1]i175[0]=i175[1]&&(>(i237[0], 0), >(+(i227[0], 1), 0))=TRUEi227[0]=i227[1]LOAD1200(i175[0], 0, i237[0], i227[0])≥COND_LOAD1200(&&(>(i237[0], 0), >(+(i227[0], 1), 0)), i175[0], i237[0], i227[0])∧(UIncreasing(COND_LOAD1200(&&(>(i237[0], 0), >(+(i227[0], 1), 0)), i175[0], i237[0], i227[0])), ≥))

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

(2)    (&&(>(i237[0], 0), >(+(i227[0], 1), 0))=TRUELOAD1200(i175[0], 0, i237[0], i227[0])≥COND_LOAD1200(&&(>(i237[0], 0), >(+(i227[0], 1), 0)), i175[0], i237[0], i227[0])∧(UIncreasing(COND_LOAD1200(&&(>(i237[0], 0), >(+(i227[0], 1), 0)), i175[0], i237[0], i227[0])), ≥))

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

(3)    (0 ≥ 0 ⇒ (UIncreasing(COND_LOAD1200(&&(>(i237[0], 0), >(+(i227[0], 1), 0)), i175[0], i237[0], i227[0])), ≥)∧[2 + (-1)bso_11] + i227[0] + i237[0] + i175[0] ≥ 0)

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

(4)    (0 ≥ 0 ⇒ (UIncreasing(COND_LOAD1200(&&(>(i237[0], 0), >(+(i227[0], 1), 0)), i175[0], i237[0], i227[0])), ≥)∧[2 + (-1)bso_11] + i227[0] + i237[0] + i175[0] ≥ 0)

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

(5)    (0 ≥ 0 ⇒ (UIncreasing(COND_LOAD1200(&&(>(i237[0], 0), >(+(i227[0], 1), 0)), i175[0], i237[0], i227[0])), ≥)∧[2 + (-1)bso_11] + i227[0] + i237[0] + i175[0] ≥ 0)

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

(6)    (0 ≥ 0 ⇒ (UIncreasing(COND_LOAD1200(&&(>(i237[0], 0), >(+(i227[0], 1), 0)), i175[0], i237[0], i227[0])), ≥)∧[1] ≥ 0∧[1] ≥ 0∧[1] ≥ 0∧[2 + (-1)bso_11] ≥ 0)

For Pair COND_LOAD1200(TRUE, i175[1], i237[1], i227[1]) → LOAD1200(i175[1], i237[1], i237[1], +(i227[1], 1)) the following chains were created:
• We consider the chain LOAD1200(i175[0], 0, i237[0], i227[0]) → COND_LOAD1200(&&(>(i237[0], 0), >(+(i227[0], 1), 0)), i175[0], i237[0], i227[0]), COND_LOAD1200(TRUE, i175[1], i237[1], i227[1]) → LOAD1200(i175[1], i237[1], i237[1], +(i227[1], 1)), LOAD1200(i175[0], 0, i237[0], i227[0]) → COND_LOAD1200(&&(>(i237[0], 0), >(+(i227[0], 1), 0)), i175[0], i237[0], i227[0]) which results in the following constraint:

(7)    (i237[0]=i237[1]i175[0]=i175[1]&&(>(i237[0], 0), >(+(i227[0], 1), 0))=TRUEi227[0]=i227[1]i237[1]=i237[0]1+(i227[1], 1)=i227[0]1i237[1]=0i175[1]=i175[0]1COND_LOAD1200(TRUE, i175[1], i237[1], i227[1])≥LOAD1200(i175[1], i237[1], i237[1], +(i227[1], 1))∧(UIncreasing(LOAD1200(i175[1], i237[1], i237[1], +(i227[1], 1))), ≥))

We solved constraint (7) using rules (I), (II), (III), (IV), (IDP_CONSTANT_FOLD).

To summarize, we get the following constraints P for the following pairs.
• LOAD1200(i175[0], 0, i237[0], i227[0]) → COND_LOAD1200(&&(>(i237[0], 0), >(+(i227[0], 1), 0)), i175[0], i237[0], i227[0])
• (0 ≥ 0 ⇒ (UIncreasing(COND_LOAD1200(&&(>(i237[0], 0), >(+(i227[0], 1), 0)), i175[0], i237[0], i227[0])), ≥)∧[1] ≥ 0∧[1] ≥ 0∧[1] ≥ 0∧[2 + (-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 with natural coefficients for all symbols [NONINF][POLO]:

POL(TRUE) = 0
POL(FALSE) = 0
POL(LOAD1200(x1, x2, x3, x4)) = [2] + [2]x4 + [2]x3 + [2]x1
POL(0) = 0
POL(COND_LOAD1200(x1, x2, x3, x4)) = x4 + x3 + x2
POL(&&(x1, x2)) = 0
POL(>(x1, x2)) = 0
POL(+(x1, x2)) = 0
POL(1) = 0

The following pairs are in P>:

LOAD1200(i175[0], 0, i237[0], i227[0]) → COND_LOAD1200(&&(>(i237[0], 0), >(+(i227[0], 1), 0)), i175[0], i237[0], i227[0])

The following pairs are in Pbound:

LOAD1200(i175[0], 0, i237[0], i227[0]) → COND_LOAD1200(&&(>(i237[0], 0), >(+(i227[0], 1), 0)), i175[0], i237[0], i227[0])

The following pairs are in P:
none

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

### (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:
none

R is empty.

The integer pair graph is empty.

The set Q consists of the following terms: