Termination proof of /tmp/tmpNhlVlI/PastaC3.jar

Termination of the given JBC could be proven:

0 JBC

↳1 JBC2FIG (⇒)

↳2 JBCTerminationGraph

↳3 FIGtoITRSProof (⇒)

↳4 IDP

↳5 IDPNonInfProof (⇒)

↳6 IDP

↳7 IDependencyGraphProof (⇔)

↳8 IDP

↳9 IDPNonInfProof (⇒)

↳10 IDP

↳11 IDependencyGraphProof (⇔)

↳12 TRUE

(0) Obligation:

JBC Problem based on JBC Program:
Manifest-Version: 1.0 Created-By: 1.6.0_16 (Sun Microsystems Inc.) Main-Class: PastaC3

/**
 * 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 PastaC3 {
    public static void main(String[] args) {
        Random.args = args;
        int x = Random.random();
        int y = Random.random();
        int z = Random.random();

        while (x < y) {
            if (x < z) {
                x++;
            } else {
                z++;
            }
        }
    }
}


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:
PastaC3.main([Ljava/lang/String;)V: Graph of 228 nodes with 1 SCC.

(3) FIGtoITRSProof (SOUND transformation)

Transformed FIGraph SCCs to IDPs. Logs:

Log for SCC 0:

Generated 16 rules for P and 2 rules for R.

Combined rules. Obtained 2 rules for P and 0 rules for R.

Filtered ground terms:

917_0_main_Load(x1, x2, x3, x4, x5) → 917_0_main_Load(x2, x3, x4, x5)
Cond_917_0_main_Load1(x1, x2, x3, x4, x5, x6) → Cond_917_0_main_Load1(x1, x3, x4, x5, x6)
Cond_917_0_main_Load(x1, x2, x3, x4, x5, x6) → Cond_917_0_main_Load(x1, x3, x4, x5, x6)

Filtered duplicate args:

917_0_main_Load(x1, x2, x3, x4) → 917_0_main_Load(x2, x3, x4)
Cond_917_0_main_Load1(x1, x2, x3, x4, x5) → Cond_917_0_main_Load1(x1, x3, x4, x5)
Cond_917_0_main_Load(x1, x2, x3, x4, x5) → Cond_917_0_main_Load(x1, x3, x4, x5)

Combined rules. Obtained 2 rules for P and 0 rules for R.

Finished conversion. Obtained 2 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:

Boolean, Integer

R is empty.

The integer pair graph contains the following rules and edges:
(0): 917_0_MAIN_LOAD(x1[0], x2[0], x0[0]) → COND_917_0_MAIN_LOAD(x2[0] >= 0 && x2[0] <= x0[0] && x1[0] > x0[0], x1[0], x2[0], x0[0])
(1): COND_917_0_MAIN_LOAD(TRUE, x1[1], x2[1], x0[1]) → 917_0_MAIN_LOAD(x1[1], x2[1] + 1, x0[1])
(2): 917_0_MAIN_LOAD(x1[2], x2[2], x0[2]) → COND_917_0_MAIN_LOAD1(x2[2] > x0[2] && x1[2] > x0[2] && x0[2] >= 0, x1[2], x2[2], x0[2])
(3): COND_917_0_MAIN_LOAD1(TRUE, x1[3], x2[3], x0[3]) → 917_0_MAIN_LOAD(x1[3], x2[3], x0[3] + 1)

(0) -> (1), if ((x2[0] >= 0 && x2[0] <= x0[0] && x1[0] > x0[0] →^* TRUE)∧(x1[0] →^* x1[1])∧(x2[0] →^* x2[1])∧(x0[0] →^* x0[1]))

(1) -> (0), if ((x1[1] →^* x1[0])∧(x2[1] + 1 →^* x2[0])∧(x0[1] →^* x0[0]))

(1) -> (2), if ((x1[1] →^* x1[2])∧(x2[1] + 1 →^* x2[2])∧(x0[1] →^* x0[2]))

(2) -> (3), if ((x2[2] > x0[2] && x1[2] > x0[2] && x0[2] >= 0 →^* TRUE)∧(x1[2] →^* x1[3])∧(x2[2] →^* x2[3])∧(x0[2] →^* x0[3]))

(3) -> (0), if ((x1[3] →^* x1[0])∧(x2[3] →^* x2[0])∧(x0[3] + 1 →^* x0[0]))

(3) -> (2), if ((x1[3] →^* x1[2])∧(x2[3] →^* x2[2])∧(x0[3] + 1 →^* x0[2]))

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 917_0_MAIN_LOAD(x1, x2, x0) → COND_917_0_MAIN_LOAD(&&(&&(>=(x2, 0), <=(x2, x0)), >(x1, x0)), x1, x2, x0) the following chains were created:

We consider the chain 917_0_MAIN_LOAD(x1[0], x2[0], x0[0]) → COND_917_0_MAIN_LOAD(&&(&&(>=(x2[0], 0), <=(x2[0], x0[0])), >(x1[0], x0[0])), x1[0], x2[0], x0[0]), COND_917_0_MAIN_LOAD(TRUE, x1[1], x2[1], x0[1]) → 917_0_MAIN_LOAD(x1[1], +(x2[1], 1), x0[1]) which results in the following constraint:
(1)    (&&(&&(>=(x2[0], 0), <=(x2[0], x0[0])), >(x1[0], x0[0]))=TRUE∧x1[0]=x1[1]∧x2[0]=x2[1]∧x0[0]=x0[1] ⇒ 917_0_MAIN_LOAD(x1[0], x2[0], x0[0])≥NonInfC∧917_0_MAIN_LOAD(x1[0], x2[0], x0[0])≥COND_917_0_MAIN_LOAD(&&(&&(>=(x2[0], 0), <=(x2[0], x0[0])), >(x1[0], x0[0])), x1[0], x2[0], x0[0])∧(U^Increasing(COND_917_0_MAIN_LOAD(&&(&&(>=(x2[0], 0), <=(x2[0], x0[0])), >(x1[0], x0[0])), x1[0], x2[0], x0[0])), ≥))

We simplified constraint (1) using rules (IV), (IDP_BOOLEAN) which results in the following new constraint:
(2)    (>(x1[0], x0[0])=TRUE∧>=(x2[0], 0)=TRUE∧<=(x2[0], x0[0])=TRUE ⇒ 917_0_MAIN_LOAD(x1[0], x2[0], x0[0])≥NonInfC∧917_0_MAIN_LOAD(x1[0], x2[0], x0[0])≥COND_917_0_MAIN_LOAD(&&(&&(>=(x2[0], 0), <=(x2[0], x0[0])), >(x1[0], x0[0])), x1[0], x2[0], x0[0])∧(U^Increasing(COND_917_0_MAIN_LOAD(&&(&&(>=(x2[0], 0), <=(x2[0], x0[0])), >(x1[0], x0[0])), x1[0], x2[0], x0[0])), ≥))

We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(3)    (x1[0] + [-1] + [-1]x0[0] ≥ 0∧x2[0] ≥ 0∧x0[0] + [-1]x2[0] ≥ 0 ⇒ (U^Increasing(COND_917_0_MAIN_LOAD(&&(&&(>=(x2[0], 0), <=(x2[0], x0[0])), >(x1[0], x0[0])), x1[0], x2[0], x0[0])), ≥)∧[(-1)bni_21 + (-1)Bound*bni_21] + [(-1)bni_21]x2[0] + [bni_21]x1[0] ≥ 0∧[(-1)bso_22] ≥ 0)

We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(4)    (x1[0] + [-1] + [-1]x0[0] ≥ 0∧x2[0] ≥ 0∧x0[0] + [-1]x2[0] ≥ 0 ⇒ (U^Increasing(COND_917_0_MAIN_LOAD(&&(&&(>=(x2[0], 0), <=(x2[0], x0[0])), >(x1[0], x0[0])), x1[0], x2[0], x0[0])), ≥)∧[(-1)bni_21 + (-1)Bound*bni_21] + [(-1)bni_21]x2[0] + [bni_21]x1[0] ≥ 0∧[(-1)bso_22] ≥ 0)

We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(5)    (x1[0] + [-1] + [-1]x0[0] ≥ 0∧x2[0] ≥ 0∧x0[0] + [-1]x2[0] ≥ 0 ⇒ (U^Increasing(COND_917_0_MAIN_LOAD(&&(&&(>=(x2[0], 0), <=(x2[0], x0[0])), >(x1[0], x0[0])), x1[0], x2[0], x0[0])), ≥)∧[(-1)bni_21 + (-1)Bound*bni_21] + [(-1)bni_21]x2[0] + [bni_21]x1[0] ≥ 0∧[(-1)bso_22] ≥ 0)

We simplified constraint (5) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(6)    (x1[0] ≥ 0∧x2[0] ≥ 0∧x0[0] + [-1]x2[0] ≥ 0 ⇒ (U^Increasing(COND_917_0_MAIN_LOAD(&&(&&(>=(x2[0], 0), <=(x2[0], x0[0])), >(x1[0], x0[0])), x1[0], x2[0], x0[0])), ≥)∧[(-1)Bound*bni_21] + [bni_21]x0[0] + [(-1)bni_21]x2[0] + [bni_21]x1[0] ≥ 0∧[(-1)bso_22] ≥ 0)

We simplified constraint (6) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(7)    (x1[0] ≥ 0∧x2[0] ≥ 0∧x0[0] ≥ 0 ⇒ (U^Increasing(COND_917_0_MAIN_LOAD(&&(&&(>=(x2[0], 0), <=(x2[0], x0[0])), >(x1[0], x0[0])), x1[0], x2[0], x0[0])), ≥)∧[(-1)Bound*bni_21] + [bni_21]x0[0] + [bni_21]x1[0] ≥ 0∧[(-1)bso_22] ≥ 0)

For Pair COND_917_0_MAIN_LOAD(TRUE, x1, x2, x0) → 917_0_MAIN_LOAD(x1, +(x2, 1), x0) the following chains were created:

We consider the chain 917_0_MAIN_LOAD(x1[0], x2[0], x0[0]) → COND_917_0_MAIN_LOAD(&&(&&(>=(x2[0], 0), <=(x2[0], x0[0])), >(x1[0], x0[0])), x1[0], x2[0], x0[0]), COND_917_0_MAIN_LOAD(TRUE, x1[1], x2[1], x0[1]) → 917_0_MAIN_LOAD(x1[1], +(x2[1], 1), x0[1]), 917_0_MAIN_LOAD(x1[0], x2[0], x0[0]) → COND_917_0_MAIN_LOAD(&&(&&(>=(x2[0], 0), <=(x2[0], x0[0])), >(x1[0], x0[0])), x1[0], x2[0], x0[0]) which results in the following constraint:
(8)    (&&(&&(>=(x2[0], 0), <=(x2[0], x0[0])), >(x1[0], x0[0]))=TRUE∧x1[0]=x1[1]∧x2[0]=x2[1]∧x0[0]=x0[1]∧x1[1]=x1[0]1∧+(x2[1], 1)=x2[0]1∧x0[1]=x0[0]1 ⇒ COND_917_0_MAIN_LOAD(TRUE, x1[1], x2[1], x0[1])≥NonInfC∧COND_917_0_MAIN_LOAD(TRUE, x1[1], x2[1], x0[1])≥917_0_MAIN_LOAD(x1[1], +(x2[1], 1), x0[1])∧(U^Increasing(917_0_MAIN_LOAD(x1[1], +(x2[1], 1), x0[1])), ≥))

We simplified constraint (8) using rules (III), (IV), (IDP_BOOLEAN) which results in the following new constraint:
(9)    (>(x1[0], x0[0])=TRUE∧>=(x2[0], 0)=TRUE∧<=(x2[0], x0[0])=TRUE ⇒ COND_917_0_MAIN_LOAD(TRUE, x1[0], x2[0], x0[0])≥NonInfC∧COND_917_0_MAIN_LOAD(TRUE, x1[0], x2[0], x0[0])≥917_0_MAIN_LOAD(x1[0], +(x2[0], 1), x0[0])∧(U^Increasing(917_0_MAIN_LOAD(x1[1], +(x2[1], 1), x0[1])), ≥))

We simplified constraint (9) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(10)    (x1[0] + [-1] + [-1]x0[0] ≥ 0∧x2[0] ≥ 0∧x0[0] + [-1]x2[0] ≥ 0 ⇒ (U^Increasing(917_0_MAIN_LOAD(x1[1], +(x2[1], 1), x0[1])), ≥)∧[(-1)bni_23 + (-1)Bound*bni_23] + [(-1)bni_23]x2[0] + [bni_23]x1[0] ≥ 0∧[1 + (-1)bso_24] ≥ 0)

We simplified constraint (10) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(11)    (x1[0] + [-1] + [-1]x0[0] ≥ 0∧x2[0] ≥ 0∧x0[0] + [-1]x2[0] ≥ 0 ⇒ (U^Increasing(917_0_MAIN_LOAD(x1[1], +(x2[1], 1), x0[1])), ≥)∧[(-1)bni_23 + (-1)Bound*bni_23] + [(-1)bni_23]x2[0] + [bni_23]x1[0] ≥ 0∧[1 + (-1)bso_24] ≥ 0)

We simplified constraint (11) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(12)    (x1[0] + [-1] + [-1]x0[0] ≥ 0∧x2[0] ≥ 0∧x0[0] + [-1]x2[0] ≥ 0 ⇒ (U^Increasing(917_0_MAIN_LOAD(x1[1], +(x2[1], 1), x0[1])), ≥)∧[(-1)bni_23 + (-1)Bound*bni_23] + [(-1)bni_23]x2[0] + [bni_23]x1[0] ≥ 0∧[1 + (-1)bso_24] ≥ 0)

We simplified constraint (12) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(13)    (x1[0] ≥ 0∧x2[0] ≥ 0∧x0[0] + [-1]x2[0] ≥ 0 ⇒ (U^Increasing(917_0_MAIN_LOAD(x1[1], +(x2[1], 1), x0[1])), ≥)∧[(-1)Bound*bni_23] + [bni_23]x0[0] + [(-1)bni_23]x2[0] + [bni_23]x1[0] ≥ 0∧[1 + (-1)bso_24] ≥ 0)

We simplified constraint (13) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(14)    (x1[0] ≥ 0∧x2[0] ≥ 0∧x0[0] ≥ 0 ⇒ (U^Increasing(917_0_MAIN_LOAD(x1[1], +(x2[1], 1), x0[1])), ≥)∧[(-1)Bound*bni_23] + [bni_23]x0[0] + [bni_23]x1[0] ≥ 0∧[1 + (-1)bso_24] ≥ 0)
We consider the chain 917_0_MAIN_LOAD(x1[0], x2[0], x0[0]) → COND_917_0_MAIN_LOAD(&&(&&(>=(x2[0], 0), <=(x2[0], x0[0])), >(x1[0], x0[0])), x1[0], x2[0], x0[0]), COND_917_0_MAIN_LOAD(TRUE, x1[1], x2[1], x0[1]) → 917_0_MAIN_LOAD(x1[1], +(x2[1], 1), x0[1]), 917_0_MAIN_LOAD(x1[2], x2[2], x0[2]) → COND_917_0_MAIN_LOAD1(&&(&&(>(x2[2], x0[2]), >(x1[2], x0[2])), >=(x0[2], 0)), x1[2], x2[2], x0[2]) which results in the following constraint:
(15)    (&&(&&(>=(x2[0], 0), <=(x2[0], x0[0])), >(x1[0], x0[0]))=TRUE∧x1[0]=x1[1]∧x2[0]=x2[1]∧x0[0]=x0[1]∧x1[1]=x1[2]∧+(x2[1], 1)=x2[2]∧x0[1]=x0[2] ⇒ COND_917_0_MAIN_LOAD(TRUE, x1[1], x2[1], x0[1])≥NonInfC∧COND_917_0_MAIN_LOAD(TRUE, x1[1], x2[1], x0[1])≥917_0_MAIN_LOAD(x1[1], +(x2[1], 1), x0[1])∧(U^Increasing(917_0_MAIN_LOAD(x1[1], +(x2[1], 1), x0[1])), ≥))

We simplified constraint (15) using rules (III), (IV), (IDP_BOOLEAN) which results in the following new constraint:
(16)    (>(x1[0], x0[0])=TRUE∧>=(x2[0], 0)=TRUE∧<=(x2[0], x0[0])=TRUE ⇒ COND_917_0_MAIN_LOAD(TRUE, x1[0], x2[0], x0[0])≥NonInfC∧COND_917_0_MAIN_LOAD(TRUE, x1[0], x2[0], x0[0])≥917_0_MAIN_LOAD(x1[0], +(x2[0], 1), x0[0])∧(U^Increasing(917_0_MAIN_LOAD(x1[1], +(x2[1], 1), x0[1])), ≥))

We simplified constraint (16) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(17)    (x1[0] + [-1] + [-1]x0[0] ≥ 0∧x2[0] ≥ 0∧x0[0] + [-1]x2[0] ≥ 0 ⇒ (U^Increasing(917_0_MAIN_LOAD(x1[1], +(x2[1], 1), x0[1])), ≥)∧[(-1)bni_23 + (-1)Bound*bni_23] + [(-1)bni_23]x2[0] + [bni_23]x1[0] ≥ 0∧[1 + (-1)bso_24] ≥ 0)

We simplified constraint (17) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(18)    (x1[0] + [-1] + [-1]x0[0] ≥ 0∧x2[0] ≥ 0∧x0[0] + [-1]x2[0] ≥ 0 ⇒ (U^Increasing(917_0_MAIN_LOAD(x1[1], +(x2[1], 1), x0[1])), ≥)∧[(-1)bni_23 + (-1)Bound*bni_23] + [(-1)bni_23]x2[0] + [bni_23]x1[0] ≥ 0∧[1 + (-1)bso_24] ≥ 0)

We simplified constraint (18) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(19)    (x1[0] + [-1] + [-1]x0[0] ≥ 0∧x2[0] ≥ 0∧x0[0] + [-1]x2[0] ≥ 0 ⇒ (U^Increasing(917_0_MAIN_LOAD(x1[1], +(x2[1], 1), x0[1])), ≥)∧[(-1)bni_23 + (-1)Bound*bni_23] + [(-1)bni_23]x2[0] + [bni_23]x1[0] ≥ 0∧[1 + (-1)bso_24] ≥ 0)

We simplified constraint (19) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(20)    (x1[0] ≥ 0∧x2[0] ≥ 0∧x0[0] + [-1]x2[0] ≥ 0 ⇒ (U^Increasing(917_0_MAIN_LOAD(x1[1], +(x2[1], 1), x0[1])), ≥)∧[(-1)Bound*bni_23] + [bni_23]x0[0] + [(-1)bni_23]x2[0] + [bni_23]x1[0] ≥ 0∧[1 + (-1)bso_24] ≥ 0)

We simplified constraint (20) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(21)    (x1[0] ≥ 0∧x2[0] ≥ 0∧x0[0] ≥ 0 ⇒ (U^Increasing(917_0_MAIN_LOAD(x1[1], +(x2[1], 1), x0[1])), ≥)∧[(-1)Bound*bni_23] + [bni_23]x0[0] + [bni_23]x1[0] ≥ 0∧[1 + (-1)bso_24] ≥ 0)

For Pair 917_0_MAIN_LOAD(x1, x2, x0) → COND_917_0_MAIN_LOAD1(&&(&&(>(x2, x0), >(x1, x0)), >=(x0, 0)), x1, x2, x0) the following chains were created:

We consider the chain 917_0_MAIN_LOAD(x1[2], x2[2], x0[2]) → COND_917_0_MAIN_LOAD1(&&(&&(>(x2[2], x0[2]), >(x1[2], x0[2])), >=(x0[2], 0)), x1[2], x2[2], x0[2]), COND_917_0_MAIN_LOAD1(TRUE, x1[3], x2[3], x0[3]) → 917_0_MAIN_LOAD(x1[3], x2[3], +(x0[3], 1)) which results in the following constraint:
(22)    (&&(&&(>(x2[2], x0[2]), >(x1[2], x0[2])), >=(x0[2], 0))=TRUE∧x1[2]=x1[3]∧x2[2]=x2[3]∧x0[2]=x0[3] ⇒ 917_0_MAIN_LOAD(x1[2], x2[2], x0[2])≥NonInfC∧917_0_MAIN_LOAD(x1[2], x2[2], x0[2])≥COND_917_0_MAIN_LOAD1(&&(&&(>(x2[2], x0[2]), >(x1[2], x0[2])), >=(x0[2], 0)), x1[2], x2[2], x0[2])∧(U^Increasing(COND_917_0_MAIN_LOAD1(&&(&&(>(x2[2], x0[2]), >(x1[2], x0[2])), >=(x0[2], 0)), x1[2], x2[2], x0[2])), ≥))

We simplified constraint (22) using rules (IV), (IDP_BOOLEAN) which results in the following new constraint:
(23)    (>=(x0[2], 0)=TRUE∧>(x2[2], x0[2])=TRUE∧>(x1[2], x0[2])=TRUE ⇒ 917_0_MAIN_LOAD(x1[2], x2[2], x0[2])≥NonInfC∧917_0_MAIN_LOAD(x1[2], x2[2], x0[2])≥COND_917_0_MAIN_LOAD1(&&(&&(>(x2[2], x0[2]), >(x1[2], x0[2])), >=(x0[2], 0)), x1[2], x2[2], x0[2])∧(U^Increasing(COND_917_0_MAIN_LOAD1(&&(&&(>(x2[2], x0[2]), >(x1[2], x0[2])), >=(x0[2], 0)), x1[2], x2[2], x0[2])), ≥))

We simplified constraint (23) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(24)    (x0[2] ≥ 0∧x2[2] + [-1] + [-1]x0[2] ≥ 0∧x1[2] + [-1] + [-1]x0[2] ≥ 0 ⇒ (U^Increasing(COND_917_0_MAIN_LOAD1(&&(&&(>(x2[2], x0[2]), >(x1[2], x0[2])), >=(x0[2], 0)), x1[2], x2[2], x0[2])), ≥)∧[(-1)bni_25 + (-1)Bound*bni_25] + [(-1)bni_25]x2[2] + [bni_25]x1[2] ≥ 0∧[(-1)bso_26] ≥ 0)

We simplified constraint (24) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(25)    (x0[2] ≥ 0∧x2[2] + [-1] + [-1]x0[2] ≥ 0∧x1[2] + [-1] + [-1]x0[2] ≥ 0 ⇒ (U^Increasing(COND_917_0_MAIN_LOAD1(&&(&&(>(x2[2], x0[2]), >(x1[2], x0[2])), >=(x0[2], 0)), x1[2], x2[2], x0[2])), ≥)∧[(-1)bni_25 + (-1)Bound*bni_25] + [(-1)bni_25]x2[2] + [bni_25]x1[2] ≥ 0∧[(-1)bso_26] ≥ 0)

We simplified constraint (25) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(26)    (x0[2] ≥ 0∧x2[2] + [-1] + [-1]x0[2] ≥ 0∧x1[2] + [-1] + [-1]x0[2] ≥ 0 ⇒ (U^Increasing(COND_917_0_MAIN_LOAD1(&&(&&(>(x2[2], x0[2]), >(x1[2], x0[2])), >=(x0[2], 0)), x1[2], x2[2], x0[2])), ≥)∧[(-1)bni_25 + (-1)Bound*bni_25] + [(-1)bni_25]x2[2] + [bni_25]x1[2] ≥ 0∧[(-1)bso_26] ≥ 0)

We simplified constraint (26) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(27)    (x0[2] ≥ 0∧x2[2] ≥ 0∧x1[2] + [-1] + [-1]x0[2] ≥ 0 ⇒ (U^Increasing(COND_917_0_MAIN_LOAD1(&&(&&(>(x2[2], x0[2]), >(x1[2], x0[2])), >=(x0[2], 0)), x1[2], x2[2], x0[2])), ≥)∧[(-2)bni_25 + (-1)Bound*bni_25] + [(-1)bni_25]x0[2] + [(-1)bni_25]x2[2] + [bni_25]x1[2] ≥ 0∧[(-1)bso_26] ≥ 0)

We simplified constraint (27) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(28)    (x0[2] ≥ 0∧x2[2] ≥ 0∧x1[2] ≥ 0 ⇒ (U^Increasing(COND_917_0_MAIN_LOAD1(&&(&&(>(x2[2], x0[2]), >(x1[2], x0[2])), >=(x0[2], 0)), x1[2], x2[2], x0[2])), ≥)∧[(-1)bni_25 + (-1)Bound*bni_25] + [(-1)bni_25]x2[2] + [bni_25]x1[2] ≥ 0∧[(-1)bso_26] ≥ 0)

For Pair COND_917_0_MAIN_LOAD1(TRUE, x1, x2, x0) → 917_0_MAIN_LOAD(x1, x2, +(x0, 1)) the following chains were created:

We consider the chain 917_0_MAIN_LOAD(x1[2], x2[2], x0[2]) → COND_917_0_MAIN_LOAD1(&&(&&(>(x2[2], x0[2]), >(x1[2], x0[2])), >=(x0[2], 0)), x1[2], x2[2], x0[2]), COND_917_0_MAIN_LOAD1(TRUE, x1[3], x2[3], x0[3]) → 917_0_MAIN_LOAD(x1[3], x2[3], +(x0[3], 1)), 917_0_MAIN_LOAD(x1[0], x2[0], x0[0]) → COND_917_0_MAIN_LOAD(&&(&&(>=(x2[0], 0), <=(x2[0], x0[0])), >(x1[0], x0[0])), x1[0], x2[0], x0[0]) which results in the following constraint:
(29)    (&&(&&(>(x2[2], x0[2]), >(x1[2], x0[2])), >=(x0[2], 0))=TRUE∧x1[2]=x1[3]∧x2[2]=x2[3]∧x0[2]=x0[3]∧x1[3]=x1[0]∧x2[3]=x2[0]∧+(x0[3], 1)=x0[0] ⇒ COND_917_0_MAIN_LOAD1(TRUE, x1[3], x2[3], x0[3])≥NonInfC∧COND_917_0_MAIN_LOAD1(TRUE, x1[3], x2[3], x0[3])≥917_0_MAIN_LOAD(x1[3], x2[3], +(x0[3], 1))∧(U^Increasing(917_0_MAIN_LOAD(x1[3], x2[3], +(x0[3], 1))), ≥))

We simplified constraint (29) using rules (III), (IV), (IDP_BOOLEAN) which results in the following new constraint:
(30)    (>=(x0[2], 0)=TRUE∧>(x2[2], x0[2])=TRUE∧>(x1[2], x0[2])=TRUE ⇒ COND_917_0_MAIN_LOAD1(TRUE, x1[2], x2[2], x0[2])≥NonInfC∧COND_917_0_MAIN_LOAD1(TRUE, x1[2], x2[2], x0[2])≥917_0_MAIN_LOAD(x1[2], x2[2], +(x0[2], 1))∧(U^Increasing(917_0_MAIN_LOAD(x1[3], x2[3], +(x0[3], 1))), ≥))

We simplified constraint (30) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(31)    (x0[2] ≥ 0∧x2[2] + [-1] + [-1]x0[2] ≥ 0∧x1[2] + [-1] + [-1]x0[2] ≥ 0 ⇒ (U^Increasing(917_0_MAIN_LOAD(x1[3], x2[3], +(x0[3], 1))), ≥)∧[(-1)bni_27 + (-1)Bound*bni_27] + [(-1)bni_27]x2[2] + [bni_27]x1[2] ≥ 0∧[(-1)bso_28] ≥ 0)

We simplified constraint (31) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(32)    (x0[2] ≥ 0∧x2[2] + [-1] + [-1]x0[2] ≥ 0∧x1[2] + [-1] + [-1]x0[2] ≥ 0 ⇒ (U^Increasing(917_0_MAIN_LOAD(x1[3], x2[3], +(x0[3], 1))), ≥)∧[(-1)bni_27 + (-1)Bound*bni_27] + [(-1)bni_27]x2[2] + [bni_27]x1[2] ≥ 0∧[(-1)bso_28] ≥ 0)

We simplified constraint (32) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(33)    (x0[2] ≥ 0∧x2[2] + [-1] + [-1]x0[2] ≥ 0∧x1[2] + [-1] + [-1]x0[2] ≥ 0 ⇒ (U^Increasing(917_0_MAIN_LOAD(x1[3], x2[3], +(x0[3], 1))), ≥)∧[(-1)bni_27 + (-1)Bound*bni_27] + [(-1)bni_27]x2[2] + [bni_27]x1[2] ≥ 0∧[(-1)bso_28] ≥ 0)

We simplified constraint (33) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(34)    (x0[2] ≥ 0∧x2[2] ≥ 0∧x1[2] + [-1] + [-1]x0[2] ≥ 0 ⇒ (U^Increasing(917_0_MAIN_LOAD(x1[3], x2[3], +(x0[3], 1))), ≥)∧[(-2)bni_27 + (-1)Bound*bni_27] + [(-1)bni_27]x0[2] + [(-1)bni_27]x2[2] + [bni_27]x1[2] ≥ 0∧[(-1)bso_28] ≥ 0)

We simplified constraint (34) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(35)    (x0[2] ≥ 0∧x2[2] ≥ 0∧x1[2] ≥ 0 ⇒ (U^Increasing(917_0_MAIN_LOAD(x1[3], x2[3], +(x0[3], 1))), ≥)∧[(-1)bni_27 + (-1)Bound*bni_27] + [(-1)bni_27]x2[2] + [bni_27]x1[2] ≥ 0∧[(-1)bso_28] ≥ 0)
We consider the chain 917_0_MAIN_LOAD(x1[2], x2[2], x0[2]) → COND_917_0_MAIN_LOAD1(&&(&&(>(x2[2], x0[2]), >(x1[2], x0[2])), >=(x0[2], 0)), x1[2], x2[2], x0[2]), COND_917_0_MAIN_LOAD1(TRUE, x1[3], x2[3], x0[3]) → 917_0_MAIN_LOAD(x1[3], x2[3], +(x0[3], 1)), 917_0_MAIN_LOAD(x1[2], x2[2], x0[2]) → COND_917_0_MAIN_LOAD1(&&(&&(>(x2[2], x0[2]), >(x1[2], x0[2])), >=(x0[2], 0)), x1[2], x2[2], x0[2]) which results in the following constraint:
(36)    (&&(&&(>(x2[2], x0[2]), >(x1[2], x0[2])), >=(x0[2], 0))=TRUE∧x1[2]=x1[3]∧x2[2]=x2[3]∧x0[2]=x0[3]∧x1[3]=x1[2]1∧x2[3]=x2[2]1∧+(x0[3], 1)=x0[2]1 ⇒ COND_917_0_MAIN_LOAD1(TRUE, x1[3], x2[3], x0[3])≥NonInfC∧COND_917_0_MAIN_LOAD1(TRUE, x1[3], x2[3], x0[3])≥917_0_MAIN_LOAD(x1[3], x2[3], +(x0[3], 1))∧(U^Increasing(917_0_MAIN_LOAD(x1[3], x2[3], +(x0[3], 1))), ≥))

We simplified constraint (36) using rules (III), (IV), (IDP_BOOLEAN) which results in the following new constraint:
(37)    (>=(x0[2], 0)=TRUE∧>(x2[2], x0[2])=TRUE∧>(x1[2], x0[2])=TRUE ⇒ COND_917_0_MAIN_LOAD1(TRUE, x1[2], x2[2], x0[2])≥NonInfC∧COND_917_0_MAIN_LOAD1(TRUE, x1[2], x2[2], x0[2])≥917_0_MAIN_LOAD(x1[2], x2[2], +(x0[2], 1))∧(U^Increasing(917_0_MAIN_LOAD(x1[3], x2[3], +(x0[3], 1))), ≥))

We simplified constraint (37) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(38)    (x0[2] ≥ 0∧x2[2] + [-1] + [-1]x0[2] ≥ 0∧x1[2] + [-1] + [-1]x0[2] ≥ 0 ⇒ (U^Increasing(917_0_MAIN_LOAD(x1[3], x2[3], +(x0[3], 1))), ≥)∧[(-1)bni_27 + (-1)Bound*bni_27] + [(-1)bni_27]x2[2] + [bni_27]x1[2] ≥ 0∧[(-1)bso_28] ≥ 0)

We simplified constraint (38) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(39)    (x0[2] ≥ 0∧x2[2] + [-1] + [-1]x0[2] ≥ 0∧x1[2] + [-1] + [-1]x0[2] ≥ 0 ⇒ (U^Increasing(917_0_MAIN_LOAD(x1[3], x2[3], +(x0[3], 1))), ≥)∧[(-1)bni_27 + (-1)Bound*bni_27] + [(-1)bni_27]x2[2] + [bni_27]x1[2] ≥ 0∧[(-1)bso_28] ≥ 0)

We simplified constraint (39) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(40)    (x0[2] ≥ 0∧x2[2] + [-1] + [-1]x0[2] ≥ 0∧x1[2] + [-1] + [-1]x0[2] ≥ 0 ⇒ (U^Increasing(917_0_MAIN_LOAD(x1[3], x2[3], +(x0[3], 1))), ≥)∧[(-1)bni_27 + (-1)Bound*bni_27] + [(-1)bni_27]x2[2] + [bni_27]x1[2] ≥ 0∧[(-1)bso_28] ≥ 0)

We simplified constraint (40) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(41)    (x0[2] ≥ 0∧x2[2] ≥ 0∧x1[2] + [-1] + [-1]x0[2] ≥ 0 ⇒ (U^Increasing(917_0_MAIN_LOAD(x1[3], x2[3], +(x0[3], 1))), ≥)∧[(-2)bni_27 + (-1)Bound*bni_27] + [(-1)bni_27]x0[2] + [(-1)bni_27]x2[2] + [bni_27]x1[2] ≥ 0∧[(-1)bso_28] ≥ 0)

We simplified constraint (41) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(42)    (x0[2] ≥ 0∧x2[2] ≥ 0∧x1[2] ≥ 0 ⇒ (U^Increasing(917_0_MAIN_LOAD(x1[3], x2[3], +(x0[3], 1))), ≥)∧[(-1)bni_27 + (-1)Bound*bni_27] + [(-1)bni_27]x2[2] + [bni_27]x1[2] ≥ 0∧[(-1)bso_28] ≥ 0)

To summarize, we get the following constraints P_≥ for the following pairs.

917_0_MAIN_LOAD(x1, x2, x0) → COND_917_0_MAIN_LOAD(&&(&&(>=(x2, 0), <=(x2, x0)), >(x1, x0)), x1, x2, x0)
- (x1[0] ≥ 0∧x2[0] ≥ 0∧x0[0] ≥ 0 ⇒ (U^Increasing(COND_917_0_MAIN_LOAD(&&(&&(>=(x2[0], 0), <=(x2[0], x0[0])), >(x1[0], x0[0])), x1[0], x2[0], x0[0])), ≥)∧[(-1)Bound*bni_21] + [bni_21]x0[0] + [bni_21]x1[0] ≥ 0∧[(-1)bso_22] ≥ 0)
COND_917_0_MAIN_LOAD(TRUE, x1, x2, x0) → 917_0_MAIN_LOAD(x1, +(x2, 1), x0)
- (x1[0] ≥ 0∧x2[0] ≥ 0∧x0[0] ≥ 0 ⇒ (U^Increasing(917_0_MAIN_LOAD(x1[1], +(x2[1], 1), x0[1])), ≥)∧[(-1)Bound*bni_23] + [bni_23]x0[0] + [bni_23]x1[0] ≥ 0∧[1 + (-1)bso_24] ≥ 0)
- (x1[0] ≥ 0∧x2[0] ≥ 0∧x0[0] ≥ 0 ⇒ (U^Increasing(917_0_MAIN_LOAD(x1[1], +(x2[1], 1), x0[1])), ≥)∧[(-1)Bound*bni_23] + [bni_23]x0[0] + [bni_23]x1[0] ≥ 0∧[1 + (-1)bso_24] ≥ 0)
917_0_MAIN_LOAD(x1, x2, x0) → COND_917_0_MAIN_LOAD1(&&(&&(>(x2, x0), >(x1, x0)), >=(x0, 0)), x1, x2, x0)
- (x0[2] ≥ 0∧x2[2] ≥ 0∧x1[2] ≥ 0 ⇒ (U^Increasing(COND_917_0_MAIN_LOAD1(&&(&&(>(x2[2], x0[2]), >(x1[2], x0[2])), >=(x0[2], 0)), x1[2], x2[2], x0[2])), ≥)∧[(-1)bni_25 + (-1)Bound*bni_25] + [(-1)bni_25]x2[2] + [bni_25]x1[2] ≥ 0∧[(-1)bso_26] ≥ 0)
COND_917_0_MAIN_LOAD1(TRUE, x1, x2, x0) → 917_0_MAIN_LOAD(x1, x2, +(x0, 1))
- (x0[2] ≥ 0∧x2[2] ≥ 0∧x1[2] ≥ 0 ⇒ (U^Increasing(917_0_MAIN_LOAD(x1[3], x2[3], +(x0[3], 1))), ≥)∧[(-1)bni_27 + (-1)Bound*bni_27] + [(-1)bni_27]x2[2] + [bni_27]x1[2] ≥ 0∧[(-1)bso_28] ≥ 0)
- (x0[2] ≥ 0∧x2[2] ≥ 0∧x1[2] ≥ 0 ⇒ (U^Increasing(917_0_MAIN_LOAD(x1[3], x2[3], +(x0[3], 1))), ≥)∧[(-1)bni_27 + (-1)Bound*bni_27] + [(-1)bni_27]x2[2] + [bni_27]x1[2] ≥ 0∧[(-1)bso_28] ≥ 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(917_0_MAIN_LOAD(x₁, x₂, x₃)) = [-1] + [-1]x₂ + x₁
POL(COND_917_0_MAIN_LOAD(x₁, x₂, x₃, x₄)) = [-1] + [-1]x₃ + x₂ + [-1]x₁
POL(&&(x₁, x₂)) = 0
POL(>=(x₁, x₂)) = [-1]
POL(0) = 0
POL(<=(x₁, x₂)) = [-1]
POL(>(x₁, x₂)) = [-1]
POL(+(x₁, x₂)) = x₁ + x₂
POL(1) = [1]
POL(COND_917_0_MAIN_LOAD1(x₁, x₂, x₃, x₄)) = [-1] + [-1]x₃ + x₂ + [-1]x₁

The following pairs are in P_>:

COND_917_0_MAIN_LOAD(TRUE, x1[1], x2[1], x0[1]) → 917_0_MAIN_LOAD(x1[1], +(x2[1], 1), x0[1])

The following pairs are in P_bound:

917_0_MAIN_LOAD(x1[0], x2[0], x0[0]) → COND_917_0_MAIN_LOAD(&&(&&(>=(x2[0], 0), <=(x2[0], x0[0])), >(x1[0], x0[0])), x1[0], x2[0], x0[0])
COND_917_0_MAIN_LOAD(TRUE, x1[1], x2[1], x0[1]) → 917_0_MAIN_LOAD(x1[1], +(x2[1], 1), x0[1])

The following pairs are in P_≥:

917_0_MAIN_LOAD(x1[0], x2[0], x0[0]) → COND_917_0_MAIN_LOAD(&&(&&(>=(x2[0], 0), <=(x2[0], x0[0])), >(x1[0], x0[0])), x1[0], x2[0], x0[0])
917_0_MAIN_LOAD(x1[2], x2[2], x0[2]) → COND_917_0_MAIN_LOAD1(&&(&&(>(x2[2], x0[2]), >(x1[2], x0[2])), >=(x0[2], 0)), x1[2], x2[2], x0[2])
COND_917_0_MAIN_LOAD1(TRUE, x1[3], x2[3], x0[3]) → 917_0_MAIN_LOAD(x1[3], x2[3], +(x0[3], 1))

At least the following rules have been oriented under context sensitive arithmetic replacement:

&&(TRUE, TRUE)¹ ↔ TRUE¹
&&(TRUE, FALSE)¹ ↔ FALSE¹
&&(FALSE, TRUE)¹ ↔ FALSE¹
&&(FALSE, FALSE)¹ ↔ FALSE¹

(6) Obligation:

IDP problem:
The following function symbols are pre-defined:

!=	~	Neq: (Integer, Integer) -> Boolean
*	~	Mul: (Integer, Integer) -> Integer
>=	~	Ge: (Integer, Integer) -> Boolean
-1	~	UnaryMinus: (Integer) -> Integer
\|	~	Bwor: (Integer, Integer) -> Integer
/	~	Div: (Integer, Integer) -> Integer
=	~	Eq: (Integer, Integer) -> Boolean
	~	Bwxor: (Integer, Integer) -> Integer
\|\|	~	Lor: (Boolean, Boolean) -> Boolean
!	~	Lnot: (Boolean) -> Boolean
<	~	Lt: (Integer, Integer) -> Boolean
-	~	Sub: (Integer, Integer) -> Integer
<=	~	Le: (Integer, Integer) -> Boolean
>	~	Gt: (Integer, Integer) -> Boolean
~	~	Bwnot: (Integer) -> Integer
%	~	Mod: (Integer, Integer) -> Integer
&	~	Bwand: (Integer, Integer) -> Integer
+	~	Add: (Integer, Integer) -> Integer
&&	~	Land: (Boolean, Boolean) -> Boolean

The following domains are used:

Boolean, Integer

R is empty.

The integer pair graph contains the following rules and edges:
(0): 917_0_MAIN_LOAD(x1[0], x2[0], x0[0]) → COND_917_0_MAIN_LOAD(x2[0] >= 0 && x2[0] <= x0[0] && x1[0] > x0[0], x1[0], x2[0], x0[0])
(2): 917_0_MAIN_LOAD(x1[2], x2[2], x0[2]) → COND_917_0_MAIN_LOAD1(x2[2] > x0[2] && x1[2] > x0[2] && x0[2] >= 0, x1[2], x2[2], x0[2])
(3): COND_917_0_MAIN_LOAD1(TRUE, x1[3], x2[3], x0[3]) → 917_0_MAIN_LOAD(x1[3], x2[3], x0[3] + 1)

(3) -> (0), if ((x1[3] →^* x1[0])∧(x2[3] →^* x2[0])∧(x0[3] + 1 →^* x0[0]))

(3) -> (2), if ((x1[3] →^* x1[2])∧(x2[3] →^* x2[2])∧(x0[3] + 1 →^* x0[2]))

(2) -> (3), if ((x2[2] > x0[2] && x1[2] > x0[2] && x0[2] >= 0 →^* TRUE)∧(x1[2] →^* x1[3])∧(x2[2] →^* x2[3])∧(x0[2] →^* x0[3]))

The set Q is empty.

(7) IDependencyGraphProof (EQUIVALENT transformation)

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

(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:
(3): COND_917_0_MAIN_LOAD1(TRUE, x1[3], x2[3], x0[3]) → 917_0_MAIN_LOAD(x1[3], x2[3], x0[3] + 1)
(2): 917_0_MAIN_LOAD(x1[2], x2[2], x0[2]) → COND_917_0_MAIN_LOAD1(x2[2] > x0[2] && x1[2] > x0[2] && x0[2] >= 0, x1[2], x2[2], x0[2])

(3) -> (2), if ((x1[3] →^* x1[2])∧(x2[3] →^* x2[2])∧(x0[3] + 1 →^* x0[2]))

(2) -> (3), if ((x2[2] > x0[2] && x1[2] > x0[2] && x0[2] >= 0 →^* TRUE)∧(x1[2] →^* x1[3])∧(x2[2] →^* x2[3])∧(x0[2] →^* x0[3]))

The set Q is empty.

(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 COND_917_0_MAIN_LOAD1(TRUE, x1[3], x2[3], x0[3]) → 917_0_MAIN_LOAD(x1[3], x2[3], +(x0[3], 1)) the following chains were created:

We consider the chain 917_0_MAIN_LOAD(x1[2], x2[2], x0[2]) → COND_917_0_MAIN_LOAD1(&&(&&(>(x2[2], x0[2]), >(x1[2], x0[2])), >=(x0[2], 0)), x1[2], x2[2], x0[2]), COND_917_0_MAIN_LOAD1(TRUE, x1[3], x2[3], x0[3]) → 917_0_MAIN_LOAD(x1[3], x2[3], +(x0[3], 1)), 917_0_MAIN_LOAD(x1[2], x2[2], x0[2]) → COND_917_0_MAIN_LOAD1(&&(&&(>(x2[2], x0[2]), >(x1[2], x0[2])), >=(x0[2], 0)), x1[2], x2[2], x0[2]) which results in the following constraint:
(1)    (&&(&&(>(x2[2], x0[2]), >(x1[2], x0[2])), >=(x0[2], 0))=TRUE∧x1[2]=x1[3]∧x2[2]=x2[3]∧x0[2]=x0[3]∧x1[3]=x1[2]1∧x2[3]=x2[2]1∧+(x0[3], 1)=x0[2]1 ⇒ COND_917_0_MAIN_LOAD1(TRUE, x1[3], x2[3], x0[3])≥NonInfC∧COND_917_0_MAIN_LOAD1(TRUE, x1[3], x2[3], x0[3])≥917_0_MAIN_LOAD(x1[3], x2[3], +(x0[3], 1))∧(U^Increasing(917_0_MAIN_LOAD(x1[3], x2[3], +(x0[3], 1))), ≥))

We simplified constraint (1) using rules (III), (IV), (IDP_BOOLEAN) which results in the following new constraint:
(2)    (>=(x0[2], 0)=TRUE∧>(x2[2], x0[2])=TRUE∧>(x1[2], x0[2])=TRUE ⇒ COND_917_0_MAIN_LOAD1(TRUE, x1[2], x2[2], x0[2])≥NonInfC∧COND_917_0_MAIN_LOAD1(TRUE, x1[2], x2[2], x0[2])≥917_0_MAIN_LOAD(x1[2], x2[2], +(x0[2], 1))∧(U^Increasing(917_0_MAIN_LOAD(x1[3], x2[3], +(x0[3], 1))), ≥))

We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(3)    (x0[2] ≥ 0∧x2[2] + [-1] + [-1]x0[2] ≥ 0∧x1[2] + [-1] + [-1]x0[2] ≥ 0 ⇒ (U^Increasing(917_0_MAIN_LOAD(x1[3], x2[3], +(x0[3], 1))), ≥)∧[(-1)bni_15 + (-1)Bound*bni_15] + [(-1)bni_15]x0[2] + [(2)bni_15]x2[2] ≥ 0∧[1 + (-1)bso_16] ≥ 0)

We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(4)    (x0[2] ≥ 0∧x2[2] + [-1] + [-1]x0[2] ≥ 0∧x1[2] + [-1] + [-1]x0[2] ≥ 0 ⇒ (U^Increasing(917_0_MAIN_LOAD(x1[3], x2[3], +(x0[3], 1))), ≥)∧[(-1)bni_15 + (-1)Bound*bni_15] + [(-1)bni_15]x0[2] + [(2)bni_15]x2[2] ≥ 0∧[1 + (-1)bso_16] ≥ 0)

We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(5)    (x0[2] ≥ 0∧x2[2] + [-1] + [-1]x0[2] ≥ 0∧x1[2] + [-1] + [-1]x0[2] ≥ 0 ⇒ (U^Increasing(917_0_MAIN_LOAD(x1[3], x2[3], +(x0[3], 1))), ≥)∧[(-1)bni_15 + (-1)Bound*bni_15] + [(-1)bni_15]x0[2] + [(2)bni_15]x2[2] ≥ 0∧[1 + (-1)bso_16] ≥ 0)

We simplified constraint (5) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(6)    (x0[2] ≥ 0∧x2[2] ≥ 0∧x1[2] + [-1] + [-1]x0[2] ≥ 0 ⇒ (U^Increasing(917_0_MAIN_LOAD(x1[3], x2[3], +(x0[3], 1))), ≥)∧[bni_15 + (-1)Bound*bni_15] + [bni_15]x0[2] + [(2)bni_15]x2[2] ≥ 0∧[1 + (-1)bso_16] ≥ 0)

We simplified constraint (6) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(7)    (x0[2] ≥ 0∧x2[2] ≥ 0∧x1[2] ≥ 0 ⇒ (U^Increasing(917_0_MAIN_LOAD(x1[3], x2[3], +(x0[3], 1))), ≥)∧[bni_15 + (-1)Bound*bni_15] + [bni_15]x0[2] + [(2)bni_15]x2[2] ≥ 0∧[1 + (-1)bso_16] ≥ 0)

For Pair 917_0_MAIN_LOAD(x1[2], x2[2], x0[2]) → COND_917_0_MAIN_LOAD1(&&(&&(>(x2[2], x0[2]), >(x1[2], x0[2])), >=(x0[2], 0)), x1[2], x2[2], x0[2]) the following chains were created:

We consider the chain 917_0_MAIN_LOAD(x1[2], x2[2], x0[2]) → COND_917_0_MAIN_LOAD1(&&(&&(>(x2[2], x0[2]), >(x1[2], x0[2])), >=(x0[2], 0)), x1[2], x2[2], x0[2]), COND_917_0_MAIN_LOAD1(TRUE, x1[3], x2[3], x0[3]) → 917_0_MAIN_LOAD(x1[3], x2[3], +(x0[3], 1)) which results in the following constraint:
(8)    (&&(&&(>(x2[2], x0[2]), >(x1[2], x0[2])), >=(x0[2], 0))=TRUE∧x1[2]=x1[3]∧x2[2]=x2[3]∧x0[2]=x0[3] ⇒ 917_0_MAIN_LOAD(x1[2], x2[2], x0[2])≥NonInfC∧917_0_MAIN_LOAD(x1[2], x2[2], x0[2])≥COND_917_0_MAIN_LOAD1(&&(&&(>(x2[2], x0[2]), >(x1[2], x0[2])), >=(x0[2], 0)), x1[2], x2[2], x0[2])∧(U^Increasing(COND_917_0_MAIN_LOAD1(&&(&&(>(x2[2], x0[2]), >(x1[2], x0[2])), >=(x0[2], 0)), x1[2], x2[2], x0[2])), ≥))

We simplified constraint (8) using rules (IV), (IDP_BOOLEAN) which results in the following new constraint:
(9)    (>=(x0[2], 0)=TRUE∧>(x2[2], x0[2])=TRUE∧>(x1[2], x0[2])=TRUE ⇒ 917_0_MAIN_LOAD(x1[2], x2[2], x0[2])≥NonInfC∧917_0_MAIN_LOAD(x1[2], x2[2], x0[2])≥COND_917_0_MAIN_LOAD1(&&(&&(>(x2[2], x0[2]), >(x1[2], x0[2])), >=(x0[2], 0)), x1[2], x2[2], x0[2])∧(U^Increasing(COND_917_0_MAIN_LOAD1(&&(&&(>(x2[2], x0[2]), >(x1[2], x0[2])), >=(x0[2], 0)), x1[2], x2[2], x0[2])), ≥))

We simplified constraint (9) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(10)    (x0[2] ≥ 0∧x2[2] + [-1] + [-1]x0[2] ≥ 0∧x1[2] + [-1] + [-1]x0[2] ≥ 0 ⇒ (U^Increasing(COND_917_0_MAIN_LOAD1(&&(&&(>(x2[2], x0[2]), >(x1[2], x0[2])), >=(x0[2], 0)), x1[2], x2[2], x0[2])), ≥)∧[(-1)bni_17 + (-1)Bound*bni_17] + [(-1)bni_17]x0[2] + [(2)bni_17]x2[2] ≥ 0∧[(-1)bso_18] ≥ 0)

We simplified constraint (10) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(11)    (x0[2] ≥ 0∧x2[2] + [-1] + [-1]x0[2] ≥ 0∧x1[2] + [-1] + [-1]x0[2] ≥ 0 ⇒ (U^Increasing(COND_917_0_MAIN_LOAD1(&&(&&(>(x2[2], x0[2]), >(x1[2], x0[2])), >=(x0[2], 0)), x1[2], x2[2], x0[2])), ≥)∧[(-1)bni_17 + (-1)Bound*bni_17] + [(-1)bni_17]x0[2] + [(2)bni_17]x2[2] ≥ 0∧[(-1)bso_18] ≥ 0)

We simplified constraint (11) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(12)    (x0[2] ≥ 0∧x2[2] + [-1] + [-1]x0[2] ≥ 0∧x1[2] + [-1] + [-1]x0[2] ≥ 0 ⇒ (U^Increasing(COND_917_0_MAIN_LOAD1(&&(&&(>(x2[2], x0[2]), >(x1[2], x0[2])), >=(x0[2], 0)), x1[2], x2[2], x0[2])), ≥)∧[(-1)bni_17 + (-1)Bound*bni_17] + [(-1)bni_17]x0[2] + [(2)bni_17]x2[2] ≥ 0∧[(-1)bso_18] ≥ 0)

We simplified constraint (12) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(13)    (x0[2] ≥ 0∧x2[2] ≥ 0∧x1[2] + [-1] + [-1]x0[2] ≥ 0 ⇒ (U^Increasing(COND_917_0_MAIN_LOAD1(&&(&&(>(x2[2], x0[2]), >(x1[2], x0[2])), >=(x0[2], 0)), x1[2], x2[2], x0[2])), ≥)∧[bni_17 + (-1)Bound*bni_17] + [bni_17]x0[2] + [(2)bni_17]x2[2] ≥ 0∧[(-1)bso_18] ≥ 0)

We simplified constraint (13) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(14)    (x0[2] ≥ 0∧x2[2] ≥ 0∧x1[2] ≥ 0 ⇒ (U^Increasing(COND_917_0_MAIN_LOAD1(&&(&&(>(x2[2], x0[2]), >(x1[2], x0[2])), >=(x0[2], 0)), x1[2], x2[2], x0[2])), ≥)∧[bni_17 + (-1)Bound*bni_17] + [bni_17]x0[2] + [(2)bni_17]x2[2] ≥ 0∧[(-1)bso_18] ≥ 0)

To summarize, we get the following constraints P_≥ for the following pairs.

COND_917_0_MAIN_LOAD1(TRUE, x1[3], x2[3], x0[3]) → 917_0_MAIN_LOAD(x1[3], x2[3], +(x0[3], 1))
- (x0[2] ≥ 0∧x2[2] ≥ 0∧x1[2] ≥ 0 ⇒ (U^Increasing(917_0_MAIN_LOAD(x1[3], x2[3], +(x0[3], 1))), ≥)∧[bni_15 + (-1)Bound*bni_15] + [bni_15]x0[2] + [(2)bni_15]x2[2] ≥ 0∧[1 + (-1)bso_16] ≥ 0)
917_0_MAIN_LOAD(x1[2], x2[2], x0[2]) → COND_917_0_MAIN_LOAD1(&&(&&(>(x2[2], x0[2]), >(x1[2], x0[2])), >=(x0[2], 0)), x1[2], x2[2], x0[2])
- (x0[2] ≥ 0∧x2[2] ≥ 0∧x1[2] ≥ 0 ⇒ (U^Increasing(COND_917_0_MAIN_LOAD1(&&(&&(>(x2[2], x0[2]), >(x1[2], x0[2])), >=(x0[2], 0)), x1[2], x2[2], x0[2])), ≥)∧[bni_17 + (-1)Bound*bni_17] + [bni_17]x0[2] + [(2)bni_17]x2[2] ≥ 0∧[(-1)bso_18] ≥ 0)

POL(TRUE) = 0
POL(FALSE) = [2]
POL(COND_917_0_MAIN_LOAD1(x₁, x₂, x₃, x₄)) = [-1] + [-1]x₄ + [2]x₃ + [-1]x₁
POL(917_0_MAIN_LOAD(x₁, x₂, x₃)) = [-1] + [-1]x₃ + [2]x₂
POL(+(x₁, x₂)) = x₁ + x₂
POL(1) = [1]
POL(&&(x₁, x₂)) = 0
POL(>(x₁, x₂)) = [-1]
POL(>=(x₁, x₂)) = [-1]
POL(0) = 0

The following pairs are in P_>:

COND_917_0_MAIN_LOAD1(TRUE, x1[3], x2[3], x0[3]) → 917_0_MAIN_LOAD(x1[3], x2[3], +(x0[3], 1))

The following pairs are in P_bound:

COND_917_0_MAIN_LOAD1(TRUE, x1[3], x2[3], x0[3]) → 917_0_MAIN_LOAD(x1[3], x2[3], +(x0[3], 1))
917_0_MAIN_LOAD(x1[2], x2[2], x0[2]) → COND_917_0_MAIN_LOAD1(&&(&&(>(x2[2], x0[2]), >(x1[2], x0[2])), >=(x0[2], 0)), x1[2], x2[2], x0[2])

The following pairs are in P_≥:

917_0_MAIN_LOAD(x1[2], x2[2], x0[2]) → COND_917_0_MAIN_LOAD1(&&(&&(>(x2[2], x0[2]), >(x1[2], x0[2])), >=(x0[2], 0)), x1[2], x2[2], x0[2])

At least the following rules have been oriented under context sensitive arithmetic replacement:

&&(TRUE, TRUE)¹ ↔ TRUE¹
FALSE¹ → &&(TRUE, FALSE)¹
FALSE¹ → &&(FALSE, TRUE)¹
FALSE¹ → &&(FALSE, FALSE)¹

(10) 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): 917_0_MAIN_LOAD(x1[2], x2[2], x0[2]) → COND_917_0_MAIN_LOAD1(x2[2] > x0[2] && x1[2] > x0[2] && x0[2] >= 0, x1[2], x2[2], x0[2])

The set Q is empty.

(11) IDependencyGraphProof (EQUIVALENT transformation)

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

(0) Obligation:

(1) JBC2FIG (SOUND transformation)

(2) Obligation:

(3) FIGtoITRSProof (SOUND transformation)

(4) Obligation:

(5) IDPNonInfProof (SOUND transformation)

(6) Obligation:

(7) IDependencyGraphProof (EQUIVALENT transformation)

(8) Obligation:

(9) IDPNonInfProof (SOUND transformation)

(10) Obligation:

(11) IDependencyGraphProof (EQUIVALENT transformation)

(12) TRUE