Termination proof of /tmp/tmpYxpEwS/PastaB8.jar

Termination of the given JBC could be proven:

0 JBC

↳1 JBC2FIG (⇐)

↳2 FIGraph

↳3 FIGtoITRSProof (⇐)

↳4 ITRS

↳5 ITRStoIDPProof (⇔)

↳6 IDP

↳7 UsableRulesProof (⇔)

↳8 IDP

↳9 IDPNonInfProof (⇐)

↳10 AND

    ↳11 IDP

        ↳12 IDependencyGraphProof (⇔)

        ↳13 TRUE

    ↳14 IDP

        ↳15 IDependencyGraphProof (⇔)

        ↳16 TRUE

(0) Obligation:

JBC Problem based on JBC Program:
Manifest-Version: 1.0 Created-By: 1.6.0_16 (Sun Microsystems Inc.) Main-Class: PastaB8

/**
 * 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 PastaB8 {
    public static void main(String[] args) {
        Random.args = args;
        int x = Random.random();

        if (x > 0) {
            while (x != 0) {
                if (x % 2 == 0) {
                    x = x/2;
                } else {
                    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:
Graph of 140 nodes with 1 SCC.

(3) FIGtoITRSProof (SOUND transformation)

Transformed FIGraph to ITRS rules

(4) Obligation:

ITRS 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 TRS R consists of the following rules:
Load644(i72) → Cond_Load644(i72 > 0 && 0 = i72 % 2, i72)
Cond_Load644(TRUE, i72) → Load644(i72 / 2)
Load644(i72) → Cond_Load6441(i72 % 2 > 0 && i72 > 0, i72)
Cond_Load6441(TRUE, i72) → Load644(i72 + -1)
The set Q consists of the following terms:
Load644(x0)
Cond_Load644(TRUE, x0)
Cond_Load6441(TRUE, x0)

(5) ITRStoIDPProof (EQUIVALENT transformation)

Added dependency pairs

(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

The ITRS R consists of the following rules:
Load644(i72) → Cond_Load644(i72 > 0 && 0 = i72 % 2, i72)
Cond_Load644(TRUE, i72) → Load644(i72 / 2)
Load644(i72) → Cond_Load6441(i72 % 2 > 0 && i72 > 0, i72)
Cond_Load6441(TRUE, i72) → Load644(i72 + -1)

The integer pair graph contains the following rules and edges:
(0): LOAD644(i72[0]) → COND_LOAD644(i72[0] > 0 && 0 = i72[0] % 2, i72[0])
(1): COND_LOAD644(TRUE, i72[1]) → LOAD644(i72[1] / 2)
(2): LOAD644(i72[2]) → COND_LOAD6441(i72[2] % 2 > 0 && i72[2] > 0, i72[2])
(3): COND_LOAD6441(TRUE, i72[3]) → LOAD644(i72[3] + -1)

(0) -> (1), if ((i72[0] > 0 && 0 = i72[0] % 2 →^* TRUE)∧(i72[0] →^* i72[1]))

(1) -> (0), if ((i72[1] / 2 →^* i72[0]))

(1) -> (2), if ((i72[1] / 2 →^* i72[2]))

(2) -> (3), if ((i72[2] % 2 > 0 && i72[2] > 0 →^* TRUE)∧(i72[2] →^* i72[3]))

(3) -> (0), if ((i72[3] + -1 →^* i72[0]))

(3) -> (2), if ((i72[3] + -1 →^* i72[2]))

The set Q consists of the following terms:
Load644(x0)
Cond_Load644(TRUE, x0)
Cond_Load6441(TRUE, x0)

(7) UsableRulesProof (EQUIVALENT transformation)

As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [LPAR04] we can delete all non-usable rules [FROCOS05] from R.

(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:

Boolean, Integer

R is empty.

