### (0) Obligation:

JBC Problem based on JBC Program:
Manifest-Version: 1.0 Created-By: 1.6.0_16 (Sun Microsystems Inc.) Main-Class: CountUpRound
`public class CountUpRound{  public static int round (int x) {    if (x % 2 == 0) return x;    else return x+1;  }  public static void main(String[] args) {    Random.args = args;    int x = Random.random();    int y = Random.random();    while (x > y) {      y = round(y+1);    }  }}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 201 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:
Load761(i11, i42) → Cond_Load761(!(i42 + 1 % 2 = 0) && i11 > i42, i11, i42)
Cond_Load761(TRUE, i11, i42) → Store1059(i11, i42 + 1 + 1)
Load761(i11, i42) → Cond_Load7611(i11 > i42 && 0 = i42 + 1 % 2, i11, i42)
The set Q consists of the following terms:
Store1059(x0, x1)

### (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:
Load761(i11, i42) → Cond_Load761(!(i42 + 1 % 2 = 0) && i11 > i42, i11, i42)
Cond_Load761(TRUE, i11, i42) → Store1059(i11, i42 + 1 + 1)
Load761(i11, i42) → Cond_Load7611(i11 > i42 && 0 = i42 + 1 % 2, i11, i42)

The integer pair graph contains the following rules and edges:
(0): LOAD761(i11[0], i42[0]) → COND_LOAD761(!(i42[0] + 1 % 2 = 0) && i11[0] > i42[0], i11[0], i42[0])
(1): COND_LOAD761(TRUE, i11[1], i42[1]) → STORE1059(i11[1], i42[1] + 1 + 1)
(2): STORE1059(i11[2], i70[2]) → LOAD761(i11[2], i70[2])
(3): LOAD761(i11[3], i42[3]) → COND_LOAD7611(i11[3] > i42[3] && 0 = i42[3] + 1 % 2, i11[3], i42[3])

(0) -> (1), if ((i42[0]* i42[1])∧(!(i42[0] + 1 % 2 = 0) && i11[0] > i42[0]* TRUE)∧(i11[0]* i11[1]))

(1) -> (2), if ((i42[1] + 1 + 1* i70[2])∧(i11[1]* i11[2]))

(2) -> (0), if ((i70[2]* i42[0])∧(i11[2]* i11[0]))

(2) -> (3), if ((i70[2]* i42[3])∧(i11[2]* i11[3]))

(3) -> (4), if ((i11[3]* i11[4])∧(i42[3]* i42[4])∧(i11[3] > i42[3] && 0 = i42[3] + 1 % 2* TRUE))

(4) -> (0), if ((i11[4]* i11[0])∧(i42[4] + 1* i42[0]))

(4) -> (3), if ((i42[4] + 1* i42[3])∧(i11[4]* i11[3]))

The set Q consists of the following terms:
Store1059(x0, x1)

### (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): LOAD761(i11[0], i42[0]) → COND_LOAD761(!(i42[0] + 1 % 2 = 0) && i11[0] > i42[0], i11[0], i42[0])
(1): COND_LOAD761(TRUE, i11[1], i42[1]) → STORE1059(i11[1], i42[1] + 1 + 1)
(2): STORE1059(i11[2], i70[2]) → LOAD761(i11[2], i70[2])
(3): LOAD761(i11[3], i42[3]) → COND_LOAD7611(i11[3] > i42[3] && 0 = i42[3] + 1 % 2, i11[3], i42[3])

(0) -> (1), if ((i42[0]* i42[1])∧(!(i42[0] + 1 % 2 = 0) && i11[0] > i42[0]* TRUE)∧(i11[0]* i11[1]))

(1) -> (2), if ((i42[1] + 1 + 1* i70[2])∧(i11[1]* i11[2]))

(2) -> (0), if ((i70[2]* i42[0])∧(i11[2]* i11[0]))

(2) -> (3), if ((i70[2]* i42[3])∧(i11[2]* i11[3]))

(3) -> (4), if ((i11[3]* i11[4])∧(i42[3]* i42[4])∧(i11[3] > i42[3] && 0 = i42[3] + 1 % 2* TRUE))

(4) -> (0), if ((i11[4]* i11[0])∧(i42[4] + 1* i42[0]))

(4) -> (3), if ((i42[4] + 1* i42[3])∧(i11[4]* i11[3]))

The set Q consists of the following terms:
Store1059(x0, x1)

### (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 LOAD761(i11, i42) → COND_LOAD761(&&(!(=(%(+(i42, 1), 2), 0)), >(i11, i42)), i11, i42) the following chains were created:
• We consider the chain LOAD761(i11[0], i42[0]) → COND_LOAD761(&&(!(=(%(+(i42[0], 1), 2), 0)), >(i11[0], i42[0])), i11[0], i42[0]), COND_LOAD761(TRUE, i11[1], i42[1]) → STORE1059(i11[1], +(+(i42[1], 1), 1)) which results in the following constraint:

(1)    (i42[0]=i42[1]&&(!(=(%(+(i42[0], 1), 2), 0)), >(i11[0], i42[0]))=TRUEi11[0]=i11[1]LOAD761(i11[0], i42[0])≥NonInfC∧LOAD761(i11[0], i42[0])≥COND_LOAD761(&&(!(=(%(+(i42[0], 1), 2), 0)), >(i11[0], i42[0])), i11[0], i42[0])∧(UIncreasing(COND_LOAD761(&&(!(=(%(+(i42[0], 1), 2), 0)), >(i11[0], i42[0])), i11[0], i42[0])), ≥))

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

(2)    (>(i11[0], i42[0])=TRUE<(%(+(i42[0], 1), 2), 0)=TRUELOAD761(i11[0], i42[0])≥NonInfC∧LOAD761(i11[0], i42[0])≥COND_LOAD761(&&(!(=(%(+(i42[0], 1), 2), 0)), >(i11[0], i42[0])), i11[0], i42[0])∧(UIncreasing(COND_LOAD761(&&(!(=(%(+(i42[0], 1), 2), 0)), >(i11[0], i42[0])), i11[0], i42[0])), ≥))

(3)    (>(i11[0], i42[0])=TRUE>(%(+(i42[0], 1), 2), 0)=TRUELOAD761(i11[0], i42[0])≥NonInfC∧LOAD761(i11[0], i42[0])≥COND_LOAD761(&&(!(=(%(+(i42[0], 1), 2), 0)), >(i11[0], i42[0])), i11[0], i42[0])∧(UIncreasing(COND_LOAD761(&&(!(=(%(+(i42[0], 1), 2), 0)), >(i11[0], i42[0])), i11[0], i42[0])), ≥))

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

(4)    (i11[0] + [-1] + [-1]i42[0] ≥ 0∧[-1] + [-1]min{[2], [-2]} ≥ 0 ⇒ (UIncreasing(COND_LOAD761(&&(!(=(%(+(i42[0], 1), 2), 0)), >(i11[0], i42[0])), i11[0], i42[0])), ≥)∧[(-1)Bound*bni_14] + [bni_14]i11[0] + [(-1)bni_14]i42[0] ≥ 0∧[1 + (-1)bso_15] ≥ 0)

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

(5)    (i11[0] + [-1] + [-1]i42[0] ≥ 0∧max{[2], [-2]} + [-1] ≥ 0 ⇒ (UIncreasing(COND_LOAD761(&&(!(=(%(+(i42[0], 1), 2), 0)), >(i11[0], i42[0])), i11[0], i42[0])), ≥)∧[(-1)Bound*bni_14] + [bni_14]i11[0] + [(-1)bni_14]i42[0] ≥ 0∧[1 + (-1)bso_15] ≥ 0)

