Termination proof of /tmp/tmplKHy1u/PastaB16.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 TRUE

    ↳12 IDP

        ↳13 IDependencyGraphProof (⇔)

        ↳14 TRUE

(0) Obligation:

JBC Problem based on JBC Program:
Manifest-Version: 1.0 Created-By: 1.6.0_16 (Sun Microsystems Inc.) Main-Class: PastaB16

/**
 * Example taken from "A Term Rewriting Approach to the Automated Termination
 * Analysis of Imperative Programs" (http://www.cs.unm.edu/~spf/papers/2009-02.pdf)
 * and converted to Java.
 */

public class PastaB16 {
    public static void main(String[] args) {
        Random.args = args;
        int x = Random.random();
        int y = Random.random();

        while (x > 0) {
            while (y > 0) {
                y--;
            }
            x--;
        }
    }
}


public class Random {
  static String[] args;
  static int index = 0;

  public static int random() {
    String string = args[index];
    index++;
    return string.length();
  }
}

(1) JBC2FIG (SOUND transformation)

Constructed FIGraph.

(2) Obligation:

FIGraph based on JBC Program:
PastaB16.main([Ljava/lang/String;)V: Graph of 159 nodes with 1 SCC.

(3) FIGtoITRSProof (SOUND transformation)

Transformed FIGraph SCCs to IDPs. Logs:

Log for SCC 0:

Generated 14 rules for P and 2 rules for R.

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

Filtered ground terms:

639_0_main_LE(x1, x2, x3, x4) → 639_0_main_LE(x2, x3, x4)
Cond_639_0_main_LE1(x1, x2, x3, x4, x5) → Cond_639_0_main_LE1(x1, x3, x4, x5)
Cond_639_0_main_LE(x1, x2, x3, x4, x5) → Cond_639_0_main_LE(x1, x3)

Filtered duplicate args:

639_0_main_LE(x1, x2, x3) → 639_0_main_LE(x1, x3)
Cond_639_0_main_LE1(x1, x2, x3, x4) → Cond_639_0_main_LE1(x1, x2, 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): 639_0_MAIN_LE(x0[0], 0) → COND_639_0_MAIN_LE(x0[0] > 0 && 0 < x0[0] + -1, x0[0], 0)
(1): COND_639_0_MAIN_LE(TRUE, x0[1], 0) → 639_0_MAIN_LE(x0[1] + -1, 0)
(2): 639_0_MAIN_LE(x0[2], x1[2]) → COND_639_0_MAIN_LE1(x1[2] > 0, x0[2], x1[2])
(3): COND_639_0_MAIN_LE1(TRUE, x0[3], x1[3]) → 639_0_MAIN_LE(x0[3], x1[3] + -1)

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

(1) -> (0), if (x0[1] + -1 →^* x0[0])

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

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

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

(3) -> (2), if ((x0[3] →^* x0[2])∧(x1[3] + -1 →^* x1[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 639_0_MAIN_LE(x0, 0) → COND_639_0_MAIN_LE(&&(>(x0, 0), <(0, +(x0, -1))), x0, 0) the following chains were created:

We consider the chain 639_0_MAIN_LE(x0[0], 0) → COND_639_0_MAIN_LE(&&(>(x0[0], 0), <(0, +(x0[0], -1))), x0[0], 0), COND_639_0_MAIN_LE(TRUE, x0[1], 0) → 639_0_MAIN_LE(+(x0[1], -1), 0) which results in the following constraint:
(1)    (&&(>(x0[0], 0), <(0, +(x0[0], -1)))=TRUE∧x0[0]=x0[1] ⇒ 639_0_MAIN_LE(x0[0], 0)≥NonInfC∧639_0_MAIN_LE(x0[0], 0)≥COND_639_0_MAIN_LE(&&(>(x0[0], 0), <(0, +(x0[0], -1))), x0[0], 0)∧(U^Increasing(COND_639_0_MAIN_LE(&&(>(x0[0], 0), <(0, +(x0[0], -1))), x0[0], 0)), ≥))

We simplified constraint (1) using rules (IV), (IDP_BOOLEAN) which results in the following new constraint:
(2)    (>(x0[0], 0)=TRUE∧<(0, +(x0[0], -1))=TRUE ⇒ 639_0_MAIN_LE(x0[0], 0)≥NonInfC∧639_0_MAIN_LE(x0[0], 0)≥COND_639_0_MAIN_LE(&&(>(x0[0], 0), <(0, +(x0[0], -1))), x0[0], 0)∧(U^Increasing(COND_639_0_MAIN_LE(&&(>(x0[0], 0), <(0, +(x0[0], -1))), x0[0], 0)), ≥))

We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(3)    (x0[0] + [-1] ≥ 0∧x0[0] + [-2] ≥ 0 ⇒ (U^Increasing(COND_639_0_MAIN_LE(&&(>(x0[0], 0), <(0, +(x0[0], -1))), x0[0], 0)), ≥)∧[(-1)Bound*bni_12] ≥ 0∧[(-1)bso_13] ≥ 0)

We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(4)    (x0[0] + [-1] ≥ 0∧x0[0] + [-2] ≥ 0 ⇒ (U^Increasing(COND_639_0_MAIN_LE(&&(>(x0[0], 0), <(0, +(x0[0], -1))), x0[0], 0)), ≥)∧[(-1)Bound*bni_12] ≥ 0∧[(-1)bso_13] ≥ 0)

We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(5)    (x0[0] + [-1] ≥ 0∧x0[0] + [-2] ≥ 0 ⇒ (U^Increasing(COND_639_0_MAIN_LE(&&(>(x0[0], 0), <(0, +(x0[0], -1))), x0[0], 0)), ≥)∧[(-1)Bound*bni_12] ≥ 0∧[(-1)bso_13] ≥ 0)

We simplified constraint (5) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(6)    (x0[0] ≥ 0∧[-1] + x0[0] ≥ 0 ⇒ (U^Increasing(COND_639_0_MAIN_LE(&&(>(x0[0], 0), <(0, +(x0[0], -1))), x0[0], 0)), ≥)∧[(-1)Bound*bni_12] ≥ 0∧[(-1)bso_13] ≥ 0)

We simplified constraint (6) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(7)    ([1] + x0[0] ≥ 0∧x0[0] ≥ 0 ⇒ (U^Increasing(COND_639_0_MAIN_LE(&&(>(x0[0], 0), <(0, +(x0[0], -1))), x0[0], 0)), ≥)∧[(-1)Bound*bni_12] ≥ 0∧[(-1)bso_13] ≥ 0)

For Pair COND_639_0_MAIN_LE(TRUE, x0, 0) → 639_0_MAIN_LE(+(x0, -1), 0) the following chains were created:

We consider the chain COND_639_0_MAIN_LE(TRUE, x0[1], 0) → 639_0_MAIN_LE(+(x0[1], -1), 0) which results in the following constraint:
(8)    (COND_639_0_MAIN_LE(TRUE, x0[1], 0)≥NonInfC∧COND_639_0_MAIN_LE(TRUE, x0[1], 0)≥639_0_MAIN_LE(+(x0[1], -1), 0)∧(U^Increasing(639_0_MAIN_LE(+(x0[1], -1), 0)), ≥))

We simplified constraint (8) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(9)    ((U^Increasing(639_0_MAIN_LE(+(x0[1], -1), 0)), ≥)∧[(-1)bso_15] ≥ 0)

We simplified constraint (9) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(10)    ((U^Increasing(639_0_MAIN_LE(+(x0[1], -1), 0)), ≥)∧[(-1)bso_15] ≥ 0)

We simplified constraint (10) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(11)    ((U^Increasing(639_0_MAIN_LE(+(x0[1], -1), 0)), ≥)∧[(-1)bso_15] ≥ 0)

We simplified constraint (11) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint:
(12)    ((U^Increasing(639_0_MAIN_LE(+(x0[1], -1), 0)), ≥)∧0 = 0∧[(-1)bso_15] ≥ 0)

For Pair 639_0_MAIN_LE(x0, x1) → COND_639_0_MAIN_LE1(>(x1, 0), x0, x1) the following chains were created:

