### (0) Obligation:

JBC Problem based on JBC Program:
Manifest-Version: 1.0 Created-By: 1.6.0_16 (Sun Microsystems Inc.) Main-Class: MinusMin
`public class MinusMin{  public static int min (int x, int y) {    if (x < y) return x;    else return y;  }  public static void main(String[] args) {    Random.args = args;    int x = Random.random();    int y = Random.random();    int res = 0;    while (min(x-1,y) == y) {      y++;      res++;    }  }}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 202 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:
Load1487(i14, i214, i215) → Cond_Load1487(i14 - 1 >= i214, i14, i214, i215)
Load1487(i14, i14 - 1, i215) → Cond_Load14871(i14 - 1 < i14 - 1 && i215 + 1 > 0, i14, i14 - 1, i215)
Cond_Load14871(TRUE, i14, i14 - 1, i215) → Load1487(i14, i14 - 1 + 1, i215 + 1)
The set Q consists of the following terms:
Cond_Load14871(TRUE, x0, x0 - 1, 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:

Integer, Boolean

The ITRS R consists of the following rules:
Load1487(i14, i214, i215) → Cond_Load1487(i14 - 1 >= i214, i14, i214, i215)
Load1487(i14, i14 - 1, i215) → Cond_Load14871(i14 - 1 < i14 - 1 && i215 + 1 > 0, i14, i14 - 1, i215)
Cond_Load14871(TRUE, i14, i14 - 1, i215) → Load1487(i14, i14 - 1 + 1, i215 + 1)

The integer pair graph contains the following rules and edges:
(0): LOAD1487(i14[0], i214[0], i215[0]) → COND_LOAD1487(i14[0] - 1 >= i214[0], i14[0], i214[0], i215[0])
(2): LOAD1538(i14[2], i215[2], i214[2]) → LOAD1487(i14[2], i214[2] + 1, i215[2] + 1)
(3): LOAD1487(i14[3], i14[3] - 1, i215[3]) → COND_LOAD14871(i14[3] - 1 < i14[3] - 1 && i215[3] + 1 > 0, i14[3], i14[3] - 1, i215[3])
(4): COND_LOAD14871(TRUE, i14[4], i14[4] - 1, i215[4]) → LOAD1487(i14[4], i14[4] - 1 + 1, i215[4] + 1)

(0) -> (1), if ((i215[0]* i215[1])∧(i214[0]* i214[1])∧(i14[0]* i14[1])∧(i14[0] - 1 >= i214[0]* TRUE))

(1) -> (2), if ((i14[1]* i14[2])∧(i214[1]* i214[2])∧(i215[1]* i215[2]))

(2) -> (0), if ((i215[2] + 1* i215[0])∧(i214[2] + 1* i214[0])∧(i14[2]* i14[0]))

(2) -> (3), if ((i14[2]* i14[3])∧(i214[2] + 1* i14[3] - 1)∧(i215[2] + 1* i215[3]))

(3) -> (4), if ((i14[3]* i14[4])∧(i14[3] - 1 < i14[3] - 1 && i215[3] + 1 > 0* TRUE)∧(i215[3]* i215[4])∧(i14[3] - 1* i14[4] - 1))

(4) -> (0), if ((i14[4] - 1 + 1* i214[0])∧(i14[4]* i14[0])∧(i215[4] + 1* i215[0]))

(4) -> (3), if ((i215[4] + 1* i215[3])∧(i14[4] - 1 + 1* i14[3] - 1)∧(i14[4]* i14[3]))

The set Q consists of the following terms:
Cond_Load14871(TRUE, x0, x0 - 1, 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:

Integer, Boolean

R is empty.

The integer pair graph contains the following rules and edges:
(0): LOAD1487(i14[0], i214[0], i215[0]) → COND_LOAD1487(i14[0] - 1 >= i214[0], i14[0], i214[0], i215[0])
(2): LOAD1538(i14[2], i215[2], i214[2]) → LOAD1487(i14[2], i214[2] + 1, i215[2] + 1)
(3): LOAD1487(i14[3], i14[3] - 1, i215[3]) → COND_LOAD14871(i14[3] - 1 < i14[3] - 1 && i215[3] + 1 > 0, i14[3], i14[3] - 1, i215[3])
(4): COND_LOAD14871(TRUE, i14[4], i14[4] - 1, i215[4]) → LOAD1487(i14[4], i14[4] - 1 + 1, i215[4] + 1)

(0) -> (1), if ((i215[0]* i215[1])∧(i214[0]* i214[1])∧(i14[0]* i14[1])∧(i14[0] - 1 >= i214[0]* TRUE))

(1) -> (2), if ((i14[1]* i14[2])∧(i214[1]* i214[2])∧(i215[1]* i215[2]))

(2) -> (0), if ((i215[2] + 1* i215[0])∧(i214[2] + 1* i214[0])∧(i14[2]* i14[0]))

(2) -> (3), if ((i14[2]* i14[3])∧(i214[2] + 1* i14[3] - 1)∧(i215[2] + 1* i215[3]))

(3) -> (4), if ((i14[3]* i14[4])∧(i14[3] - 1 < i14[3] - 1 && i215[3] + 1 > 0* TRUE)∧(i215[3]* i215[4])∧(i14[3] - 1* i14[4] - 1))

(4) -> (0), if ((i14[4] - 1 + 1* i214[0])∧(i14[4]* i14[0])∧(i215[4] + 1* i215[0]))

(4) -> (3), if ((i215[4] + 1* i215[3])∧(i14[4] - 1 + 1* i14[3] - 1)∧(i14[4]* i14[3]))

The set Q consists of the following terms:
Cond_Load14871(TRUE, x0, x0 - 1, 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 LOAD1487(i14, i214, i215) → COND_LOAD1487(>=(-(i14, 1), i214), i14, i214, i215) the following chains were created:
• We consider the chain LOAD1487(i14[0], i214[0], i215[0]) → COND_LOAD1487(>=(-(i14[0], 1), i214[0]), i14[0], i214[0], i215[0]), COND_LOAD1487(TRUE, i14[1], i214[1], i215[1]) → LOAD1538(i14[1], i215[1], i214[1]) which results in the following constraint:

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)    (i14[0] + [-1] + [-1]i214[0] ≥ 0 ⇒ (UIncreasing(COND_LOAD1487(>=(-(i14[0], 1), i214[0]), i14[0], i214[0], i215[0])), ≥)∧[(-1)bni_25 + (-1)Bound*bni_25] + [(-1)bni_25]i215[0] + [(-1)bni_25]i214[0] + [(-1)bni_25]i14[0] ≥ 0∧[(-1)bso_26] ≥ 0)

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

(4)    (i14[0] + [-1] + [-1]i214[0] ≥ 0 ⇒ (UIncreasing(COND_LOAD1487(>=(-(i14[0], 1), i214[0]), i14[0], i214[0], i215[0])), ≥)∧[(-1)bni_25 + (-1)Bound*bni_25] + [(-1)bni_25]i215[0] + [(-1)bni_25]i214[0] + [(-1)bni_25]i14[0] ≥ 0∧[(-1)bso_26] ≥ 0)

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

(5)    (i14[0] + [-1] + [-1]i214[0] ≥ 0 ⇒ (UIncreasing(COND_LOAD1487(>=(-(i14[0], 1), i214[0]), i14[0], i214[0], i215[0])), ≥)∧[(-1)bni_25 + (-1)Bound*bni_25] + [(-1)bni_25]i215[0] + [(-1)bni_25]i214[0] + [(-1)bni_25]i14[0] ≥ 0∧[(-1)bso_26] ≥ 0)

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

(6)    (i14[0] + [-1] + [-1]i214[0] ≥ 0 ⇒ (UIncreasing(COND_LOAD1487(>=(-(i14[0], 1), i214[0]), i14[0], i214[0], i215[0])), ≥)∧[(-1)bni_25] = 0∧[(-1)bni_25 + (-1)Bound*bni_25] + [(-1)bni_25]i214[0] + [(-1)bni_25]i14[0] ≥ 0∧0 = 0∧[(-1)bso_26] ≥ 0)

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

(7)    (i14[0] ≥ 0 ⇒ (UIncreasing(COND_LOAD1487(>=(-(i14[0], 1), i214[0]), i14[0], i214[0], i215[0])), ≥)∧[(-1)bni_25] = 0∧[(-2)bni_25 + (-1)Bound*bni_25] + [(-2)bni_25]i214[0] + [(-1)bni_25]i14[0] ≥ 0∧0 = 0∧[(-1)bso_26] ≥ 0)

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

(8)    (i14[0] ≥ 0∧i214[0] ≥ 0 ⇒ (UIncreasing(COND_LOAD1487(>=(-(i14[0], 1), i214[0]), i14[0], i214[0], i215[0])), ≥)∧[(-1)bni_25] = 0∧[(-2)bni_25 + (-1)Bound*bni_25] + [(2)bni_25]i214[0] + [(-1)bni_25]i14[0] ≥ 0∧0 = 0∧[(-1)bso_26] ≥ 0)

(9)    (i14[0] ≥ 0∧i214[0] ≥ 0 ⇒ (UIncreasing(COND_LOAD1487(>=(-(i14[0], 1), i214[0]), i14[0], i214[0], i215[0])), ≥)∧[(-1)bni_25] = 0∧[(-2)bni_25 + (-1)Bound*bni_25] + [(-2)bni_25]i214[0] + [(-1)bni_25]i14[0] ≥ 0∧0 = 0∧[(-1)bso_26] ≥ 0)

For Pair COND_LOAD1487(TRUE, i14, i214, i215) → LOAD1538(i14, i215, i214) the following chains were created:
• We consider the chain COND_LOAD1487(TRUE, i14[1], i214[1], i215[1]) → LOAD1538(i14[1], i215[1], i214[1]), LOAD1538(i14[2], i215[2], i214[2]) → LOAD1487(i14[2], +(i214[2], 1), +(i215[2], 1)) which results in the following constraint:

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

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

(12)    ((UIncreasing(LOAD1538(i14[1], i215[1], i214[1])), ≥)∧[(-1)bso_28] ≥ 0)

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

(13)    ((UIncreasing(LOAD1538(i14[1], i215[1], i214[1])), ≥)∧[(-1)bso_28] ≥ 0)

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

(14)    ((UIncreasing(LOAD1538(i14[1], i215[1], i214[1])), ≥)∧[(-1)bso_28] ≥ 0)

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

(15)    ((UIncreasing(LOAD1538(i14[1], i215[1], i214[1])), ≥)∧0 = 0∧0 = 0∧0 = 0∧[(-1)bso_28] ≥ 0)

For Pair LOAD1538(i14, i215, i214) → LOAD1487(i14, +(i214, 1), +(i215, 1)) the following chains were created:
• We consider the chain COND_LOAD1487(TRUE, i14[1], i214[1], i215[1]) → LOAD1538(i14[1], i215[1], i214[1]), LOAD1538(i14[2], i215[2], i214[2]) → LOAD1487(i14[2], +(i214[2], 1), +(i215[2], 1)), LOAD1487(i14[0], i214[0], i215[0]) → COND_LOAD1487(>=(-(i14[0], 1), i214[0]), i14[0], i214[0], i215[0]) which results in the following constraint:

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

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

(18)    ((UIncreasing(LOAD1487(i14[2], +(i214[2], 1), +(i215[2], 1))), ≥)∧[2 + (-1)bso_30] ≥ 0)

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

(19)    ((UIncreasing(LOAD1487(i14[2], +(i214[2], 1), +(i215[2], 1))), ≥)∧[2 + (-1)bso_30] ≥ 0)

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

(20)    ((UIncreasing(LOAD1487(i14[2], +(i214[2], 1), +(i215[2], 1))), ≥)∧[2 + (-1)bso_30] ≥ 0)

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

(21)    ((UIncreasing(LOAD1487(i14[2], +(i214[2], 1), +(i215[2], 1))), ≥)∧0 = 0∧0 = 0∧0 = 0∧[2 + (-1)bso_30] ≥ 0)

• We consider the chain COND_LOAD1487(TRUE, i14[1], i214[1], i215[1]) → LOAD1538(i14[1], i215[1], i214[1]), LOAD1538(i14[2], i215[2], i214[2]) → LOAD1487(i14[2], +(i214[2], 1), +(i215[2], 1)), LOAD1487(i14[3], -(i14[3], 1), i215[3]) → COND_LOAD14871(&&(<(-(i14[3], 1), -(i14[3], 1)), >(+(i215[3], 1), 0)), i14[3], -(i14[3], 1), i215[3]) which results in the following constraint:

We simplified constraint (22) using rules (III), (IV) which results in the following new constraint:

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

(24)    (i214[2] + [2] + [-1]i14[1] ≥ 0 ⇒ (UIncreasing(LOAD1487(i14[2], +(i214[2], 1), +(i215[2], 1))), ≥)∧0 ≥ 0∧[2 + (-1)bso_30] ≥ 0)

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

(25)    (i214[2] + [2] + [-1]i14[1] ≥ 0 ⇒ (UIncreasing(LOAD1487(i14[2], +(i214[2], 1), +(i215[2], 1))), ≥)∧0 ≥ 0∧[2 + (-1)bso_30] ≥ 0)

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

(26)    (i214[2] + [2] + [-1]i14[1] ≥ 0 ⇒ (UIncreasing(LOAD1487(i14[2], +(i214[2], 1), +(i215[2], 1))), ≥)∧0 ≥ 0∧[2 + (-1)bso_30] ≥ 0)

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

(27)    (i214[2] + [2] + [-1]i14[1] ≥ 0 ⇒ (UIncreasing(LOAD1487(i14[2], +(i214[2], 1), +(i215[2], 1))), ≥)∧0 = 0∧0 ≥ 0∧0 = 0∧[2 + (-1)bso_30] ≥ 0)

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

(28)    (i14[1] ≥ 0 ⇒ (UIncreasing(LOAD1487(i14[2], +(i214[2], 1), +(i215[2], 1))), ≥)∧0 = 0∧0 ≥ 0∧0 = 0∧[2 + (-1)bso_30] ≥ 0)

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

(29)    (i14[1] ≥ 0∧i214[2] ≥ 0 ⇒ (UIncreasing(LOAD1487(i14[2], +(i214[2], 1), +(i215[2], 1))), ≥)∧0 = 0∧0 ≥ 0∧0 = 0∧[2 + (-1)bso_30] ≥ 0)

(30)    (i14[1] ≥ 0∧i214[2] ≥ 0 ⇒ (UIncreasing(LOAD1487(i14[2], +(i214[2], 1), +(i215[2], 1))), ≥)∧0 = 0∧0 ≥ 0∧0 = 0∧[2 + (-1)bso_30] ≥ 0)

For Pair LOAD1487(i14, -(i14, 1), i215) → COND_LOAD14871(&&(<(-(i14, 1), -(i14, 1)), >(+(i215, 1), 0)), i14, -(i14, 1), i215) the following chains were created:
• We consider the chain LOAD1487(i14[3], -(i14[3], 1), i215[3]) → COND_LOAD14871(&&(<(-(i14[3], 1), -(i14[3], 1)), >(+(i215[3], 1), 0)), i14[3], -(i14[3], 1), i215[3]), COND_LOAD14871(TRUE, i14[4], -(i14[4], 1), i215[4]) → LOAD1487(i14[4], +(-(i14[4], 1), 1), +(i215[4], 1)) which results in the following constraint:

(31)    (i14[3]=i14[4]&&(<(-(i14[3], 1), -(i14[3], 1)), >(+(i215[3], 1), 0))=TRUEi215[3]=i215[4]-(i14[3], 1)=-(i14[4], 1) ⇒ LOAD1487(i14[3], -(i14[3], 1), i215[3])≥NonInfC∧LOAD1487(i14[3], -(i14[3], 1), i215[3])≥COND_LOAD14871(&&(<(-(i14[3], 1), -(i14[3], 1)), >(+(i215[3], 1), 0)), i14[3], -(i14[3], 1), i215[3])∧(UIncreasing(COND_LOAD14871(&&(<(-(i14[3], 1), -(i14[3], 1)), >(+(i215[3], 1), 0)), i14[3], -(i14[3], 1), i215[3])), ≥))

We simplified constraint (31) using rules (III), (IV), (DELETE_TRIVIAL_REDUCESTO), (IDP_BOOLEAN) which results in the following new constraint:

(32)    (<(-(i14[3], 1), -(i14[3], 1))=TRUE>(+(i215[3], 1), 0)=TRUELOAD1487(i14[3], -(i14[3], 1), i215[3])≥NonInfC∧LOAD1487(i14[3], -(i14[3], 1), i215[3])≥COND_LOAD14871(&&(<(-(i14[3], 1), -(i14[3], 1)), >(+(i215[3], 1), 0)), i14[3], -(i14[3], 1), i215[3])∧(UIncreasing(COND_LOAD14871(&&(<(-(i14[3], 1), -(i14[3], 1)), >(+(i215[3], 1), 0)), i14[3], -(i14[3], 1), i215[3])), ≥))

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

(33)    ([-1] ≥ 0∧i215[3] ≥ 0 ⇒ (UIncreasing(COND_LOAD14871(&&(<(-(i14[3], 1), -(i14[3], 1)), >(+(i215[3], 1), 0)), i14[3], -(i14[3], 1), i215[3])), ≥)∧[(-1)Bound*bni_31] + [(-1)bni_31]i215[3] + [(-2)bni_31]i14[3] ≥ 0∧[-1 + (-1)bso_32] ≥ 0)

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

(34)    ([-1] ≥ 0∧i215[3] ≥ 0 ⇒ (UIncreasing(COND_LOAD14871(&&(<(-(i14[3], 1), -(i14[3], 1)), >(+(i215[3], 1), 0)), i14[3], -(i14[3], 1), i215[3])), ≥)∧[(-1)Bound*bni_31] + [(-1)bni_31]i215[3] + [(-2)bni_31]i14[3] ≥ 0∧[-1 + (-1)bso_32] ≥ 0)

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

(35)    ([-1] ≥ 0∧i215[3] ≥ 0 ⇒ (UIncreasing(COND_LOAD14871(&&(<(-(i14[3], 1), -(i14[3], 1)), >(+(i215[3], 1), 0)), i14[3], -(i14[3], 1), i215[3])), ≥)∧[(-1)Bound*bni_31] + [(-1)bni_31]i215[3] + [(-2)bni_31]i14[3] ≥ 0∧[-1 + (-1)bso_32] ≥ 0)

We solved constraint (35) using rule (IDP_SMT_SPLIT).

For Pair COND_LOAD14871(TRUE, i14, -(i14, 1), i215) → LOAD1487(i14, +(-(i14, 1), 1), +(i215, 1)) the following chains were created:
• We consider the chain LOAD1487(i14[3], -(i14[3], 1), i215[3]) → COND_LOAD14871(&&(<(-(i14[3], 1), -(i14[3], 1)), >(+(i215[3], 1), 0)), i14[3], -(i14[3], 1), i215[3]), COND_LOAD14871(TRUE, i14[4], -(i14[4], 1), i215[4]) → LOAD1487(i14[4], +(-(i14[4], 1), 1), +(i215[4], 1)), LOAD1487(i14[0], i214[0], i215[0]) → COND_LOAD1487(>=(-(i14[0], 1), i214[0]), i14[0], i214[0], i215[0]) which results in the following constraint:

(36)    (i14[3]=i14[4]&&(<(-(i14[3], 1), -(i14[3], 1)), >(+(i215[3], 1), 0))=TRUEi215[3]=i215[4]-(i14[3], 1)=-(i14[4], 1)∧+(-(i14[4], 1), 1)=i214[0]i14[4]=i14[0]+(i215[4], 1)=i215[0]COND_LOAD14871(TRUE, i14[4], -(i14[4], 1), i215[4])≥NonInfC∧COND_LOAD14871(TRUE, i14[4], -(i14[4], 1), i215[4])≥LOAD1487(i14[4], +(-(i14[4], 1), 1), +(i215[4], 1))∧(UIncreasing(LOAD1487(i14[4], +(-(i14[4], 1), 1), +(i215[4], 1))), ≥))