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

(6)    (i11[0] + [-1] + [-1]i42[0] ≥ 0∧[-1] + [-1]min{[2], [-2]} ≥ 0 ⇒ (UIncreasing(COND_LOAD761(&&(!(=(%(+(i42[0], 1), 2), 0)), >(i11[0], i42[0])), i11[0], i42[0])), ≥)∧[(-1)Bound*bni_14] + [bni_14]i11[0] + [(-1)bni_14]i42[0] ≥ 0∧[1 + (-1)bso_15] ≥ 0)

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

(7)    (i11[0] + [-1] + [-1]i42[0] ≥ 0∧max{[2], [-2]} + [-1] ≥ 0 ⇒ (UIncreasing(COND_LOAD761(&&(!(=(%(+(i42[0], 1), 2), 0)), >(i11[0], i42[0])), i11[0], i42[0])), ≥)∧[(-1)Bound*bni_14] + [bni_14]i11[0] + [(-1)bni_14]i42[0] ≥ 0∧[1 + (-1)bso_15] ≥ 0)

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

(8)    (i11[0] + [-1] + [-1]i42[0] ≥ 0∧[4] ≥ 0∧[1] ≥ 0 ⇒ (UIncreasing(COND_LOAD761(&&(!(=(%(+(i42[0], 1), 2), 0)), >(i11[0], i42[0])), i11[0], i42[0])), ≥)∧[(-1)Bound*bni_14] + [bni_14]i11[0] + [(-1)bni_14]i42[0] ≥ 0∧[1 + (-1)bso_15] ≥ 0)

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

(9)    (i11[0] + [-1] + [-1]i42[0] ≥ 0∧[4] ≥ 0∧[1] ≥ 0 ⇒ (UIncreasing(COND_LOAD761(&&(!(=(%(+(i42[0], 1), 2), 0)), >(i11[0], i42[0])), i11[0], i42[0])), ≥)∧[(-1)Bound*bni_14] + [bni_14]i11[0] + [(-1)bni_14]i42[0] ≥ 0∧[1 + (-1)bso_15] ≥ 0)

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

(10)    (i11[0] ≥ 0∧[4] ≥ 0∧[1] ≥ 0 ⇒ (UIncreasing(COND_LOAD761(&&(!(=(%(+(i42[0], 1), 2), 0)), >(i11[0], i42[0])), i11[0], i42[0])), ≥)∧[(-1)Bound*bni_14 + bni_14] + [bni_14]i11[0] ≥ 0∧[1 + (-1)bso_15] ≥ 0)

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

(11)    (i11[0] ≥ 0∧[4] ≥ 0∧[1] ≥ 0 ⇒ (UIncreasing(COND_LOAD761(&&(!(=(%(+(i42[0], 1), 2), 0)), >(i11[0], i42[0])), i11[0], i42[0])), ≥)∧[(-1)Bound*bni_14 + bni_14] + [bni_14]i11[0] ≥ 0∧[1 + (-1)bso_15] ≥ 0)

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

(12)    (i11[0] ≥ 0∧[4] ≥ 0∧[1] ≥ 0∧i42[0] ≥ 0 ⇒ (UIncreasing(COND_LOAD761(&&(!(=(%(+(i42[0], 1), 2), 0)), >(i11[0], i42[0])), i11[0], i42[0])), ≥)∧[(-1)Bound*bni_14 + bni_14] + [bni_14]i11[0] ≥ 0∧[1 + (-1)bso_15] ≥ 0)

(13)    (i11[0] ≥ 0∧[4] ≥ 0∧[1] ≥ 0∧i42[0] ≥ 0 ⇒ (UIncreasing(COND_LOAD761(&&(!(=(%(+(i42[0], 1), 2), 0)), >(i11[0], i42[0])), i11[0], i42[0])), ≥)∧[(-1)Bound*bni_14 + bni_14] + [bni_14]i11[0] ≥ 0∧[1 + (-1)bso_15] ≥ 0)

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

(14)    (i11[0] ≥ 0∧[1] ≥ 0∧i42[0] ≥ 0∧[1] ≥ 0 ⇒ (UIncreasing(COND_LOAD761(&&(!(=(%(+(i42[0], 1), 2), 0)), >(i11[0], i42[0])), i11[0], i42[0])), ≥)∧[(-1)Bound*bni_14 + bni_14] + [bni_14]i11[0] ≥ 0∧[1 + (-1)bso_15] ≥ 0)

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

(15)    (i11[0] ≥ 0∧[1] ≥ 0∧i42[0] ≥ 0∧[1] ≥ 0 ⇒ (UIncreasing(COND_LOAD761(&&(!(=(%(+(i42[0], 1), 2), 0)), >(i11[0], i42[0])), i11[0], i42[0])), ≥)∧[(-1)Bound*bni_14 + bni_14] + [bni_14]i11[0] ≥ 0∧[1 + (-1)bso_15] ≥ 0)

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

(16)    (i11[0] ≥ 0∧[4] ≥ 0∧[1] ≥ 0∧i42[0] ≥ 0 ⇒ (UIncreasing(COND_LOAD761(&&(!(=(%(+(i42[0], 1), 2), 0)), >(i11[0], i42[0])), i11[0], i42[0])), ≥)∧[(-1)Bound*bni_14 + bni_14] + [bni_14]i11[0] ≥ 0∧[1 + (-1)bso_15] ≥ 0)

(17)    (i11[0] ≥ 0∧[4] ≥ 0∧[1] ≥ 0∧i42[0] ≥ 0 ⇒ (UIncreasing(COND_LOAD761(&&(!(=(%(+(i42[0], 1), 2), 0)), >(i11[0], i42[0])), i11[0], i42[0])), ≥)∧[(-1)Bound*bni_14 + bni_14] + [bni_14]i11[0] ≥ 0∧[1 + (-1)bso_15] ≥ 0)

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

(18)    (i11[0] ≥ 0∧[1] ≥ 0∧i42[0] ≥ 0∧[1] ≥ 0 ⇒ (UIncreasing(COND_LOAD761(&&(!(=(%(+(i42[0], 1), 2), 0)), >(i11[0], i42[0])), i11[0], i42[0])), ≥)∧[(-1)Bound*bni_14 + bni_14] + [bni_14]i11[0] ≥ 0∧[1 + (-1)bso_15] ≥ 0)

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

(19)    (i11[0] ≥ 0∧[1] ≥ 0∧i42[0] ≥ 0∧[1] ≥ 0 ⇒ (UIncreasing(COND_LOAD761(&&(!(=(%(+(i42[0], 1), 2), 0)), >(i11[0], i42[0])), i11[0], i42[0])), ≥)∧[(-1)Bound*bni_14 + bni_14] + [bni_14]i11[0] ≥ 0∧[1 + (-1)bso_15] ≥ 0)

For Pair COND_LOAD761(TRUE, i11, i42) → STORE1059(i11, +(+(i42, 1), 1)) the following chains were created:
• We consider the chain COND_LOAD761(TRUE, i11[1], i42[1]) → STORE1059(i11[1], +(+(i42[1], 1), 1)) which results in the following constraint:

(20)    (COND_LOAD761(TRUE, i11[1], i42[1])≥NonInfC∧COND_LOAD761(TRUE, i11[1], i42[1])≥STORE1059(i11[1], +(+(i42[1], 1), 1))∧(UIncreasing(STORE1059(i11[1], +(+(i42[1], 1), 1))), ≥))

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

(21)    ((UIncreasing(STORE1059(i11[1], +(+(i42[1], 1), 1))), ≥)∧[(-1)bso_17] ≥ 0)

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

(22)    ((UIncreasing(STORE1059(i11[1], +(+(i42[1], 1), 1))), ≥)∧[(-1)bso_17] ≥ 0)

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

(23)    ((UIncreasing(STORE1059(i11[1], +(+(i42[1], 1), 1))), ≥)∧[(-1)bso_17] ≥ 0)

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

(24)    ((UIncreasing(STORE1059(i11[1], +(+(i42[1], 1), 1))), ≥)∧0 = 0∧0 = 0∧[(-1)bso_17] ≥ 0)

For Pair STORE1059(i11, i70) → LOAD761(i11, i70) the following chains were created:
• We consider the chain STORE1059(i11[2], i70[2]) → LOAD761(i11[2], i70[2]), LOAD761(i11[0], i42[0]) → COND_LOAD761(&&(!(=(%(+(i42[0], 1), 2), 0)), >(i11[0], i42[0])), i11[0], i42[0]) 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)    ((UIncreasing(LOAD761(i11[2], i70[2])), ≥)∧[1 + (-1)bso_19] ≥ 0)

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

(28)    ((UIncreasing(LOAD761(i11[2], i70[2])), ≥)∧[1 + (-1)bso_19] ≥ 0)

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

(29)    ((UIncreasing(LOAD761(i11[2], i70[2])), ≥)∧[1 + (-1)bso_19] ≥ 0)

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

(30)    ((UIncreasing(LOAD761(i11[2], i70[2])), ≥)∧0 = 0∧0 = 0∧[1 + (-1)bso_19] ≥ 0)

• We consider the chain STORE1059(i11[2], i70[2]) → LOAD761(i11[2], i70[2]), LOAD761(i11[3], i42[3]) → COND_LOAD7611(&&(>(i11[3], i42[3]), =(0, %(+(i42[3], 1), 2))), i11[3], i42[3]) which results in the following constraint:

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

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

(33)    ((UIncreasing(LOAD761(i11[2], i70[2])), ≥)∧[1 + (-1)bso_19] ≥ 0)

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

(34)    ((UIncreasing(LOAD761(i11[2], i70[2])), ≥)∧[1 + (-1)bso_19] ≥ 0)

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

(35)    ((UIncreasing(LOAD761(i11[2], i70[2])), ≥)∧[1 + (-1)bso_19] ≥ 0)

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

(36)    ((UIncreasing(LOAD761(i11[2], i70[2])), ≥)∧0 = 0∧0 = 0∧[1 + (-1)bso_19] ≥ 0)

For Pair LOAD761(i11, i42) → COND_LOAD7611(&&(>(i11, i42), =(0, %(+(i42, 1), 2))), i11, i42) the following chains were created:
• We consider the chain LOAD761(i11[3], i42[3]) → COND_LOAD7611(&&(>(i11[3], i42[3]), =(0, %(+(i42[3], 1), 2))), i11[3], i42[3]), COND_LOAD7611(TRUE, i11[4], i42[4]) → LOAD761(i11[4], +(i42[4], 1)) which results in the following constraint:

(37)    (i11[3]=i11[4]i42[3]=i42[4]&&(>(i11[3], i42[3]), =(0, %(+(i42[3], 1), 2)))=TRUELOAD761(i11[3], i42[3])≥NonInfC∧LOAD761(i11[3], i42[3])≥COND_LOAD7611(&&(>(i11[3], i42[3]), =(0, %(+(i42[3], 1), 2))), i11[3], i42[3])∧(UIncreasing(COND_LOAD7611(&&(>(i11[3], i42[3]), =(0, %(+(i42[3], 1), 2))), i11[3], i42[3])), ≥))

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

(38)    (>(i11[3], i42[3])=TRUE>=(0, %(+(i42[3], 1), 2))=TRUE<=(0, %(+(i42[3], 1), 2))=TRUELOAD761(i11[3], i42[3])≥NonInfC∧LOAD761(i11[3], i42[3])≥COND_LOAD7611(&&(>(i11[3], i42[3]), =(0, %(+(i42[3], 1), 2))), i11[3], i42[3])∧(UIncreasing(COND_LOAD7611(&&(>(i11[3], i42[3]), =(0, %(+(i42[3], 1), 2))), i11[3], i42[3])), ≥))

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

(39)    (i11[3] + [-1] + [-1]i42[3] ≥ 0∧[-1]min{[2], [-2]} ≥ 0∧max{[2], [-2]} ≥ 0 ⇒ (UIncreasing(COND_LOAD7611(&&(>(i11[3], i42[3]), =(0, %(+(i42[3], 1), 2))), i11[3], i42[3])), ≥)∧[(-1)Bound*bni_20] + [bni_20]i11[3] + [(-1)bni_20]i42[3] ≥ 0∧[1 + (-1)bso_21] ≥ 0)

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

(40)    (i11[3] + [-1] + [-1]i42[3] ≥ 0∧[-1]min{[2], [-2]} ≥ 0∧max{[2], [-2]} ≥ 0 ⇒ (UIncreasing(COND_LOAD7611(&&(>(i11[3], i42[3]), =(0, %(+(i42[3], 1), 2))), i11[3], i42[3])), ≥)∧[(-1)Bound*bni_20] + [bni_20]i11[3] + [(-1)bni_20]i42[3] ≥ 0∧[1 + (-1)bso_21] ≥ 0)

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

(41)    (i11[3] + [-1] + [-1]i42[3] ≥ 0∧[4] ≥ 0∧[2] ≥ 0∧[2] ≥ 0 ⇒ (UIncreasing(COND_LOAD7611(&&(>(i11[3], i42[3]), =(0, %(+(i42[3], 1), 2))), i11[3], i42[3])), ≥)∧[(-1)Bound*bni_20] + [bni_20]i11[3] + [(-1)bni_20]i42[3] ≥ 0∧[1 + (-1)bso_21] ≥ 0)

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

(42)    (i11[3] ≥ 0∧[4] ≥ 0∧[2] ≥ 0∧[2] ≥ 0 ⇒ (UIncreasing(COND_LOAD7611(&&(>(i11[3], i42[3]), =(0, %(+(i42[3], 1), 2))), i11[3], i42[3])), ≥)∧[(-1)Bound*bni_20 + bni_20] + [bni_20]i11[3] ≥ 0∧[1 + (-1)bso_21] ≥ 0)

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

(43)    (i11[3] ≥ 0∧[4] ≥ 0∧[2] ≥ 0∧[2] ≥ 0∧i42[3] ≥ 0 ⇒ (UIncreasing(COND_LOAD7611(&&(>(i11[3], i42[3]), =(0, %(+(i42[3], 1), 2))), i11[3], i42[3])), ≥)∧[(-1)Bound*bni_20 + bni_20] + [bni_20]i11[3] ≥ 0∧[1 + (-1)bso_21] ≥ 0)

(44)    (i11[3] ≥ 0∧[4] ≥ 0∧[2] ≥ 0∧[2] ≥ 0∧i42[3] ≥ 0 ⇒ (UIncreasing(COND_LOAD7611(&&(>(i11[3], i42[3]), =(0, %(+(i42[3], 1), 2))), i11[3], i42[3])), ≥)∧[(-1)Bound*bni_20 + bni_20] + [bni_20]i11[3] ≥ 0∧[1 + (-1)bso_21] ≥ 0)

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

(45)    (i11[3] ≥ 0∧i42[3] ≥ 0∧[1] ≥ 0∧[1] ≥ 0∧[1] ≥ 0 ⇒ (UIncreasing(COND_LOAD7611(&&(>(i11[3], i42[3]), =(0, %(+(i42[3], 1), 2))), i11[3], i42[3])), ≥)∧[(-1)Bound*bni_20 + bni_20] + [bni_20]i11[3] ≥ 0∧[1 + (-1)bso_21] ≥ 0)

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

(46)    (i11[3] ≥ 0∧i42[3] ≥ 0∧[1] ≥ 0∧[1] ≥ 0∧[1] ≥ 0 ⇒ (UIncreasing(COND_LOAD7611(&&(>(i11[3], i42[3]), =(0, %(+(i42[3], 1), 2))), i11[3], i42[3])), ≥)∧[(-1)Bound*bni_20 + bni_20] + [bni_20]i11[3] ≥ 0∧[1 + (-1)bso_21] ≥ 0)

For Pair COND_LOAD7611(TRUE, i11, i42) → LOAD761(i11, +(i42, 1)) the following chains were created:
• We consider the chain COND_LOAD7611(TRUE, i11[4], i42[4]) → LOAD761(i11[4], +(i42[4], 1)) which results in the following constraint:

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

(48)    ((UIncreasing(LOAD761(i11[4], +(i42[4], 1))), ≥)∧[(-1)bso_23] ≥ 0)

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

(49)    ((UIncreasing(LOAD761(i11[4], +(i42[4], 1))), ≥)∧[(-1)bso_23] ≥ 0)

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

(50)    ((UIncreasing(LOAD761(i11[4], +(i42[4], 1))), ≥)∧[(-1)bso_23] ≥ 0)

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

(51)    ((UIncreasing(LOAD761(i11[4], +(i42[4], 1))), ≥)∧0 = 0∧0 = 0∧[(-1)bso_23] ≥ 0)

To summarize, we get the following constraints P for the following pairs.
• LOAD761(i11, i42) → COND_LOAD761(&&(!(=(%(+(i42, 1), 2), 0)), >(i11, i42)), i11, i42)
• (i11[0] ≥ 0∧[1] ≥ 0∧i42[0] ≥ 0∧[1] ≥ 0 ⇒ (UIncreasing(COND_LOAD761(&&(!(=(%(+(i42[0], 1), 2), 0)), >(i11[0], i42[0])), i11[0], i42[0])), ≥)∧[(-1)Bound*bni_14 + bni_14] + [bni_14]i11[0] ≥ 0∧[1 + (-1)bso_15] ≥ 0)
• (i11[0] ≥ 0∧[1] ≥ 0∧i42[0] ≥ 0∧[1] ≥ 0 ⇒ (UIncreasing(COND_LOAD761(&&(!(=(%(+(i42[0], 1), 2), 0)), >(i11[0], i42[0])), i11[0], i42[0])), ≥)∧[(-1)Bound*bni_14 + bni_14] + [bni_14]i11[0] ≥ 0∧[1 + (-1)bso_15] ≥ 0)
• (i11[0] ≥ 0∧[1] ≥ 0∧i42[0] ≥ 0∧[1] ≥ 0 ⇒ (UIncreasing(COND_LOAD761(&&(!(=(%(+(i42[0], 1), 2), 0)), >(i11[0], i42[0])), i11[0], i42[0])), ≥)∧[(-1)Bound*bni_14 + bni_14] + [bni_14]i11[0] ≥ 0∧[1 + (-1)bso_15] ≥ 0)
• (i11[0] ≥ 0∧[1] ≥ 0∧i42[0] ≥ 0∧[1] ≥ 0 ⇒ (UIncreasing(COND_LOAD761(&&(!(=(%(+(i42[0], 1), 2), 0)), >(i11[0], i42[0])), i11[0], i42[0])), ≥)∧[(-1)Bound*bni_14 + bni_14] + [bni_14]i11[0] ≥ 0∧[1 + (-1)bso_15] ≥ 0)

• COND_LOAD761(TRUE, i11, i42) → STORE1059(i11, +(+(i42, 1), 1))
• ((UIncreasing(STORE1059(i11[1], +(+(i42[1], 1), 1))), ≥)∧0 = 0∧0 = 0∧[(-1)bso_17] ≥ 0)

• STORE1059(i11, i70) → LOAD761(i11, i70)
• ((UIncreasing(LOAD761(i11[2], i70[2])), ≥)∧0 = 0∧0 = 0∧[1 + (-1)bso_19] ≥ 0)
• ((UIncreasing(LOAD761(i11[2], i70[2])), ≥)∧0 = 0∧0 = 0∧[1 + (-1)bso_19] ≥ 0)