We consider the chain 639_0_MAIN_LE(x0[2], x1[2]) → COND_639_0_MAIN_LE1(>(x1[2], 0), x0[2], x1[2]), COND_639_0_MAIN_LE1(TRUE, x0[3], x1[3]) → 639_0_MAIN_LE(x0[3], +(x1[3], -1)) which results in the following constraint:
(13)    (>(x1[2], 0)=TRUE∧x0[2]=x0[3]∧x1[2]=x1[3] ⇒ 639_0_MAIN_LE(x0[2], x1[2])≥NonInfC∧639_0_MAIN_LE(x0[2], x1[2])≥COND_639_0_MAIN_LE1(>(x1[2], 0), x0[2], x1[2])∧(U^Increasing(COND_639_0_MAIN_LE1(>(x1[2], 0), x0[2], x1[2])), ≥))

We simplified constraint (13) using rule (IV) which results in the following new constraint:
(14)    (>(x1[2], 0)=TRUE ⇒ 639_0_MAIN_LE(x0[2], x1[2])≥NonInfC∧639_0_MAIN_LE(x0[2], x1[2])≥COND_639_0_MAIN_LE1(>(x1[2], 0), x0[2], x1[2])∧(U^Increasing(COND_639_0_MAIN_LE1(>(x1[2], 0), x0[2], x1[2])), ≥))

We simplified constraint (14) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(15)    (x1[2] + [-1] ≥ 0 ⇒ (U^Increasing(COND_639_0_MAIN_LE1(>(x1[2], 0), x0[2], x1[2])), ≥)∧[(-1)Bound*bni_16] + [(2)bni_16]x1[2] ≥ 0∧[1 + (-1)bso_17] ≥ 0)

We simplified constraint (15) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(16)    (x1[2] + [-1] ≥ 0 ⇒ (U^Increasing(COND_639_0_MAIN_LE1(>(x1[2], 0), x0[2], x1[2])), ≥)∧[(-1)Bound*bni_16] + [(2)bni_16]x1[2] ≥ 0∧[1 + (-1)bso_17] ≥ 0)

We simplified constraint (16) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(17)    (x1[2] + [-1] ≥ 0 ⇒ (U^Increasing(COND_639_0_MAIN_LE1(>(x1[2], 0), x0[2], x1[2])), ≥)∧[(-1)Bound*bni_16] + [(2)bni_16]x1[2] ≥ 0∧[1 + (-1)bso_17] ≥ 0)

We simplified constraint (17) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint:
(18)    (x1[2] + [-1] ≥ 0 ⇒ (U^Increasing(COND_639_0_MAIN_LE1(>(x1[2], 0), x0[2], x1[2])), ≥)∧0 = 0∧[(-1)Bound*bni_16] + [(2)bni_16]x1[2] ≥ 0∧0 = 0∧[1 + (-1)bso_17] ≥ 0)

We simplified constraint (18) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(19)    (x1[2] ≥ 0 ⇒ (U^Increasing(COND_639_0_MAIN_LE1(>(x1[2], 0), x0[2], x1[2])), ≥)∧0 = 0∧[(-1)Bound*bni_16 + (2)bni_16] + [(2)bni_16]x1[2] ≥ 0∧0 = 0∧[1 + (-1)bso_17] ≥ 0)

For Pair COND_639_0_MAIN_LE1(TRUE, x0, x1) → 639_0_MAIN_LE(x0, +(x1, -1)) the following chains were created:

We consider the chain COND_639_0_MAIN_LE1(TRUE, x0[3], x1[3]) → 639_0_MAIN_LE(x0[3], +(x1[3], -1)) which results in the following constraint:
(20)    (COND_639_0_MAIN_LE1(TRUE, x0[3], x1[3])≥NonInfC∧COND_639_0_MAIN_LE1(TRUE, x0[3], x1[3])≥639_0_MAIN_LE(x0[3], +(x1[3], -1))∧(U^Increasing(639_0_MAIN_LE(x0[3], +(x1[3], -1))), ≥))

We simplified constraint (20) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(21)    ((U^Increasing(639_0_MAIN_LE(x0[3], +(x1[3], -1))), ≥)∧[1 + (-1)bso_19] ≥ 0)

We simplified constraint (21) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(22)    ((U^Increasing(639_0_MAIN_LE(x0[3], +(x1[3], -1))), ≥)∧[1 + (-1)bso_19] ≥ 0)

We simplified constraint (22) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(23)    ((U^Increasing(639_0_MAIN_LE(x0[3], +(x1[3], -1))), ≥)∧[1 + (-1)bso_19] ≥ 0)

We simplified constraint (23) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint:
(24)    ((U^Increasing(639_0_MAIN_LE(x0[3], +(x1[3], -1))), ≥)∧0 = 0∧0 = 0∧[1 + (-1)bso_19] ≥ 0)

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

639_0_MAIN_LE(x0, 0) → COND_639_0_MAIN_LE(&&(>(x0, 0), <(0, +(x0, -1))), x0, 0)
- ([1] + x0[0] ≥ 0∧x0[0] ≥ 0 ⇒ (U^Increasing(COND_639_0_MAIN_LE(&&(>(x0[0], 0), <(0, +(x0[0], -1))), x0[0], 0)), ≥)∧[(-1)Bound*bni_12] ≥ 0∧[(-1)bso_13] ≥ 0)
COND_639_0_MAIN_LE(TRUE, x0, 0) → 639_0_MAIN_LE(+(x0, -1), 0)
- ((U^Increasing(639_0_MAIN_LE(+(x0[1], -1), 0)), ≥)∧0 = 0∧[(-1)bso_15] ≥ 0)
639_0_MAIN_LE(x0, x1) → COND_639_0_MAIN_LE1(>(x1, 0), x0, x1)
- (x1[2] ≥ 0 ⇒ (U^Increasing(COND_639_0_MAIN_LE1(>(x1[2], 0), x0[2], x1[2])), ≥)∧0 = 0∧[(-1)Bound*bni_16 + (2)bni_16] + [(2)bni_16]x1[2] ≥ 0∧0 = 0∧[1 + (-1)bso_17] ≥ 0)
COND_639_0_MAIN_LE1(TRUE, x0, x1) → 639_0_MAIN_LE(x0, +(x1, -1))
- ((U^Increasing(639_0_MAIN_LE(x0[3], +(x1[3], -1))), ≥)∧0 = 0∧0 = 0∧[1 + (-1)bso_19] ≥ 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(639_0_MAIN_LE(x₁, x₂)) = [2]x₂
POL(0) = 0
POL(COND_639_0_MAIN_LE(x₁, x₂, x₃)) = 0
POL(&&(x₁, x₂)) = [-1]
POL(>(x₁, x₂)) = [-1]
POL(<(x₁, x₂)) = [-1]
POL(+(x₁, x₂)) = x₁ + x₂
POL(-1) = [-1]
POL(COND_639_0_MAIN_LE1(x₁, x₂, x₃)) = [-1] + [2]x₃

The following pairs are in P_>:

639_0_MAIN_LE(x0[2], x1[2]) → COND_639_0_MAIN_LE1(>(x1[2], 0), x0[2], x1[2])
COND_639_0_MAIN_LE1(TRUE, x0[3], x1[3]) → 639_0_MAIN_LE(x0[3], +(x1[3], -1))

The following pairs are in P_bound:

639_0_MAIN_LE(x0[0], 0) → COND_639_0_MAIN_LE(&&(>(x0[0], 0), <(0, +(x0[0], -1))), x0[0], 0)
639_0_MAIN_LE(x0[2], x1[2]) → COND_639_0_MAIN_LE1(>(x1[2], 0), x0[2], x1[2])

The following pairs are in P_≥:

639_0_MAIN_LE(x0[0], 0) → COND_639_0_MAIN_LE(&&(>(x0[0], 0), <(0, +(x0[0], -1))), x0[0], 0)
COND_639_0_MAIN_LE(TRUE, x0[1], 0) → 639_0_MAIN_LE(+(x0[1], -1), 0)

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

(1) -> (0), if (x0[1] + -1 →^* x0[0])

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

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 639_0_MAIN_LE(x0[0], 0) → COND_639_0_MAIN_LE(&&(>(x0[0], 0), <(0, +(x0[0], -1))), x0[0], 0) the following chains were created:

We consider the chain 639_0_MAIN_LE(x0[0], 0) → COND_639_0_MAIN_LE(&&(>(x0[0], 0), <(0, +(x0[0], -1))), x0[0], 0), COND_639_0_MAIN_LE(TRUE, x0[1], 0) → 639_0_MAIN_LE(+(x0[1], -1), 0) which results in the following constraint:
(1)    (&&(>(x0[0], 0), <(0, +(x0[0], -1)))=TRUE∧x0[0]=x0[1] ⇒ 639_0_MAIN_LE(x0[0], 0)≥NonInfC∧639_0_MAIN_LE(x0[0], 0)≥COND_639_0_MAIN_LE(&&(>(x0[0], 0), <(0, +(x0[0], -1))), x0[0], 0)∧(U^Increasing(COND_639_0_MAIN_LE(&&(>(x0[0], 0), <(0, +(x0[0], -1))), x0[0], 0)), ≥))

We simplified constraint (1) using rules (IV), (IDP_BOOLEAN) which results in the following new constraint:
(2)    (>(x0[0], 0)=TRUE∧<(0, +(x0[0], -1))=TRUE ⇒ 639_0_MAIN_LE(x0[0], 0)≥NonInfC∧639_0_MAIN_LE(x0[0], 0)≥COND_639_0_MAIN_LE(&&(>(x0[0], 0), <(0, +(x0[0], -1))), x0[0], 0)∧(U^Increasing(COND_639_0_MAIN_LE(&&(>(x0[0], 0), <(0, +(x0[0], -1))), x0[0], 0)), ≥))

We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(3)    (x0[0] + [-1] ≥ 0∧x0[0] + [-2] ≥ 0 ⇒ (U^Increasing(COND_639_0_MAIN_LE(&&(>(x0[0], 0), <(0, +(x0[0], -1))), x0[0], 0)), ≥)∧[(-1)Bound*bni_8] + [(2)bni_8]x0[0] ≥ 0∧[1 + (-1)bso_9] ≥ 0)

We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(4)    (x0[0] + [-1] ≥ 0∧x0[0] + [-2] ≥ 0 ⇒ (U^Increasing(COND_639_0_MAIN_LE(&&(>(x0[0], 0), <(0, +(x0[0], -1))), x0[0], 0)), ≥)∧[(-1)Bound*bni_8] + [(2)bni_8]x0[0] ≥ 0∧[1 + (-1)bso_9] ≥ 0)

We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(5)    (x0[0] + [-1] ≥ 0∧x0[0] + [-2] ≥ 0 ⇒ (U^Increasing(COND_639_0_MAIN_LE(&&(>(x0[0], 0), <(0, +(x0[0], -1))), x0[0], 0)), ≥)∧[(-1)Bound*bni_8] + [(2)bni_8]x0[0] ≥ 0∧[1 + (-1)bso_9] ≥ 0)

We simplified constraint (5) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(6)    (x0[0] ≥ 0∧[-1] + x0[0] ≥ 0 ⇒ (U^Increasing(COND_639_0_MAIN_LE(&&(>(x0[0], 0), <(0, +(x0[0], -1))), x0[0], 0)), ≥)∧[(-1)Bound*bni_8 + (2)bni_8] + [(2)bni_8]x0[0] ≥ 0∧[1 + (-1)bso_9] ≥ 0)

We simplified constraint (6) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(7)    ([1] + x0[0] ≥ 0∧x0[0] ≥ 0 ⇒ (U^Increasing(COND_639_0_MAIN_LE(&&(>(x0[0], 0), <(0, +(x0[0], -1))), x0[0], 0)), ≥)∧[(-1)Bound*bni_8 + (4)bni_8] + [(2)bni_8]x0[0] ≥ 0∧[1 + (-1)bso_9] ≥ 0)

For Pair COND_639_0_MAIN_LE(TRUE, x0[1], 0) → 639_0_MAIN_LE(+(x0[1], -1), 0) the following chains were created:

We consider the chain COND_639_0_MAIN_LE(TRUE, x0[1], 0) → 639_0_MAIN_LE(+(x0[1], -1), 0) which results in the following constraint:
(8)    (COND_639_0_MAIN_LE(TRUE, x0[1], 0)≥NonInfC∧COND_639_0_MAIN_LE(TRUE, x0[1], 0)≥639_0_MAIN_LE(+(x0[1], -1), 0)∧(U^Increasing(639_0_MAIN_LE(+(x0[1], -1), 0)), ≥))

We simplified constraint (8) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(9)    ((U^Increasing(639_0_MAIN_LE(+(x0[1], -1), 0)), ≥)∧[1 + (-1)bso_11] ≥ 0)

We simplified constraint (9) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(10)    ((U^Increasing(639_0_MAIN_LE(+(x0[1], -1), 0)), ≥)∧[1 + (-1)bso_11] ≥ 0)

We simplified constraint (10) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(11)    ((U^Increasing(639_0_MAIN_LE(+(x0[1], -1), 0)), ≥)∧[1 + (-1)bso_11] ≥ 0)

We simplified constraint (11) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint:
(12)    ((U^Increasing(639_0_MAIN_LE(+(x0[1], -1), 0)), ≥)∧0 = 0∧[1 + (-1)bso_11] ≥ 0)

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

639_0_MAIN_LE(x0[0], 0) → COND_639_0_MAIN_LE(&&(>(x0[0], 0), <(0, +(x0[0], -1))), x0[0], 0)
- ([1] + x0[0] ≥ 0∧x0[0] ≥ 0 ⇒ (U^Increasing(COND_639_0_MAIN_LE(&&(>(x0[0], 0), <(0, +(x0[0], -1))), x0[0], 0)), ≥)∧[(-1)Bound*bni_8 + (4)bni_8] + [(2)bni_8]x0[0] ≥ 0∧[1 + (-1)bso_9] ≥ 0)
COND_639_0_MAIN_LE(TRUE, x0[1], 0) → 639_0_MAIN_LE(+(x0[1], -1), 0)
- ((U^Increasing(639_0_MAIN_LE(+(x0[1], -1), 0)), ≥)∧0 = 0∧[1 + (-1)bso_11] ≥ 0)

POL(TRUE) = 0
POL(FALSE) = 0
POL(639_0_MAIN_LE(x₁, x₂)) = [2]x₁
POL(0) = 0
POL(COND_639_0_MAIN_LE(x₁, x₂, x₃)) = [-1] + [2]x₂
POL(&&(x₁, x₂)) = [2]
POL(>(x₁, x₂)) = [-1]
POL(<(x₁, x₂)) = [2]
POL(+(x₁, x₂)) = x₁ + x₂
POL(-1) = [-1]

The following pairs are in P_>:

639_0_MAIN_LE(x0[0], 0) → COND_639_0_MAIN_LE(&&(>(x0[0], 0), <(0, +(x0[0], -1))), x0[0], 0)
COND_639_0_MAIN_LE(TRUE, x0[1], 0) → 639_0_MAIN_LE(+(x0[1], -1), 0)

The following pairs are in P_bound:

639_0_MAIN_LE(x0[0], 0) → COND_639_0_MAIN_LE(&&(>(x0[0], 0), <(0, +(x0[0], -1))), x0[0], 0)

The following pairs are in P_≥:
none

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:

Integer

R is empty.

The integer pair graph contains the following rules and edges:
(1): COND_639_0_MAIN_LE(TRUE, x0[1], 0) → 639_0_MAIN_LE(x0[1] + -1, 0)

The set Q is empty.

(10) IDependencyGraphProof (EQUIVALENT transformation)

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

(11) TRUE

(12) 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_639_0_MAIN_LE(TRUE, x0[1], 0) → 639_0_MAIN_LE(x0[1] + -1, 0)
(3): COND_639_0_MAIN_LE1(TRUE, x0[3], x1[3]) → 639_0_MAIN_LE(x0[3], x1[3] + -1)

The set Q is empty.

(13) 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) IDPNonInfProof (SOUND transformation)

(9) Obligation:

(10) IDependencyGraphProof (EQUIVALENT transformation)

(11) TRUE

(12) Obligation:

(13) IDependencyGraphProof (EQUIVALENT transformation)

(14) TRUE