The integer pair graph contains the following rules and edges:
(0): LOAD644(i72[0]) → COND_LOAD644(i72[0] > 0 && 0 = i72[0] % 2, i72[0])
(1): COND_LOAD644(TRUE, i72[1]) → LOAD644(i72[1] / 2)
(2): LOAD644(i72[2]) → COND_LOAD6441(i72[2] % 2 > 0 && i72[2] > 0, i72[2])
(3): COND_LOAD6441(TRUE, i72[3]) → LOAD644(i72[3] + -1)

(0) -> (1), if ((i72[0] > 0 && 0 = i72[0] % 2 →^* TRUE)∧(i72[0] →^* i72[1]))

(1) -> (0), if ((i72[1] / 2 →^* i72[0]))

(1) -> (2), if ((i72[1] / 2 →^* i72[2]))

(2) -> (3), if ((i72[2] % 2 > 0 && i72[2] > 0 →^* TRUE)∧(i72[2] →^* i72[3]))

(3) -> (0), if ((i72[3] + -1 →^* i72[0]))

(3) -> (2), if ((i72[3] + -1 →^* i72[2]))

The set Q consists of the following terms:
Load644(x0)
Cond_Load644(TRUE, x0)
Cond_Load6441(TRUE, x0)

(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 LOAD644(i72) → COND_LOAD644(&&(>(i72, 0), =(0, %(i72, 2))), i72) the following chains were created:

We consider the chain LOAD644(i72[0]) → COND_LOAD644(&&(>(i72[0], 0), =(0, %(i72[0], 2))), i72[0]), COND_LOAD644(TRUE, i72[1]) → LOAD644(/(i72[1], 2)) which results in the following constraint:
(1)    (&&(>(i72[0], 0), =(0, %(i72[0], 2)))=TRUE∧i72[0]=i72[1] ⇒ LOAD644(i72[0])≥NonInfC∧LOAD644(i72[0])≥COND_LOAD644(&&(>(i72[0], 0), =(0, %(i72[0], 2))), i72[0])∧(U^Increasing(COND_LOAD644(&&(>(i72[0], 0), =(0, %(i72[0], 2))), i72[0])), ≥))

We simplified constraint (1) using rules (IV), (IDP_BOOLEAN) which results in the following new constraint:
(2)    (>(i72[0], 0)=TRUE∧>=(0, %(i72[0], 2))=TRUE∧<=(0, %(i72[0], 2))=TRUE ⇒ LOAD644(i72[0])≥NonInfC∧LOAD644(i72[0])≥COND_LOAD644(&&(>(i72[0], 0), =(0, %(i72[0], 2))), i72[0])∧(U^Increasing(COND_LOAD644(&&(>(i72[0], 0), =(0, %(i72[0], 2))), i72[0])), ≥))

We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(3)    (i72[0] + [-1] ≥ 0∧[-1]min{[2], [-2]} ≥ 0∧max{[2], [-2]} ≥ 0 ⇒ (U^Increasing(COND_LOAD644(&&(>(i72[0], 0), =(0, %(i72[0], 2))), i72[0])), ≥)∧[(-1)bni_14 + (-1)Bound*bni_14] + [bni_14]i72[0] ≥ 0∧[(-1)bso_15] ≥ 0)

We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(4)    (i72[0] + [-1] ≥ 0∧[-1]min{[2], [-2]} ≥ 0∧max{[2], [-2]} ≥ 0 ⇒ (U^Increasing(COND_LOAD644(&&(>(i72[0], 0), =(0, %(i72[0], 2))), i72[0])), ≥)∧[(-1)bni_14 + (-1)Bound*bni_14] + [bni_14]i72[0] ≥ 0∧[(-1)bso_15] ≥ 0)

We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(5)    (i72[0] + [-1] ≥ 0∧[4] ≥ 0∧[2] ≥ 0∧[2] ≥ 0 ⇒ (U^Increasing(COND_LOAD644(&&(>(i72[0], 0), =(0, %(i72[0], 2))), i72[0])), ≥)∧[(-1)bni_14 + (-1)Bound*bni_14] + [bni_14]i72[0] ≥ 0∧[(-1)bso_15] ≥ 0)

We simplified constraint (5) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(6)    (i72[0] ≥ 0∧[4] ≥ 0∧[2] ≥ 0∧[2] ≥ 0 ⇒ (U^Increasing(COND_LOAD644(&&(>(i72[0], 0), =(0, %(i72[0], 2))), i72[0])), ≥)∧[(-1)Bound*bni_14] + [bni_14]i72[0] ≥ 0∧[(-1)bso_15] ≥ 0)

We simplified constraint (6) using rule (IDP_POLY_GCD) which results in the following new constraint:
(7)    (i72[0] ≥ 0∧[1] ≥ 0∧[1] ≥ 0∧[1] ≥ 0 ⇒ (U^Increasing(COND_LOAD644(&&(>(i72[0], 0), =(0, %(i72[0], 2))), i72[0])), ≥)∧[(-1)Bound*bni_14] + [bni_14]i72[0] ≥ 0∧[(-1)bso_15] ≥ 0)

For Pair COND_LOAD644(TRUE, i72) → LOAD644(/(i72, 2)) the following chains were created:

We consider the chain LOAD644(i72[0]) → COND_LOAD644(&&(>(i72[0], 0), =(0, %(i72[0], 2))), i72[0]), COND_LOAD644(TRUE, i72[1]) → LOAD644(/(i72[1], 2)) which results in the following constraint:
(8)    (&&(>(i72[0], 0), =(0, %(i72[0], 2)))=TRUE∧i72[0]=i72[1] ⇒ COND_LOAD644(TRUE, i72[1])≥NonInfC∧COND_LOAD644(TRUE, i72[1])≥LOAD644(/(i72[1], 2))∧(U^Increasing(LOAD644(/(i72[1], 2))), ≥))

We simplified constraint (8) using rules (III), (IDP_BOOLEAN) which results in the following new constraint:
(9)    (>(i72[0], 0)=TRUE∧>=(0, %(i72[0], 2))=TRUE∧<=(0, %(i72[0], 2))=TRUE ⇒ COND_LOAD644(TRUE, i72[0])≥NonInfC∧COND_LOAD644(TRUE, i72[0])≥LOAD644(/(i72[0], 2))∧(U^Increasing(LOAD644(/(i72[1], 2))), ≥))

We simplified constraint (9) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(10)    (i72[0] + [-1] ≥ 0∧[-1]min{[2], [-2]} ≥ 0∧max{[2], [-2]} ≥ 0 ⇒ (U^Increasing(LOAD644(/(i72[1], 2))), ≥)∧[(-1)bni_16 + (-1)Bound*bni_16] + [bni_16]i72[0] ≥ 0∧[1 + (-1)bso_20] + i72[0] + [-1]max{i72[0], [-1]i72[0]} ≥ 0)

We simplified constraint (10) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(11)    (i72[0] + [-1] ≥ 0∧[-1]min{[2], [-2]} ≥ 0∧max{[2], [-2]} ≥ 0 ⇒ (U^Increasing(LOAD644(/(i72[1], 2))), ≥)∧[(-1)bni_16 + (-1)Bound*bni_16] + [bni_16]i72[0] ≥ 0∧[1 + (-1)bso_20] + i72[0] + [-1]max{i72[0], [-1]i72[0]} ≥ 0)

We simplified constraint (11) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(12)    (i72[0] + [-1] ≥ 0∧[4] ≥ 0∧[2] ≥ 0∧[2] ≥ 0∧[2]i72[0] ≥ 0 ⇒ (U^Increasing(LOAD644(/(i72[1], 2))), ≥)∧[(-1)bni_16 + (-1)Bound*bni_16] + [bni_16]i72[0] ≥ 0∧[1 + (-1)bso_20] ≥ 0)

We simplified constraint (12) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(13)    (i72[0] ≥ 0∧[4] ≥ 0∧[2] ≥ 0∧[2] ≥ 0∧[2] + [2]i72[0] ≥ 0 ⇒ (U^Increasing(LOAD644(/(i72[1], 2))), ≥)∧[(-1)Bound*bni_16] + [bni_16]i72[0] ≥ 0∧[1 + (-1)bso_20] ≥ 0)

We simplified constraint (13) using rule (IDP_POLY_GCD) which results in the following new constraint:
(14)    (i72[0] ≥ 0∧[1] ≥ 0∧[1] ≥ 0∧[1] ≥ 0∧[1] + i72[0] ≥ 0 ⇒ (U^Increasing(LOAD644(/(i72[1], 2))), ≥)∧[(-1)Bound*bni_16] + [bni_16]i72[0] ≥ 0∧[1 + (-1)bso_20] ≥ 0)

For Pair LOAD644(i72) → COND_LOAD6441(&&(>(%(i72, 2), 0), >(i72, 0)), i72) the following chains were created:

We consider the chain LOAD644(i72[2]) → COND_LOAD6441(&&(>(%(i72[2], 2), 0), >(i72[2], 0)), i72[2]), COND_LOAD6441(TRUE, i72[3]) → LOAD644(+(i72[3], -1)) which results in the following constraint:
(15)    (&&(>(%(i72[2], 2), 0), >(i72[2], 0))=TRUE∧i72[2]=i72[3] ⇒ LOAD644(i72[2])≥NonInfC∧LOAD644(i72[2])≥COND_LOAD6441(&&(>(%(i72[2], 2), 0), >(i72[2], 0)), i72[2])∧(U^Increasing(COND_LOAD6441(&&(>(%(i72[2], 2), 0), >(i72[2], 0)), i72[2])), ≥))

We simplified constraint (15) using rules (IV), (IDP_BOOLEAN) which results in the following new constraint:
(16)    (>(%(i72[2], 2), 0)=TRUE∧>(i72[2], 0)=TRUE ⇒ LOAD644(i72[2])≥NonInfC∧LOAD644(i72[2])≥COND_LOAD6441(&&(>(%(i72[2], 2), 0), >(i72[2], 0)), i72[2])∧(U^Increasing(COND_LOAD6441(&&(>(%(i72[2], 2), 0), >(i72[2], 0)), i72[2])), ≥))

We simplified constraint (16) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(17)    (max{[2], [-2]} + [-1] ≥ 0∧i72[2] + [-1] ≥ 0 ⇒ (U^Increasing(COND_LOAD6441(&&(>(%(i72[2], 2), 0), >(i72[2], 0)), i72[2])), ≥)∧[(-1)bni_21 + (-1)Bound*bni_21] + [bni_21]i72[2] ≥ 0∧[(-1)bso_22] ≥ 0)

We simplified constraint (17) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(18)    (max{[2], [-2]} + [-1] ≥ 0∧i72[2] + [-1] ≥ 0 ⇒ (U^Increasing(COND_LOAD6441(&&(>(%(i72[2], 2), 0), >(i72[2], 0)), i72[2])), ≥)∧[(-1)bni_21 + (-1)Bound*bni_21] + [bni_21]i72[2] ≥ 0∧[(-1)bso_22] ≥ 0)

We simplified constraint (18) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(19)    (i72[2] + [-1] ≥ 0∧[4] ≥ 0∧[1] ≥ 0 ⇒ (U^Increasing(COND_LOAD6441(&&(>(%(i72[2], 2), 0), >(i72[2], 0)), i72[2])), ≥)∧[(-1)bni_21 + (-1)Bound*bni_21] + [bni_21]i72[2] ≥ 0∧[(-1)bso_22] ≥ 0)

We simplified constraint (19) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(20)    (i72[2] ≥ 0∧[4] ≥ 0∧[1] ≥ 0 ⇒ (U^Increasing(COND_LOAD6441(&&(>(%(i72[2], 2), 0), >(i72[2], 0)), i72[2])), ≥)∧[(-1)Bound*bni_21] + [bni_21]i72[2] ≥ 0∧[(-1)bso_22] ≥ 0)

We simplified constraint (20) using rule (IDP_POLY_GCD) which results in the following new constraint:
(21)    (i72[2] ≥ 0∧[1] ≥ 0∧[1] ≥ 0 ⇒ (U^Increasing(COND_LOAD6441(&&(>(%(i72[2], 2), 0), >(i72[2], 0)), i72[2])), ≥)∧[(-1)Bound*bni_21] + [bni_21]i72[2] ≥ 0∧[(-1)bso_22] ≥ 0)

For Pair COND_LOAD6441(TRUE, i72) → LOAD644(+(i72, -1)) the following chains were created:

We consider the chain LOAD644(i72[2]) → COND_LOAD6441(&&(>(%(i72[2], 2), 0), >(i72[2], 0)), i72[2]), COND_LOAD6441(TRUE, i72[3]) → LOAD644(+(i72[3], -1)), LOAD644(i72[0]) → COND_LOAD644(&&(>(i72[0], 0), =(0, %(i72[0], 2))), i72[0]) which results in the following constraint:
(22)    (&&(>(%(i72[2], 2), 0), >(i72[2], 0))=TRUE∧i72[2]=i72[3]∧+(i72[3], -1)=i72[0] ⇒ COND_LOAD6441(TRUE, i72[3])≥NonInfC∧COND_LOAD6441(TRUE, i72[3])≥LOAD644(+(i72[3], -1))∧(U^Increasing(LOAD644(+(i72[3], -1))), ≥))

We simplified constraint (22) using rules (III), (IV), (IDP_BOOLEAN) which results in the following new constraint:
(23)    (>(%(i72[2], 2), 0)=TRUE∧>(i72[2], 0)=TRUE ⇒ COND_LOAD6441(TRUE, i72[2])≥NonInfC∧COND_LOAD6441(TRUE, i72[2])≥LOAD644(+(i72[2], -1))∧(U^Increasing(LOAD644(+(i72[3], -1))), ≥))

We simplified constraint (23) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(24)    (max{[2], [-2]} + [-1] ≥ 0∧i72[2] + [-1] ≥ 0 ⇒ (U^Increasing(LOAD644(+(i72[3], -1))), ≥)∧[(-1)bni_23 + (-1)Bound*bni_23] + [bni_23]i72[2] ≥ 0∧[1 + (-1)bso_24] ≥ 0)

We simplified constraint (24) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(25)    (max{[2], [-2]} + [-1] ≥ 0∧i72[2] + [-1] ≥ 0 ⇒ (U^Increasing(LOAD644(+(i72[3], -1))), ≥)∧[(-1)bni_23 + (-1)Bound*bni_23] + [bni_23]i72[2] ≥ 0∧[1 + (-1)bso_24] ≥ 0)

We simplified constraint (25) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(26)    (i72[2] + [-1] ≥ 0∧[4] ≥ 0∧[1] ≥ 0 ⇒ (U^Increasing(LOAD644(+(i72[3], -1))), ≥)∧[(-1)bni_23 + (-1)Bound*bni_23] + [bni_23]i72[2] ≥ 0∧[1 + (-1)bso_24] ≥ 0)

We simplified constraint (26) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(27)    (i72[2] ≥ 0∧[4] ≥ 0∧[1] ≥ 0 ⇒ (U^Increasing(LOAD644(+(i72[3], -1))), ≥)∧[(-1)Bound*bni_23] + [bni_23]i72[2] ≥ 0∧[1 + (-1)bso_24] ≥ 0)

We simplified constraint (27) using rule (IDP_POLY_GCD) which results in the following new constraint:
(28)    (i72[2] ≥ 0∧[1] ≥ 0∧[1] ≥ 0 ⇒ (U^Increasing(LOAD644(+(i72[3], -1))), ≥)∧[(-1)Bound*bni_23] + [bni_23]i72[2] ≥ 0∧[1 + (-1)bso_24] ≥ 0)
We consider the chain LOAD644(i72[2]) → COND_LOAD6441(&&(>(%(i72[2], 2), 0), >(i72[2], 0)), i72[2]), COND_LOAD6441(TRUE, i72[3]) → LOAD644(+(i72[3], -1)), LOAD644(i72[2]) → COND_LOAD6441(&&(>(%(i72[2], 2), 0), >(i72[2], 0)), i72[2]) which results in the following constraint:
(29)    (&&(>(%(i72[2], 2), 0), >(i72[2], 0))=TRUE∧i72[2]=i72[3]∧+(i72[3], -1)=i72[2]1 ⇒ COND_LOAD6441(TRUE, i72[3])≥NonInfC∧COND_LOAD6441(TRUE, i72[3])≥LOAD644(+(i72[3], -1))∧(U^Increasing(LOAD644(+(i72[3], -1))), ≥))

We simplified constraint (29) using rules (III), (IV), (IDP_BOOLEAN) which results in the following new constraint:
(30)    (>(%(i72[2], 2), 0)=TRUE∧>(i72[2], 0)=TRUE ⇒ COND_LOAD6441(TRUE, i72[2])≥NonInfC∧COND_LOAD6441(TRUE, i72[2])≥LOAD644(+(i72[2], -1))∧(U^Increasing(LOAD644(+(i72[3], -1))), ≥))

We simplified constraint (30) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(31)    (max{[2], [-2]} + [-1] ≥ 0∧i72[2] + [-1] ≥ 0 ⇒ (U^Increasing(LOAD644(+(i72[3], -1))), ≥)∧[(-1)bni_23 + (-1)Bound*bni_23] + [bni_23]i72[2] ≥ 0∧[1 + (-1)bso_24] ≥ 0)

We simplified constraint (31) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(32)    (max{[2], [-2]} + [-1] ≥ 0∧i72[2] + [-1] ≥ 0 ⇒ (U^Increasing(LOAD644(+(i72[3], -1))), ≥)∧[(-1)bni_23 + (-1)Bound*bni_23] + [bni_23]i72[2] ≥ 0∧[1 + (-1)bso_24] ≥ 0)

We simplified constraint (32) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(33)    (i72[2] + [-1] ≥ 0∧[4] ≥ 0∧[1] ≥ 0 ⇒ (U^Increasing(LOAD644(+(i72[3], -1))), ≥)∧[(-1)bni_23 + (-1)Bound*bni_23] + [bni_23]i72[2] ≥ 0∧[1 + (-1)bso_24] ≥ 0)

We simplified constraint (33) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(34)    (i72[2] ≥ 0∧[4] ≥ 0∧[1] ≥ 0 ⇒ (U^Increasing(LOAD644(+(i72[3], -1))), ≥)∧[(-1)Bound*bni_23] + [bni_23]i72[2] ≥ 0∧[1 + (-1)bso_24] ≥ 0)

We simplified constraint (34) using rule (IDP_POLY_GCD) which results in the following new constraint:
(35)    (i72[2] ≥ 0∧[1] ≥ 0∧[1] ≥ 0 ⇒ (U^Increasing(LOAD644(+(i72[3], -1))), ≥)∧[(-1)Bound*bni_23] + [bni_23]i72[2] ≥ 0∧[1 + (-1)bso_24] ≥ 0)

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

LOAD644(i72) → COND_LOAD644(&&(>(i72, 0), =(0, %(i72, 2))), i72)
- (i72[0] ≥ 0∧[1] ≥ 0∧[1] ≥ 0∧[1] ≥ 0 ⇒ (U^Increasing(COND_LOAD644(&&(>(i72[0], 0), =(0, %(i72[0], 2))), i72[0])), ≥)∧[(-1)Bound*bni_14] + [bni_14]i72[0] ≥ 0∧[(-1)bso_15] ≥ 0)
COND_LOAD644(TRUE, i72) → LOAD644(/(i72, 2))
- (i72[0] ≥ 0∧[1] ≥ 0∧[1] ≥ 0∧[1] ≥ 0∧[1] + i72[0] ≥ 0 ⇒ (U^Increasing(LOAD644(/(i72[1], 2))), ≥)∧[(-1)Bound*bni_16] + [bni_16]i72[0] ≥ 0∧[1 + (-1)bso_20] ≥ 0)
LOAD644(i72) → COND_LOAD6441(&&(>(%(i72, 2), 0), >(i72, 0)), i72)
- (i72[2] ≥ 0∧[1] ≥ 0∧[1] ≥ 0 ⇒ (U^Increasing(COND_LOAD6441(&&(>(%(i72[2], 2), 0), >(i72[2], 0)), i72[2])), ≥)∧[(-1)Bound*bni_21] + [bni_21]i72[2] ≥ 0∧[(-1)bso_22] ≥ 0)
COND_LOAD6441(TRUE, i72) → LOAD644(+(i72, -1))
- (i72[2] ≥ 0∧[1] ≥ 0∧[1] ≥ 0 ⇒ (U^Increasing(LOAD644(+(i72[3], -1))), ≥)∧[(-1)Bound*bni_23] + [bni_23]i72[2] ≥ 0∧[1 + (-1)bso_24] ≥ 0)
- (i72[2] ≥ 0∧[1] ≥ 0∧[1] ≥ 0 ⇒ (U^Increasing(LOAD644(+(i72[3], -1))), ≥)∧[(-1)Bound*bni_23] + [bni_23]i72[2] ≥ 0∧[1 + (-1)bso_24] ≥ 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) = [3]
POL(LOAD644(x₁)) = [-1] + x₁
POL(COND_LOAD644(x₁, x₂)) = [-1] + x₂
POL(&&(x₁, x₂)) = [-1]
POL(>(x₁, x₂)) = [-1]
POL(0) = 0
POL(=(x₁, x₂)) = [-1]
POL(2) = [2]
POL(COND_LOAD6441(x₁, x₂)) = [-1] + x₂
POL(+(x₁, x₂)) = x₁ + x₂
POL(-1) = [-1]