• LOAD761(i11, i42) → COND_LOAD7611(&&(>(i11, i42), =(0, %(+(i42, 1), 2))), i11, i42)
• (i11[3] ≥ 0∧i42[3] ≥ 0∧[1] ≥ 0∧[1] ≥ 0∧[1] ≥ 0 ⇒ (UIncreasing(COND_LOAD7611(&&(>(i11[3], i42[3]), =(0, %(+(i42[3], 1), 2))), i11[3], i42[3])), ≥)∧[(-1)Bound*bni_20 + bni_20] + [bni_20]i11[3] ≥ 0∧[1 + (-1)bso_21] ≥ 0)
• (i11[3] ≥ 0∧i42[3] ≥ 0∧[1] ≥ 0∧[1] ≥ 0∧[1] ≥ 0 ⇒ (UIncreasing(COND_LOAD7611(&&(>(i11[3], i42[3]), =(0, %(+(i42[3], 1), 2))), i11[3], i42[3])), ≥)∧[(-1)Bound*bni_20 + bni_20] + [bni_20]i11[3] ≥ 0∧[1 + (-1)bso_21] ≥ 0)

• ((UIncreasing(LOAD761(i11[4], +(i42[4], 1))), ≥)∧0 = 0∧0 = 0∧[(-1)bso_23] ≥ 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(LOAD761(x1, x2)) = x1 + [-1]x2
POL(COND_LOAD761(x1, x2, x3)) = [-1] + x2 + [-1]x3
POL(&&(x1, x2)) = [-1]
POL(!(x1)) = [-1]
POL(=(x1, x2)) = [-1]
POL(+(x1, x2)) = x1 + x2
POL(1) = [1]
POL(2) = [2]
POL(0) = 0
POL(>(x1, x2)) = [-1]
POL(STORE1059(x1, x2)) = [1] + [-1]x2 + x1
POL(COND_LOAD7611(x1, x2, x3)) = [-1] + [-1]x3 + x2

Polynomial Interpretations with Context Sensitive Arithemetic Replacement
POL(TermCSAR-Mode @ Context)

POL(%(x1, 2)-1 @ {}) = min{x2, [-1]x2}
POL(%(x1, 2)1 @ {}) = max{x2, [-1]x2}

The following pairs are in P>:

The following pairs are in Pbound:

The following pairs are in P:

COND_LOAD761(TRUE, i11[1], i42[1]) → STORE1059(i11[1], +(+(i42[1], 1), 1))

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:

Integer

R is empty.

The integer pair graph contains the following rules and edges:
(1): COND_LOAD761(TRUE, i11[1], i42[1]) → STORE1059(i11[1], i42[1] + 1 + 1)

The set Q consists of the following terms:
Store1059(x0, x1)

### (12) IDependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 2 less nodes.

### (14) Obligation:

IDP problem:
The following function symbols are pre-defined:
 != ~ Neq: (Integer, Integer) -> Boolean * ~ Mul: (Integer, Integer) -> Integer >= ~ Ge: (Integer, Integer) -> Boolean -1 ~ UnaryMinus: (Integer) -> Integer | ~ Bwor: (Integer, Integer) -> Integer / ~ Div: (Integer, Integer) -> Integer = ~ Eq: (Integer, Integer) -> Boolean ~ Bwxor: (Integer, Integer) -> Integer || ~ Lor: (Boolean, Boolean) -> Boolean ! ~ Lnot: (Boolean) -> Boolean < ~ Lt: (Integer, Integer) -> Boolean - ~ Sub: (Integer, Integer) -> Integer <= ~ Le: (Integer, Integer) -> Boolean > ~ Gt: (Integer, Integer) -> Boolean ~ ~ Bwnot: (Integer) -> Integer % ~ Mod: (Integer, Integer) -> Integer & ~ Bwand: (Integer, Integer) -> Integer + ~ Add: (Integer, Integer) -> Integer && ~ Land: (Boolean, Boolean) -> Boolean

The following domains are used:

Integer

R is empty.

The integer pair graph contains the following rules and edges:
(1): COND_LOAD761(TRUE, i11[1], i42[1]) → STORE1059(i11[1], i42[1] + 1 + 1)
(2): STORE1059(i11[2], i70[2]) → LOAD761(i11[2], i70[2])

(1) -> (2), if ((i42[1] + 1 + 1* i70[2])∧(i11[1]* i11[2]))

The set Q consists of the following terms: