### (0) Obligation:

JBC Problem based on JBC Program:
Manifest-Version: 1.0 Created-By: 1.6.0_16 (Sun Microsystems Inc.) Main-Class: PastaB10
`/** * 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 PastaB10 {    public static void main(String[] args) {        Random.args = args;        int x = Random.random();        int y = Random.random();        while (x + y > 0) {            if (x > 0) {                x--;            } else if (y > 0) {                y--;            } else {                continue;            }                    }    }}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:
PastaB10.main([Ljava/lang/String;)V: Graph of 166 nodes with 1 SCC.

### (3) FIGtoITRSProof (SOUND transformation)

Transformed FIGraph SCCs to IDPs. Logs:

Log for SCC 0:

Generated 22 rules for P and 2 rules for R.

Combined rules. Obtained 3 rules for P and 0 rules for R.

Filtered ground terms:

579_0_main_Load(x1, x2, x3, x4) → 579_0_main_Load(x2, x3, x4)
Cond_579_0_main_Load2(x1, x2, x3, x4, x5) → Cond_579_0_main_Load2(x1, x3, x4, x5)
Cond_579_0_main_Load1(x1, x2, x3, x4, x5) → Cond_579_0_main_Load1(x1, x4)
Cond_579_0_main_Load(x1, x2, x3, x4, x5) → Cond_579_0_main_Load(x1)

Filtered duplicate args:

Cond_579_0_main_Load2(x1, x2, x3, x4) → Cond_579_0_main_Load2(x1, x3, x4)

Combined rules. Obtained 3 rules for P and 0 rules for R.

Finished conversion. Obtained 3 rules for P and 0 rules for R. System has predefined symbols.

### (4) 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): 579_0_MAIN_LOAD(0, 0) → COND_579_0_MAIN_LOAD(0 < 0 + 0, 0, 0)
(1): COND_579_0_MAIN_LOAD(TRUE, 0, 0) → 579_0_MAIN_LOAD(0, 0)
(2): 579_0_MAIN_LOAD(x1[2], 0) → COND_579_0_MAIN_LOAD1(x1[2] > 0 && 0 < 0 + x1[2], x1[2], 0)
(3): COND_579_0_MAIN_LOAD1(TRUE, x1[3], 0) → 579_0_MAIN_LOAD(x1[3] + -1, 0)
(4): 579_0_MAIN_LOAD(x1[4], x0[4]) → COND_579_0_MAIN_LOAD2(x1[4] >= 0 && x0[4] > 0 && 0 < x0[4] + x1[4], x1[4], x0[4])
(5): COND_579_0_MAIN_LOAD2(TRUE, x1[5], x0[5]) → 579_0_MAIN_LOAD(x1[5], x0[5] + -1)

(0) -> (1), if (0 < 0 + 0* TRUE)

(1) -> (0), if true

(1) -> (2), if (0* x1[2])

(1) -> (4), if ((0* x1[4])∧(0* x0[4]))

(2) -> (3), if ((x1[2] > 0 && 0 < 0 + x1[2]* TRUE)∧(x1[2]* x1[3]))

(3) -> (0), if (x1[3] + -1* 0)

(3) -> (2), if (x1[3] + -1* x1[2])

(3) -> (4), if ((x1[3] + -1* x1[4])∧(0* x0[4]))

(4) -> (5), if ((x1[4] >= 0 && x0[4] > 0 && 0 < x0[4] + x1[4]* TRUE)∧(x1[4]* x1[5])∧(x0[4]* x0[5]))

(5) -> (0), if ((x1[5]* 0)∧(x0[5] + -1* 0))

(5) -> (2), if ((x1[5]* x1[2])∧(x0[5] + -1* 0))

(5) -> (4), if ((x1[5]* x1[4])∧(x0[5] + -1* x0[4]))

The set Q is empty.

### (5) 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 579_0_MAIN_LOAD(0, 0) → COND_579_0_MAIN_LOAD(<(0, +(0, 0)), 0, 0) the following chains were created:
• We consider the chain 579_0_MAIN_LOAD(0, 0) → COND_579_0_MAIN_LOAD(<(0, +(0, 0)), 0, 0), COND_579_0_MAIN_LOAD(TRUE, 0, 0) → 579_0_MAIN_LOAD(0, 0) which results in the following constraint:

(1)    (<(0, +(0, 0))=TRUE579_0_MAIN_LOAD(0, 0)≥NonInfC∧579_0_MAIN_LOAD(0, 0)≥COND_579_0_MAIN_LOAD(<(0, +(0, 0)), 0, 0)∧(UIncreasing(COND_579_0_MAIN_LOAD(<(0, +(0, 0)), 0, 0)), ≥))

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

For Pair COND_579_0_MAIN_LOAD(TRUE, 0, 0) → 579_0_MAIN_LOAD(0, 0) the following chains were created:
• We consider the chain COND_579_0_MAIN_LOAD(TRUE, 0, 0) → 579_0_MAIN_LOAD(0, 0), 579_0_MAIN_LOAD(0, 0) → COND_579_0_MAIN_LOAD(<(0, +(0, 0)), 0, 0) which results in the following constraint:

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

(3)    ((UIncreasing(579_0_MAIN_LOAD(0, 0)), ≥)∧[2 + (-1)bso_15] ≥ 0)

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

(4)    ((UIncreasing(579_0_MAIN_LOAD(0, 0)), ≥)∧[2 + (-1)bso_15] ≥ 0)

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

(5)    ((UIncreasing(579_0_MAIN_LOAD(0, 0)), ≥)∧[2 + (-1)bso_15] ≥ 0)

• We consider the chain COND_579_0_MAIN_LOAD(TRUE, 0, 0) → 579_0_MAIN_LOAD(0, 0), 579_0_MAIN_LOAD(x1[2], 0) → COND_579_0_MAIN_LOAD1(&&(>(x1[2], 0), <(0, +(0, x1[2]))), x1[2], 0) 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)    ((UIncreasing(579_0_MAIN_LOAD(0, 0)), ≥)∧[2 + (-1)bso_15] ≥ 0)

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

(9)    ((UIncreasing(579_0_MAIN_LOAD(0, 0)), ≥)∧[2 + (-1)bso_15] ≥ 0)

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

(10)    ((UIncreasing(579_0_MAIN_LOAD(0, 0)), ≥)∧[2 + (-1)bso_15] ≥ 0)

• We consider the chain COND_579_0_MAIN_LOAD(TRUE, 0, 0) → 579_0_MAIN_LOAD(0, 0), 579_0_MAIN_LOAD(x1[4], x0[4]) → COND_579_0_MAIN_LOAD2(&&(&&(>=(x1[4], 0), >(x0[4], 0)), <(0, +(x0[4], x1[4]))), x1[4], x0[4]) which results in the following constraint:

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

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

(13)    ((UIncreasing(579_0_MAIN_LOAD(0, 0)), ≥)∧[2 + (-1)bso_15] ≥ 0)

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

(14)    ((UIncreasing(579_0_MAIN_LOAD(0, 0)), ≥)∧[2 + (-1)bso_15] ≥ 0)

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

(15)    ((UIncreasing(579_0_MAIN_LOAD(0, 0)), ≥)∧[2 + (-1)bso_15] ≥ 0)

For Pair 579_0_MAIN_LOAD(x1, 0) → COND_579_0_MAIN_LOAD1(&&(>(x1, 0), <(0, +(0, x1))), x1, 0) the following chains were created:
• We consider the chain 579_0_MAIN_LOAD(x1[2], 0) → COND_579_0_MAIN_LOAD1(&&(>(x1[2], 0), <(0, +(0, x1[2]))), x1[2], 0), COND_579_0_MAIN_LOAD1(TRUE, x1[3], 0) → 579_0_MAIN_LOAD(+(x1[3], -1), 0) which results in the following constraint:

(16)    (&&(>(x1[2], 0), <(0, +(0, x1[2])))=TRUEx1[2]=x1[3]579_0_MAIN_LOAD(x1[2], 0)≥NonInfC∧579_0_MAIN_LOAD(x1[2], 0)≥COND_579_0_MAIN_LOAD1(&&(>(x1[2], 0), <(0, +(0, x1[2]))), x1[2], 0)∧(UIncreasing(COND_579_0_MAIN_LOAD1(&&(>(x1[2], 0), <(0, +(0, x1[2]))), x1[2], 0)), ≥))

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

(17)    (>(x1[2], 0)=TRUE<(0, +(0, x1[2]))=TRUE579_0_MAIN_LOAD(x1[2], 0)≥NonInfC∧579_0_MAIN_LOAD(x1[2], 0)≥COND_579_0_MAIN_LOAD1(&&(>(x1[2], 0), <(0, +(0, x1[2]))), x1[2], 0)∧(UIncreasing(COND_579_0_MAIN_LOAD1(&&(>(x1[2], 0), <(0, +(0, x1[2]))), x1[2], 0)), ≥))

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

(18)    (x1[2] + [-1] ≥ 0∧[-1] + x1[2] ≥ 0 ⇒ (UIncreasing(COND_579_0_MAIN_LOAD1(&&(>(x1[2], 0), <(0, +(0, x1[2]))), x1[2], 0)), ≥)∧[(-1)bni_16 + (-1)Bound*bni_16] ≥ 0∧[(-1)bso_17] ≥ 0)

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

(19)    (x1[2] + [-1] ≥ 0∧[-1] + x1[2] ≥ 0 ⇒ (UIncreasing(COND_579_0_MAIN_LOAD1(&&(>(x1[2], 0), <(0, +(0, x1[2]))), x1[2], 0)), ≥)∧[(-1)bni_16 + (-1)Bound*bni_16] ≥ 0∧[(-1)bso_17] ≥ 0)

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

(20)    (x1[2] + [-1] ≥ 0∧[-1] + x1[2] ≥ 0 ⇒ (UIncreasing(COND_579_0_MAIN_LOAD1(&&(>(x1[2], 0), <(0, +(0, x1[2]))), x1[2], 0)), ≥)∧[(-1)bni_16 + (-1)Bound*bni_16] ≥ 0∧[(-1)bso_17] ≥ 0)

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

(21)    (x1[2] ≥ 0∧x1[2] ≥ 0 ⇒ (UIncreasing(COND_579_0_MAIN_LOAD1(&&(>(x1[2], 0), <(0, +(0, x1[2]))), x1[2], 0)), ≥)∧[(-1)bni_16 + (-1)Bound*bni_16] ≥ 0∧[(-1)bso_17] ≥ 0)

For Pair COND_579_0_MAIN_LOAD1(TRUE, x1, 0) → 579_0_MAIN_LOAD(+(x1, -1), 0) the following chains were created:
• We consider the chain COND_579_0_MAIN_LOAD1(TRUE, x1[3], 0) → 579_0_MAIN_LOAD(+(x1[3], -1), 0) which results in the following constraint:

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

(23)    ((UIncreasing(579_0_MAIN_LOAD(+(x1[3], -1), 0)), ≥)∧[(-1)bso_19] ≥ 0)

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

(24)    ((UIncreasing(579_0_MAIN_LOAD(+(x1[3], -1), 0)), ≥)∧[(-1)bso_19] ≥ 0)

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

(25)    ((UIncreasing(579_0_MAIN_LOAD(+(x1[3], -1), 0)), ≥)∧[(-1)bso_19] ≥ 0)

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

(26)    ((UIncreasing(579_0_MAIN_LOAD(+(x1[3], -1), 0)), ≥)∧0 = 0∧[(-1)bso_19] ≥ 0)

For Pair 579_0_MAIN_LOAD(x1, x0) → COND_579_0_MAIN_LOAD2(&&(&&(>=(x1, 0), >(x0, 0)), <(0, +(x0, x1))), x1, x0) the following chains were created:
• We consider the chain 579_0_MAIN_LOAD(x1[4], x0[4]) → COND_579_0_MAIN_LOAD2(&&(&&(>=(x1[4], 0), >(x0[4], 0)), <(0, +(x0[4], x1[4]))), x1[4], x0[4]), COND_579_0_MAIN_LOAD2(TRUE, x1[5], x0[5]) → 579_0_MAIN_LOAD(x1[5], +(x0[5], -1)) which results in the following constraint:

(27)    (&&(&&(>=(x1[4], 0), >(x0[4], 0)), <(0, +(x0[4], x1[4])))=TRUEx1[4]=x1[5]x0[4]=x0[5]579_0_MAIN_LOAD(x1[4], x0[4])≥NonInfC∧579_0_MAIN_LOAD(x1[4], x0[4])≥COND_579_0_MAIN_LOAD2(&&(&&(>=(x1[4], 0), >(x0[4], 0)), <(0, +(x0[4], x1[4]))), x1[4], x0[4])∧(UIncreasing(COND_579_0_MAIN_LOAD2(&&(&&(>=(x1[4], 0), >(x0[4], 0)), <(0, +(x0[4], x1[4]))), x1[4], x0[4])), ≥))

We simplified constraint (27) using rules (IV), (IDP_BOOLEAN) which results in the following new constraint:

(28)    (<(0, +(x0[4], x1[4]))=TRUE>=(x1[4], 0)=TRUE>(x0[4], 0)=TRUE579_0_MAIN_LOAD(x1[4], x0[4])≥NonInfC∧579_0_MAIN_LOAD(x1[4], x0[4])≥COND_579_0_MAIN_LOAD2(&&(&&(>=(x1[4], 0), >(x0[4], 0)), <(0, +(x0[4], x1[4]))), x1[4], x0[4])∧(UIncreasing(COND_579_0_MAIN_LOAD2(&&(&&(>=(x1[4], 0), >(x0[4], 0)), <(0, +(x0[4], x1[4]))), x1[4], x0[4])), ≥))

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

(29)    (x0[4] + [-1] + x1[4] ≥ 0∧x1[4] ≥ 0∧x0[4] + [-1] ≥ 0 ⇒ (UIncreasing(COND_579_0_MAIN_LOAD2(&&(&&(>=(x1[4], 0), >(x0[4], 0)), <(0, +(x0[4], x1[4]))), x1[4], x0[4])), ≥)∧[(-1)bni_20 + (-1)Bound*bni_20] ≥ 0∧[(-1)bso_21] ≥ 0)

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

(30)    (x0[4] + [-1] + x1[4] ≥ 0∧x1[4] ≥ 0∧x0[4] + [-1] ≥ 0 ⇒ (UIncreasing(COND_579_0_MAIN_LOAD2(&&(&&(>=(x1[4], 0), >(x0[4], 0)), <(0, +(x0[4], x1[4]))), x1[4], x0[4])), ≥)∧[(-1)bni_20 + (-1)Bound*bni_20] ≥ 0∧[(-1)bso_21] ≥ 0)

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

(31)    (x0[4] + [-1] + x1[4] ≥ 0∧x1[4] ≥ 0∧x0[4] + [-1] ≥ 0 ⇒ (UIncreasing(COND_579_0_MAIN_LOAD2(&&(&&(>=(x1[4], 0), >(x0[4], 0)), <(0, +(x0[4], x1[4]))), x1[4], x0[4])), ≥)∧[(-1)bni_20 + (-1)Bound*bni_20] ≥ 0∧[(-1)bso_21] ≥ 0)

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

(32)    (x0[4] + x1[4] ≥ 0∧x1[4] ≥ 0∧x0[4] ≥ 0 ⇒ (UIncreasing(COND_579_0_MAIN_LOAD2(&&(&&(>=(x1[4], 0), >(x0[4], 0)), <(0, +(x0[4], x1[4]))), x1[4], x0[4])), ≥)∧[(-1)bni_20 + (-1)Bound*bni_20] ≥ 0∧[(-1)bso_21] ≥ 0)

For Pair COND_579_0_MAIN_LOAD2(TRUE, x1, x0) → 579_0_MAIN_LOAD(x1, +(x0, -1)) the following chains were created:
• We consider the chain COND_579_0_MAIN_LOAD2(TRUE, x1[5], x0[5]) → 579_0_MAIN_LOAD(x1[5], +(x0[5], -1)) which results in the following constraint:

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

(34)    ((UIncreasing(579_0_MAIN_LOAD(x1[5], +(x0[5], -1))), ≥)∧[(-1)bso_23] ≥ 0)

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

(35)    ((UIncreasing(579_0_MAIN_LOAD(x1[5], +(x0[5], -1))), ≥)∧[(-1)bso_23] ≥ 0)

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

(36)    ((UIncreasing(579_0_MAIN_LOAD(x1[5], +(x0[5], -1))), ≥)∧[(-1)bso_23] ≥ 0)

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

(37)    ((UIncreasing(579_0_MAIN_LOAD(x1[5], +(x0[5], -1))), ≥)∧0 = 0∧0 = 0∧[(-1)bso_23] ≥ 0)

To summarize, we get the following constraints P for the following pairs.
• 579_0_MAIN_LOAD(0, 0) → COND_579_0_MAIN_LOAD(<(0, +(0, 0)), 0, 0)

• COND_579_0_MAIN_LOAD(TRUE, 0, 0) → 579_0_MAIN_LOAD(0, 0)
• ((UIncreasing(579_0_MAIN_LOAD(0, 0)), ≥)∧[2 + (-1)bso_15] ≥ 0)
• ((UIncreasing(579_0_MAIN_LOAD(0, 0)), ≥)∧[2 + (-1)bso_15] ≥ 0)
• ((UIncreasing(579_0_MAIN_LOAD(0, 0)), ≥)∧[2 + (-1)bso_15] ≥ 0)

• 579_0_MAIN_LOAD(x1, 0) → COND_579_0_MAIN_LOAD1(&&(>(x1, 0), <(0, +(0, x1))), x1, 0)
• (x1[2] ≥ 0∧x1[2] ≥ 0 ⇒ (UIncreasing(COND_579_0_MAIN_LOAD1(&&(>(x1[2], 0), <(0, +(0, x1[2]))), x1[2], 0)), ≥)∧[(-1)bni_16 + (-1)Bound*bni_16] ≥ 0∧[(-1)bso_17] ≥ 0)

• COND_579_0_MAIN_LOAD1(TRUE, x1, 0) → 579_0_MAIN_LOAD(+(x1, -1), 0)
• ((UIncreasing(579_0_MAIN_LOAD(+(x1[3], -1), 0)), ≥)∧0 = 0∧[(-1)bso_19] ≥ 0)

• 579_0_MAIN_LOAD(x1, x0) → COND_579_0_MAIN_LOAD2(&&(&&(>=(x1, 0), >(x0, 0)), <(0, +(x0, x1))), x1, x0)
• (x0[4] + x1[4] ≥ 0∧x1[4] ≥ 0∧x0[4] ≥ 0 ⇒ (UIncreasing(COND_579_0_MAIN_LOAD2(&&(&&(>=(x1[4], 0), >(x0[4], 0)), <(0, +(x0[4], x1[4]))), x1[4], x0[4])), ≥)∧[(-1)bni_20 + (-1)Bound*bni_20] ≥ 0∧[(-1)bso_21] ≥ 0)

• COND_579_0_MAIN_LOAD2(TRUE, x1, x0) → 579_0_MAIN_LOAD(x1, +(x0, -1))
• ((UIncreasing(579_0_MAIN_LOAD(x1[5], +(x0[5], -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(579_0_MAIN_LOAD(x1, x2)) = [-1]
POL(0) = 0
POL(COND_579_0_MAIN_LOAD(x1, x2, x3)) = [1]
POL(<(x1, x2)) = [-1]
POL(+(x1, x2)) = x1 + x2
POL(COND_579_0_MAIN_LOAD1(x1, x2, x3)) = [-1]
POL(&&(x1, x2)) = [-1]
POL(>(x1, x2)) = [-1]
POL(-1) = [-1]
POL(COND_579_0_MAIN_LOAD2(x1, x2, x3)) = [-1]
POL(>=(x1, x2)) = [-1]

The following pairs are in P>:

579_0_MAIN_LOAD(0, 0) → COND_579_0_MAIN_LOAD(<(0, +(0, 0)), 0, 0)

The following pairs are in Pbound:

579_0_MAIN_LOAD(0, 0) → COND_579_0_MAIN_LOAD(<(0, +(0, 0)), 0, 0)
579_0_MAIN_LOAD(x1[2], 0) → COND_579_0_MAIN_LOAD1(&&(>(x1[2], 0), <(0, +(0, x1[2]))), x1[2], 0)
579_0_MAIN_LOAD(x1[4], x0[4]) → COND_579_0_MAIN_LOAD2(&&(&&(>=(x1[4], 0), >(x0[4], 0)), <(0, +(x0[4], x1[4]))), x1[4], x0[4])

The following pairs are in P:

579_0_MAIN_LOAD(x1[2], 0) → COND_579_0_MAIN_LOAD1(&&(>(x1[2], 0), <(0, +(0, x1[2]))), x1[2], 0)
COND_579_0_MAIN_LOAD1(TRUE, x1[3], 0) → 579_0_MAIN_LOAD(+(x1[3], -1), 0)
579_0_MAIN_LOAD(x1[4], x0[4]) → COND_579_0_MAIN_LOAD2(&&(&&(>=(x1[4], 0), >(x0[4], 0)), <(0, +(x0[4], x1[4]))), x1[4], x0[4])
COND_579_0_MAIN_LOAD2(TRUE, x1[5], x0[5]) → 579_0_MAIN_LOAD(x1[5], +(x0[5], -1))

There are no usable rules.

### (7) Obligation:

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

The following domains are used:

Boolean, Integer

R is empty.

The integer pair graph contains the following rules and edges:
(2): 579_0_MAIN_LOAD(x1[2], 0) → COND_579_0_MAIN_LOAD1(x1[2] > 0 && 0 < 0 + x1[2], x1[2], 0)
(3): COND_579_0_MAIN_LOAD1(TRUE, x1[3], 0) → 579_0_MAIN_LOAD(x1[3] + -1, 0)
(4): 579_0_MAIN_LOAD(x1[4], x0[4]) → COND_579_0_MAIN_LOAD2(x1[4] >= 0 && x0[4] > 0 && 0 < x0[4] + x1[4], x1[4], x0[4])
(5): COND_579_0_MAIN_LOAD2(TRUE, x1[5], x0[5]) → 579_0_MAIN_LOAD(x1[5], x0[5] + -1)

(3) -> (2), if (x1[3] + -1* x1[2])

(5) -> (2), if ((x1[5]* x1[2])∧(x0[5] + -1* 0))

(2) -> (3), if ((x1[2] > 0 && 0 < 0 + x1[2]* TRUE)∧(x1[2]* x1[3]))

(3) -> (4), if ((x1[3] + -1* x1[4])∧(0* x0[4]))

(5) -> (4), if ((x1[5]* x1[4])∧(x0[5] + -1* x0[4]))

(4) -> (5), if ((x1[4] >= 0 && x0[4] > 0 && 0 < x0[4] + x1[4]* TRUE)∧(x1[4]* x1[5])∧(x0[4]* x0[5]))

The set Q is empty.

### (8) 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 579_0_MAIN_LOAD(x1[2], 0) → COND_579_0_MAIN_LOAD1(&&(>(x1[2], 0), <(0, +(0, x1[2]))), x1[2], 0) the following chains were created:
• We consider the chain 579_0_MAIN_LOAD(x1[2], 0) → COND_579_0_MAIN_LOAD1(&&(>(x1[2], 0), <(0, +(0, x1[2]))), x1[2], 0), COND_579_0_MAIN_LOAD1(TRUE, x1[3], 0) → 579_0_MAIN_LOAD(+(x1[3], -1), 0) which results in the following constraint:

(1)    (&&(>(x1[2], 0), <(0, +(0, x1[2])))=TRUEx1[2]=x1[3]579_0_MAIN_LOAD(x1[2], 0)≥NonInfC∧579_0_MAIN_LOAD(x1[2], 0)≥COND_579_0_MAIN_LOAD1(&&(>(x1[2], 0), <(0, +(0, x1[2]))), x1[2], 0)∧(UIncreasing(COND_579_0_MAIN_LOAD1(&&(>(x1[2], 0), <(0, +(0, x1[2]))), x1[2], 0)), ≥))

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

(2)    (>(x1[2], 0)=TRUE<(0, +(0, x1[2]))=TRUE579_0_MAIN_LOAD(x1[2], 0)≥NonInfC∧579_0_MAIN_LOAD(x1[2], 0)≥COND_579_0_MAIN_LOAD1(&&(>(x1[2], 0), <(0, +(0, x1[2]))), x1[2], 0)∧(UIncreasing(COND_579_0_MAIN_LOAD1(&&(>(x1[2], 0), <(0, +(0, x1[2]))), x1[2], 0)), ≥))

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

(3)    (x1[2] + [-1] ≥ 0∧[-1] + x1[2] ≥ 0 ⇒ (UIncreasing(COND_579_0_MAIN_LOAD1(&&(>(x1[2], 0), <(0, +(0, x1[2]))), x1[2], 0)), ≥)∧[bni_10 + (-1)Bound*bni_10] ≥ 0∧[(-1)bso_11] ≥ 0)

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

(4)    (x1[2] + [-1] ≥ 0∧[-1] + x1[2] ≥ 0 ⇒ (UIncreasing(COND_579_0_MAIN_LOAD1(&&(>(x1[2], 0), <(0, +(0, x1[2]))), x1[2], 0)), ≥)∧[bni_10 + (-1)Bound*bni_10] ≥ 0∧[(-1)bso_11] ≥ 0)

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

(5)    (x1[2] + [-1] ≥ 0∧[-1] + x1[2] ≥ 0 ⇒ (UIncreasing(COND_579_0_MAIN_LOAD1(&&(>(x1[2], 0), <(0, +(0, x1[2]))), x1[2], 0)), ≥)∧[bni_10 + (-1)Bound*bni_10] ≥ 0∧[(-1)bso_11] ≥ 0)

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

(6)    (x1[2] ≥ 0∧x1[2] ≥ 0 ⇒ (UIncreasing(COND_579_0_MAIN_LOAD1(&&(>(x1[2], 0), <(0, +(0, x1[2]))), x1[2], 0)), ≥)∧[bni_10 + (-1)Bound*bni_10] ≥ 0∧[(-1)bso_11] ≥ 0)

For Pair COND_579_0_MAIN_LOAD1(TRUE, x1[3], 0) → 579_0_MAIN_LOAD(+(x1[3], -1), 0) the following chains were created:
• We consider the chain COND_579_0_MAIN_LOAD1(TRUE, x1[3], 0) → 579_0_MAIN_LOAD(+(x1[3], -1), 0) which results in the following constraint:

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

(8)    ((UIncreasing(579_0_MAIN_LOAD(+(x1[3], -1), 0)), ≥)∧[(-1)bso_13] ≥ 0)

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

(9)    ((UIncreasing(579_0_MAIN_LOAD(+(x1[3], -1), 0)), ≥)∧[(-1)bso_13] ≥ 0)

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

(10)    ((UIncreasing(579_0_MAIN_LOAD(+(x1[3], -1), 0)), ≥)∧[(-1)bso_13] ≥ 0)

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

(11)    ((UIncreasing(579_0_MAIN_LOAD(+(x1[3], -1), 0)), ≥)∧0 = 0∧[(-1)bso_13] ≥ 0)

For Pair 579_0_MAIN_LOAD(x1[4], x0[4]) → COND_579_0_MAIN_LOAD2(&&(&&(>=(x1[4], 0), >(x0[4], 0)), <(0, +(x0[4], x1[4]))), x1[4], x0[4]) the following chains were created:
• We consider the chain 579_0_MAIN_LOAD(x1[4], x0[4]) → COND_579_0_MAIN_LOAD2(&&(&&(>=(x1[4], 0), >(x0[4], 0)), <(0, +(x0[4], x1[4]))), x1[4], x0[4]), COND_579_0_MAIN_LOAD2(TRUE, x1[5], x0[5]) → 579_0_MAIN_LOAD(x1[5], +(x0[5], -1)) which results in the following constraint:

(12)    (&&(&&(>=(x1[4], 0), >(x0[4], 0)), <(0, +(x0[4], x1[4])))=TRUEx1[4]=x1[5]x0[4]=x0[5]579_0_MAIN_LOAD(x1[4], x0[4])≥NonInfC∧579_0_MAIN_LOAD(x1[4], x0[4])≥COND_579_0_MAIN_LOAD2(&&(&&(>=(x1[4], 0), >(x0[4], 0)), <(0, +(x0[4], x1[4]))), x1[4], x0[4])∧(UIncreasing(COND_579_0_MAIN_LOAD2(&&(&&(>=(x1[4], 0), >(x0[4], 0)), <(0, +(x0[4], x1[4]))), x1[4], x0[4])), ≥))

We simplified constraint (12) using rules (IV), (IDP_BOOLEAN) which results in the following new constraint:

(13)    (<(0, +(x0[4], x1[4]))=TRUE>=(x1[4], 0)=TRUE>(x0[4], 0)=TRUE579_0_MAIN_LOAD(x1[4], x0[4])≥NonInfC∧579_0_MAIN_LOAD(x1[4], x0[4])≥COND_579_0_MAIN_LOAD2(&&(&&(>=(x1[4], 0), >(x0[4], 0)), <(0, +(x0[4], x1[4]))), x1[4], x0[4])∧(UIncreasing(COND_579_0_MAIN_LOAD2(&&(&&(>=(x1[4], 0), >(x0[4], 0)), <(0, +(x0[4], x1[4]))), x1[4], x0[4])), ≥))

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

(14)    (x0[4] + [-1] + x1[4] ≥ 0∧x1[4] ≥ 0∧x0[4] + [-1] ≥ 0 ⇒ (UIncreasing(COND_579_0_MAIN_LOAD2(&&(&&(>=(x1[4], 0), >(x0[4], 0)), <(0, +(x0[4], x1[4]))), x1[4], x0[4])), ≥)∧[bni_14 + (-1)Bound*bni_14] + [bni_14]x0[4] ≥ 0∧[1 + (-1)bso_15] ≥ 0)

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

(15)    (x0[4] + [-1] + x1[4] ≥ 0∧x1[4] ≥ 0∧x0[4] + [-1] ≥ 0 ⇒ (UIncreasing(COND_579_0_MAIN_LOAD2(&&(&&(>=(x1[4], 0), >(x0[4], 0)), <(0, +(x0[4], x1[4]))), x1[4], x0[4])), ≥)∧[bni_14 + (-1)Bound*bni_14] + [bni_14]x0[4] ≥ 0∧[1 + (-1)bso_15] ≥ 0)

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

(16)    (x0[4] + [-1] + x1[4] ≥ 0∧x1[4] ≥ 0∧x0[4] + [-1] ≥ 0 ⇒ (UIncreasing(COND_579_0_MAIN_LOAD2(&&(&&(>=(x1[4], 0), >(x0[4], 0)), <(0, +(x0[4], x1[4]))), x1[4], x0[4])), ≥)∧[bni_14 + (-1)Bound*bni_14] + [bni_14]x0[4] ≥ 0∧[1 + (-1)bso_15] ≥ 0)

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

(17)    (x0[4] + x1[4] ≥ 0∧x1[4] ≥ 0∧x0[4] ≥ 0 ⇒ (UIncreasing(COND_579_0_MAIN_LOAD2(&&(&&(>=(x1[4], 0), >(x0[4], 0)), <(0, +(x0[4], x1[4]))), x1[4], x0[4])), ≥)∧[(2)bni_14 + (-1)Bound*bni_14] + [bni_14]x0[4] ≥ 0∧[1 + (-1)bso_15] ≥ 0)

For Pair COND_579_0_MAIN_LOAD2(TRUE, x1[5], x0[5]) → 579_0_MAIN_LOAD(x1[5], +(x0[5], -1)) the following chains were created:
• We consider the chain COND_579_0_MAIN_LOAD2(TRUE, x1[5], x0[5]) → 579_0_MAIN_LOAD(x1[5], +(x0[5], -1)) which results in the following constraint:

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

(19)    ((UIncreasing(579_0_MAIN_LOAD(x1[5], +(x0[5], -1))), ≥)∧[(-1)bso_17] ≥ 0)

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

(20)    ((UIncreasing(579_0_MAIN_LOAD(x1[5], +(x0[5], -1))), ≥)∧[(-1)bso_17] ≥ 0)

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

(21)    ((UIncreasing(579_0_MAIN_LOAD(x1[5], +(x0[5], -1))), ≥)∧[(-1)bso_17] ≥ 0)

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

(22)    ((UIncreasing(579_0_MAIN_LOAD(x1[5], +(x0[5], -1))), ≥)∧0 = 0∧0 = 0∧[(-1)bso_17] ≥ 0)

To summarize, we get the following constraints P for the following pairs.
• 579_0_MAIN_LOAD(x1[2], 0) → COND_579_0_MAIN_LOAD1(&&(>(x1[2], 0), <(0, +(0, x1[2]))), x1[2], 0)
• (x1[2] ≥ 0∧x1[2] ≥ 0 ⇒ (UIncreasing(COND_579_0_MAIN_LOAD1(&&(>(x1[2], 0), <(0, +(0, x1[2]))), x1[2], 0)), ≥)∧[bni_10 + (-1)Bound*bni_10] ≥ 0∧[(-1)bso_11] ≥ 0)

• COND_579_0_MAIN_LOAD1(TRUE, x1[3], 0) → 579_0_MAIN_LOAD(+(x1[3], -1), 0)
• ((UIncreasing(579_0_MAIN_LOAD(+(x1[3], -1), 0)), ≥)∧0 = 0∧[(-1)bso_13] ≥ 0)

• 579_0_MAIN_LOAD(x1[4], x0[4]) → COND_579_0_MAIN_LOAD2(&&(&&(>=(x1[4], 0), >(x0[4], 0)), <(0, +(x0[4], x1[4]))), x1[4], x0[4])
• (x0[4] + x1[4] ≥ 0∧x1[4] ≥ 0∧x0[4] ≥ 0 ⇒ (UIncreasing(COND_579_0_MAIN_LOAD2(&&(&&(>=(x1[4], 0), >(x0[4], 0)), <(0, +(x0[4], x1[4]))), x1[4], x0[4])), ≥)∧[(2)bni_14 + (-1)Bound*bni_14] + [bni_14]x0[4] ≥ 0∧[1 + (-1)bso_15] ≥ 0)

• COND_579_0_MAIN_LOAD2(TRUE, x1[5], x0[5]) → 579_0_MAIN_LOAD(x1[5], +(x0[5], -1))
• ((UIncreasing(579_0_MAIN_LOAD(x1[5], +(x0[5], -1))), ≥)∧0 = 0∧0 = 0∧[(-1)bso_17] ≥ 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(579_0_MAIN_LOAD(x1, x2)) = [1] + x2
POL(0) = 0
POL(COND_579_0_MAIN_LOAD1(x1, x2, x3)) = [1]
POL(&&(x1, x2)) = 0
POL(>(x1, x2)) = [-1]
POL(<(x1, x2)) = [-1]
POL(+(x1, x2)) = x1 + x2
POL(-1) = [-1]
POL(COND_579_0_MAIN_LOAD2(x1, x2, x3)) = x3
POL(>=(x1, x2)) = 0

The following pairs are in P>:

579_0_MAIN_LOAD(x1[4], x0[4]) → COND_579_0_MAIN_LOAD2(&&(&&(>=(x1[4], 0), >(x0[4], 0)), <(0, +(x0[4], x1[4]))), x1[4], x0[4])

The following pairs are in Pbound:

579_0_MAIN_LOAD(x1[2], 0) → COND_579_0_MAIN_LOAD1(&&(>(x1[2], 0), <(0, +(0, x1[2]))), x1[2], 0)
579_0_MAIN_LOAD(x1[4], x0[4]) → COND_579_0_MAIN_LOAD2(&&(&&(>=(x1[4], 0), >(x0[4], 0)), <(0, +(x0[4], x1[4]))), x1[4], x0[4])

The following pairs are in P:

579_0_MAIN_LOAD(x1[2], 0) → COND_579_0_MAIN_LOAD1(&&(>(x1[2], 0), <(0, +(0, x1[2]))), x1[2], 0)
COND_579_0_MAIN_LOAD1(TRUE, x1[3], 0) → 579_0_MAIN_LOAD(+(x1[3], -1), 0)
COND_579_0_MAIN_LOAD2(TRUE, x1[5], x0[5]) → 579_0_MAIN_LOAD(x1[5], +(x0[5], -1))

There are no usable rules.

### (9) Obligation:

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

The following domains are used:

Boolean, Integer

R is empty.

The integer pair graph contains the following rules and edges:
(2): 579_0_MAIN_LOAD(x1[2], 0) → COND_579_0_MAIN_LOAD1(x1[2] > 0 && 0 < 0 + x1[2], x1[2], 0)
(3): COND_579_0_MAIN_LOAD1(TRUE, x1[3], 0) → 579_0_MAIN_LOAD(x1[3] + -1, 0)
(5): COND_579_0_MAIN_LOAD2(TRUE, x1[5], x0[5]) → 579_0_MAIN_LOAD(x1[5], x0[5] + -1)

(3) -> (2), if (x1[3] + -1* x1[2])

(5) -> (2), if ((x1[5]* x1[2])∧(x0[5] + -1* 0))

(2) -> (3), if ((x1[2] > 0 && 0 < 0 + x1[2]* TRUE)∧(x1[2]* x1[3]))

The set Q is empty.

### (10) IDependencyGraphProof (EQUIVALENT transformation)

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

### (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:
(3): COND_579_0_MAIN_LOAD1(TRUE, x1[3], 0) → 579_0_MAIN_LOAD(x1[3] + -1, 0)
(2): 579_0_MAIN_LOAD(x1[2], 0) → COND_579_0_MAIN_LOAD1(x1[2] > 0 && 0 < 0 + x1[2], x1[2], 0)

(3) -> (2), if (x1[3] + -1* x1[2])

(2) -> (3), if ((x1[2] > 0 && 0 < 0 + x1[2]* TRUE)∧(x1[2]* x1[3]))

The set Q is empty.

### (12) 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_579_0_MAIN_LOAD1(TRUE, x1[3], 0) → 579_0_MAIN_LOAD(+(x1[3], -1), 0) the following chains were created:
• We consider the chain COND_579_0_MAIN_LOAD1(TRUE, x1[3], 0) → 579_0_MAIN_LOAD(+(x1[3], -1), 0) which results in the following constraint:

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

(2)    ((UIncreasing(579_0_MAIN_LOAD(+(x1[3], -1), 0)), ≥)∧[1 + (-1)bso_9] ≥ 0)

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

(3)    ((UIncreasing(579_0_MAIN_LOAD(+(x1[3], -1), 0)), ≥)∧[1 + (-1)bso_9] ≥ 0)

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

(4)    ((UIncreasing(579_0_MAIN_LOAD(+(x1[3], -1), 0)), ≥)∧[1 + (-1)bso_9] ≥ 0)

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

(5)    ((UIncreasing(579_0_MAIN_LOAD(+(x1[3], -1), 0)), ≥)∧0 = 0∧[1 + (-1)bso_9] ≥ 0)

For Pair 579_0_MAIN_LOAD(x1[2], 0) → COND_579_0_MAIN_LOAD1(&&(>(x1[2], 0), <(0, +(0, x1[2]))), x1[2], 0) the following chains were created:
• We consider the chain 579_0_MAIN_LOAD(x1[2], 0) → COND_579_0_MAIN_LOAD1(&&(>(x1[2], 0), <(0, +(0, x1[2]))), x1[2], 0), COND_579_0_MAIN_LOAD1(TRUE, x1[3], 0) → 579_0_MAIN_LOAD(+(x1[3], -1), 0) which results in the following constraint:

(6)    (&&(>(x1[2], 0), <(0, +(0, x1[2])))=TRUEx1[2]=x1[3]579_0_MAIN_LOAD(x1[2], 0)≥NonInfC∧579_0_MAIN_LOAD(x1[2], 0)≥COND_579_0_MAIN_LOAD1(&&(>(x1[2], 0), <(0, +(0, x1[2]))), x1[2], 0)∧(UIncreasing(COND_579_0_MAIN_LOAD1(&&(>(x1[2], 0), <(0, +(0, x1[2]))), x1[2], 0)), ≥))

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

(7)    (>(x1[2], 0)=TRUE<(0, +(0, x1[2]))=TRUE579_0_MAIN_LOAD(x1[2], 0)≥NonInfC∧579_0_MAIN_LOAD(x1[2], 0)≥COND_579_0_MAIN_LOAD1(&&(>(x1[2], 0), <(0, +(0, x1[2]))), x1[2], 0)∧(UIncreasing(COND_579_0_MAIN_LOAD1(&&(>(x1[2], 0), <(0, +(0, x1[2]))), x1[2], 0)), ≥))

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

(8)    (x1[2] + [-1] ≥ 0∧[-1] + x1[2] ≥ 0 ⇒ (UIncreasing(COND_579_0_MAIN_LOAD1(&&(>(x1[2], 0), <(0, +(0, x1[2]))), x1[2], 0)), ≥)∧[(-1)bni_10 + (-1)Bound*bni_10] + [bni_10]x1[2] ≥ 0∧[(-1)bso_11] ≥ 0)

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

(9)    (x1[2] + [-1] ≥ 0∧[-1] + x1[2] ≥ 0 ⇒ (UIncreasing(COND_579_0_MAIN_LOAD1(&&(>(x1[2], 0), <(0, +(0, x1[2]))), x1[2], 0)), ≥)∧[(-1)bni_10 + (-1)Bound*bni_10] + [bni_10]x1[2] ≥ 0∧[(-1)bso_11] ≥ 0)

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

(10)    (x1[2] + [-1] ≥ 0∧[-1] + x1[2] ≥ 0 ⇒ (UIncreasing(COND_579_0_MAIN_LOAD1(&&(>(x1[2], 0), <(0, +(0, x1[2]))), x1[2], 0)), ≥)∧[(-1)bni_10 + (-1)Bound*bni_10] + [bni_10]x1[2] ≥ 0∧[(-1)bso_11] ≥ 0)

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

(11)    (x1[2] ≥ 0∧x1[2] ≥ 0 ⇒ (UIncreasing(COND_579_0_MAIN_LOAD1(&&(>(x1[2], 0), <(0, +(0, x1[2]))), x1[2], 0)), ≥)∧[(-1)Bound*bni_10] + [bni_10]x1[2] ≥ 0∧[(-1)bso_11] ≥ 0)

To summarize, we get the following constraints P for the following pairs.
• COND_579_0_MAIN_LOAD1(TRUE, x1[3], 0) → 579_0_MAIN_LOAD(+(x1[3], -1), 0)
• ((UIncreasing(579_0_MAIN_LOAD(+(x1[3], -1), 0)), ≥)∧0 = 0∧[1 + (-1)bso_9] ≥ 0)

• 579_0_MAIN_LOAD(x1[2], 0) → COND_579_0_MAIN_LOAD1(&&(>(x1[2], 0), <(0, +(0, x1[2]))), x1[2], 0)
• (x1[2] ≥ 0∧x1[2] ≥ 0 ⇒ (UIncreasing(COND_579_0_MAIN_LOAD1(&&(>(x1[2], 0), <(0, +(0, x1[2]))), x1[2], 0)), ≥)∧[(-1)Bound*bni_10] + [bni_10]x1[2] ≥ 0∧[(-1)bso_11] ≥ 0)

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

POL(TRUE) = 0
POL(FALSE) = 0
POL(COND_579_0_MAIN_LOAD1(x1, x2, x3)) = [-1] + x2
POL(0) = 0
POL(579_0_MAIN_LOAD(x1, x2)) = [-1] + x1
POL(+(x1, x2)) = x1 + x2
POL(-1) = [-1]
POL(&&(x1, x2)) = [1]
POL(>(x1, x2)) = [-1]
POL(<(x1, x2)) = [-1]

The following pairs are in P>:

COND_579_0_MAIN_LOAD1(TRUE, x1[3], 0) → 579_0_MAIN_LOAD(+(x1[3], -1), 0)

The following pairs are in Pbound:

579_0_MAIN_LOAD(x1[2], 0) → COND_579_0_MAIN_LOAD1(&&(>(x1[2], 0), <(0, +(0, x1[2]))), x1[2], 0)

The following pairs are in P:

579_0_MAIN_LOAD(x1[2], 0) → COND_579_0_MAIN_LOAD1(&&(>(x1[2], 0), <(0, +(0, x1[2]))), x1[2], 0)

There are no usable rules.

### (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:

Boolean, Integer

R is empty.

The integer pair graph contains the following rules and edges:
(2): 579_0_MAIN_LOAD(x1[2], 0) → COND_579_0_MAIN_LOAD1(x1[2] > 0 && 0 < 0 + x1[2], x1[2], 0)

The set Q is empty.

### (15) IDependencyGraphProof (EQUIVALENT transformation)

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

### (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:
(3): COND_579_0_MAIN_LOAD1(TRUE, x1[3], 0) → 579_0_MAIN_LOAD(x1[3] + -1, 0)

The set Q is empty.

### (18) IDependencyGraphProof (EQUIVALENT transformation)

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

### (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:
(1): COND_579_0_MAIN_LOAD(TRUE, 0, 0) → 579_0_MAIN_LOAD(0, 0)
(3): COND_579_0_MAIN_LOAD1(TRUE, x1[3], 0) → 579_0_MAIN_LOAD(x1[3] + -1, 0)
(5): COND_579_0_MAIN_LOAD2(TRUE, x1[5], x0[5]) → 579_0_MAIN_LOAD(x1[5], x0[5] + -1)

The set Q is empty.

### (21) IDependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 3 less nodes.