### (0) Obligation:

JBC Problem based on JBC Program:
Manifest-Version: 1.0 Created-By: 1.6.0_16 (Sun Microsystems Inc.) Main-Class: PastaB16
`/** * Example taken from "A Term Rewriting Approach to the Automated Termination * Analysis of Imperative Programs" (http://www.cs.unm.edu/~spf/papers/2009-02.pdf) * and converted to Java. */public class PastaB16 {    public static void main(String[] args) {        Random.args = args;        int x = Random.random();        int y = Random.random();        while (x > 0) {            while (y > 0) {                y--;            }            x--;        }    }}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 190 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:
Load683(i50, i61) → Cond_Load683(i61 > 0, i50, i61)
Cond_Load683(TRUE, i50, i61) → Load683(i50, i61 + -1)
Load622(i50, i61) → Cond_Load622(i61 > 0 && i50 > 0, i50, i61)
Cond_Load622(TRUE, i50, i61) → Load683(i50, i61 + -1)
Load683(i50, i62) → Cond_Load6831(i62 <= 0 && i50 > 0, i50, i62)
Cond_Load6831(TRUE, i50, i62) → Load622(i50 + -1, i62)
Load622(i50, i62) → Cond_Load6221(i62 <= 0 && i50 > 0, i50, i62)
Cond_Load6221(TRUE, i50, i62) → Load622(i50 + -1, i62)
The set Q consists of the following terms:

### (6) Obligation:

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

The following domains are used:

Integer, Boolean

The ITRS R consists of the following rules:
Load683(i50, i61) → Cond_Load683(i61 > 0, i50, i61)
Cond_Load683(TRUE, i50, i61) → Load683(i50, i61 + -1)
Load622(i50, i61) → Cond_Load622(i61 > 0 && i50 > 0, i50, i61)
Cond_Load622(TRUE, i50, i61) → Load683(i50, i61 + -1)
Load683(i50, i62) → Cond_Load6831(i62 <= 0 && i50 > 0, i50, i62)
Cond_Load6831(TRUE, i50, i62) → Load622(i50 + -1, i62)
Load622(i50, i62) → Cond_Load6221(i62 <= 0 && i50 > 0, i50, i62)
Cond_Load6221(TRUE, i50, i62) → Load622(i50 + -1, i62)

The integer pair graph contains the following rules and edges:
(0): LOAD683(i50[0], i61[0]) → COND_LOAD683(i61[0] > 0, i50[0], i61[0])
(1): COND_LOAD683(TRUE, i50[1], i61[1]) → LOAD683(i50[1], i61[1] + -1)
(2): LOAD622(i50[2], i61[2]) → COND_LOAD622(i61[2] > 0 && i50[2] > 0, i50[2], i61[2])
(3): COND_LOAD622(TRUE, i50[3], i61[3]) → LOAD683(i50[3], i61[3] + -1)
(4): LOAD683(i50[4], i62[4]) → COND_LOAD6831(i62[4] <= 0 && i50[4] > 0, i50[4], i62[4])
(5): COND_LOAD6831(TRUE, i50[5], i62[5]) → LOAD622(i50[5] + -1, i62[5])
(6): LOAD622(i50[6], i62[6]) → COND_LOAD6221(i62[6] <= 0 && i50[6] > 0, i50[6], i62[6])
(7): COND_LOAD6221(TRUE, i50[7], i62[7]) → LOAD622(i50[7] + -1, i62[7])

(0) -> (1), if ((i61[0]* i61[1])∧(i61[0] > 0* TRUE)∧(i50[0]* i50[1]))

(1) -> (0), if ((i61[1] + -1* i61[0])∧(i50[1]* i50[0]))

(1) -> (4), if ((i50[1]* i50[4])∧(i61[1] + -1* i62[4]))

(2) -> (3), if ((i61[2]* i61[3])∧(i50[2]* i50[3])∧(i61[2] > 0 && i50[2] > 0* TRUE))

(3) -> (0), if ((i61[3] + -1* i61[0])∧(i50[3]* i50[0]))

(3) -> (4), if ((i61[3] + -1* i62[4])∧(i50[3]* i50[4]))

(4) -> (5), if ((i62[4] <= 0 && i50[4] > 0* TRUE)∧(i50[4]* i50[5])∧(i62[4]* i62[5]))

(5) -> (2), if ((i62[5]* i61[2])∧(i50[5] + -1* i50[2]))

(5) -> (6), if ((i50[5] + -1* i50[6])∧(i62[5]* i62[6]))

(6) -> (7), if ((i62[6]* i62[7])∧(i62[6] <= 0 && i50[6] > 0* TRUE)∧(i50[6]* i50[7]))

(7) -> (2), if ((i50[7] + -1* i50[2])∧(i62[7]* i61[2]))

(7) -> (6), if ((i50[7] + -1* i50[6])∧(i62[7]* i62[6]))

The set Q consists of the following terms:

### (7) UsableRulesProof (EQUIVALENT transformation)

As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [LPAR04] we can delete all non-usable rules [FROCOS05] from R.

### (8) Obligation:

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

The following domains are used:

Integer, Boolean

R is empty.

The integer pair graph contains the following rules and edges:
(0): LOAD683(i50[0], i61[0]) → COND_LOAD683(i61[0] > 0, i50[0], i61[0])
(1): COND_LOAD683(TRUE, i50[1], i61[1]) → LOAD683(i50[1], i61[1] + -1)
(2): LOAD622(i50[2], i61[2]) → COND_LOAD622(i61[2] > 0 && i50[2] > 0, i50[2], i61[2])
(3): COND_LOAD622(TRUE, i50[3], i61[3]) → LOAD683(i50[3], i61[3] + -1)
(4): LOAD683(i50[4], i62[4]) → COND_LOAD6831(i62[4] <= 0 && i50[4] > 0, i50[4], i62[4])
(5): COND_LOAD6831(TRUE, i50[5], i62[5]) → LOAD622(i50[5] + -1, i62[5])
(6): LOAD622(i50[6], i62[6]) → COND_LOAD6221(i62[6] <= 0 && i50[6] > 0, i50[6], i62[6])
(7): COND_LOAD6221(TRUE, i50[7], i62[7]) → LOAD622(i50[7] + -1, i62[7])

(0) -> (1), if ((i61[0]* i61[1])∧(i61[0] > 0* TRUE)∧(i50[0]* i50[1]))

(1) -> (0), if ((i61[1] + -1* i61[0])∧(i50[1]* i50[0]))

(1) -> (4), if ((i50[1]* i50[4])∧(i61[1] + -1* i62[4]))

(2) -> (3), if ((i61[2]* i61[3])∧(i50[2]* i50[3])∧(i61[2] > 0 && i50[2] > 0* TRUE))

(3) -> (0), if ((i61[3] + -1* i61[0])∧(i50[3]* i50[0]))

(3) -> (4), if ((i61[3] + -1* i62[4])∧(i50[3]* i50[4]))

(4) -> (5), if ((i62[4] <= 0 && i50[4] > 0* TRUE)∧(i50[4]* i50[5])∧(i62[4]* i62[5]))

(5) -> (2), if ((i62[5]* i61[2])∧(i50[5] + -1* i50[2]))

(5) -> (6), if ((i50[5] + -1* i50[6])∧(i62[5]* i62[6]))

(6) -> (7), if ((i62[6]* i62[7])∧(i62[6] <= 0 && i50[6] > 0* TRUE)∧(i50[6]* i50[7]))

(7) -> (2), if ((i50[7] + -1* i50[2])∧(i62[7]* i61[2]))

(7) -> (6), if ((i50[7] + -1* i50[6])∧(i62[7]* i62[6]))

The set Q consists of the following terms:

### (9) IDPNonInfProof (SOUND transformation)

The constraints were generated the following way:
The DP Problem is simplified using the Induction Calculus [NONINF] with the following steps:
Note that final constraints are written in bold face.

For Pair LOAD683(i50, i61) → COND_LOAD683(>(i61, 0), i50, i61) the following chains were created:
• We consider the chain LOAD683(i50[0], i61[0]) → COND_LOAD683(>(i61[0], 0), i50[0], i61[0]), COND_LOAD683(TRUE, i50[1], i61[1]) → LOAD683(i50[1], +(i61[1], -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)    (i61[0] + [-1] ≥ 0 ⇒ (UIncreasing(COND_LOAD683(>(i61[0], 0), i50[0], i61[0])), ≥)∧[bni_15 + (-1)Bound*bni_15] + [bni_15]i61[0] + [bni_15]i50[0] ≥ 0∧[1 + (-1)bso_16] ≥ 0)

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

(4)    (i61[0] + [-1] ≥ 0 ⇒ (UIncreasing(COND_LOAD683(>(i61[0], 0), i50[0], i61[0])), ≥)∧[bni_15 + (-1)Bound*bni_15] + [bni_15]i61[0] + [bni_15]i50[0] ≥ 0∧[1 + (-1)bso_16] ≥ 0)

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

(5)    (i61[0] + [-1] ≥ 0 ⇒ (UIncreasing(COND_LOAD683(>(i61[0], 0), i50[0], i61[0])), ≥)∧[bni_15 + (-1)Bound*bni_15] + [bni_15]i61[0] + [bni_15]i50[0] ≥ 0∧[1 + (-1)bso_16] ≥ 0)

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

(6)    (i61[0] + [-1] ≥ 0 ⇒ (UIncreasing(COND_LOAD683(>(i61[0], 0), i50[0], i61[0])), ≥)∧[bni_15] = 0∧[bni_15 + (-1)Bound*bni_15] + [bni_15]i61[0] ≥ 0∧0 = 0∧[1 + (-1)bso_16] ≥ 0)

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

(7)    (i61[0] ≥ 0 ⇒ (UIncreasing(COND_LOAD683(>(i61[0], 0), i50[0], i61[0])), ≥)∧[bni_15] = 0∧[(2)bni_15 + (-1)Bound*bni_15] + [bni_15]i61[0] ≥ 0∧0 = 0∧[1 + (-1)bso_16] ≥ 0)

For Pair COND_LOAD683(TRUE, i50, i61) → LOAD683(i50, +(i61, -1)) the following chains were created:
• We consider the chain COND_LOAD683(TRUE, i50[1], i61[1]) → LOAD683(i50[1], +(i61[1], -1)) which results in the following constraint:

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

(9)    ((UIncreasing(LOAD683(i50[1], +(i61[1], -1))), ≥)∧[(-1)bso_18] ≥ 0)

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

(10)    ((UIncreasing(LOAD683(i50[1], +(i61[1], -1))), ≥)∧[(-1)bso_18] ≥ 0)

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

(11)    ((UIncreasing(LOAD683(i50[1], +(i61[1], -1))), ≥)∧[(-1)bso_18] ≥ 0)

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

(12)    ((UIncreasing(LOAD683(i50[1], +(i61[1], -1))), ≥)∧0 = 0∧0 = 0∧[(-1)bso_18] ≥ 0)

For Pair LOAD622(i50, i61) → COND_LOAD622(&&(>(i61, 0), >(i50, 0)), i50, i61) the following chains were created:
• We consider the chain LOAD622(i50[2], i61[2]) → COND_LOAD622(&&(>(i61[2], 0), >(i50[2], 0)), i50[2], i61[2]), COND_LOAD622(TRUE, i50[3], i61[3]) → LOAD683(i50[3], +(i61[3], -1)) which results in the following constraint:

(13)    (i61[2]=i61[3]i50[2]=i50[3]&&(>(i61[2], 0), >(i50[2], 0))=TRUELOAD622(i50[2], i61[2])≥NonInfC∧LOAD622(i50[2], i61[2])≥COND_LOAD622(&&(>(i61[2], 0), >(i50[2], 0)), i50[2], i61[2])∧(UIncreasing(COND_LOAD622(&&(>(i61[2], 0), >(i50[2], 0)), i50[2], i61[2])), ≥))

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

(14)    (>(i61[2], 0)=TRUE>(i50[2], 0)=TRUELOAD622(i50[2], i61[2])≥NonInfC∧LOAD622(i50[2], i61[2])≥COND_LOAD622(&&(>(i61[2], 0), >(i50[2], 0)), i50[2], i61[2])∧(UIncreasing(COND_LOAD622(&&(>(i61[2], 0), >(i50[2], 0)), i50[2], i61[2])), ≥))

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

(15)    (i61[2] + [-1] ≥ 0∧i50[2] + [-1] ≥ 0 ⇒ (UIncreasing(COND_LOAD622(&&(>(i61[2], 0), >(i50[2], 0)), i50[2], i61[2])), ≥)∧[(2)bni_19 + (-1)Bound*bni_19] + [bni_19]i50[2] + [bni_19]i61[2] ≥ 0∧[1 + (-1)bso_20] ≥ 0)

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

(16)    (i61[2] + [-1] ≥ 0∧i50[2] + [-1] ≥ 0 ⇒ (UIncreasing(COND_LOAD622(&&(>(i61[2], 0), >(i50[2], 0)), i50[2], i61[2])), ≥)∧[(2)bni_19 + (-1)Bound*bni_19] + [bni_19]i50[2] + [bni_19]i61[2] ≥ 0∧[1 + (-1)bso_20] ≥ 0)

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

(17)    (i61[2] + [-1] ≥ 0∧i50[2] + [-1] ≥ 0 ⇒ (UIncreasing(COND_LOAD622(&&(>(i61[2], 0), >(i50[2], 0)), i50[2], i61[2])), ≥)∧[(2)bni_19 + (-1)Bound*bni_19] + [bni_19]i50[2] + [bni_19]i61[2] ≥ 0∧[1 + (-1)bso_20] ≥ 0)

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

(18)    (i61[2] ≥ 0∧i50[2] + [-1] ≥ 0 ⇒ (UIncreasing(COND_LOAD622(&&(>(i61[2], 0), >(i50[2], 0)), i50[2], i61[2])), ≥)∧[(3)bni_19 + (-1)Bound*bni_19] + [bni_19]i50[2] + [bni_19]i61[2] ≥ 0∧[1 + (-1)bso_20] ≥ 0)

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

(19)    (i61[2] ≥ 0∧i50[2] ≥ 0 ⇒ (UIncreasing(COND_LOAD622(&&(>(i61[2], 0), >(i50[2], 0)), i50[2], i61[2])), ≥)∧[(4)bni_19 + (-1)Bound*bni_19] + [bni_19]i50[2] + [bni_19]i61[2] ≥ 0∧[1 + (-1)bso_20] ≥ 0)

For Pair COND_LOAD622(TRUE, i50, i61) → LOAD683(i50, +(i61, -1)) the following chains were created:
• We consider the chain COND_LOAD622(TRUE, i50[3], i61[3]) → LOAD683(i50[3], +(i61[3], -1)) which results in the following constraint:

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

(21)    ((UIncreasing(LOAD683(i50[3], +(i61[3], -1))), ≥)∧[1 + (-1)bso_22] ≥ 0)

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

(22)    ((UIncreasing(LOAD683(i50[3], +(i61[3], -1))), ≥)∧[1 + (-1)bso_22] ≥ 0)

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

(23)    ((UIncreasing(LOAD683(i50[3], +(i61[3], -1))), ≥)∧[1 + (-1)bso_22] ≥ 0)

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

(24)    ((UIncreasing(LOAD683(i50[3], +(i61[3], -1))), ≥)∧0 = 0∧0 = 0∧[1 + (-1)bso_22] ≥ 0)

For Pair LOAD683(i50, i62) → COND_LOAD6831(&&(<=(i62, 0), >(i50, 0)), i50, i62) the following chains were created:
• We consider the chain LOAD683(i50[4], i62[4]) → COND_LOAD6831(&&(<=(i62[4], 0), >(i50[4], 0)), i50[4], i62[4]), COND_LOAD6831(TRUE, i50[5], i62[5]) → LOAD622(+(i50[5], -1), i62[5]) which results in the following constraint:

(25)    (&&(<=(i62[4], 0), >(i50[4], 0))=TRUEi50[4]=i50[5]i62[4]=i62[5]LOAD683(i50[4], i62[4])≥NonInfC∧LOAD683(i50[4], i62[4])≥COND_LOAD6831(&&(<=(i62[4], 0), >(i50[4], 0)), i50[4], i62[4])∧(UIncreasing(COND_LOAD6831(&&(<=(i62[4], 0), >(i50[4], 0)), i50[4], i62[4])), ≥))

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

(26)    (<=(i62[4], 0)=TRUE>(i50[4], 0)=TRUELOAD683(i50[4], i62[4])≥NonInfC∧LOAD683(i50[4], i62[4])≥COND_LOAD6831(&&(<=(i62[4], 0), >(i50[4], 0)), i50[4], i62[4])∧(UIncreasing(COND_LOAD6831(&&(<=(i62[4], 0), >(i50[4], 0)), i50[4], i62[4])), ≥))

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

(27)    ([-1]i62[4] ≥ 0∧i50[4] + [-1] ≥ 0 ⇒ (UIncreasing(COND_LOAD6831(&&(<=(i62[4], 0), >(i50[4], 0)), i50[4], i62[4])), ≥)∧[bni_23 + (-1)Bound*bni_23] + [bni_23]i62[4] + [bni_23]i50[4] ≥ 0∧[(-1)bso_24] ≥ 0)

