Termination of the given JBC could be proven:

0 JBC
↳1 JBC2FIG (⇐)
↳2 FIGraph
↳3 FIGtoITRSProof (⇐)
↳4 ITRS
↳5 GroundTermsRemoverProof (⇔)
↳6 ITRS
↳7 ITRStoIDPProof (⇔)
↳8 IDP
↳9 UsableRulesProof (⇔)
↳10 IDP
↳11 IDPNonInfProof (⇐)
↳12 AND
    ↳13 IDP
        ↳14 IDependencyGraphProof (⇔)
        ↳15 IDP
        ↳16 IDPNonInfProof (⇐)
        ↳17 AND
            ↳18 IDP
                ↳19 IDependencyGraphProof (⇔)
                ↳20 TRUE
            ↳21 IDP
                ↳22 IDependencyGraphProof (⇔)
                ↳23 TRUE
    ↳24 IDP
        ↳25 IDependencyGraphProof (⇔)
        ↳26 TRUE

(0) Obligation:

JBC Problem based on JBC Program:
Manifest-Version: 1.0 Created-By: 1.6.0_16 (Sun Microsystems Inc.) Main-Class: PastaC1 
/**
 * 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 PastaC1 {
    public static void main(String[] args) {
        Random.args = args;
        int x = Random.random();

		while (x >= 0) {
			int y = 1;
			while (x > y) {
				y = 2*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:
Graph of 136 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:
Load288(1, i20) → Cond_Load288(i20 >= 0, 1, i20)
Cond_Load288(TRUE, 1, i20) → Load671(1, i20, 1)
Load671(1, i47, i48) → Cond_Load671(i48 > 0 && i47 > i48, 1, i47, i48)
Cond_Load671(TRUE, 1, i47, i48) → Load671(1, i47, 2 * i48)
Load671(1, i47, i48) → Cond_Load6711(i47 >= 0 && i48 > 0 && i47 <= i48, 1, i47, i48)
Cond_Load6711(TRUE, 1, i47, i48) → Load288(1, i47 + -1)
The set Q consists of the following terms:
Load288(1, x0)
Cond_Load288(TRUE, 1, x0)
Load671(1, x0, x1)
Cond_Load671(TRUE, 1, x0, x1)
Cond_Load6711(TRUE, 1, x0, x1)

(5) GroundTermsRemoverProof (EQUIVALENT transformation)

Some arguments are removed because they always contain the same ground term.
We removed the following ground terms:
  • 1

We removed arguments according to the following replacements:

Load288(x1, x2) → Load288(x2)
Cond_Load6711(x1, x2, x3, x4) → Cond_Load6711(x1, x3, x4)
Load671(x1, x2, x3) → Load671(x2, x3)
Cond_Load671(x1, x2, x3, x4) → Cond_Load671(x1, x3, x4)
Cond_Load288(x1, x2, x3) → Cond_Load288(x1, x3)

(6) 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:
Load288(i20) → Cond_Load288(i20 >= 0, i20)
Cond_Load288(TRUE, i20) → Load671(i20, 1)
Load671(i47, i48) → Cond_Load671(i48 > 0 && i47 > i48, i47, i48)
Cond_Load671(TRUE, i47, i48) → Load671(i47, 2 * i48)
Load671(i47, i48) → Cond_Load6711(i47 >= 0 && i48 > 0 && i47 <= i48, i47, i48)
Cond_Load6711(TRUE, i47, i48) → Load288(i47 + -1)
The set Q consists of the following terms:
Load288(x0)
Cond_Load288(TRUE, x0)
Load671(x0, x1)
Cond_Load671(TRUE, x0, x1)
Cond_Load6711(TRUE, x0, x1)

(7) ITRStoIDPProof (EQUIVALENT transformation)

Added dependency pairs

(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


The ITRS R consists of the following rules:
Load288(i20) → Cond_Load288(i20 >= 0, i20)
Cond_Load288(TRUE, i20) → Load671(i20, 1)
Load671(i47, i48) → Cond_Load671(i48 > 0 && i47 > i48, i47, i48)
Cond_Load671(TRUE, i47, i48) → Load671(i47, 2 * i48)
Load671(i47, i48) → Cond_Load6711(i47 >= 0 && i48 > 0 && i47 <= i48, i47, i48)
Cond_Load6711(TRUE, i47, i48) → Load288(i47 + -1)

The integer pair graph contains the following rules and edges:
(0): LOAD288(i20[0]) → COND_LOAD288(i20[0] >= 0, i20[0])
(1): COND_LOAD288(TRUE, i20[1]) → LOAD671(i20[1], 1)
(2): LOAD671(i47[2], i48[2]) → COND_LOAD671(i48[2] > 0 && i47[2] > i48[2], i47[2], i48[2])
(3): COND_LOAD671(TRUE, i47[3], i48[3]) → LOAD671(i47[3], 2 * i48[3])
(4): LOAD671(i47[4], i48[4]) → COND_LOAD6711(i47[4] >= 0 && i48[4] > 0 && i47[4] <= i48[4], i47[4], i48[4])
(5): COND_LOAD6711(TRUE, i47[5], i48[5]) → LOAD288(i47[5] + -1)

(0) -> (1), if ((i20[0] >= 0 →* TRUE)∧(i20[0] →* i20[1]))


(1) -> (2), if ((i20[1] →* i47[2])∧(1 →* i48[2]))


(1) -> (4), if ((1 →* i48[4])∧(i20[1] →* i47[4]))


(2) -> (3), if ((i48[2] →* i48[3])∧(i48[2] > 0 && i47[2] > i48[2] →* TRUE)∧(i47[2] →* i47[3]))


(3) -> (2), if ((i47[3] →* i47[2])∧(2 * i48[3] →* i48[2]))


(3) -> (4), if ((i47[3] →* i47[4])∧(2 * i48[3] →* i48[4]))


(4) -> (5), if ((i47[4] >= 0 && i48[4] > 0 && i47[4] <= i48[4] →* TRUE)∧(i48[4] →* i48[5])∧(i47[4] →* i47[5]))


(5) -> (0), if ((i47[5] + -1 →* i20[0]))



The set Q consists of the following terms:
Load288(x0)
Cond_Load288(TRUE, x0)
Load671(x0, x1)
Cond_Load671(TRUE, x0, x1)
Cond_Load6711(TRUE, x0, x1)

(9) 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.

(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, Boolean


R is empty.

The integer pair graph contains the following rules and edges:
(0): LOAD288(i20[0]) → COND_LOAD288(i20[0] >= 0, i20[0])
(1): COND_LOAD288(TRUE, i20[1]) → LOAD671(i20[1], 1)
(2): LOAD671(i47[2], i48[2]) → COND_LOAD671(i48[2] > 0 && i47[2] > i48[2], i47[2], i48[2])
(3): COND_LOAD671(TRUE, i47[3], i48[3]) → LOAD671(i47[3], 2 * i48[3])
(4): LOAD671(i47[4], i48[4]) → COND_LOAD6711(i47[4] >= 0 && i48[4] > 0 && i47[4] <= i48[4], i47[4], i48[4])
(5): COND_LOAD6711(TRUE, i47[5], i48[5]) → LOAD288(i47[5] + -1)

(0) -> (1), if ((i20[0] >= 0 →* TRUE)∧(i20[0] →* i20[1]))


(1) -> (2), if ((i20[1] →* i47[2])∧(1 →* i48[2]))


(1) -> (4), if ((1 →* i48[4])∧(i20[1] →* i47[4]))


(2) -> (3), if ((i48[2] →* i48[3])∧(i48[2] > 0 && i47[2] > i48[2] →* TRUE)∧(i47[2] →* i47[3]))


(3) -> (2), if ((i47[3] →* i47[2])∧(2 * i48[3] →* i48[2]))


(3) -> (4), if ((i47[3] →* i47[4])∧(2 * i48[3] →* i48[4]))


(4) -> (5), if ((i47[4] >= 0 && i48[4] > 0 && i47[4] <= i48[4] →* TRUE)∧(i48[4] →* i48[5])∧(i47[4] →* i47[5]))


(5) -> (0), if ((i47[5] + -1 →* i20[0]))



The set Q consists of the following terms:
Load288(x0)
Cond_Load288(TRUE, x0)
Load671(x0, x1)
Cond_Load671(TRUE, x0, x1)
Cond_Load6711(TRUE, x0, x1)

(11) 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 LOAD288(i20) → COND_LOAD288(>=(i20, 0), i20) the following chains were created:
  • We consider the chain LOAD288(i20[0]) → COND_LOAD288(>=(i20[0], 0), i20[0]), COND_LOAD288(TRUE, i20[1]) → LOAD671(i20[1], 1) which results in the following constraint:

    (1)    (>=(i20[0], 0)=TRUE∧i20[0]=i20[1] ⇒ LOAD288(i20[0])≥NonInfC∧LOAD288(i20[0])≥COND_LOAD288(>=(i20[0], 0), i20[0])∧(UIncreasing(COND_LOAD288(>=(i20[0], 0), i20[0])), ≥))



    We simplified constraint (1) using rule (IV) which results in the following new constraint:

    (2)    (>=(i20[0], 0)=TRUE ⇒ LOAD288(i20[0])≥NonInfC∧LOAD288(i20[0])≥COND_LOAD288(>=(i20[0], 0), i20[0])∧(UIncreasing(COND_LOAD288(>=(i20[0], 0), i20[0])), ≥))



    We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint:

    (3)    (i20[0] ≥ 0 ⇒ (UIncreasing(COND_LOAD288(>=(i20[0], 0), i20[0])), ≥)∧[bni_23 + (-1)Bound*bni_23] + [(2)bni_23]i20[0] ≥ 0∧[1 + (-1)bso_24] ≥ 0)



    We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:

    (4)    (i20[0] ≥ 0 ⇒ (UIncreasing(COND_LOAD288(>=(i20[0], 0), i20[0])), ≥)∧[bni_23 + (-1)Bound*bni_23] + [(2)bni_23]i20[0] ≥ 0∧[1 + (-1)bso_24] ≥ 0)



    We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:

    (5)    (i20[0] ≥ 0 ⇒ (UIncreasing(COND_LOAD288(>=(i20[0], 0), i20[0])), ≥)∧[bni_23 + (-1)Bound*bni_23] + [(2)bni_23]i20[0] ≥ 0∧[1 + (-1)bso_24] ≥ 0)







For Pair COND_LOAD288(TRUE, i20) → LOAD671(i20, 1) the following chains were created:
  • We consider the chain LOAD288(i20[0]) → COND_LOAD288(>=(i20[0], 0), i20[0]), COND_LOAD288(TRUE, i20[1]) → LOAD671(i20[1], 1), LOAD671(i47[2], i48[2]) → COND_LOAD671(&&(>(i48[2], 0), >(i47[2], i48[2])), i47[2], i48[2]) which results in the following constraint:

    (6)    (>=(i20[0], 0)=TRUE∧i20[0]=i20[1]∧i20[1]=i47[2]∧1=i48[2] ⇒ COND_LOAD288(TRUE, i20[1])≥NonInfC∧COND_LOAD288(TRUE, i20[1])≥LOAD671(i20[1], 1)∧(UIncreasing(LOAD671(i20[1], 1)), ≥))



    We simplified constraint (6) using rules (III), (IV) which results in the following new constraint:

    (7)    (>=(i20[0], 0)=TRUE ⇒ COND_LOAD288(TRUE, i20[0])≥NonInfC∧COND_LOAD288(TRUE, i20[0])≥LOAD671(i20[0], 1)∧(UIncreasing(LOAD671(i20[1], 1)), ≥))



    We simplified constraint (7) using rule (POLY_CONSTRAINTS) which results in the following new constraint:

    (8)    (i20[0] ≥ 0 ⇒ (UIncreasing(LOAD671(i20[1], 1)), ≥)∧[(-1)Bound*bni_25] + [(2)bni_25]i20[0] ≥ 0∧[(-1)bso_26] ≥ 0)



    We simplified constraint (8) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:

    (9)    (i20[0] ≥ 0 ⇒ (UIncreasing(LOAD671(i20[1], 1)), ≥)∧[(-1)Bound*bni_25] + [(2)bni_25]i20[0] ≥ 0∧[(-1)bso_26] ≥ 0)



    We simplified constraint (9) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:

    (10)    (i20[0] ≥ 0 ⇒ (UIncreasing(LOAD671(i20[1], 1)), ≥)∧[(-1)Bound*bni_25] + [(2)bni_25]i20[0] ≥ 0∧[(-1)bso_26] ≥ 0)



  • We consider the chain LOAD288(i20[0]) → COND_LOAD288(>=(i20[0], 0), i20[0]), COND_LOAD288(TRUE, i20[1]) → LOAD671(i20[1], 1), LOAD671(i47[4], i48[4]) → COND_LOAD6711(&&(&&(>=(i47[4], 0), >(i48[4], 0)), <=(i47[4], i48[4])), i47[4], i48[4]) which results in the following constraint:

    (11)    (>=(i20[0], 0)=TRUE∧i20[0]=i20[1]∧1=i48[4]∧i20[1]=i47[4] ⇒ COND_LOAD288(TRUE, i20[1])≥NonInfC∧COND_LOAD288(TRUE, i20[1])≥LOAD671(i20[1], 1)∧(UIncreasing(LOAD671(i20[1], 1)), ≥))



    We simplified constraint (11) using rules (III), (IV) which results in the following new constraint:

    (12)    (>=(i20[0], 0)=TRUE ⇒ COND_LOAD288(TRUE, i20[0])≥NonInfC∧COND_LOAD288(TRUE, i20[0])≥LOAD671(i20[0], 1)∧(UIncreasing(LOAD671(i20[1], 1)), ≥))



    We simplified constraint (12) using rule (POLY_CONSTRAINTS) which results in the following new constraint:

    (13)    (i20[0] ≥ 0 ⇒ (UIncreasing(LOAD671(i20[1], 1)), ≥)∧[(-1)Bound*bni_25] + [(2)bni_25]i20[0] ≥ 0∧[(-1)bso_26] ≥ 0)



    We simplified constraint (13) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:

    (14)    (i20[0] ≥ 0 ⇒ (UIncreasing(LOAD671(i20[1], 1)), ≥)∧[(-1)Bound*bni_25] + [(2)bni_25]i20[0] ≥ 0∧[(-1)bso_26] ≥ 0)



    We simplified constraint (14) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:

    (15)    (i20[0] ≥ 0 ⇒ (UIncreasing(LOAD671(i20[1], 1)), ≥)∧[(-1)Bound*bni_25] + [(2)bni_25]i20[0] ≥ 0∧[(-1)bso_26] ≥ 0)







For Pair LOAD671(i47, i48) → COND_LOAD671(&&(>(i48, 0), >(i47, i48)), i47, i48) the following chains were created:
  • We consider the chain LOAD671(i47[2], i48[2]) → COND_LOAD671(&&(>(i48[2], 0), >(i47[2], i48[2])), i47[2], i48[2]), COND_LOAD671(TRUE, i47[3], i48[3]) → LOAD671(i47[3], *(2, i48[3])) which results in the following constraint:

    (16)    (i48[2]=i48[3]∧&&(>(i48[2], 0), >(i47[2], i48[2]))=TRUE∧i47[2]=i47[3] ⇒ LOAD671(i47[2], i48[2])≥NonInfC∧LOAD671(i47[2], i48[2])≥COND_LOAD671(&&(>(i48[2], 0), >(i47[2], i48[2])), i47[2], i48[2])∧(UIncreasing(COND_LOAD671(&&(>(i48[2], 0), >(i47[2], i48[2])), i47[2], i48[2])), ≥))



    We simplified constraint (16) using rules (IV), (IDP_BOOLEAN) which results in the following new constraint:

    (17)    (>(i48[2], 0)=TRUE∧>(i47[2], i48[2])=TRUE ⇒ LOAD671(i47[2], i48[2])≥NonInfC∧LOAD671(i47[2], i48[2])≥COND_LOAD671(&&(>(i48[2], 0), >(i47[2], i48[2])), i47[2], i48[2])∧(UIncreasing(COND_LOAD671(&&(>(i48[2], 0), >(i47[2], i48[2])), i47[2], i48[2])), ≥))



    We simplified constraint (17) using rule (POLY_CONSTRAINTS) which results in the following new constraint:

    (18)    (i48[2] + [-1] ≥ 0∧i47[2] + [-1] + [-1]i48[2] ≥ 0 ⇒ (UIncreasing(COND_LOAD671(&&(>(i48[2], 0), >(i47[2], i48[2])), i47[2], i48[2])), ≥)∧[(-1)Bound*bni_27] + [(2)bni_27]i47[2] ≥ 0∧[(-1)bso_28] ≥ 0)



    We simplified constraint (18) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:

    (19)    (i48[2] + [-1] ≥ 0∧i47[2] + [-1] + [-1]i48[2] ≥ 0 ⇒ (UIncreasing(COND_LOAD671(&&(>(i48[2], 0), >(i47[2], i48[2])), i47[2], i48[2])), ≥)∧[(-1)Bound*bni_27] + [(2)bni_27]i47[2] ≥ 0∧[(-1)bso_28] ≥ 0)



    We simplified constraint (19) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:

    (20)    (i48[2] + [-1] ≥ 0∧i47[2] + [-1] + [-1]i48[2] ≥ 0 ⇒ (UIncreasing(COND_LOAD671(&&(>(i48[2], 0), >(i47[2], i48[2])), i47[2], i48[2])), ≥)∧[(-1)Bound*bni_27] + [(2)bni_27]i47[2] ≥ 0∧[(-1)bso_28] ≥ 0)



    We simplified constraint (20) using rule (IDP_SMT_SPLIT) which results in the following new constraint:

    (21)    (i48[2] ≥ 0∧i47[2] + [-2] + [-1]i48[2] ≥ 0 ⇒ (UIncreasing(COND_LOAD671(&&(>(i48[2], 0), >(i47[2], i48[2])), i47[2], i48[2])), ≥)∧[(-1)Bound*bni_27] + [(2)bni_27]i47[2] ≥ 0∧[(-1)bso_28] ≥ 0)



    We simplified constraint (21) using rule (IDP_SMT_SPLIT) which results in the following new constraint:

    (22)    (i48[2] ≥ 0∧i47[2] ≥ 0 ⇒ (UIncreasing(COND_LOAD671(&&(>(i48[2], 0), >(i47[2], i48[2])), i47[2], i48[2])), ≥)∧[(-1)Bound*bni_27 + (4)bni_27] + [(2)bni_27]i48[2] + [(2)bni_27]i47[2] ≥ 0∧[(-1)bso_28] ≥ 0)







For Pair COND_LOAD671(TRUE, i47, i48) → LOAD671(i47, *(2, i48)) the following chains were created:
  • We consider the chain LOAD671(i47[2], i48[2]) → COND_LOAD671(&&(>(i48[2], 0), >(i47[2], i48[2])), i47[2], i48[2]), COND_LOAD671(TRUE, i47[3], i48[3]) → LOAD671(i47[3], *(2, i48[3])), LOAD671(i47[2], i48[2]) → COND_LOAD671(&&(>(i48[2], 0), >(i47[2], i48[2])), i47[2], i48[2]) which results in the following constraint:

    (23)    (i48[2]=i48[3]∧&&(>(i48[2], 0), >(i47[2], i48[2]))=TRUE∧i47[2]=i47[3]∧i47[3]=i47[2]1∧*(2, i48[3])=i48[2]1 ⇒ COND_LOAD671(TRUE, i47[3], i48[3])≥NonInfC∧COND_LOAD671(TRUE, i47[3], i48[3])≥LOAD671(i47[3], *(2, i48[3]))∧(UIncreasing(LOAD671(i47[3], *(2, i48[3]))), ≥))



    We simplified constraint (23) using rules (III), (IV), (IDP_BOOLEAN) which results in the following new constraint:

    (24)    (>(i48[2], 0)=TRUE∧>(i47[2], i48[2])=TRUE ⇒ COND_LOAD671(TRUE, i47[2], i48[2])≥NonInfC∧COND_LOAD671(TRUE, i47[2], i48[2])≥LOAD671(i47[2], *(2, i48[2]))∧(UIncreasing(LOAD671(i47[3], *(2, i48[3]))), ≥))



    We simplified constraint (24) using rule (POLY_CONSTRAINTS) which results in the following new constraint:

    (25)    (i48[2] + [-1] ≥ 0∧i47[2] + [-1] + [-1]i48[2] ≥ 0 ⇒ (UIncreasing(LOAD671(i47[3], *(2, i48[3]))), ≥)∧[(-1)Bound*bni_29] + [(2)bni_29]i47[2] ≥ 0∧[(-1)bso_30] ≥ 0)



    We simplified constraint (25) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:

    (26)    (i48[2] + [-1] ≥ 0∧i47[2] + [-1] + [-1]i48[2] ≥ 0 ⇒ (UIncreasing(LOAD671(i47[3], *(2, i48[3]))), ≥)∧[(-1)Bound*bni_29] + [(2)bni_29]i47[2] ≥ 0∧[(-1)bso_30] ≥ 0)



    We simplified constraint (26) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:

    (27)    (i48[2] + [-1] ≥ 0∧i47[2] + [-1] + [-1]i48[2] ≥ 0 ⇒ (UIncreasing(LOAD671(i47[3], *(2, i48[3]))), ≥)∧[(-1)Bound*bni_29] + [(2)bni_29]i47[2] ≥ 0∧[(-1)bso_30] ≥ 0)



    We simplified constraint (27) using rule (IDP_SMT_SPLIT) which results in the following new constraint:

    (28)    (i48[2] ≥ 0∧i47[2] + [-2] + [-1]i48[2] ≥ 0 ⇒ (UIncreasing(LOAD671(i47[3], *(2, i48[3]))), ≥)∧[(-1)Bound*bni_29] + [(2)bni_29]i47[2] ≥ 0∧[(-1)bso_30] ≥ 0)



    We simplified constraint (28) using rule (IDP_SMT_SPLIT) which results in the following new constraint:

    (29)    (i48[2] ≥ 0∧i47[2] ≥ 0 ⇒ (UIncreasing(LOAD671(i47[3], *(2, i48[3]))), ≥)∧[(-1)Bound*bni_29 + (4)bni_29] + [(2)bni_29]i48[2] + [(2)bni_29]i47[2] ≥ 0∧[(-1)bso_30] ≥ 0)



  • We consider the chain LOAD671(i47[2], i48[2]) → COND_LOAD671(&&(>(i48[2], 0), >(i47[2], i48[2])), i47[2], i48[2]), COND_LOAD671(TRUE, i47[3], i48[3]) → LOAD671(i47[3], *(2, i48[3])), LOAD671(i47[4], i48[4]) → COND_LOAD6711(&&(&&(>=(i47[4], 0), >(i48[4], 0)), <=(i47[4], i48[4])), i47[4], i48[4]) which results in the following constraint:

    (30)    (i48[2]=i48[3]∧&&(>(i48[2], 0), >(i47[2], i48[2]))=TRUE∧i47[2]=i47[3]∧i47[3]=i47[4]∧*(2, i48[3])=i48[4] ⇒ COND_LOAD671(TRUE, i47[3], i48[3])≥NonInfC∧COND_LOAD671(TRUE, i47[3], i48[3])≥LOAD671(i47[3], *(2, i48[3]))∧(UIncreasing(LOAD671(i47[3], *(2, i48[3]))), ≥))



    We simplified constraint (30) using rules (III), (IV), (IDP_BOOLEAN) which results in the following new constraint:

    (31)    (>(i48[2], 0)=TRUE∧>(i47[2], i48[2])=TRUE ⇒ COND_LOAD671(TRUE, i47[2], i48[2])≥NonInfC∧COND_LOAD671(TRUE, i47[2], i48[2])≥LOAD671(i47[2], *(2, i48[2]))∧(UIncreasing(LOAD671(i47[3], *(2, i48[3]))), ≥))



    We simplified constraint (31) using rule (POLY_CONSTRAINTS) which results in the following new constraint:

    (32)    (i48[2] + [-1] ≥ 0∧i47[2] + [-1] + [-1]i48[2] ≥ 0 ⇒ (UIncreasing(LOAD671(i47[3], *(2, i48[3]))), ≥)∧[(-1)Bound*bni_29] + [(2)bni_29]i47[2] ≥ 0∧[(-1)bso_30] ≥ 0)



    We simplified constraint (32) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:

    (33)    (i48[2] + [-1] ≥ 0∧i47[2] + [-1] + [-1]i48[2] ≥ 0 ⇒ (UIncreasing(LOAD671(i47[3], *(2, i48[3]))), ≥)∧[(-1)Bound*bni_29] + [(2)bni_29]i47[2] ≥ 0∧[(-1)bso_30] ≥ 0)



    We simplified constraint (33) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:

    (34)    (i48[2] + [-1] ≥ 0∧i47[2] + [-1] + [-1]i48[2] ≥ 0 ⇒ (UIncreasing(LOAD671(i47[3], *(2, i48[3]))), ≥)∧[(-1)Bound*bni_29] + [(2)bni_29]i47[2] ≥ 0∧[(-1)bso_30] ≥ 0)



    We simplified constraint (34) using rule (IDP_SMT_SPLIT) which results in the following new constraint:

    (35)    (i48[2] ≥ 0∧i47[2] + [-2] + [-1]i48[2] ≥ 0 ⇒ (UIncreasing(LOAD671(i47[3], *(2, i48[3]))), ≥)∧[(-1)Bound*bni_29] + [(2)bni_29]i47[2] ≥ 0∧[(-1)bso_30] ≥ 0)



    We simplified constraint (35) using rule (IDP_SMT_SPLIT) which results in the following new constraint:

    (36)    (i48[2] ≥ 0∧i47[2] ≥ 0 ⇒ (UIncreasing(LOAD671(i47[3], *(2, i48[3]))), ≥)∧[(-1)Bound*bni_29 + (4)bni_29] + [(2)bni_29]i48[2] + [(2)bni_29]i47[2] ≥ 0∧[(-1)bso_30] ≥ 0)







For Pair LOAD671(i47, i48) → COND_LOAD6711(&&(&&(>=(i47, 0), >(i48, 0)), <=(i47, i48)), i47, i48) the following chains were created:
  • We consider the chain LOAD671(i47[4], i48[4]) → COND_LOAD6711(&&(&&(>=(i47[4], 0), >(i48[4], 0)), <=(i47[4], i48[4])), i47[4], i48[4]), COND_LOAD6711(TRUE, i47[5], i48[5]) → LOAD288(+(i47[5], -1)) which results in the following constraint:

    (37)    (&&(&&(>=(i47[4], 0), >(i48[4], 0)), <=(i47[4], i48[4]))=TRUE∧i48[4]=i48[5]∧i47[4]=i47[5] ⇒ LOAD671(i47[4], i48[4])≥NonInfC∧LOAD671(i47[4], i48[4])≥COND_LOAD6711(&&(&&(>=(i47[4], 0), >(i48[4], 0)), <=(i47[4], i48[4])), i47[4], i48[4])∧(UIncreasing(COND_LOAD6711(&&(&&(>=(i47[4], 0), >(i48[4], 0)), <=(i47[4], i48[4])), i47[4], i48[4])), ≥))



    We simplified constraint (37) using rules (IV), (IDP_BOOLEAN) which results in the following new constraint:

    (38)    (<=(i47[4], i48[4])=TRUE∧>=(i47[4], 0)=TRUE∧>(i48[4], 0)=TRUE ⇒ LOAD671(i47[4], i48[4])≥NonInfC∧LOAD671(i47[4], i48[4])≥COND_LOAD6711(&&(&&(>=(i47[4], 0), >(i48[4], 0)), <=(i47[4], i48[4])), i47[4], i48[4])∧(UIncreasing(COND_LOAD6711(&&(&&(>=(i47[4], 0), >(i48[4], 0)), <=(i47[4], i48[4])), i47[4], i48[4])), ≥))



    We simplified constraint (38) using rule (POLY_CONSTRAINTS) which results in the following new constraint:

    (39)    (i48[4] + [-1]i47[4] ≥ 0∧i47[4] ≥ 0∧i48[4] + [-1] ≥ 0 ⇒ (UIncreasing(COND_LOAD6711(&&(&&(>=(i47[4], 0), >(i48[4], 0)), <=(i47[4], i48[4])), i47[4], i48[4])), ≥)∧[(-1)Bound*bni_31] + [(2)bni_31]i47[4] ≥ 0∧[(-1)bso_32] ≥ 0)



    We simplified constraint (39) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:

    (40)    (i48[4] + [-1]i47[4] ≥ 0∧i47[4] ≥ 0∧i48[4] + [-1] ≥ 0 ⇒ (UIncreasing(COND_LOAD6711(&&(&&(>=(i47[4], 0), >(i48[4], 0)), <=(i47[4], i48[4])), i47[4], i48[4])), ≥)∧[(-1)Bound*bni_31] + [(2)bni_31]i47[4] ≥ 0∧[(-1)bso_32] ≥ 0)



    We simplified constraint (40) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:

    (41)    (i48[4] + [-1]i47[4] ≥ 0∧i47[4] ≥ 0∧i48[4] + [-1] ≥ 0 ⇒ (UIncreasing(COND_LOAD6711(&&(&&(>=(i47[4], 0), >(i48[4], 0)), <=(i47[4], i48[4])), i47[4], i48[4])), ≥)∧[(-1)Bound*bni_31] + [(2)bni_31]i47[4] ≥ 0∧[(-1)bso_32] ≥ 0)



    We simplified constraint (41) using rule (IDP_SMT_SPLIT) which results in the following new constraint:

    (42)    (i48[4] ≥ 0∧i47[4] ≥ 0∧i47[4] + [-1] + i48[4] ≥ 0 ⇒ (UIncreasing(COND_LOAD6711(&&(&&(>=(i47[4], 0), >(i48[4], 0)), <=(i47[4], i48[4])), i47[4], i48[4])), ≥)∧[(-1)Bound*bni_31] + [(2)bni_31]i47[4] ≥ 0∧[(-1)bso_32] ≥ 0)







For Pair COND_LOAD6711(TRUE, i47, i48) → LOAD288(+(i47, -1)) the following chains were created:
  • We consider the chain LOAD671(i47[4], i48[4]) → COND_LOAD6711(&&(&&(>=(i47[4], 0), >(i48[4], 0)), <=(i47[4], i48[4])), i47[4], i48[4]), COND_LOAD6711(TRUE, i47[5], i48[5]) → LOAD288(+(i47[5], -1)), LOAD288(i20[0]) → COND_LOAD288(>=(i20[0], 0), i20[0]) which results in the following constraint:

    (43)    (&&(&&(>=(i47[4], 0), >(i48[4], 0)), <=(i47[4], i48[4]))=TRUE∧i48[4]=i48[5]∧i47[4]=i47[5]∧+(i47[5], -1)=i20[0] ⇒ COND_LOAD6711(TRUE, i47[5], i48[5])≥NonInfC∧COND_LOAD6711(TRUE, i47[5], i48[5])≥LOAD288(+(i47[5], -1))∧(UIncreasing(LOAD288(+(i47[5], -1))), ≥))



    We simplified constraint (43) using rules (III), (IV), (IDP_BOOLEAN) which results in the following new constraint:

    (44)    (<=(i47[4], i48[4])=TRUE∧>=(i47[4], 0)=TRUE∧>(i48[4], 0)=TRUE ⇒ COND_LOAD6711(TRUE, i47[4], i48[4])≥NonInfC∧COND_LOAD6711(TRUE, i47[4], i48[4])≥LOAD288(+(i47[4], -1))∧(UIncreasing(LOAD288(+(i47[5], -1))), ≥))



    We simplified constraint (44) using rule (POLY_CONSTRAINTS) which results in the following new constraint:

    (45)    (i48[4] + [-1]i47[4] ≥ 0∧i47[4] ≥ 0∧i48[4] + [-1] ≥ 0 ⇒ (UIncreasing(LOAD288(+(i47[5], -1))), ≥)∧[(-1)bni_33 + (-1)Bound*bni_33] + [(2)bni_33]i47[4] ≥ 0∧[(-1)bso_34] ≥ 0)



    We simplified constraint (45) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:

    (46)    (i48[4] + [-1]i47[4] ≥ 0∧i47[4] ≥ 0∧i48[4] + [-1] ≥ 0 ⇒ (UIncreasing(LOAD288(+(i47[5], -1))), ≥)∧[(-1)bni_33 + (-1)Bound*bni_33] + [(2)bni_33]i47[4] ≥ 0∧[(-1)bso_34] ≥ 0)



    We simplified constraint (46) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:

    (47)    (i48[4] + [-1]i47[4] ≥ 0∧i47[4] ≥ 0∧i48[4] + [-1] ≥ 0 ⇒ (UIncreasing(LOAD288(+(i47[5], -1))), ≥)∧[(-1)bni_33 + (-1)Bound*bni_33] + [(2)bni_33]i47[4] ≥ 0∧[(-1)bso_34] ≥ 0)



    We simplified constraint (47) using rule (IDP_SMT_SPLIT) which results in the following new constraint:

    (48)    (i48[4] ≥ 0∧i47[4] ≥ 0∧i47[4] + [-1] + i48[4] ≥ 0 ⇒ (UIncreasing(LOAD288(+(i47[5], -1))), ≥)∧[(-1)bni_33 + (-1)Bound*bni_33] + [(2)bni_33]i47[4] ≥ 0∧[(-1)bso_34] ≥ 0)