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}
POL(/(x₁, 2)¹ @ {LOAD644_1/0}) = max{x₁, [-1]x₁} + [-1]

The following pairs are in P_>:

COND_LOAD644(TRUE, i72[1]) → LOAD644(/(i72[1], 2))
COND_LOAD6441(TRUE, i72[3]) → LOAD644(+(i72[3], -1))

The following pairs are in P_bound:

LOAD644(i72[0]) → COND_LOAD644(&&(>(i72[0], 0), =(0, %(i72[0], 2))), i72[0])
COND_LOAD644(TRUE, i72[1]) → LOAD644(/(i72[1], 2))
LOAD644(i72[2]) → COND_LOAD6441(&&(>(%(i72[2], 2), 0), >(i72[2], 0)), i72[2])
COND_LOAD6441(TRUE, i72[3]) → LOAD644(+(i72[3], -1))

The following pairs are in P_≥:

LOAD644(i72[0]) → COND_LOAD644(&&(>(i72[0], 0), =(0, %(i72[0], 2))), i72[0])
LOAD644(i72[2]) → COND_LOAD6441(&&(>(%(i72[2], 2), 0), >(i72[2], 0)), i72[2])

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

FALSE¹ → &&(FALSE, FALSE)¹
/¹ →

(10) Complex Obligation (AND)

(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:

Boolean, Integer

R is empty.

The integer pair graph contains the following rules and edges:
(0): LOAD644(i72[0]) → COND_LOAD644(i72[0] > 0 && 0 = i72[0] % 2, i72[0])
(2): LOAD644(i72[2]) → COND_LOAD6441(i72[2] % 2 > 0 && i72[2] > 0, i72[2])

The set Q consists of the following terms:
Load644(x0)
Cond_Load644(TRUE, x0)
Cond_Load6441(TRUE, x0)

(12) IDependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 2 less nodes.

(13) TRUE

(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:
none

R is empty.

The integer pair graph is empty.

The set Q consists of the following terms:
Load644(x0)
Cond_Load644(TRUE, x0)
Cond_Load6441(TRUE, x0)

(15) IDependencyGraphProof (EQUIVALENT transformation)

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

(0) Obligation:

(1) JBC2FIG (SOUND transformation)

(2) Obligation:

(3) FIGtoITRSProof (SOUND transformation)

(4) Obligation:

(5) ITRStoIDPProof (EQUIVALENT transformation)

(6) Obligation:

(7) UsableRulesProof (EQUIVALENT transformation)

(8) Obligation:

(9) IDPNonInfProof (SOUND transformation)

(10) Complex Obligation (AND)

(11) Obligation:

(12) IDependencyGraphProof (EQUIVALENT transformation)

(13) TRUE

(14) Obligation:

(15) IDependencyGraphProof (EQUIVALENT transformation)

(16) TRUE