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

(28)    ([-1]i62[4] ≥ 0∧i50[4] + [-1] ≥ 0 ⇒ (UIncreasing(COND_LOAD6831(&&(<=(i62[4], 0), >(i50[4], 0)), i50[4], i62[4])), ≥)∧[bni_23 + (-1)Bound*bni_23] + [bni_23]i62[4] + [bni_23]i50[4] ≥ 0∧[(-1)bso_24] ≥ 0)

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

(29)    ([-1]i62[4] ≥ 0∧i50[4] + [-1] ≥ 0 ⇒ (UIncreasing(COND_LOAD6831(&&(<=(i62[4], 0), >(i50[4], 0)), i50[4], i62[4])), ≥)∧[bni_23 + (-1)Bound*bni_23] + [bni_23]i62[4] + [bni_23]i50[4] ≥ 0∧[(-1)bso_24] ≥ 0)

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

(30)    (i62[4] ≥ 0∧i50[4] + [-1] ≥ 0 ⇒ (UIncreasing(COND_LOAD6831(&&(<=(i62[4], 0), >(i50[4], 0)), i50[4], i62[4])), ≥)∧[bni_23 + (-1)Bound*bni_23] + [(-1)bni_23]i62[4] + [bni_23]i50[4] ≥ 0∧[(-1)bso_24] ≥ 0)

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

(31)    (i62[4] ≥ 0∧i50[4] ≥ 0 ⇒ (UIncreasing(COND_LOAD6831(&&(<=(i62[4], 0), >(i50[4], 0)), i50[4], i62[4])), ≥)∧[(2)bni_23 + (-1)Bound*bni_23] + [(-1)bni_23]i62[4] + [bni_23]i50[4] ≥ 0∧[(-1)bso_24] ≥ 0)

For Pair COND_LOAD6831(TRUE, i50, i62) → LOAD622(+(i50, -1), i62) the following chains were created:
• We consider the chain COND_LOAD6831(TRUE, i50[5], i62[5]) → LOAD622(+(i50[5], -1), i62[5]) which results in the following constraint:

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

(33)    ((UIncreasing(LOAD622(+(i50[5], -1), i62[5])), ≥)∧[(-1)bso_26] ≥ 0)

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

(34)    ((UIncreasing(LOAD622(+(i50[5], -1), i62[5])), ≥)∧[(-1)bso_26] ≥ 0)

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

(35)    ((UIncreasing(LOAD622(+(i50[5], -1), i62[5])), ≥)∧[(-1)bso_26] ≥ 0)

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

(36)    ((UIncreasing(LOAD622(+(i50[5], -1), i62[5])), ≥)∧0 = 0∧0 = 0∧[(-1)bso_26] ≥ 0)

For Pair LOAD622(i50, i62) → COND_LOAD6221(&&(<=(i62, 0), >(i50, 0)), i50, i62) the following chains were created:
• We consider the chain LOAD622(i50[6], i62[6]) → COND_LOAD6221(&&(<=(i62[6], 0), >(i50[6], 0)), i50[6], i62[6]), COND_LOAD6221(TRUE, i50[7], i62[7]) → LOAD622(+(i50[7], -1), i62[7]) which results in the following constraint:

(37)    (i62[6]=i62[7]&&(<=(i62[6], 0), >(i50[6], 0))=TRUEi50[6]=i50[7]LOAD622(i50[6], i62[6])≥NonInfC∧LOAD622(i50[6], i62[6])≥COND_LOAD6221(&&(<=(i62[6], 0), >(i50[6], 0)), i50[6], i62[6])∧(UIncreasing(COND_LOAD6221(&&(<=(i62[6], 0), >(i50[6], 0)), i50[6], i62[6])), ≥))

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

(38)    (<=(i62[6], 0)=TRUE>(i50[6], 0)=TRUELOAD622(i50[6], i62[6])≥NonInfC∧LOAD622(i50[6], i62[6])≥COND_LOAD6221(&&(<=(i62[6], 0), >(i50[6], 0)), i50[6], i62[6])∧(UIncreasing(COND_LOAD6221(&&(<=(i62[6], 0), >(i50[6], 0)), i50[6], i62[6])), ≥))

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

(39)    ([-1]i62[6] ≥ 0∧i50[6] + [-1] ≥ 0 ⇒ (UIncreasing(COND_LOAD6221(&&(<=(i62[6], 0), >(i50[6], 0)), i50[6], i62[6])), ≥)∧[(2)bni_27 + (-1)Bound*bni_27] + [bni_27]i50[6] + [bni_27]i62[6] ≥ 0∧[1 + (-1)bso_28] ≥ 0)

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

(40)    ([-1]i62[6] ≥ 0∧i50[6] + [-1] ≥ 0 ⇒ (UIncreasing(COND_LOAD6221(&&(<=(i62[6], 0), >(i50[6], 0)), i50[6], i62[6])), ≥)∧[(2)bni_27 + (-1)Bound*bni_27] + [bni_27]i50[6] + [bni_27]i62[6] ≥ 0∧[1 + (-1)bso_28] ≥ 0)

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

(41)    ([-1]i62[6] ≥ 0∧i50[6] + [-1] ≥ 0 ⇒ (UIncreasing(COND_LOAD6221(&&(<=(i62[6], 0), >(i50[6], 0)), i50[6], i62[6])), ≥)∧[(2)bni_27 + (-1)Bound*bni_27] + [bni_27]i50[6] + [bni_27]i62[6] ≥ 0∧[1 + (-1)bso_28] ≥ 0)

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

(42)    (i62[6] ≥ 0∧i50[6] + [-1] ≥ 0 ⇒ (UIncreasing(COND_LOAD6221(&&(<=(i62[6], 0), >(i50[6], 0)), i50[6], i62[6])), ≥)∧[(2)bni_27 + (-1)Bound*bni_27] + [bni_27]i50[6] + [(-1)bni_27]i62[6] ≥ 0∧[1 + (-1)bso_28] ≥ 0)

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

(43)    (i62[6] ≥ 0∧i50[6] ≥ 0 ⇒ (UIncreasing(COND_LOAD6221(&&(<=(i62[6], 0), >(i50[6], 0)), i50[6], i62[6])), ≥)∧[(3)bni_27 + (-1)Bound*bni_27] + [bni_27]i50[6] + [(-1)bni_27]i62[6] ≥ 0∧[1 + (-1)bso_28] ≥ 0)

For Pair COND_LOAD6221(TRUE, i50, i62) → LOAD622(+(i50, -1), i62) the following chains were created:
• We consider the chain COND_LOAD6221(TRUE, i50[7], i62[7]) → LOAD622(+(i50[7], -1), i62[7]) which results in the following constraint:

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

(45)    ((UIncreasing(LOAD622(+(i50[7], -1), i62[7])), ≥)∧[(-1)bso_30] ≥ 0)

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

(46)    ((UIncreasing(LOAD622(+(i50[7], -1), i62[7])), ≥)∧[(-1)bso_30] ≥ 0)

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

(47)    ((UIncreasing(LOAD622(+(i50[7], -1), i62[7])), ≥)∧[(-1)bso_30] ≥ 0)

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

(48)    ((UIncreasing(LOAD622(+(i50[7], -1), i62[7])), ≥)∧0 = 0∧0 = 0∧[(-1)bso_30] ≥ 0)

To summarize, we get the following constraints P for the following pairs.
• LOAD683(i50, i61) → COND_LOAD683(>(i61, 0), i50, i61)
• (i61[0] ≥ 0 ⇒ (UIncreasing(COND_LOAD683(>(i61[0], 0), i50[0], i61[0])), ≥)∧[bni_15] = 0∧[(2)bni_15 + (-1)Bound*bni_15] + [bni_15]i61[0] ≥ 0∧0 = 0∧[1 + (-1)bso_16] ≥ 0)

• COND_LOAD683(TRUE, i50, i61) → LOAD683(i50, +(i61, -1))
• ((UIncreasing(LOAD683(i50[1], +(i61[1], -1))), ≥)∧0 = 0∧0 = 0∧[(-1)bso_18] ≥ 0)

• LOAD622(i50, i61) → COND_LOAD622(&&(>(i61, 0), >(i50, 0)), i50, i61)
• (i61[2] ≥ 0∧i50[2] ≥ 0 ⇒ (UIncreasing(COND_LOAD622(&&(>(i61[2], 0), >(i50[2], 0)), i50[2], i61[2])), ≥)∧[(4)bni_19 + (-1)Bound*bni_19] + [bni_19]i50[2] + [bni_19]i61[2] ≥ 0∧[1 + (-1)bso_20] ≥ 0)

• COND_LOAD622(TRUE, i50, i61) → LOAD683(i50, +(i61, -1))
• ((UIncreasing(LOAD683(i50[3], +(i61[3], -1))), ≥)∧0 = 0∧0 = 0∧[1 + (-1)bso_22] ≥ 0)

• LOAD683(i50, i62) → COND_LOAD6831(&&(<=(i62, 0), >(i50, 0)), i50, i62)
• (i62[4] ≥ 0∧i50[4] ≥ 0 ⇒ (UIncreasing(COND_LOAD6831(&&(<=(i62[4], 0), >(i50[4], 0)), i50[4], i62[4])), ≥)∧[(2)bni_23 + (-1)Bound*bni_23] + [(-1)bni_23]i62[4] + [bni_23]i50[4] ≥ 0∧[(-1)bso_24] ≥ 0)

• COND_LOAD6831(TRUE, i50, i62) → LOAD622(+(i50, -1), i62)
• ((UIncreasing(LOAD622(+(i50[5], -1), i62[5])), ≥)∧0 = 0∧0 = 0∧[(-1)bso_26] ≥ 0)

• LOAD622(i50, i62) → COND_LOAD6221(&&(<=(i62, 0), >(i50, 0)), i50, i62)
• (i62[6] ≥ 0∧i50[6] ≥ 0 ⇒ (UIncreasing(COND_LOAD6221(&&(<=(i62[6], 0), >(i50[6], 0)), i50[6], i62[6])), ≥)∧[(3)bni_27 + (-1)Bound*bni_27] + [bni_27]i50[6] + [(-1)bni_27]i62[6] ≥ 0∧[1 + (-1)bso_28] ≥ 0)

• COND_LOAD6221(TRUE, i50, i62) → LOAD622(+(i50, -1), i62)
• ((UIncreasing(LOAD622(+(i50[7], -1), i62[7])), ≥)∧0 = 0∧0 = 0∧[(-1)bso_30] ≥ 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(LOAD683(x1, x2)) = [1] + x2 + x1
POL(COND_LOAD683(x1, x2, x3)) = x3 + x2
POL(>(x1, x2)) = [-1]
POL(0) = 0
POL(+(x1, x2)) = x1 + x2
POL(-1) = [-1]
POL(LOAD622(x1, x2)) = [2] + x1 + x2
POL(COND_LOAD622(x1, x2, x3)) = [1] + x2 + x3
POL(&&(x1, x2)) = [-1]
POL(COND_LOAD6831(x1, x2, x3)) = [1] + x3 + x2
POL(<=(x1, x2)) = [-1]
POL(COND_LOAD6221(x1, x2, x3)) = [1] + x3 + x2

The following pairs are in P>:

LOAD683(i50[0], i61[0]) → COND_LOAD683(>(i61[0], 0), i50[0], i61[0])
LOAD622(i50[2], i61[2]) → COND_LOAD622(&&(>(i61[2], 0), >(i50[2], 0)), i50[2], i61[2])
COND_LOAD622(TRUE, i50[3], i61[3]) → LOAD683(i50[3], +(i61[3], -1))
LOAD622(i50[6], i62[6]) → COND_LOAD6221(&&(<=(i62[6], 0), >(i50[6], 0)), i50[6], i62[6])

The following pairs are in Pbound:

LOAD622(i50[2], i61[2]) → COND_LOAD622(&&(>(i61[2], 0), >(i50[2], 0)), i50[2], i61[2])

The following pairs are in P:

COND_LOAD683(TRUE, i50[1], i61[1]) → LOAD683(i50[1], +(i61[1], -1))
LOAD683(i50[4], i62[4]) → COND_LOAD6831(&&(<=(i62[4], 0), >(i50[4], 0)), i50[4], i62[4])
COND_LOAD6831(TRUE, i50[5], i62[5]) → LOAD622(+(i50[5], -1), i62[5])
COND_LOAD6221(TRUE, i50[7], i62[7]) → LOAD622(+(i50[7], -1), i62[7])

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, Boolean

R is empty.

The integer pair graph contains the following rules and edges:
(1): COND_LOAD683(TRUE, i50[1], i61[1]) → LOAD683(i50[1], i61[1] + -1)
(4): LOAD683(i50[4], i62[4]) → COND_LOAD6831(i62[4] <= 0 && i50[4] > 0, i50[4], i62[4])
(5): COND_LOAD6831(TRUE, i50[5], i62[5]) → LOAD622(i50[5] + -1, i62[5])
(7): COND_LOAD6221(TRUE, i50[7], i62[7]) → LOAD622(i50[7] + -1, i62[7])

(1) -> (4), if ((i50[1]* i50[4])∧(i61[1] + -1* i62[4]))

(4) -> (5), if ((i62[4] <= 0 && i50[4] > 0* TRUE)∧(i50[4]* i50[5])∧(i62[4]* i62[5]))

The set Q consists of the following terms:

### (12) IDependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 4 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, Boolean

R is empty.

The integer pair graph contains the following rules and edges:
(0): LOAD683(i50[0], i61[0]) → COND_LOAD683(i61[0] > 0, i50[0], i61[0])
(1): COND_LOAD683(TRUE, i50[1], i61[1]) → LOAD683(i50[1], i61[1] + -1)
(3): COND_LOAD622(TRUE, i50[3], i61[3]) → LOAD683(i50[3], i61[3] + -1)
(4): LOAD683(i50[4], i62[4]) → COND_LOAD6831(i62[4] <= 0 && i50[4] > 0, i50[4], i62[4])
(5): COND_LOAD6831(TRUE, i50[5], i62[5]) → LOAD622(i50[5] + -1, i62[5])
(6): LOAD622(i50[6], i62[6]) → COND_LOAD6221(i62[6] <= 0 && i50[6] > 0, i50[6], i62[6])
(7): COND_LOAD6221(TRUE, i50[7], i62[7]) → LOAD622(i50[7] + -1, i62[7])

(1) -> (0), if ((i61[1] + -1* i61[0])∧(i50[1]* i50[0]))

(3) -> (0), if ((i61[3] + -1* i61[0])∧(i50[3]* i50[0]))

(0) -> (1), if ((i61[0]* i61[1])∧(i61[0] > 0* TRUE)∧(i50[0]* i50[1]))

(1) -> (4), if ((i50[1]* i50[4])∧(i61[1] + -1* i62[4]))

(3) -> (4), if ((i61[3] + -1* i62[4])∧(i50[3]* i50[4]))

(4) -> (5), if ((i62[4] <= 0 && i50[4] > 0* TRUE)∧(i50[4]* i50[5])∧(i62[4]* i62[5]))

(5) -> (6), if ((i50[5] + -1* i50[6])∧(i62[5]* i62[6]))

(7) -> (6), if ((i50[7] + -1* i50[6])∧(i62[7]* i62[6]))

(6) -> (7), if ((i62[6]* i62[7])∧(i62[6] <= 0 && i50[6] > 0* TRUE)∧(i50[6]* i50[7]))

The set Q consists of the following terms:

### (15) IDependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 2 SCCs with 3 less nodes.

### (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, Boolean

R is empty.

The integer pair graph contains the following rules and edges:
(7): COND_LOAD6221(TRUE, i50[7], i62[7]) → LOAD622(i50[7] + -1, i62[7])
(6): LOAD622(i50[6], i62[6]) → COND_LOAD6221(i62[6] <= 0 && i50[6] > 0, i50[6], i62[6])

(7) -> (6), if ((i50[7] + -1* i50[6])∧(i62[7]* i62[6]))

(6) -> (7), if ((i62[6]* i62[7])∧(i62[6] <= 0 && i50[6] > 0* TRUE)∧(i50[6]* i50[7]))

The set Q consists of the following terms:

### (18) 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_LOAD6221(TRUE, i50[7], i62[7]) → LOAD622(+(i50[7], -1), i62[7]) the following chains were created:
• We consider the chain COND_LOAD6221(TRUE, i50[7], i62[7]) → LOAD622(+(i50[7], -1), i62[7]) which results in the following constraint:

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

(2)    ((UIncreasing(LOAD622(+(i50[7], -1), i62[7])), ≥)∧[(-1)bso_11] ≥ 0)

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

(3)    ((UIncreasing(LOAD622(+(i50[7], -1), i62[7])), ≥)∧[(-1)bso_11] ≥ 0)

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

(4)    ((UIncreasing(LOAD622(+(i50[7], -1), i62[7])), ≥)∧[(-1)bso_11] ≥ 0)

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

(5)    ((UIncreasing(LOAD622(+(i50[7], -1), i62[7])), ≥)∧0 = 0∧0 = 0∧[(-1)bso_11] ≥ 0)

For Pair LOAD622(i50[6], i62[6]) → COND_LOAD6221(&&(<=(i62[6], 0), >(i50[6], 0)), i50[6], i62[6]) the following chains were created:
• We consider the chain LOAD622(i50[6], i62[6]) → COND_LOAD6221(&&(<=(i62[6], 0), >(i50[6], 0)), i50[6], i62[6]), COND_LOAD6221(TRUE, i50[7], i62[7]) → LOAD622(+(i50[7], -1), i62[7]) which results in the following constraint:

(6)    (i62[6]=i62[7]&&(<=(i62[6], 0), >(i50[6], 0))=TRUEi50[6]=i50[7]LOAD622(i50[6], i62[6])≥NonInfC∧LOAD622(i50[6], i62[6])≥COND_LOAD6221(&&(<=(i62[6], 0), >(i50[6], 0)), i50[6], i62[6])∧(UIncreasing(COND_LOAD6221(&&(<=(i62[6], 0), >(i50[6], 0)), i50[6], i62[6])), ≥))

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

(7)    (<=(i62[6], 0)=TRUE>(i50[6], 0)=TRUELOAD622(i50[6], i62[6])≥NonInfC∧LOAD622(i50[6], i62[6])≥COND_LOAD6221(&&(<=(i62[6], 0), >(i50[6], 0)), i50[6], i62[6])∧(UIncreasing(COND_LOAD6221(&&(<=(i62[6], 0), >(i50[6], 0)), i50[6], i62[6])), ≥))

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

(8)    ([-1]i62[6] ≥ 0∧i50[6] + [-1] ≥ 0 ⇒ (UIncreasing(COND_LOAD6221(&&(<=(i62[6], 0), >(i50[6], 0)), i50[6], i62[6])), ≥)∧[bni_12 + (-1)Bound*bni_12] + [(2)bni_12]i50[6] + [(-1)bni_12]i62[6] ≥ 0∧[2 + (-1)bso_13] ≥ 0)

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

(9)    ([-1]i62[6] ≥ 0∧i50[6] + [-1] ≥ 0 ⇒ (UIncreasing(COND_LOAD6221(&&(<=(i62[6], 0), >(i50[6], 0)), i50[6], i62[6])), ≥)∧[bni_12 + (-1)Bound*bni_12] + [(2)bni_12]i50[6] + [(-1)bni_12]i62[6] ≥ 0∧[2 + (-1)bso_13] ≥ 0)

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

(10)    ([-1]i62[6] ≥ 0∧i50[6] + [-1] ≥ 0 ⇒ (UIncreasing(COND_LOAD6221(&&(<=(i62[6], 0), >(i50[6], 0)), i50[6], i62[6])), ≥)∧[bni_12 + (-1)Bound*bni_12] + [(2)bni_12]i50[6] + [(-1)bni_12]i62[6] ≥ 0∧[2 + (-1)bso_13] ≥ 0)

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

(11)    (i62[6] ≥ 0∧i50[6] + [-1] ≥ 0 ⇒ (UIncreasing(COND_LOAD6221(&&(<=(i62[6], 0), >(i50[6], 0)), i50[6], i62[6])), ≥)∧[bni_12 + (-1)Bound*bni_12] + [(2)bni_12]i50[6] + [bni_12]i62[6] ≥ 0∧[2 + (-1)bso_13] ≥ 0)

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

(12)    (i62[6] ≥ 0∧i50[6] ≥ 0 ⇒ (UIncreasing(COND_LOAD6221(&&(<=(i62[6], 0), >(i50[6], 0)), i50[6], i62[6])), ≥)∧[(3)bni_12 + (-1)Bound*bni_12] + [(2)bni_12]i50[6] + [bni_12]i62[6] ≥ 0∧[2 + (-1)bso_13] ≥ 0)

To summarize, we get the following constraints P for the following pairs.
• COND_LOAD6221(TRUE, i50[7], i62[7]) → LOAD622(+(i50[7], -1), i62[7])
• ((UIncreasing(LOAD622(+(i50[7], -1), i62[7])), ≥)∧0 = 0∧0 = 0∧[(-1)bso_11] ≥ 0)

• LOAD622(i50[6], i62[6]) → COND_LOAD6221(&&(<=(i62[6], 0), >(i50[6], 0)), i50[6], i62[6])
• (i62[6] ≥ 0∧i50[6] ≥ 0 ⇒ (UIncreasing(COND_LOAD6221(&&(<=(i62[6], 0), >(i50[6], 0)), i50[6], i62[6])), ≥)∧[(3)bni_12 + (-1)Bound*bni_12] + [(2)bni_12]i50[6] + [bni_12]i62[6] ≥ 0∧[2 + (-1)bso_13] ≥ 0)

The constraints for P> respective Pbound are constructed from P where we just replace every occurence of "t ≥ s" in P by "t > s" respective "t ≥ c". Here c stands for the fresh constant used for Pbound.
Using the following integer polynomial ordering the resulting constraints can be solved
Polynomial interpretation over integers[POLO]:

POL(TRUE) = 0
POL(FALSE) = 0
POL(COND_LOAD6221(x1, x2, x3)) = [-1] + [-1]x3 + [2]x2
POL(LOAD622(x1, x2)) = [1] + [2]x1 + [-1]x2
POL(+(x1, x2)) = x1 + x2
POL(-1) = [-1]
POL(&&(x1, x2)) = [-1]
POL(<=(x1, x2)) = [-1]
POL(0) = 0
POL(>(x1, x2)) = [2]

The following pairs are in P>:

LOAD622(i50[6], i62[6]) → COND_LOAD6221(&&(<=(i62[6], 0), >(i50[6], 0)), i50[6], i62[6])

The following pairs are in Pbound:

LOAD622(i50[6], i62[6]) → COND_LOAD6221(&&(<=(i62[6], 0), >(i50[6], 0)), i50[6], i62[6])

The following pairs are in P:

COND_LOAD6221(TRUE, i50[7], i62[7]) → LOAD622(+(i50[7], -1), i62[7])

There are no usable rules.

### (19) Obligation:

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

The following domains are used:

Integer

R is empty.

The integer pair graph contains the following rules and edges:
(7): COND_LOAD6221(TRUE, i50[7], i62[7]) → LOAD622(i50[7] + -1, i62[7])

The set Q consists of the following terms:

### (20) IDependencyGraphProof (EQUIVALENT transformation)

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

### (22) Obligation:

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

The following domains are used:

Integer

R is empty.

The integer pair graph contains the following rules and edges:
(1): COND_LOAD683(TRUE, i50[1], i61[1]) → LOAD683(i50[1], i61[1] + -1)
(0): LOAD683(i50[0], i61[0]) → COND_LOAD683(i61[0] > 0, i50[0], i61[0])

(1) -> (0), if ((i61[1] + -1* i61[0])∧(i50[1]* i50[0]))

(0) -> (1), if ((i61[0]* i61[1])∧(i61[0] > 0* TRUE)∧(i50[0]* i50[1]))

The set Q consists of the following terms:

### (23) 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_LOAD683(TRUE, i50[1], i61[1]) → LOAD683(i50[1], +(i61[1], -1)) the following chains were created:
• We consider the chain COND_LOAD683(TRUE, i50[1], i61[1]) → LOAD683(i50[1], +(i61[1], -1)) which results in the following constraint:

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

(2)    ((UIncreasing(LOAD683(i50[1], +(i61[1], -1))), ≥)∧[(-1)bso_8] ≥ 0)

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

(3)    ((UIncreasing(LOAD683(i50[1], +(i61[1], -1))), ≥)∧[(-1)bso_8] ≥ 0)

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

(4)    ((UIncreasing(LOAD683(i50[1], +(i61[1], -1))), ≥)∧[(-1)bso_8] ≥ 0)

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

(5)    ((UIncreasing(LOAD683(i50[1], +(i61[1], -1))), ≥)∧0 = 0∧[(-1)bso_8] ≥ 0)

For Pair LOAD683(i50[0], i61[0]) → COND_LOAD683(>(i61[0], 0), i50[0], i61[0]) the following chains were created:
• We consider the chain LOAD683(i50[0], i61[0]) → COND_LOAD683(>(i61[0], 0), i50[0], i61[0]), COND_LOAD683(TRUE, i50[1], i61[1]) → LOAD683(i50[1], +(i61[1], -1)) which results in the following constraint:

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

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

(8)    (i61[0] + [-1] ≥ 0 ⇒ (UIncreasing(COND_LOAD683(>(i61[0], 0), i50[0], i61[0])), ≥)∧[bni_9 + (-1)Bound*bni_9] + [(2)bni_9]i61[0] ≥ 0∧[2 + (-1)bso_10] ≥ 0)

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

(9)    (i61[0] + [-1] ≥ 0 ⇒ (UIncreasing(COND_LOAD683(>(i61[0], 0), i50[0], i61[0])), ≥)∧[bni_9 + (-1)Bound*bni_9] + [(2)bni_9]i61[0] ≥ 0∧[2 + (-1)bso_10] ≥ 0)

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

(10)    (i61[0] + [-1] ≥ 0 ⇒ (UIncreasing(COND_LOAD683(>(i61[0], 0), i50[0], i61[0])), ≥)∧[bni_9 + (-1)Bound*bni_9] + [(2)bni_9]i61[0] ≥ 0∧[2 + (-1)bso_10] ≥ 0)

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

(11)    (i61[0] ≥ 0 ⇒ (UIncreasing(COND_LOAD683(>(i61[0], 0), i50[0], i61[0])), ≥)∧[(3)bni_9 + (-1)Bound*bni_9] + [(2)bni_9]i61[0] ≥ 0∧[2 + (-1)bso_10] ≥ 0)

To summarize, we get the following constraints P for the following pairs.
• COND_LOAD683(TRUE, i50[1], i61[1]) → LOAD683(i50[1], +(i61[1], -1))
• ((UIncreasing(LOAD683(i50[1], +(i61[1], -1))), ≥)∧0 = 0∧[(-1)bso_8] ≥ 0)

• LOAD683(i50[0], i61[0]) → COND_LOAD683(>(i61[0], 0), i50[0], i61[0])
• (i61[0] ≥ 0 ⇒ (UIncreasing(COND_LOAD683(>(i61[0], 0), i50[0], i61[0])), ≥)∧[(3)bni_9 + (-1)Bound*bni_9] + [(2)bni_9]i61[0] ≥ 0∧[2 + (-1)bso_10] ≥ 0)

The constraints for P> respective Pbound are constructed from P where we just replace every occurence of "t ≥ s" in P by "t > s" respective "t ≥ c". Here c stands for the fresh constant used for Pbound.
Using the following integer polynomial ordering the resulting constraints can be solved
Polynomial interpretation over integers[POLO]:

POL(TRUE) = 0
POL(FALSE) = 0
POL(COND_LOAD683(x1, x2, x3)) = [-1] + [2]x3
POL(LOAD683(x1, x2)) = [1] + [2]x2
POL(+(x1, x2)) = x1 + x2
POL(-1) = [-1]
POL(>(x1, x2)) = [-1]
POL(0) = 0

The following pairs are in P>:

LOAD683(i50[0], i61[0]) → COND_LOAD683(>(i61[0], 0), i50[0], i61[0])

The following pairs are in Pbound:

LOAD683(i50[0], i61[0]) → COND_LOAD683(>(i61[0], 0), i50[0], i61[0])

The following pairs are in P:

COND_LOAD683(TRUE, i50[1], i61[1]) → LOAD683(i50[1], +(i61[1], -1))

There are no usable rules.

### (24) Obligation:

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

The following domains are used:

Integer

R is empty.

The integer pair graph contains the following rules and edges:
(1): COND_LOAD683(TRUE, i50[1], i61[1]) → LOAD683(i50[1], i61[1] + -1)

The set Q consists of the following terms: