Termination proof of /tmp/tmpQMzoYS/CountUpRound.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 IDependencyGraphProof (⇔)

        ↳9 TRUE

    ↳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: CountUpRound

public class CountUpRound{
  public static int round (int x) {

    if (x % 2 == 0) return x;
    else return x+1;
  }


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



    while (x > y) {

      y = round(y+1);

    }


  }

}


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:
CountUpRound.main([Ljava/lang/String;)V: Graph of 170 nodes with 1 SCC.

(3) FIGtoITRSProof (SOUND transformation)

Transformed FIGraph SCCs to IDPs. Logs:

Log for SCC 0:

Generated 25 rules for P and 2 rules for R.

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

Filtered ground terms:

551_0_main_Load(x1, x2, x3, x4) → 551_0_main_Load(x2, x3, x4)
Cond_551_0_main_Load1(x1, x2, x3, x4, x5) → Cond_551_0_main_Load1(x1, x3, x4, x5)
Cond_551_0_main_Load(x1, x2, x3, x4, x5) → Cond_551_0_main_Load(x1, x3, x4, x5)

Filtered duplicate args:

551_0_main_Load(x1, x2, x3) → 551_0_main_Load(x2, x3)
Cond_551_0_main_Load1(x1, x2, x3, x4) → Cond_551_0_main_Load1(x1, x3, x4)
Cond_551_0_main_Load(x1, x2, x3, x4) → Cond_551_0_main_Load(x1, x3, x4)

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): 551_0_MAIN_LOAD(x1[0], x0[0]) → COND_551_0_MAIN_LOAD(x1[0] >= 0 && x1[0] < x0[0] && 0 < x1[0] + 1 && !(x1[0] + 1 % 2 = 0), x1[0], x0[0])
(1): COND_551_0_MAIN_LOAD(TRUE, x1[1], x0[1]) → 551_0_MAIN_LOAD(x1[1] + 1 + 1, x0[1])
(2): 551_0_MAIN_LOAD(x1[2], x0[2]) → COND_551_0_MAIN_LOAD1(x1[2] >= 0 && x1[2] < x0[2] && 0 = x1[2] + 1 % 2, x1[2], x0[2])
(3): COND_551_0_MAIN_LOAD1(TRUE, x1[3], x0[3]) → 551_0_MAIN_LOAD(x1[3] + 1, x0[3])

(0) -> (1), if ((x1[0] >= 0 && x1[0] < x0[0] && 0 < x1[0] + 1 && !(x1[0] + 1 % 2 = 0) →^* TRUE)∧(x1[0] →^* x1[1])∧(x0[0] →^* x0[1]))

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

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

(2) -> (3), if ((x1[2] >= 0 && x1[2] < x0[2] && 0 = x1[2] + 1 % 2 →^* TRUE)∧(x1[2] →^* x1[3])∧(x0[2] →^* x0[3]))

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

(3) -> (2), if ((x1[3] + 1 →^* x1[2])∧(x0[3] →^* 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 551_0_MAIN_LOAD(x1, x0) → COND_551_0_MAIN_LOAD(&&(&&(&&(>=(x1, 0), <(x1, x0)), <(0, +(x1, 1))), !(=(%(+(x1, 1), 2), 0))), x1, x0) the following chains were created:

We consider the chain 551_0_MAIN_LOAD(x1[0], x0[0]) → COND_551_0_MAIN_LOAD(&&(&&(&&(>=(x1[0], 0), <(x1[0], x0[0])), <(0, +(x1[0], 1))), !(=(%(+(x1[0], 1), 2), 0))), x1[0], x0[0]), COND_551_0_MAIN_LOAD(TRUE, x1[1], x0[1]) → 551_0_MAIN_LOAD(+(+(x1[1], 1), 1), x0[1]) which results in the following constraint:
(1)    (&&(&&(&&(>=(x1[0], 0), <(x1[0], x0[0])), <(0, +(x1[0], 1))), !(=(%(+(x1[0], 1), 2), 0)))=TRUE∧x1[0]=x1[1]∧x0[0]=x0[1] ⇒ 551_0_MAIN_LOAD(x1[0], x0[0])≥NonInfC∧551_0_MAIN_LOAD(x1[0], x0[0])≥COND_551_0_MAIN_LOAD(&&(&&(&&(>=(x1[0], 0), <(x1[0], x0[0])), <(0, +(x1[0], 1))), !(=(%(+(x1[0], 1), 2), 0))), x1[0], x0[0])∧(U^Increasing(COND_551_0_MAIN_LOAD(&&(&&(&&(>=(x1[0], 0), <(x1[0], x0[0])), <(0, +(x1[0], 1))), !(=(%(+(x1[0], 1), 2), 0))), x1[0], x0[0])), ≥))

We simplified constraint (1) using rules (IV), (IDP_BOOLEAN) which results in the following new constraints:
(2)    (<(0, +(x1[0], 1))=TRUE∧>=(x1[0], 0)=TRUE∧<(x1[0], x0[0])=TRUE∧<(%(+(x1[0], 1), 2), 0)=TRUE ⇒ 551_0_MAIN_LOAD(x1[0], x0[0])≥NonInfC∧551_0_MAIN_LOAD(x1[0], x0[0])≥COND_551_0_MAIN_LOAD(&&(&&(&&(>=(x1[0], 0), <(x1[0], x0[0])), <(0, +(x1[0], 1))), !(=(%(+(x1[0], 1), 2), 0))), x1[0], x0[0])∧(U^Increasing(COND_551_0_MAIN_LOAD(&&(&&(&&(>=(x1[0], 0), <(x1[0], x0[0])), <(0, +(x1[0], 1))), !(=(%(+(x1[0], 1), 2), 0))), x1[0], x0[0])), ≥))

(3)    (<(0, +(x1[0], 1))=TRUE∧>=(x1[0], 0)=TRUE∧<(x1[0], x0[0])=TRUE∧>(%(+(x1[0], 1), 2), 0)=TRUE ⇒ 551_0_MAIN_LOAD(x1[0], x0[0])≥NonInfC∧551_0_MAIN_LOAD(x1[0], x0[0])≥COND_551_0_MAIN_LOAD(&&(&&(&&(>=(x1[0], 0), <(x1[0], x0[0])), <(0, +(x1[0], 1))), !(=(%(+(x1[0], 1), 2), 0))), x1[0], x0[0])∧(U^Increasing(COND_551_0_MAIN_LOAD(&&(&&(&&(>=(x1[0], 0), <(x1[0], x0[0])), <(0, +(x1[0], 1))), !(=(%(+(x1[0], 1), 2), 0))), x1[0], x0[0])), ≥))

We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(4)    (x1[0] ≥ 0∧x1[0] ≥ 0∧x0[0] + [-1] + [-1]x1[0] ≥ 0∧[-1] + [-1]min{[2], [-2]} ≥ 0 ⇒ (U^Increasing(COND_551_0_MAIN_LOAD(&&(&&(&&(>=(x1[0], 0), <(x1[0], x0[0])), <(0, +(x1[0], 1))), !(=(%(+(x1[0], 1), 2), 0))), x1[0], x0[0])), ≥)∧[(2)bni_13 + (-1)Bound*bni_13] + [bni_13]x0[0] + [(-1)bni_13]x1[0] ≥ 0∧[1 + (-1)bso_14] ≥ 0)

We simplified constraint (3) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(5)    (x1[0] ≥ 0∧x1[0] ≥ 0∧x0[0] + [-1] + [-1]x1[0] ≥ 0∧max{[2], [-2]} + [-1] ≥ 0 ⇒ (U^Increasing(COND_551_0_MAIN_LOAD(&&(&&(&&(>=(x1[0], 0), <(x1[0], x0[0])), <(0, +(x1[0], 1))), !(=(%(+(x1[0], 1), 2), 0))), x1[0], x0[0])), ≥)∧[(2)bni_13 + (-1)Bound*bni_13] + [bni_13]x0[0] + [(-1)bni_13]x1[0] ≥ 0∧[1 + (-1)bso_14] ≥ 0)

We simplified constraint (4) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(6)    (x1[0] ≥ 0∧x1[0] ≥ 0∧x0[0] + [-1] + [-1]x1[0] ≥ 0∧[-1] + [-1]min{[2], [-2]} ≥ 0 ⇒ (U^Increasing(COND_551_0_MAIN_LOAD(&&(&&(&&(>=(x1[0], 0), <(x1[0], x0[0])), <(0, +(x1[0], 1))), !(=(%(+(x1[0], 1), 2), 0))), x1[0], x0[0])), ≥)∧[(2)bni_13 + (-1)Bound*bni_13] + [bni_13]x0[0] + [(-1)bni_13]x1[0] ≥ 0∧[1 + (-1)bso_14] ≥ 0)

We simplified constraint (5) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(7)    (x1[0] ≥ 0∧x1[0] ≥ 0∧x0[0] + [-1] + [-1]x1[0] ≥ 0∧max{[2], [-2]} + [-1] ≥ 0 ⇒ (U^Increasing(COND_551_0_MAIN_LOAD(&&(&&(&&(>=(x1[0], 0), <(x1[0], x0[0])), <(0, +(x1[0], 1))), !(=(%(+(x1[0], 1), 2), 0))), x1[0], x0[0])), ≥)∧[(2)bni_13 + (-1)Bound*bni_13] + [bni_13]x0[0] + [(-1)bni_13]x1[0] ≥ 0∧[1 + (-1)bso_14] ≥ 0)

We simplified constraint (6) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(8)    (x1[0] ≥ 0∧x1[0] ≥ 0∧x0[0] + [-1] + [-1]x1[0] ≥ 0∧[4] ≥ 0∧[1] ≥ 0 ⇒ (U^Increasing(COND_551_0_MAIN_LOAD(&&(&&(&&(>=(x1[0], 0), <(x1[0], x0[0])), <(0, +(x1[0], 1))), !(=(%(+(x1[0], 1), 2), 0))), x1[0], x0[0])), ≥)∧[(2)bni_13 + (-1)Bound*bni_13] + [bni_13]x0[0] + [(-1)bni_13]x1[0] ≥ 0∧[1 + (-1)bso_14] ≥ 0)

We simplified constraint (7) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(9)    (x1[0] ≥ 0∧x1[0] ≥ 0∧x0[0] + [-1] + [-1]x1[0] ≥ 0∧[4] ≥ 0∧[1] ≥ 0 ⇒ (U^Increasing(COND_551_0_MAIN_LOAD(&&(&&(&&(>=(x1[0], 0), <(x1[0], x0[0])), <(0, +(x1[0], 1))), !(=(%(+(x1[0], 1), 2), 0))), x1[0], x0[0])), ≥)∧[(2)bni_13 + (-1)Bound*bni_13] + [bni_13]x0[0] + [(-1)bni_13]x1[0] ≥ 0∧[1 + (-1)bso_14] ≥ 0)

We simplified constraint (8) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(10)    (x1[0] ≥ 0∧x1[0] ≥ 0∧x0[0] ≥ 0∧[4] ≥ 0∧[1] ≥ 0 ⇒ (U^Increasing(COND_551_0_MAIN_LOAD(&&(&&(&&(>=(x1[0], 0), <(x1[0], x0[0])), <(0, +(x1[0], 1))), !(=(%(+(x1[0], 1), 2), 0))), x1[0], x0[0])), ≥)∧[(3)bni_13 + (-1)Bound*bni_13] + [bni_13]x0[0] ≥ 0∧[1 + (-1)bso_14] ≥ 0)

We simplified constraint (9) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(11)    (x1[0] ≥ 0∧x1[0] ≥ 0∧x0[0] ≥ 0∧[4] ≥ 0∧[1] ≥ 0 ⇒ (U^Increasing(COND_551_0_MAIN_LOAD(&&(&&(&&(>=(x1[0], 0), <(x1[0], x0[0])), <(0, +(x1[0], 1))), !(=(%(+(x1[0], 1), 2), 0))), x1[0], x0[0])), ≥)∧[(3)bni_13 + (-1)Bound*bni_13] + [bni_13]x0[0] ≥ 0∧[1 + (-1)bso_14] ≥ 0)

We simplified constraint (10) using rule (IDP_POLY_GCD) which results in the following new constraint:
(12)    (x1[0] ≥ 0∧x1[0] ≥ 0∧x0[0] ≥ 0∧[1] ≥ 0∧[1] ≥ 0 ⇒ (U^Increasing(COND_551_0_MAIN_LOAD(&&(&&(&&(>=(x1[0], 0), <(x1[0], x0[0])), <(0, +(x1[0], 1))), !(=(%(+(x1[0], 1), 2), 0))), x1[0], x0[0])), ≥)∧[(3)bni_13 + (-1)Bound*bni_13] + [bni_13]x0[0] ≥ 0∧[1 + (-1)bso_14] ≥ 0)

We simplified constraint (11) using rule (IDP_POLY_GCD) which results in the following new constraint:
(13)    (x1[0] ≥ 0∧x1[0] ≥ 0∧x0[0] ≥ 0∧[1] ≥ 0∧[1] ≥ 0 ⇒ (U^Increasing(COND_551_0_MAIN_LOAD(&&(&&(&&(>=(x1[0], 0), <(x1[0], x0[0])), <(0, +(x1[0], 1))), !(=(%(+(x1[0], 1), 2), 0))), x1[0], x0[0])), ≥)∧[(3)bni_13 + (-1)Bound*bni_13] + [bni_13]x0[0] ≥ 0∧[1 + (-1)bso_14] ≥ 0)

For Pair COND_551_0_MAIN_LOAD(TRUE, x1, x0) → 551_0_MAIN_LOAD(+(+(x1, 1), 1), x0) the following chains were created:

We consider the chain COND_551_0_MAIN_LOAD(TRUE, x1[1], x0[1]) → 551_0_MAIN_LOAD(+(+(x1[1], 1), 1), x0[1]) which results in the following constraint:
(14)    (COND_551_0_MAIN_LOAD(TRUE, x1[1], x0[1])≥NonInfC∧COND_551_0_MAIN_LOAD(TRUE, x1[1], x0[1])≥551_0_MAIN_LOAD(+(+(x1[1], 1), 1), x0[1])∧(U^Increasing(551_0_MAIN_LOAD(+(+(x1[1], 1), 1), x0[1])), ≥))

We simplified constraint (14) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(15)    ((U^Increasing(551_0_MAIN_LOAD(+(+(x1[1], 1), 1), x0[1])), ≥)∧[1 + (-1)bso_16] ≥ 0)

We simplified constraint (15) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(16)    ((U^Increasing(551_0_MAIN_LOAD(+(+(x1[1], 1), 1), x0[1])), ≥)∧[1 + (-1)bso_16] ≥ 0)

We simplified constraint (16) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(17)    ((U^Increasing(551_0_MAIN_LOAD(+(+(x1[1], 1), 1), x0[1])), ≥)∧[1 + (-1)bso_16] ≥ 0)

We simplified constraint (17) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint:
(18)    ((U^Increasing(551_0_MAIN_LOAD(+(+(x1[1], 1), 1), x0[1])), ≥)∧0 = 0∧0 = 0∧[1 + (-1)bso_16] ≥ 0)

For Pair 551_0_MAIN_LOAD(x1, x0) → COND_551_0_MAIN_LOAD1(&&(&&(>=(x1, 0), <(x1, x0)), =(0, %(+(x1, 1), 2))), x1, x0) the following chains were created:

We consider the chain 551_0_MAIN_LOAD(x1[2], x0[2]) → COND_551_0_MAIN_LOAD1(&&(&&(>=(x1[2], 0), <(x1[2], x0[2])), =(0, %(+(x1[2], 1), 2))), x1[2], x0[2]), COND_551_0_MAIN_LOAD1(TRUE, x1[3], x0[3]) → 551_0_MAIN_LOAD(+(x1[3], 1), x0[3]) which results in the following constraint:
(19)    (&&(&&(>=(x1[2], 0), <(x1[2], x0[2])), =(0, %(+(x1[2], 1), 2)))=TRUE∧x1[2]=x1[3]∧x0[2]=x0[3] ⇒ 551_0_MAIN_LOAD(x1[2], x0[2])≥NonInfC∧551_0_MAIN_LOAD(x1[2], x0[2])≥COND_551_0_MAIN_LOAD1(&&(&&(>=(x1[2], 0), <(x1[2], x0[2])), =(0, %(+(x1[2], 1), 2))), x1[2], x0[2])∧(U^Increasing(COND_551_0_MAIN_LOAD1(&&(&&(>=(x1[2], 0), <(x1[2], x0[2])), =(0, %(+(x1[2], 1), 2))), x1[2], x0[2])), ≥))

We simplified constraint (19) using rules (IV), (IDP_BOOLEAN) which results in the following new constraint:
(20)    (>=(x1[2], 0)=TRUE∧<(x1[2], x0[2])=TRUE∧>=(0, %(+(x1[2], 1), 2))=TRUE∧<=(0, %(+(x1[2], 1), 2))=TRUE ⇒ 551_0_MAIN_LOAD(x1[2], x0[2])≥NonInfC∧551_0_MAIN_LOAD(x1[2], x0[2])≥COND_551_0_MAIN_LOAD1(&&(&&(>=(x1[2], 0), <(x1[2], x0[2])), =(0, %(+(x1[2], 1), 2))), x1[2], x0[2])∧(U^Increasing(COND_551_0_MAIN_LOAD1(&&(&&(>=(x1[2], 0), <(x1[2], x0[2])), =(0, %(+(x1[2], 1), 2))), x1[2], x0[2])), ≥))

We simplified constraint (20) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(21)    (x1[2] ≥ 0∧x0[2] + [-1] + [-1]x1[2] ≥ 0∧[-1]min{[2], [-2]} ≥ 0∧max{[2], [-2]} ≥ 0 ⇒ (U^Increasing(COND_551_0_MAIN_LOAD1(&&(&&(>=(x1[2], 0), <(x1[2], x0[2])), =(0, %(+(x1[2], 1), 2))), x1[2], x0[2])), ≥)∧[(2)bni_17 + (-1)Bound*bni_17] + [bni_17]x0[2] + [(-1)bni_17]x1[2] ≥ 0∧[(-1)bso_18] ≥ 0)

We simplified constraint (21) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(22)    (x1[2] ≥ 0∧x0[2] + [-1] + [-1]x1[2] ≥ 0∧[-1]min{[2], [-2]} ≥ 0∧max{[2], [-2]} ≥ 0 ⇒ (U^Increasing(COND_551_0_MAIN_LOAD1(&&(&&(>=(x1[2], 0), <(x1[2], x0[2])), =(0, %(+(x1[2], 1), 2))), x1[2], x0[2])), ≥)∧[(2)bni_17 + (-1)Bound*bni_17] + [bni_17]x0[2] + [(-1)bni_17]x1[2] ≥ 0∧[(-1)bso_18] ≥ 0)

We simplified constraint (22) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(23)    (x1[2] ≥ 0∧x0[2] + [-1] + [-1]x1[2] ≥ 0∧[4] ≥ 0∧[2] ≥ 0∧[2] ≥ 0 ⇒ (U^Increasing(COND_551_0_MAIN_LOAD1(&&(&&(>=(x1[2], 0), <(x1[2], x0[2])), =(0, %(+(x1[2], 1), 2))), x1[2], x0[2])), ≥)∧[(2)bni_17 + (-1)Bound*bni_17] + [bni_17]x0[2] + [(-1)bni_17]x1[2] ≥ 0∧[(-1)bso_18] ≥ 0)

We simplified constraint (23) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(24)    (x1[2] ≥ 0∧x0[2] ≥ 0∧[4] ≥ 0∧[2] ≥ 0∧[2] ≥ 0 ⇒ (U^Increasing(COND_551_0_MAIN_LOAD1(&&(&&(>=(x1[2], 0), <(x1[2], x0[2])), =(0, %(+(x1[2], 1), 2))), x1[2], x0[2])), ≥)∧[(3)bni_17 + (-1)Bound*bni_17] + [bni_17]x0[2] ≥ 0∧[(-1)bso_18] ≥ 0)

We simplified constraint (24) using rule (IDP_POLY_GCD) which results in the following new constraint:
(25)    (x1[2] ≥ 0∧x0[2] ≥ 0∧[1] ≥ 0∧[1] ≥ 0∧[1] ≥ 0 ⇒ (U^Increasing(COND_551_0_MAIN_LOAD1(&&(&&(>=(x1[2], 0), <(x1[2], x0[2])), =(0, %(+(x1[2], 1), 2))), x1[2], x0[2])), ≥)∧[(3)bni_17 + (-1)Bound*bni_17] + [bni_17]x0[2] ≥ 0∧[(-1)bso_18] ≥ 0)

For Pair COND_551_0_MAIN_LOAD1(TRUE, x1, x0) → 551_0_MAIN_LOAD(+(x1, 1), x0) the following chains were created:

We consider the chain COND_551_0_MAIN_LOAD1(TRUE, x1[3], x0[3]) → 551_0_MAIN_LOAD(+(x1[3], 1), x0[3]) which results in the following constraint:
(26)    (COND_551_0_MAIN_LOAD1(TRUE, x1[3], x0[3])≥NonInfC∧COND_551_0_MAIN_LOAD1(TRUE, x1[3], x0[3])≥551_0_MAIN_LOAD(+(x1[3], 1), x0[3])∧(U^Increasing(551_0_MAIN_LOAD(+(x1[3], 1), x0[3])), ≥))

We simplified constraint (26) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(27)    ((U^Increasing(551_0_MAIN_LOAD(+(x1[3], 1), x0[3])), ≥)∧[1 + (-1)bso_20] ≥ 0)

We simplified constraint (27) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(28)    ((U^Increasing(551_0_MAIN_LOAD(+(x1[3], 1), x0[3])), ≥)∧[1 + (-1)bso_20] ≥ 0)

We simplified constraint (28) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(29)    ((U^Increasing(551_0_MAIN_LOAD(+(x1[3], 1), x0[3])), ≥)∧[1 + (-1)bso_20] ≥ 0)

We simplified constraint (29) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint:
(30)    ((U^Increasing(551_0_MAIN_LOAD(+(x1[3], 1), x0[3])), ≥)∧0 = 0∧0 = 0∧[1 + (-1)bso_20] ≥ 0)

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

551_0_MAIN_LOAD(x1, x0) → COND_551_0_MAIN_LOAD(&&(&&(&&(>=(x1, 0), <(x1, x0)), <(0, +(x1, 1))), !(=(%(+(x1, 1), 2), 0))), x1, x0)
- (x1[0] ≥ 0∧x1[0] ≥ 0∧x0[0] ≥ 0∧[1] ≥ 0∧[1] ≥ 0 ⇒ (U^Increasing(COND_551_0_MAIN_LOAD(&&(&&(&&(>=(x1[0], 0), <(x1[0], x0[0])), <(0, +(x1[0], 1))), !(=(%(+(x1[0], 1), 2), 0))), x1[0], x0[0])), ≥)∧[(3)bni_13 + (-1)Bound*bni_13] + [bni_13]x0[0] ≥ 0∧[1 + (-1)bso_14] ≥ 0)
- (x1[0] ≥ 0∧x1[0] ≥ 0∧x0[0] ≥ 0∧[1] ≥ 0∧[1] ≥ 0 ⇒ (U^Increasing(COND_551_0_MAIN_LOAD(&&(&&(&&(>=(x1[0], 0), <(x1[0], x0[0])), <(0, +(x1[0], 1))), !(=(%(+(x1[0], 1), 2), 0))), x1[0], x0[0])), ≥)∧[(3)bni_13 + (-1)Bound*bni_13] + [bni_13]x0[0] ≥ 0∧[1 + (-1)bso_14] ≥ 0)
COND_551_0_MAIN_LOAD(TRUE, x1, x0) → 551_0_MAIN_LOAD(+(+(x1, 1), 1), x0)
- ((U^Increasing(551_0_MAIN_LOAD(+(+(x1[1], 1), 1), x0[1])), ≥)∧0 = 0∧0 = 0∧[1 + (-1)bso_16] ≥ 0)
551_0_MAIN_LOAD(x1, x0) → COND_551_0_MAIN_LOAD1(&&(&&(>=(x1, 0), <(x1, x0)), =(0, %(+(x1, 1), 2))), x1, x0)
- (x1[2] ≥ 0∧x0[2] ≥ 0∧[1] ≥ 0∧[1] ≥ 0∧[1] ≥ 0 ⇒ (U^Increasing(COND_551_0_MAIN_LOAD1(&&(&&(>=(x1[2], 0), <(x1[2], x0[2])), =(0, %(+(x1[2], 1), 2))), x1[2], x0[2])), ≥)∧[(3)bni_17 + (-1)Bound*bni_17] + [bni_17]x0[2] ≥ 0∧[(-1)bso_18] ≥ 0)
COND_551_0_MAIN_LOAD1(TRUE, x1, x0) → 551_0_MAIN_LOAD(+(x1, 1), x0)
- ((U^Increasing(551_0_MAIN_LOAD(+(x1[3], 1), x0[3])), ≥)∧0 = 0∧0 = 0∧[1 + (-1)bso_20] ≥ 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(551_0_MAIN_LOAD(x₁, x₂)) = [2] + x₂ + [-1]x₁
POL(COND_551_0_MAIN_LOAD(x₁, x₂, x₃)) = [1] + x₃ + [-1]x₂
POL(&&(x₁, x₂)) = [-1]
POL(>=(x₁, x₂)) = [-1]
POL(0) = 0
POL(<(x₁, x₂)) = [-1]
POL(+(x₁, x₂)) = x₁ + x₂
POL(1) = [1]
POL(!(x₁)) = [-1]
POL(=(x₁, x₂)) = [-1]
POL(2) = [2]
POL(COND_551_0_MAIN_LOAD1(x₁, x₂, x₃)) = [2] + x₃ + [-1]x₂

Polynomial Interpretations with Context Sensitive Arithemetic Replacement
POL(Term^CSAR-Mode @ Context)

POL(%(x₁, 2)^-1 @ {}) = min{x₂, [-1]x₂}
POL(%(x₁, 2)¹ @ {}) = max{x₂, [-1]x₂}

The following pairs are in P_>:

551_0_MAIN_LOAD(x1[0], x0[0]) → COND_551_0_MAIN_LOAD(&&(&&(&&(>=(x1[0], 0), <(x1[0], x0[0])), <(0, +(x1[0], 1))), !(=(%(+(x1[0], 1), 2), 0))), x1[0], x0[0])
COND_551_0_MAIN_LOAD(TRUE, x1[1], x0[1]) → 551_0_MAIN_LOAD(+(+(x1[1], 1), 1), x0[1])
COND_551_0_MAIN_LOAD1(TRUE, x1[3], x0[3]) → 551_0_MAIN_LOAD(+(x1[3], 1), x0[3])

The following pairs are in P_bound:

551_0_MAIN_LOAD(x1[0], x0[0]) → COND_551_0_MAIN_LOAD(&&(&&(&&(>=(x1[0], 0), <(x1[0], x0[0])), <(0, +(x1[0], 1))), !(=(%(+(x1[0], 1), 2), 0))), x1[0], x0[0])
551_0_MAIN_LOAD(x1[2], x0[2]) → COND_551_0_MAIN_LOAD1(&&(&&(>=(x1[2], 0), <(x1[2], x0[2])), =(0, %(+(x1[2], 1), 2))), x1[2], x0[2])

The following pairs are in P_≥:

551_0_MAIN_LOAD(x1[2], x0[2]) → COND_551_0_MAIN_LOAD1(&&(&&(>=(x1[2], 0), <(x1[2], x0[2])), =(0, %(+(x1[2], 1), 2))), x1[2], x0[2])

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): 551_0_MAIN_LOAD(x1[2], x0[2]) → COND_551_0_MAIN_LOAD1(x1[2] >= 0 && x1[2] < x0[2] && 0 = x1[2] + 1 % 2, x1[2], x0[2])

The set Q is empty.

(8) IDependencyGraphProof (EQUIVALENT transformation)

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

(9) TRUE

(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:

Integer

R is empty.

The integer pair graph contains the following rules and edges:
(1): COND_551_0_MAIN_LOAD(TRUE, x1[1], x0[1]) → 551_0_MAIN_LOAD(x1[1] + 1 + 1, x0[1])
(3): COND_551_0_MAIN_LOAD1(TRUE, x1[3], x0[3]) → 551_0_MAIN_LOAD(x1[3] + 1, x0[3])

The set Q is empty.

(11) IDependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 2 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) IDependencyGraphProof (EQUIVALENT transformation)

(9) TRUE

(10) Obligation:

(11) IDependencyGraphProof (EQUIVALENT transformation)

(12) TRUE