To summarize, we get the following constraints P≥ for the following pairs.
  • LOAD288(i20) → COND_LOAD288(>=(i20, 0), i20)
    • (i20[0] ≥ 0 ⇒ (UIncreasing(COND_LOAD288(>=(i20[0], 0), i20[0])), ≥)∧[bni_23 + (-1)Bound*bni_23] + [(2)bni_23]i20[0] ≥ 0∧[1 + (-1)bso_24] ≥ 0)

  • COND_LOAD288(TRUE, i20) → LOAD671(i20, 1)
    • (i20[0] ≥ 0 ⇒ (UIncreasing(LOAD671(i20[1], 1)), ≥)∧[(-1)Bound*bni_25] + [(2)bni_25]i20[0] ≥ 0∧[(-1)bso_26] ≥ 0)
    • (i20[0] ≥ 0 ⇒ (UIncreasing(LOAD671(i20[1], 1)), ≥)∧[(-1)Bound*bni_25] + [(2)bni_25]i20[0] ≥ 0∧[(-1)bso_26] ≥ 0)

  • LOAD671(i47, i48) → COND_LOAD671(&&(>(i48, 0), >(i47, i48)), i47, i48)
    • (i48[2] ≥ 0∧i47[2] ≥ 0 ⇒ (UIncreasing(COND_LOAD671(&&(>(i48[2], 0), >(i47[2], i48[2])), i47[2], i48[2])), ≥)∧[(-1)Bound*bni_27 + (4)bni_27] + [(2)bni_27]i48[2] + [(2)bni_27]i47[2] ≥ 0∧[(-1)bso_28] ≥ 0)

  • COND_LOAD671(TRUE, i47, i48) → LOAD671(i47, *(2, i48))
    • (i48[2] ≥ 0∧i47[2] ≥ 0 ⇒ (UIncreasing(LOAD671(i47[3], *(2, i48[3]))), ≥)∧[(-1)Bound*bni_29 + (4)bni_29] + [(2)bni_29]i48[2] + [(2)bni_29]i47[2] ≥ 0∧[(-1)bso_30] ≥ 0)
    • (i48[2] ≥ 0∧i47[2] ≥ 0 ⇒ (UIncreasing(LOAD671(i47[3], *(2, i48[3]))), ≥)∧[(-1)Bound*bni_29 + (4)bni_29] + [(2)bni_29]i48[2] + [(2)bni_29]i47[2] ≥ 0∧[(-1)bso_30] ≥ 0)

  • LOAD671(i47, i48) → COND_LOAD6711(&&(&&(>=(i47, 0), >(i48, 0)), <=(i47, i48)), i47, i48)
    • (i48[4] ≥ 0∧i47[4] ≥ 0∧i47[4] + [-1] + i48[4] ≥ 0 ⇒ (UIncreasing(COND_LOAD6711(&&(&&(>=(i47[4], 0), >(i48[4], 0)), <=(i47[4], i48[4])), i47[4], i48[4])), ≥)∧[(-1)Bound*bni_31] + [(2)bni_31]i47[4] ≥ 0∧[(-1)bso_32] ≥ 0)

  • COND_LOAD6711(TRUE, i47, i48) → LOAD288(+(i47, -1))
    • (i48[4] ≥ 0∧i47[4] ≥ 0∧i47[4] + [-1] + i48[4] ≥ 0 ⇒ (UIncreasing(LOAD288(+(i47[5], -1))), ≥)∧[(-1)bni_33 + (-1)Bound*bni_33] + [(2)bni_33]i47[4] ≥ 0∧[(-1)bso_34] ≥ 0)




The constraints for P> respective Pbound 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 Pbound.
Using the following integer polynomial ordering the resulting constraints can be solved
Polynomial interpretation over integers[POLO]:

POL(TRUE) = 0   
POL(FALSE) = [3]   
POL(LOAD288(x1)) = [1] + [2]x1   
POL(COND_LOAD288(x1, x2)) = [2]x2   
POL(>=(x1, x2)) = [-1]   
POL(0) = 0   
POL(LOAD671(x1, x2)) = [2]x1   
POL(1) = [1]   
POL(COND_LOAD671(x1, x2, x3)) = [2]x2   
POL(&&(x1, x2)) = [-1]   
POL(>(x1, x2)) = [-1]   
POL(*(x1, x2)) = x1·x2   
POL(2) = [2]   
POL(COND_LOAD6711(x1, x2, x3)) = [-1] + [2]x2 + [-1]x1   
POL(<=(x1, x2)) = [-1]   
POL(+(x1, x2)) = x1 + x2   
POL(-1) = [-1]   

The following pairs are in P>:

LOAD288(i20[0]) → COND_LOAD288(>=(i20[0], 0), i20[0])

The following pairs are in Pbound:

LOAD288(i20[0]) → COND_LOAD288(>=(i20[0], 0), i20[0])
COND_LOAD288(TRUE, i20[1]) → LOAD671(i20[1], 1)
LOAD671(i47[2], i48[2]) → COND_LOAD671(&&(>(i48[2], 0), >(i47[2], i48[2])), i47[2], i48[2])
COND_LOAD671(TRUE, i47[3], i48[3]) → LOAD671(i47[3], *(2, i48[3]))
LOAD671(i47[4], i48[4]) → COND_LOAD6711(&&(&&(>=(i47[4], 0), >(i48[4], 0)), <=(i47[4], i48[4])), i47[4], i48[4])
COND_LOAD6711(TRUE, i47[5], i48[5]) → LOAD288(+(i47[5], -1))

The following pairs are in P≥:

COND_LOAD288(TRUE, i20[1]) → LOAD671(i20[1], 1)
LOAD671(i47[2], i48[2]) → COND_LOAD671(&&(>(i48[2], 0), >(i47[2], i48[2])), i47[2], i48[2])
COND_LOAD671(TRUE, i47[3], i48[3]) → LOAD671(i47[3], *(2, i48[3]))
LOAD671(i47[4], i48[4]) → COND_LOAD6711(&&(&&(>=(i47[4], 0), >(i48[4], 0)), <=(i47[4], i48[4])), i47[4], i48[4])
COND_LOAD6711(TRUE, i47[5], i48[5]) → LOAD288(+(i47[5], -1))

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

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

(12) Complex Obligation (AND)

(13) 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:
(1): COND_LOAD288(TRUE, i20[1]) → LOAD671(i20[1], 1)
(2): LOAD671(i47[2], i48[2]) → COND_LOAD671(i48[2] > 0 && i47[2] > i48[2], i47[2], i48[2])
(3): COND_LOAD671(TRUE, i47[3], i48[3]) → LOAD671(i47[3], 2 * i48[3])
(4): LOAD671(i47[4], i48[4]) → COND_LOAD6711(i47[4] >= 0 && i48[4] > 0 && i47[4] <= i48[4], i47[4], i48[4])
(5): COND_LOAD6711(TRUE, i47[5], i48[5]) → LOAD288(i47[5] + -1)

(1) -> (2), if ((i20[1] →* i47[2])∧(1 →* i48[2]))


(3) -> (2), if ((i47[3] →* i47[2])∧(2 * i48[3] →* i48[2]))


(2) -> (3), if ((i48[2] →* i48[3])∧(i48[2] > 0 && i47[2] > i48[2] →* TRUE)∧(i47[2] →* i47[3]))


(1) -> (4), if ((1 →* i48[4])∧(i20[1] →* i47[4]))


(3) -> (4), if ((i47[3] →* i47[4])∧(2 * i48[3] →* i48[4]))


(4) -> (5), if ((i47[4] >= 0 && i48[4] > 0 && i47[4] <= i48[4] →* TRUE)∧(i48[4] →* i48[5])∧(i47[4] →* i47[5]))



The set Q consists of the following terms:
Load288(x0)
Cond_Load288(TRUE, x0)
Load671(x0, x1)
Cond_Load671(TRUE, x0, x1)
Cond_Load6711(TRUE, x0, x1)

(14) IDependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 3 less nodes.

(15) 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_LOAD671(TRUE, i47[3], i48[3]) → LOAD671(i47[3], 2 * i48[3])
(2): LOAD671(i47[2], i48[2]) → COND_LOAD671(i48[2] > 0 && i47[2] > i48[2], i47[2], i48[2])

(3) -> (2), if ((i47[3] →* i47[2])∧(2 * i48[3] →* i48[2]))


(2) -> (3), if ((i48[2] →* i48[3])∧(i48[2] > 0 && i47[2] > i48[2] →* TRUE)∧(i47[2] →* i47[3]))



The set Q consists of the following terms:
Load288(x0)
Cond_Load288(TRUE, x0)
Load671(x0, x1)
Cond_Load671(TRUE, x0, x1)
Cond_Load6711(TRUE, x0, x1)

(16) 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_LOAD671(TRUE, i47[3], i48[3]) → LOAD671(i47[3], *(2, i48[3])) the following chains were created:
  • We consider the chain LOAD671(i47[2], i48[2]) → COND_LOAD671(&&(>(i48[2], 0), >(i47[2], i48[2])), i47[2], i48[2]), COND_LOAD671(TRUE, i47[3], i48[3]) → LOAD671(i47[3], *(2, i48[3])), LOAD671(i47[2], i48[2]) → COND_LOAD671(&&(>(i48[2], 0), >(i47[2], i48[2])), i47[2], i48[2]) which results in the following constraint:

    (1)    (i48[2]=i48[3]∧&&(>(i48[2], 0), >(i47[2], i48[2]))=TRUE∧i47[2]=i47[3]∧i47[3]=i47[2]1∧*(2, i48[3])=i48[2]1 ⇒ COND_LOAD671(TRUE, i47[3], i48[3])≥NonInfC∧COND_LOAD671(TRUE, i47[3], i48[3])≥LOAD671(i47[3], *(2, i48[3]))∧(UIncreasing(LOAD671(i47[3], *(2, i48[3]))), ≥))



    We simplified constraint (1) using rules (III), (IV), (IDP_BOOLEAN) which results in the following new constraint:

    (2)    (>(i48[2], 0)=TRUE∧>(i47[2], i48[2])=TRUE ⇒ COND_LOAD671(TRUE, i47[2], i48[2])≥NonInfC∧COND_LOAD671(TRUE, i47[2], i48[2])≥LOAD671(i47[2], *(2, i48[2]))∧(UIncreasing(LOAD671(i47[3], *(2, i48[3]))), ≥))



    We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint:

    (3)    (i48[2] + [-1] ≥ 0∧i47[2] + [-1] + [-1]i48[2] ≥ 0 ⇒ (UIncreasing(LOAD671(i47[3], *(2, i48[3]))), ≥)∧[(-1)bni_12 + (-1)Bound*bni_12] + [(-1)bni_12]i48[2] + [bni_12]i47[2] ≥ 0∧[(-1)bso_13] + i48[2] ≥ 0)



    We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:

    (4)    (i48[2] + [-1] ≥ 0∧i47[2] + [-1] + [-1]i48[2] ≥ 0 ⇒ (UIncreasing(LOAD671(i47[3], *(2, i48[3]))), ≥)∧[(-1)bni_12 + (-1)Bound*bni_12] + [(-1)bni_12]i48[2] + [bni_12]i47[2] ≥ 0∧[(-1)bso_13] + i48[2] ≥ 0)



    We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:

    (5)    (i48[2] + [-1] ≥ 0∧i47[2] + [-1] + [-1]i48[2] ≥ 0 ⇒ (UIncreasing(LOAD671(i47[3], *(2, i48[3]))), ≥)∧[(-1)bni_12 + (-1)Bound*bni_12] + [(-1)bni_12]i48[2] + [bni_12]i47[2] ≥ 0∧[(-1)bso_13] + i48[2] ≥ 0)



    We simplified constraint (5) using rule (IDP_SMT_SPLIT) which results in the following new constraint:

    (6)    (i48[2] ≥ 0∧i47[2] + [-2] + [-1]i48[2] ≥ 0 ⇒ (UIncreasing(LOAD671(i47[3], *(2, i48[3]))), ≥)∧[(-2)bni_12 + (-1)Bound*bni_12] + [(-1)bni_12]i48[2] + [bni_12]i47[2] ≥ 0∧[1 + (-1)bso_13] + i48[2] ≥ 0)



    We simplified constraint (6) using rule (IDP_SMT_SPLIT) which results in the following new constraint:

    (7)    (i48[2] ≥ 0∧i47[2] ≥ 0 ⇒ (UIncreasing(LOAD671(i47[3], *(2, i48[3]))), ≥)∧[(-1)Bound*bni_12] + [bni_12]i47[2] ≥ 0∧[1 + (-1)bso_13] + i48[2] ≥ 0)







For Pair LOAD671(i47[2], i48[2]) → COND_LOAD671(&&(>(i48[2], 0), >(i47[2], i48[2])), i47[2], i48[2]) the following chains were created:
  • We consider the chain LOAD671(i47[2], i48[2]) → COND_LOAD671(&&(>(i48[2], 0), >(i47[2], i48[2])), i47[2], i48[2]), COND_LOAD671(TRUE, i47[3], i48[3]) → LOAD671(i47[3], *(2, i48[3])) which results in the following constraint:

    (8)    (i48[2]=i48[3]∧&&(>(i48[2], 0), >(i47[2], i48[2]))=TRUE∧i47[2]=i47[3] ⇒ LOAD671(i47[2], i48[2])≥NonInfC∧LOAD671(i47[2], i48[2])≥COND_LOAD671(&&(>(i48[2], 0), >(i47[2], i48[2])), i47[2], i48[2])∧(UIncreasing(COND_LOAD671(&&(>(i48[2], 0), >(i47[2], i48[2])), i47[2], i48[2])), ≥))



    We simplified constraint (8) using rules (IV), (IDP_BOOLEAN) which results in the following new constraint:

    (9)    (>(i48[2], 0)=TRUE∧>(i47[2], i48[2])=TRUE ⇒ LOAD671(i47[2], i48[2])≥NonInfC∧LOAD671(i47[2], i48[2])≥COND_LOAD671(&&(>(i48[2], 0), >(i47[2], i48[2])), i47[2], i48[2])∧(UIncreasing(COND_LOAD671(&&(>(i48[2], 0), >(i47[2], i48[2])), i47[2], i48[2])), ≥))



    We simplified constraint (9) using rule (POLY_CONSTRAINTS) which results in the following new constraint:

    (10)    (i48[2] + [-1] ≥ 0∧i47[2] + [-1] + [-1]i48[2] ≥ 0 ⇒ (UIncreasing(COND_LOAD671(&&(>(i48[2], 0), >(i47[2], i48[2])), i47[2], i48[2])), ≥)∧[(-1)bni_14 + (-1)Bound*bni_14] + [(-1)bni_14]i48[2] + [bni_14]i47[2] ≥ 0∧[(-1)bso_15] ≥ 0)



    We simplified constraint (10) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:

    (11)    (i48[2] + [-1] ≥ 0∧i47[2] + [-1] + [-1]i48[2] ≥ 0 ⇒ (UIncreasing(COND_LOAD671(&&(>(i48[2], 0), >(i47[2], i48[2])), i47[2], i48[2])), ≥)∧[(-1)bni_14 + (-1)Bound*bni_14] + [(-1)bni_14]i48[2] + [bni_14]i47[2] ≥ 0∧[(-1)bso_15] ≥ 0)



    We simplified constraint (11) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:

    (12)    (i48[2] + [-1] ≥ 0∧i47[2] + [-1] + [-1]i48[2] ≥ 0 ⇒ (UIncreasing(COND_LOAD671(&&(>(i48[2], 0), >(i47[2], i48[2])), i47[2], i48[2])), ≥)∧[(-1)bni_14 + (-1)Bound*bni_14] + [(-1)bni_14]i48[2] + [bni_14]i47[2] ≥ 0∧[(-1)bso_15] ≥ 0)



    We simplified constraint (12) using rule (IDP_SMT_SPLIT) which results in the following new constraint:

    (13)    (i48[2] ≥ 0∧i47[2] + [-2] + [-1]i48[2] ≥ 0 ⇒ (UIncreasing(COND_LOAD671(&&(>(i48[2], 0), >(i47[2], i48[2])), i47[2], i48[2])), ≥)∧[(-2)bni_14 + (-1)Bound*bni_14] + [(-1)bni_14]i48[2] + [bni_14]i47[2] ≥ 0∧[(-1)bso_15] ≥ 0)



    We simplified constraint (13) using rule (IDP_SMT_SPLIT) which results in the following new constraint:

    (14)    (i48[2] ≥ 0∧i47[2] ≥ 0 ⇒ (UIncreasing(COND_LOAD671(&&(>(i48[2], 0), >(i47[2], i48[2])), i47[2], i48[2])), ≥)∧[(-1)Bound*bni_14] + [bni_14]i47[2] ≥ 0∧[(-1)bso_15] ≥ 0)







To summarize, we get the following constraints P≥ for the following pairs.
  • COND_LOAD671(TRUE, i47[3], i48[3]) → LOAD671(i47[3], *(2, i48[3]))
    • (i48[2] ≥ 0∧i47[2] ≥ 0 ⇒ (UIncreasing(LOAD671(i47[3], *(2, i48[3]))), ≥)∧[(-1)Bound*bni_12] + [bni_12]i47[2] ≥ 0∧[1 + (-1)bso_13] + i48[2] ≥ 0)

  • LOAD671(i47[2], i48[2]) → COND_LOAD671(&&(>(i48[2], 0), >(i47[2], i48[2])), i47[2], i48[2])
    • (i48[2] ≥ 0∧i47[2] ≥ 0 ⇒ (UIncreasing(COND_LOAD671(&&(>(i48[2], 0), >(i47[2], i48[2])), i47[2], i48[2])), ≥)∧[(-1)Bound*bni_14] + [bni_14]i47[2] ≥ 0∧[(-1)bso_15] ≥ 0)




The constraints for P> respective Pbound 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 Pbound.
Using the following integer polynomial ordering the resulting constraints can be solved
Polynomial interpretation over integers[POLO]:

POL(TRUE) = 0   
POL(FALSE) = [1]   
POL(COND_LOAD671(x1, x2, x3)) = [-1] + [-1]x3 + x2   
POL(LOAD671(x1, x2)) = [-1] + [-1]x2 + x1   
POL(*(x1, x2)) = x1·x2   
POL(2) = [2]   
POL(&&(x1, x2)) = [-1]   
POL(>(x1, x2)) = [-1]   
POL(0) = 0   

The following pairs are in P>:

COND_LOAD671(TRUE, i47[3], i48[3]) → LOAD671(i47[3], *(2, i48[3]))

The following pairs are in Pbound:

COND_LOAD671(TRUE, i47[3], i48[3]) → LOAD671(i47[3], *(2, i48[3]))
LOAD671(i47[2], i48[2]) → COND_LOAD671(&&(>(i48[2], 0), >(i47[2], i48[2])), i47[2], i48[2])

The following pairs are in P≥:

LOAD671(i47[2], i48[2]) → COND_LOAD671(&&(>(i48[2], 0), >(i47[2], i48[2])), i47[2], i48[2])

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

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

(17) Complex Obligation (AND)

(18) 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): LOAD671(i47[2], i48[2]) → COND_LOAD671(i48[2] > 0 && i47[2] > i48[2], i47[2], i48[2])


The set Q consists of the following terms:
Load288(x0)
Cond_Load288(TRUE, x0)
Load671(x0, x1)
Cond_Load671(TRUE, x0, x1)
Cond_Load6711(TRUE, x0, x1)

(19) IDependencyGraphProof (EQUIVALENT transformation)

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

(20) TRUE

(21) 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:
Load288(x0)
Cond_Load288(TRUE, x0)
Load671(x0, x1)
Cond_Load671(TRUE, x0, x1)
Cond_Load6711(TRUE, x0, x1)

(22) IDependencyGraphProof (EQUIVALENT transformation)

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

(23) TRUE

(24) 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:
Load288(x0)
Cond_Load288(TRUE, x0)
Load671(x0, x1)
Cond_Load671(TRUE, x0, x1)
Cond_Load6711(TRUE, x0, x1)

(25) IDependencyGraphProof (EQUIVALENT transformation)

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

(26) TRUE