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

(37)    (<(-(i14[3], 1), -(i14[3], 1))=TRUE>(+(i215[3], 1), 0)=TRUECOND_LOAD14871(TRUE, i14[3], -(i14[3], 1), i215[3])≥NonInfC∧COND_LOAD14871(TRUE, i14[3], -(i14[3], 1), i215[3])≥LOAD1487(i14[3], +(-(i14[3], 1), 1), +(i215[3], 1))∧(UIncreasing(LOAD1487(i14[4], +(-(i14[4], 1), 1), +(i215[4], 1))), ≥))

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

(38)    ([-1] ≥ 0∧i215[3] ≥ 0 ⇒ (UIncreasing(LOAD1487(i14[4], +(-(i14[4], 1), 1), +(i215[4], 1))), ≥)∧[(-1)Bound*bni_33] + [(-1)bni_33]i215[3] + [(-2)bni_33]i14[3] ≥ 0∧[2 + (-1)bso_34] ≥ 0)

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

(39)    ([-1] ≥ 0∧i215[3] ≥ 0 ⇒ (UIncreasing(LOAD1487(i14[4], +(-(i14[4], 1), 1), +(i215[4], 1))), ≥)∧[(-1)Bound*bni_33] + [(-1)bni_33]i215[3] + [(-2)bni_33]i14[3] ≥ 0∧[2 + (-1)bso_34] ≥ 0)

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

(40)    ([-1] ≥ 0∧i215[3] ≥ 0 ⇒ (UIncreasing(LOAD1487(i14[4], +(-(i14[4], 1), 1), +(i215[4], 1))), ≥)∧[(-1)Bound*bni_33] + [(-1)bni_33]i215[3] + [(-2)bni_33]i14[3] ≥ 0∧[2 + (-1)bso_34] ≥ 0)

We solved constraint (40) using rule (IDP_SMT_SPLIT).
• We consider the chain LOAD1487(i14[3], -(i14[3], 1), i215[3]) → COND_LOAD14871(&&(<(-(i14[3], 1), -(i14[3], 1)), >(+(i215[3], 1), 0)), i14[3], -(i14[3], 1), i215[3]), COND_LOAD14871(TRUE, i14[4], -(i14[4], 1), i215[4]) → LOAD1487(i14[4], +(-(i14[4], 1), 1), +(i215[4], 1)), LOAD1487(i14[3], -(i14[3], 1), i215[3]) → COND_LOAD14871(&&(<(-(i14[3], 1), -(i14[3], 1)), >(+(i215[3], 1), 0)), i14[3], -(i14[3], 1), i215[3]) which results in the following constraint:

(41)    (i14[3]=i14[4]&&(<(-(i14[3], 1), -(i14[3], 1)), >(+(i215[3], 1), 0))=TRUEi215[3]=i215[4]-(i14[3], 1)=-(i14[4], 1)∧+(i215[4], 1)=i215[3]1+(-(i14[4], 1), 1)=-(i14[3]1, 1)∧i14[4]=i14[3]1COND_LOAD14871(TRUE, i14[4], -(i14[4], 1), i215[4])≥NonInfC∧COND_LOAD14871(TRUE, i14[4], -(i14[4], 1), i215[4])≥LOAD1487(i14[4], +(-(i14[4], 1), 1), +(i215[4], 1))∧(UIncreasing(LOAD1487(i14[4], +(-(i14[4], 1), 1), +(i215[4], 1))), ≥))

We simplified constraint (41) using rules (III), (IV), (DELETE_TRIVIAL_REDUCESTO), (IDP_BOOLEAN) which results in the following new constraint:

(42)    (+(-(i14[3], 1), 1)=-(i14[3], 1)∧<(-(i14[3], 1), -(i14[3], 1))=TRUE>(+(i215[3], 1), 0)=TRUECOND_LOAD14871(TRUE, i14[3], -(i14[3], 1), i215[3])≥NonInfC∧COND_LOAD14871(TRUE, i14[3], -(i14[3], 1), i215[3])≥LOAD1487(i14[3], +(-(i14[3], 1), 1), +(i215[3], 1))∧(UIncreasing(LOAD1487(i14[4], +(-(i14[4], 1), 1), +(i215[4], 1))), ≥))

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

(43)    ([1] ≥ 0∧[-1] ≥ 0∧i215[3] ≥ 0 ⇒ (UIncreasing(LOAD1487(i14[4], +(-(i14[4], 1), 1), +(i215[4], 1))), ≥)∧[(-1)Bound*bni_33] + [(-1)bni_33]i215[3] + [(-2)bni_33]i14[3] ≥ 0∧[2 + (-1)bso_34] ≥ 0)

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

(44)    ([1] ≥ 0∧[-1] ≥ 0∧i215[3] ≥ 0 ⇒ (UIncreasing(LOAD1487(i14[4], +(-(i14[4], 1), 1), +(i215[4], 1))), ≥)∧[(-1)Bound*bni_33] + [(-1)bni_33]i215[3] + [(-2)bni_33]i14[3] ≥ 0∧[2 + (-1)bso_34] ≥ 0)

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

(45)    ([1] ≥ 0∧[-1] ≥ 0∧i215[3] ≥ 0 ⇒ (UIncreasing(LOAD1487(i14[4], +(-(i14[4], 1), 1), +(i215[4], 1))), ≥)∧[(-1)Bound*bni_33] + [(-1)bni_33]i215[3] + [(-2)bni_33]i14[3] ≥ 0∧[2 + (-1)bso_34] ≥ 0)

We solved constraint (45) using rule (IDP_SMT_SPLIT).

To summarize, we get the following constraints P for the following pairs.
• (i14[0] ≥ 0∧i214[0] ≥ 0 ⇒ (UIncreasing(COND_LOAD1487(>=(-(i14[0], 1), i214[0]), i14[0], i214[0], i215[0])), ≥)∧[(-1)bni_25] = 0∧[(-2)bni_25 + (-1)Bound*bni_25] + [(2)bni_25]i214[0] + [(-1)bni_25]i14[0] ≥ 0∧0 = 0∧[(-1)bso_26] ≥ 0)
• (i14[0] ≥ 0∧i214[0] ≥ 0 ⇒ (UIncreasing(COND_LOAD1487(>=(-(i14[0], 1), i214[0]), i14[0], i214[0], i215[0])), ≥)∧[(-1)bni_25] = 0∧[(-2)bni_25 + (-1)Bound*bni_25] + [(-2)bni_25]i214[0] + [(-1)bni_25]i14[0] ≥ 0∧0 = 0∧[(-1)bso_26] ≥ 0)

• ((UIncreasing(LOAD1538(i14[1], i215[1], i214[1])), ≥)∧0 = 0∧0 = 0∧0 = 0∧[(-1)bso_28] ≥ 0)

• ((UIncreasing(LOAD1487(i14[2], +(i214[2], 1), +(i215[2], 1))), ≥)∧0 = 0∧0 = 0∧0 = 0∧[2 + (-1)bso_30] ≥ 0)
• (i14[1] ≥ 0∧i214[2] ≥ 0 ⇒ (UIncreasing(LOAD1487(i14[2], +(i214[2], 1), +(i215[2], 1))), ≥)∧0 = 0∧0 ≥ 0∧0 = 0∧[2 + (-1)bso_30] ≥ 0)
• (i14[1] ≥ 0∧i214[2] ≥ 0 ⇒ (UIncreasing(LOAD1487(i14[2], +(i214[2], 1), +(i215[2], 1))), ≥)∧0 = 0∧0 ≥ 0∧0 = 0∧[2 + (-1)bso_30] ≥ 0)

• LOAD1487(i14, -(i14, 1), i215) → COND_LOAD14871(&&(<(-(i14, 1), -(i14, 1)), >(+(i215, 1), 0)), i14, -(i14, 1), i215)

• COND_LOAD14871(TRUE, i14, -(i14, 1), i215) → LOAD1487(i14, +(-(i14, 1), 1), +(i215, 1))

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(LOAD1487(x1, x2, x3)) = [-1] + [-1]x3 + [-1]x2 + [-1]x1
POL(COND_LOAD1487(x1, x2, x3, x4)) = [-1] + [-1]x4 + [-1]x3 + [-1]x2
POL(>=(x1, x2)) = [-1]
POL(-(x1, x2)) = x1 + [-1]x2
POL(1) = [1]
POL(LOAD1538(x1, x2, x3)) = [-1] + [-1]x3 + [-1]x2 + [-1]x1
POL(+(x1, x2)) = x1 + x2
POL(COND_LOAD14871(x1, x2, x3, x4)) = [-1] + [-1]x4 + [-1]x3 + [-1]x2 + [-1]x1
POL(&&(x1, x2)) = [-1]
POL(<(x1, x2)) = [-1]
POL(>(x1, x2)) = [-1]
POL(0) = 0

The following pairs are in P>:

LOAD1487(i14[3], -(i14[3], 1), i215[3]) → COND_LOAD14871(&&(<(-(i14[3], 1), -(i14[3], 1)), >(+(i215[3], 1), 0)), i14[3], -(i14[3], 1), i215[3])
COND_LOAD14871(TRUE, i14[4], -(i14[4], 1), i215[4]) → LOAD1487(i14[4], +(-(i14[4], 1), 1), +(i215[4], 1))

The following pairs are in Pbound:

LOAD1487(i14[3], -(i14[3], 1), i215[3]) → COND_LOAD14871(&&(<(-(i14[3], 1), -(i14[3], 1)), >(+(i215[3], 1), 0)), i14[3], -(i14[3], 1), i215[3])
COND_LOAD14871(TRUE, i14[4], -(i14[4], 1), i215[4]) → LOAD1487(i14[4], +(-(i14[4], 1), 1), +(i215[4], 1))

The following pairs are in P:

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:
(0): LOAD1487(i14[0], i214[0], i215[0]) → COND_LOAD1487(i14[0] - 1 >= i214[0], i14[0], i214[0], i215[0])

(0) -> (1), if ((i215[0]* i215[1])∧(i214[0]* i214[1])∧(i14[0]* i14[1])∧(i14[0] - 1 >= i214[0]* TRUE))

The set Q consists of the following terms:
Cond_Load14871(TRUE, x0, x0 - 1, 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:
(0): LOAD1487(i14[0], i214[0], i215[0]) → COND_LOAD1487(i14[0] - 1 >= i214[0], i14[0], i214[0], i215[0])
(2): LOAD1538(i14[2], i215[2], i214[2]) → LOAD1487(i14[2], i214[2] + 1, i215[2] + 1)

(2) -> (0), if ((i215[2] + 1* i215[0])∧(i214[2] + 1* i214[0])∧(i14[2]* i14[0]))

(0) -> (1), if ((i215[0]* i215[1])∧(i214[0]* i214[1])∧(i14[0]* i14[1])∧(i14[0] - 1 >= i214[0]* TRUE))

(1) -> (2), if ((i14[1]* i14[2])∧(i214[1]* i214[2])∧(i215[1]* i215[2]))

The set Q consists of the following terms:
Cond_Load14871(TRUE, x0, x0 - 1, x1)

### (15) 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 LOAD1487(i14[0], i214[0], i215[0]) → COND_LOAD1487(>=(-(i14[0], 1), i214[0]), i14[0], i214[0], i215[0]) the following chains were created:
• We consider the chain LOAD1487(i14[0], i214[0], i215[0]) → COND_LOAD1487(>=(-(i14[0], 1), i214[0]), i14[0], i214[0], i215[0]), COND_LOAD1487(TRUE, i14[1], i214[1], i215[1]) → LOAD1538(i14[1], i215[1], i214[1]) which results in the following constraint:

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)    (i14[0] + [-1] + [-1]i214[0] ≥ 0 ⇒ (UIncreasing(COND_LOAD1487(>=(-(i14[0], 1), i214[0]), i14[0], i214[0], i215[0])), ≥)∧[(-1)bni_17 + (-1)Bound*bni_17] + [(-1)bni_17]i214[0] + [bni_17]i14[0] ≥ 0∧[(-1)bso_18] ≥ 0)

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

(4)    (i14[0] + [-1] + [-1]i214[0] ≥ 0 ⇒ (UIncreasing(COND_LOAD1487(>=(-(i14[0], 1), i214[0]), i14[0], i214[0], i215[0])), ≥)∧[(-1)bni_17 + (-1)Bound*bni_17] + [(-1)bni_17]i214[0] + [bni_17]i14[0] ≥ 0∧[(-1)bso_18] ≥ 0)

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

(5)    (i14[0] + [-1] + [-1]i214[0] ≥ 0 ⇒ (UIncreasing(COND_LOAD1487(>=(-(i14[0], 1), i214[0]), i14[0], i214[0], i215[0])), ≥)∧[(-1)bni_17 + (-1)Bound*bni_17] + [(-1)bni_17]i214[0] + [bni_17]i14[0] ≥ 0∧[(-1)bso_18] ≥ 0)

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

(6)    (i14[0] + [-1] + [-1]i214[0] ≥ 0 ⇒ (UIncreasing(COND_LOAD1487(>=(-(i14[0], 1), i214[0]), i14[0], i214[0], i215[0])), ≥)∧0 = 0∧[(-1)bni_17 + (-1)Bound*bni_17] + [(-1)bni_17]i214[0] + [bni_17]i14[0] ≥ 0∧0 = 0∧[(-1)bso_18] ≥ 0)

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

(7)    (i14[0] ≥ 0 ⇒ (UIncreasing(COND_LOAD1487(>=(-(i14[0], 1), i214[0]), i14[0], i214[0], i215[0])), ≥)∧0 = 0∧[(-1)Bound*bni_17] + [bni_17]i14[0] ≥ 0∧0 = 0∧[(-1)bso_18] ≥ 0)

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

(8)    (i14[0] ≥ 0∧i214[0] ≥ 0 ⇒ (UIncreasing(COND_LOAD1487(>=(-(i14[0], 1), i214[0]), i14[0], i214[0], i215[0])), ≥)∧0 = 0∧[(-1)Bound*bni_17] + [bni_17]i14[0] ≥ 0∧0 = 0∧[(-1)bso_18] ≥ 0)

(9)    (i14[0] ≥ 0∧i214[0] ≥ 0 ⇒ (UIncreasing(COND_LOAD1487(>=(-(i14[0], 1), i214[0]), i14[0], i214[0], i215[0])), ≥)∧0 = 0∧[(-1)Bound*bni_17] + [bni_17]i14[0] ≥ 0∧0 = 0∧[(-1)bso_18] ≥ 0)

For Pair COND_LOAD1487(TRUE, i14[1], i214[1], i215[1]) → LOAD1538(i14[1], i215[1], i214[1]) the following chains were created:
• We consider the chain COND_LOAD1487(TRUE, i14[1], i214[1], i215[1]) → LOAD1538(i14[1], i215[1], i214[1]), LOAD1538(i14[2], i215[2], i214[2]) → LOAD1487(i14[2], +(i214[2], 1), +(i215[2], 1)) which results in the following constraint:

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

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

(12)    ((UIncreasing(LOAD1538(i14[1], i215[1], i214[1])), ≥)∧[(-1)bso_20] ≥ 0)

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

(13)    ((UIncreasing(LOAD1538(i14[1], i215[1], i214[1])), ≥)∧[(-1)bso_20] ≥ 0)

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

(14)    ((UIncreasing(LOAD1538(i14[1], i215[1], i214[1])), ≥)∧[(-1)bso_20] ≥ 0)

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

(15)    ((UIncreasing(LOAD1538(i14[1], i215[1], i214[1])), ≥)∧0 = 0∧0 = 0∧0 = 0∧[(-1)bso_20] ≥ 0)

For Pair LOAD1538(i14[2], i215[2], i214[2]) → LOAD1487(i14[2], +(i214[2], 1), +(i215[2], 1)) the following chains were created:
• We consider the chain COND_LOAD1487(TRUE, i14[1], i214[1], i215[1]) → LOAD1538(i14[1], i215[1], i214[1]), LOAD1538(i14[2], i215[2], i214[2]) → LOAD1487(i14[2], +(i214[2], 1), +(i215[2], 1)), LOAD1487(i14[0], i214[0], i215[0]) → COND_LOAD1487(>=(-(i14[0], 1), i214[0]), i14[0], i214[0], i215[0]) which results in the following constraint:

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

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

(18)    ((UIncreasing(LOAD1487(i14[2], +(i214[2], 1), +(i215[2], 1))), ≥)∧[1 + (-1)bso_22] ≥ 0)

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

(19)    ((UIncreasing(LOAD1487(i14[2], +(i214[2], 1), +(i215[2], 1))), ≥)∧[1 + (-1)bso_22] ≥ 0)

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

(20)    ((UIncreasing(LOAD1487(i14[2], +(i214[2], 1), +(i215[2], 1))), ≥)∧[1 + (-1)bso_22] ≥ 0)

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

(21)    ((UIncreasing(LOAD1487(i14[2], +(i214[2], 1), +(i215[2], 1))), ≥)∧0 = 0∧0 = 0∧0 = 0∧[1 + (-1)bso_22] ≥ 0)

To summarize, we get the following constraints P for the following pairs.
• (i14[0] ≥ 0∧i214[0] ≥ 0 ⇒ (UIncreasing(COND_LOAD1487(>=(-(i14[0], 1), i214[0]), i14[0], i214[0], i215[0])), ≥)∧0 = 0∧[(-1)Bound*bni_17] + [bni_17]i14[0] ≥ 0∧0 = 0∧[(-1)bso_18] ≥ 0)
• (i14[0] ≥ 0∧i214[0] ≥ 0 ⇒ (UIncreasing(COND_LOAD1487(>=(-(i14[0], 1), i214[0]), i14[0], i214[0], i215[0])), ≥)∧0 = 0∧[(-1)Bound*bni_17] + [bni_17]i14[0] ≥ 0∧0 = 0∧[(-1)bso_18] ≥ 0)

• ((UIncreasing(LOAD1538(i14[1], i215[1], i214[1])), ≥)∧0 = 0∧0 = 0∧0 = 0∧[(-1)bso_20] ≥ 0)

• ((UIncreasing(LOAD1487(i14[2], +(i214[2], 1), +(i215[2], 1))), ≥)∧0 = 0∧0 = 0∧0 = 0∧[1 + (-1)bso_22] ≥ 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(LOAD1487(x1, x2, x3)) = [-1] + [-1]x2 + x1
POL(COND_LOAD1487(x1, x2, x3, x4)) = [-1] + [-1]x3 + x2
POL(>=(x1, x2)) = [-1]
POL(-(x1, x2)) = x1 + [-1]x2
POL(1) = [1]
POL(LOAD1538(x1, x2, x3)) = [-1] + [-1]x3 + x1
POL(+(x1, x2)) = x1 + x2

The following pairs are in P>:

The following pairs are in Pbound:

The following pairs are in P:

There are no usable rules.

### (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:

Integer

R is empty.

The integer pair graph contains the following rules and edges:
(0): LOAD1487(i14[0], i214[0], i215[0]) → COND_LOAD1487(i14[0] - 1 >= i214[0], i14[0], i214[0], i215[0])

(0) -> (1), if ((i215[0]* i215[1])∧(i214[0]* i214[1])∧(i14[0]* i14[1])∧(i14[0] - 1 >= i214[0]* TRUE))

The set Q consists of the following terms:
Cond_Load14871(TRUE, x0, x0 - 1, x1)

### (18) IDependencyGraphProof (EQUIVALENT transformation)

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

### (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:

Integer

R is empty.

The integer pair graph contains the following rules and edges:
(2): LOAD1538(i14[2], i215[2], i214[2]) → LOAD1487(i14[2], i214[2] + 1, i215[2] + 1)

(1) -> (2), if ((i14[1]* i14[2])∧(i214[1]* i214[2])∧(i215[1]* i215[2]))

The set Q consists of the following terms: