Termination proof of /tmp/tmp6iRhiF/PastaB10.jar

Termination of the given JBC could be proven:

0 JBC

↳1 JBC2FIG (⇒)

↳2 JBCTerminationGraph

↳3 FIGtoITRSProof (⇒)

↳4 IDP

↳5 IDPNonInfProof (⇒)

↳6 AND

    ↳7 IDP

        ↳8 IDPNonInfProof (⇒)

        ↳9 IDP

        ↳10 IDependencyGraphProof (⇔)

        ↳11 IDP

        ↳12 IDPNonInfProof (⇒)

        ↳13 AND

            ↳14 IDP

                ↳15 IDependencyGraphProof (⇔)

                ↳16 TRUE

            ↳17 IDP

                ↳18 IDependencyGraphProof (⇔)

                ↳19 TRUE

    ↳20 IDP

        ↳21 IDependencyGraphProof (⇔)

        ↳22 TRUE

(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:

579_0_main_Load(x1, x2, x3) → 579_0_main_Load(x2, x3)
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))=TRUE ⇒ 579_0_MAIN_LOAD(0, 0)≥NonInfC∧579_0_MAIN_LOAD(0, 0)≥COND_579_0_MAIN_LOAD(<(0, +(0, 0)), 0, 0)∧(U^Increasing(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:
(2)    (COND_579_0_MAIN_LOAD(TRUE, 0, 0)≥NonInfC∧COND_579_0_MAIN_LOAD(TRUE, 0, 0)≥579_0_MAIN_LOAD(0, 0)∧(U^Increasing(579_0_MAIN_LOAD(0, 0)), ≥))

We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(3)    ((U^Increasing(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)    ((U^Increasing(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)    ((U^Increasing(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:
(6)    (0=x1[2] ⇒ COND_579_0_MAIN_LOAD(TRUE, 0, 0)≥NonInfC∧COND_579_0_MAIN_LOAD(TRUE, 0, 0)≥579_0_MAIN_LOAD(0, 0)∧(U^Increasing(579_0_MAIN_LOAD(0, 0)), ≥))

We simplified constraint (6) using rule (IV) which results in the following new constraint:
(7)    (COND_579_0_MAIN_LOAD(TRUE, 0, 0)≥NonInfC∧COND_579_0_MAIN_LOAD(TRUE, 0, 0)≥579_0_MAIN_LOAD(0, 0)∧(U^Increasing(579_0_MAIN_LOAD(0, 0)), ≥))

We simplified constraint (7) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(8)    ((U^Increasing(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)    ((U^Increasing(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)    ((U^Increasing(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:
(11)    (0=x1[4]∧0=x0[4] ⇒ COND_579_0_MAIN_LOAD(TRUE, 0, 0)≥NonInfC∧COND_579_0_MAIN_LOAD(TRUE, 0, 0)≥579_0_MAIN_LOAD(0, 0)∧(U^Increasing(579_0_MAIN_LOAD(0, 0)), ≥))

We simplified constraint (11) using rule (IV) which results in the following new constraint:
(12)    (COND_579_0_MAIN_LOAD(TRUE, 0, 0)≥NonInfC∧COND_579_0_MAIN_LOAD(TRUE, 0, 0)≥579_0_MAIN_LOAD(0, 0)∧(U^Increasing(579_0_MAIN_LOAD(0, 0)), ≥))

We simplified constraint (12) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(13)    ((U^Increasing(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)    ((U^Increasing(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)    ((U^Increasing(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])))=TRUE∧x1[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)∧(U^Increasing(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]))=TRUE ⇒ 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)∧(U^Increasing(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 ⇒ (U^Increasing(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 ⇒ (U^Increasing(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 ⇒ (U^Increasing(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 ⇒ (U^Increasing(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:
(22)    (COND_579_0_MAIN_LOAD1(TRUE, x1[3], 0)≥NonInfC∧COND_579_0_MAIN_LOAD1(TRUE, x1[3], 0)≥579_0_MAIN_LOAD(+(x1[3], -1), 0)∧(U^Increasing(579_0_MAIN_LOAD(+(x1[3], -1), 0)), ≥))

We simplified constraint (22) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(23)    ((U^Increasing(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)    ((U^Increasing(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)    ((U^Increasing(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)    ((U^Increasing(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])))=TRUE∧x1[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])∧(U^Increasing(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)=TRUE ⇒ 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])∧(U^Increasing(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 ⇒ (U^Increasing(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 ⇒ (U^Increasing(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 ⇒ (U^Increasing(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 ⇒ (U^Increasing(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:
(33)    (COND_579_0_MAIN_LOAD2(TRUE, x1[5], x0[5])≥NonInfC∧COND_579_0_MAIN_LOAD2(TRUE, x1[5], x0[5])≥579_0_MAIN_LOAD(x1[5], +(x0[5], -1))∧(U^Increasing(579_0_MAIN_LOAD(x1[5], +(x0[5], -1))), ≥))

We simplified constraint (33) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(34)    ((U^Increasing(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)    ((U^Increasing(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)    ((U^Increasing(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)    ((U^Increasing(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)
- ((U^Increasing(579_0_MAIN_LOAD(0, 0)), ≥)∧[2 + (-1)bso_15] ≥ 0)
- ((U^Increasing(579_0_MAIN_LOAD(0, 0)), ≥)∧[2 + (-1)bso_15] ≥ 0)
- ((U^Increasing(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 ⇒ (U^Increasing(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)
- ((U^Increasing(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 ⇒ (U^Increasing(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))
- ((U^Increasing(579_0_MAIN_LOAD(x1[5], +(x0[5], -1))), ≥)∧0 = 0∧0 = 0∧[(-1)bso_23] ≥ 0)

The constraints for P_> respective P_bound 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 P_bound.
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)
COND_579_0_MAIN_LOAD(TRUE, 0, 0) → 579_0_MAIN_LOAD(0, 0)

The following pairs are in P_bound:

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.

(6) Complex Obligation (AND)

(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])))=TRUE∧x1[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)∧(U^Increasing(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]))=TRUE ⇒ 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)∧(U^Increasing(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 ⇒ (U^Increasing(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 ⇒ (U^Increasing(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 ⇒ (U^Increasing(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 ⇒ (U^Increasing(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:
(7)    (COND_579_0_MAIN_LOAD1(TRUE, x1[3], 0)≥NonInfC∧COND_579_0_MAIN_LOAD1(TRUE, x1[3], 0)≥579_0_MAIN_LOAD(+(x1[3], -1), 0)∧(U^Increasing(579_0_MAIN_LOAD(+(x1[3], -1), 0)), ≥))

We simplified constraint (7) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(8)    ((U^Increasing(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)    ((U^Increasing(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)    ((U^Increasing(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)    ((U^Increasing(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])))=TRUE∧x1[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])∧(U^Increasing(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)=TRUE ⇒ 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])∧(U^Increasing(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 ⇒ (U^Increasing(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 ⇒ (U^Increasing(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 ⇒ (U^Increasing(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 ⇒ (U^Increasing(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:
(18)    (COND_579_0_MAIN_LOAD2(TRUE, x1[5], x0[5])≥NonInfC∧COND_579_0_MAIN_LOAD2(TRUE, x1[5], x0[5])≥579_0_MAIN_LOAD(x1[5], +(x0[5], -1))∧(U^Increasing(579_0_MAIN_LOAD(x1[5], +(x0[5], -1))), ≥))

We simplified constraint (18) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(19)    ((U^Increasing(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)    ((U^Increasing(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)    ((U^Increasing(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)    ((U^Increasing(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 ⇒ (U^Increasing(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)
- ((U^Increasing(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 ⇒ (U^Increasing(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))
- ((U^Increasing(579_0_MAIN_LOAD(x1[5], +(x0[5], -1))), ≥)∧0 = 0∧0 = 0∧[(-1)bso_17] ≥ 0)

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 P_bound:

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:
(1)    (COND_579_0_MAIN_LOAD1(TRUE, x1[3], 0)≥NonInfC∧COND_579_0_MAIN_LOAD1(TRUE, x1[3], 0)≥579_0_MAIN_LOAD(+(x1[3], -1), 0)∧(U^Increasing(579_0_MAIN_LOAD(+(x1[3], -1), 0)), ≥))

We simplified constraint (1) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(2)    ((U^Increasing(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)    ((U^Increasing(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)    ((U^Increasing(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)    ((U^Increasing(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])))=TRUE∧x1[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)∧(U^Increasing(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]))=TRUE ⇒ 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)∧(U^Increasing(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 ⇒ (U^Increasing(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 ⇒ (U^Increasing(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 ⇒ (U^Increasing(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 ⇒ (U^Increasing(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)
- ((U^Increasing(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 ⇒ (U^Increasing(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)

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 P_bound:

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.

(13) Complex Obligation (AND)

(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.

(16) TRUE

(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.

(19) TRUE

(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.

(0) Obligation:

(1) JBC2FIG (SOUND transformation)

(2) Obligation:

(3) FIGtoITRSProof (SOUND transformation)

(4) Obligation:

(5) IDPNonInfProof (SOUND transformation)

(6) Complex Obligation (AND)

(7) Obligation:

(8) IDPNonInfProof (SOUND transformation)

(9) Obligation:

(10) IDependencyGraphProof (EQUIVALENT transformation)

(11) Obligation:

(12) IDPNonInfProof (SOUND transformation)

(13) Complex Obligation (AND)

(14) Obligation:

(15) IDependencyGraphProof (EQUIVALENT transformation)

(16) TRUE

(17) Obligation:

(18) IDependencyGraphProof (EQUIVALENT transformation)

(19) TRUE

(20) Obligation:

(21) IDependencyGraphProof (EQUIVALENT transformation)

(22) TRUE