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: PastaC2 
/**
 * 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 PastaC2 {
    public static void main(String[] args) {
        Random.args = args;
        int x = Random.random();

        while (x >= 0) {
            x = x+1;
            int y = 1;
            while (x >= y) {
                y++;
            }
            x = x-2;
        }
    }
}


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:
Load291(1, i21) → Cond_Load291(i21 >= 0 && i21 + 1 > 0, 1, i21)
Cond_Load291(TRUE, 1, i21) → Load680(1, i21 + 1, 1)
Load680(1, i33, i49) → Cond_Load680(i49 > 0 && i33 >= i49, 1, i33, i49)
Cond_Load680(TRUE, 1, i33, i49) → Load680(1, i33, i49 + 1)
Load680(1, i33, i49) → Cond_Load6801(i33 > 0 && i33 < i49, 1, i33, i49)
Cond_Load6801(TRUE, 1, i33, i49) → Load291(1, i33 - 2)
The set Q consists of the following terms:
Load291(1, x0)
Cond_Load291(TRUE, 1, x0)
Load680(1, x0, x1)
Cond_Load680(TRUE, 1, x0, x1)
Cond_Load6801(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:

Load291(x1, x2) → Load291(x2)
Cond_Load6801(x1, x2, x3, x4) → Cond_Load6801(x1, x3, x4)
Load680(x1, x2, x3) → Load680(x2, x3)
Cond_Load680(x1, x2, x3, x4) → Cond_Load680(x1, x3, x4)
Cond_Load291(x1, x2, x3) → Cond_Load291(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:
Load291(i21) → Cond_Load291(i21 >= 0 && i21 + 1 > 0, i21)
Cond_Load291(TRUE, i21) → Load680(i21 + 1, 1)
Load680(i33, i49) → Cond_Load680(i49 > 0 && i33 >= i49, i33, i49)
Cond_Load680(TRUE, i33, i49) → Load680(i33, i49 + 1)
Load680(i33, i49) → Cond_Load6801(i33 > 0 && i33 < i49, i33, i49)
Cond_Load6801(TRUE, i33, i49) → Load291(i33 - 2)
The set Q consists of the following terms:
Load291(x0)
Cond_Load291(TRUE, x0)
Load680(x0, x1)
Cond_Load680(TRUE, x0, x1)
Cond_Load6801(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:

Boolean, Integer


The ITRS R consists of the following rules:
Load291(i21) → Cond_Load291(i21 >= 0 && i21 + 1 > 0, i21)
Cond_Load291(TRUE, i21) → Load680(i21 + 1, 1)
Load680(i33, i49) → Cond_Load680(i49 > 0 && i33 >= i49, i33, i49)
Cond_Load680(TRUE, i33, i49) → Load680(i33, i49 + 1)
Load680(i33, i49) → Cond_Load6801(i33 > 0 && i33 < i49, i33, i49)
Cond_Load6801(TRUE, i33, i49) → Load291(i33 - 2)

The integer pair graph contains the following rules and edges:
(0): LOAD291(i21[0]) → COND_LOAD291(i21[0] >= 0 && i21[0] + 1 > 0, i21[0])
(1): COND_LOAD291(TRUE, i21[1]) → LOAD680(i21[1] + 1, 1)
(2): LOAD680(i33[2], i49[2]) → COND_LOAD680(i49[2] > 0 && i33[2] >= i49[2], i33[2], i49[2])
(3): COND_LOAD680(TRUE, i33[3], i49[3]) → LOAD680(i33[3], i49[3] + 1)
(4): LOAD680(i33[4], i49[4]) → COND_LOAD6801(i33[4] > 0 && i33[4] < i49[4], i33[4], i49[4])
(5): COND_LOAD6801(TRUE, i33[5], i49[5]) → LOAD291(i33[5] - 2)

(0) -> (1), if ((i21[0] >= 0 && i21[0] + 1 > 0 →* TRUE)∧(i21[0] →* i21[1]))


(1) -> (2), if ((1 →* i49[2])∧(i21[1] + 1 →* i33[2]))


(1) -> (4), if ((1 →* i49[4])∧(i21[1] + 1 →* i33[4]))


(2) -> (3), if ((i49[2] →* i49[3])∧(i33[2] →* i33[3])∧(i49[2] > 0 && i33[2] >= i49[2] →* TRUE))


(3) -> (2), if ((i49[3] + 1 →* i49[2])∧(i33[3] →* i33[2]))


(3) -> (4), if ((i49[3] + 1 →* i49[4])∧(i33[3] →* i33[4]))


(4) -> (5), if ((i33[4] > 0 && i33[4] < i49[4] →* TRUE)∧(i49[4] →* i49[5])∧(i33[4] →* i33[5]))


(5) -> (0), if ((i33[5] - 2 →* i21[0]))



The set Q consists of the following terms:
Load291(x0)
Cond_Load291(TRUE, x0)
Load680(x0, x1)
Cond_Load680(TRUE, x0, x1)
Cond_Load6801(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:

Boolean, Integer


R is empty.

The integer pair graph contains the following rules and edges:
(0): LOAD291(i21[0]) → COND_LOAD291(i21[0] >= 0 && i21[0] + 1 > 0, i21[0])
(1): COND_LOAD291(TRUE, i21[1]) → LOAD680(i21[1] + 1, 1)
(2): LOAD680(i33[2], i49[2]) → COND_LOAD680(i49[2] > 0 && i33[2] >= i49[2], i33[2], i49[2])
(3): COND_LOAD680(TRUE, i33[3], i49[3]) → LOAD680(i33[3], i49[3] + 1)
(4): LOAD680(i33[4], i49[4]) → COND_LOAD6801(i33[4] > 0 && i33[4] < i49[4], i33[4], i49[4])
(5): COND_LOAD6801(TRUE, i33[5], i49[5]) → LOAD291(i33[5] - 2)

(0) -> (1), if ((i21[0] >= 0 && i21[0] + 1 > 0 →* TRUE)∧(i21[0] →* i21[1]))


(1) -> (2), if ((1 →* i49[2])∧(i21[1] + 1 →* i33[2]))


(1) -> (4), if ((1 →* i49[4])∧(i21[1] + 1 →* i33[4]))


(2) -> (3), if ((i49[2] →* i49[3])∧(i33[2] →* i33[3])∧(i49[2] > 0 && i33[2] >= i49[2] →* TRUE))


(3) -> (2), if ((i49[3] + 1 →* i49[2])∧(i33[3] →* i33[2]))


(3) -> (4), if ((i49[3] + 1 →* i49[4])∧(i33[3] →* i33[4]))


(4) -> (5), if ((i33[4] > 0 && i33[4] < i49[4] →* TRUE)∧(i49[4] →* i49[5])∧(i33[4] →* i33[5]))


(5) -> (0), if ((i33[5] - 2 →* i21[0]))



The set Q consists of the following terms:
Load291(x0)
Cond_Load291(TRUE, x0)
Load680(x0, x1)
Cond_Load680(TRUE, x0, x1)
Cond_Load6801(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 LOAD291(i21) → COND_LOAD291(&&(>=(i21, 0), >(+(i21, 1), 0)), i21) the following chains were created:
  • We consider the chain LOAD291(i21[0]) → COND_LOAD291(&&(>=(i21[0], 0), >(+(i21[0], 1), 0)), i21[0]), COND_LOAD291(TRUE, i21[1]) → LOAD680(+(i21[1], 1), 1) which results in the following constraint:

    (1)    (&&(>=(i21[0], 0), >(+(i21[0], 1), 0))=TRUE∧i21[0]=i21[1] ⇒ LOAD291(i21[0])≥NonInfC∧LOAD291(i21[0])≥COND_LOAD291(&&(>=(i21[0], 0), >(+(i21[0], 1), 0)), i21[0])∧(UIncreasing(COND_LOAD291(&&(>=(i21[0], 0), >(+(i21[0], 1), 0)), i21[0])), ≥))



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

    (2)    (>=(i21[0], 0)=TRUE∧>(+(i21[0], 1), 0)=TRUE ⇒ LOAD291(i21[0])≥NonInfC∧LOAD291(i21[0])≥COND_LOAD291(&&(>=(i21[0], 0), >(+(i21[0], 1), 0)), i21[0])∧(UIncreasing(COND_LOAD291(&&(>=(i21[0], 0), >(+(i21[0], 1), 0)), i21[0])), ≥))



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

    (3)    (i21[0] ≥ 0∧i21[0] ≥ 0 ⇒ (UIncreasing(COND_LOAD291(&&(>=(i21[0], 0), >(+(i21[0], 1), 0)), i21[0])), ≥)∧[(-1)Bound*bni_23] + [bni_23]i21[0] ≥ 0∧[(-1)bso_24] ≥ 0)



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

    (4)    (i21[0] ≥ 0∧i21[0] ≥ 0 ⇒ (UIncreasing(COND_LOAD291(&&(>=(i21[0], 0), >(+(i21[0], 1), 0)), i21[0])), ≥)∧[(-1)Bound*bni_23] + [bni_23]i21[0] ≥ 0∧[(-1)bso_24] ≥ 0)



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

    (5)    (i21[0] ≥ 0∧i21[0] ≥ 0 ⇒ (UIncreasing(COND_LOAD291(&&(>=(i21[0], 0), >(+(i21[0], 1), 0)), i21[0])), ≥)∧[(-1)Bound*bni_23] + [bni_23]i21[0] ≥ 0∧[(-1)bso_24] ≥ 0)







For Pair COND_LOAD291(TRUE, i21) → LOAD680(+(i21, 1), 1) the following chains were created:
  • We consider the chain LOAD291(i21[0]) → COND_LOAD291(&&(>=(i21[0], 0), >(+(i21[0], 1), 0)), i21[0]), COND_LOAD291(TRUE, i21[1]) → LOAD680(+(i21[1], 1), 1), LOAD680(i33[2], i49[2]) → COND_LOAD680(&&(>(i49[2], 0), >=(i33[2], i49[2])), i33[2], i49[2]) which results in the following constraint:

    (6)    (&&(>=(i21[0], 0), >(+(i21[0], 1), 0))=TRUE∧i21[0]=i21[1]∧1=i49[2]∧+(i21[1], 1)=i33[2] ⇒ COND_LOAD291(TRUE, i21[1])≥NonInfC∧COND_LOAD291(TRUE, i21[1])≥LOAD680(+(i21[1], 1), 1)∧(UIncreasing(LOAD680(+(i21[1], 1), 1)), ≥))



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

    (7)    (>=(i21[0], 0)=TRUE∧>(+(i21[0], 1), 0)=TRUE ⇒ COND_LOAD291(TRUE, i21[0])≥NonInfC∧COND_LOAD291(TRUE, i21[0])≥LOAD680(+(i21[0], 1), 1)∧(UIncreasing(LOAD680(+(i21[1], 1), 1)), ≥))



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

    (8)    (i21[0] ≥ 0∧i21[0] ≥ 0 ⇒ (UIncreasing(LOAD680(+(i21[1], 1), 1)), ≥)∧[(-1)Bound*bni_25] + [bni_25]i21[0] ≥ 0∧[(-1)bso_26] ≥ 0)



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

    (9)    (i21[0] ≥ 0∧i21[0] ≥ 0 ⇒ (UIncreasing(LOAD680(+(i21[1], 1), 1)), ≥)∧[(-1)Bound*bni_25] + [bni_25]i21[0] ≥ 0∧[(-1)bso_26] ≥ 0)



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

    (10)    (i21[0] ≥ 0∧i21[0] ≥ 0 ⇒ (UIncreasing(LOAD680(+(i21[1], 1), 1)), ≥)∧[(-1)Bound*bni_25] + [bni_25]i21[0] ≥ 0∧[(-1)bso_26] ≥ 0)



  • We consider the chain LOAD291(i21[0]) → COND_LOAD291(&&(>=(i21[0], 0), >(+(i21[0], 1), 0)), i21[0]), COND_LOAD291(TRUE, i21[1]) → LOAD680(+(i21[1], 1), 1), LOAD680(i33[4], i49[4]) → COND_LOAD6801(&&(>(i33[4], 0), <(i33[4], i49[4])), i33[4], i49[4]) which results in the following constraint:

    (11)    (&&(>=(i21[0], 0), >(+(i21[0], 1), 0))=TRUE∧i21[0]=i21[1]∧1=i49[4]∧+(i21[1], 1)=i33[4] ⇒ COND_LOAD291(TRUE, i21[1])≥NonInfC∧COND_LOAD291(TRUE, i21[1])≥LOAD680(+(i21[1], 1), 1)∧(UIncreasing(LOAD680(+(i21[1], 1), 1)), ≥))



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

    (12)    (>=(i21[0], 0)=TRUE∧>(+(i21[0], 1), 0)=TRUE ⇒ COND_LOAD291(TRUE, i21[0])≥NonInfC∧COND_LOAD291(TRUE, i21[0])≥LOAD680(+(i21[0], 1), 1)∧(UIncreasing(LOAD680(+(i21[1], 1), 1)), ≥))



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

    (13)    (i21[0] ≥ 0∧i21[0] ≥ 0 ⇒ (UIncreasing(LOAD680(+(i21[1], 1), 1)), ≥)∧[(-1)Bound*bni_25] + [bni_25]i21[0] ≥ 0∧[(-1)bso_26] ≥ 0)



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

    (14)    (i21[0] ≥ 0∧i21[0] ≥ 0 ⇒ (UIncreasing(LOAD680(+(i21[1], 1), 1)), ≥)∧[(-1)Bound*bni_25] + [bni_25]i21[0] ≥ 0∧[(-1)bso_26] ≥ 0)



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

    (15)    (i21[0] ≥ 0∧i21[0] ≥ 0 ⇒ (UIncreasing(LOAD680(+(i21[1], 1), 1)), ≥)∧[(-1)Bound*bni_25] + [bni_25]i21[0] ≥ 0∧[(-1)bso_26] ≥ 0)







For Pair LOAD680(i33, i49) → COND_LOAD680(&&(>(i49, 0), >=(i33, i49)), i33, i49) the following chains were created:
  • We consider the chain LOAD680(i33[2], i49[2]) → COND_LOAD680(&&(>(i49[2], 0), >=(i33[2], i49[2])), i33[2], i49[2]), COND_LOAD680(TRUE, i33[3], i49[3]) → LOAD680(i33[3], +(i49[3], 1)) which results in the following constraint:

    (16)    (i49[2]=i49[3]∧i33[2]=i33[3]∧&&(>(i49[2], 0), >=(i33[2], i49[2]))=TRUE ⇒ LOAD680(i33[2], i49[2])≥NonInfC∧LOAD680(i33[2], i49[2])≥COND_LOAD680(&&(>(i49[2], 0), >=(i33[2], i49[2])), i33[2], i49[2])∧(UIncreasing(COND_LOAD680(&&(>(i49[2], 0), >=(i33[2], i49[2])), i33[2], i49[2])), ≥))



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

    (17)    (>(i49[2], 0)=TRUE∧>=(i33[2], i49[2])=TRUE ⇒ LOAD680(i33[2], i49[2])≥NonInfC∧LOAD680(i33[2], i49[2])≥COND_LOAD680(&&(>(i49[2], 0), >=(i33[2], i49[2])), i33[2], i49[2])∧(UIncreasing(COND_LOAD680(&&(>(i49[2], 0), >=(i33[2], i49[2])), i33[2], i49[2])), ≥))



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

    (18)    (i49[2] + [-1] ≥ 0∧i33[2] + [-1]i49[2] ≥ 0 ⇒ (UIncreasing(COND_LOAD680(&&(>(i49[2], 0), >=(i33[2], i49[2])), i33[2], i49[2])), ≥)∧[(-1)bni_27 + (-1)Bound*bni_27] + [bni_27]i33[2] ≥ 0∧[(-1)bso_28] ≥ 0)



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

    (19)    (i49[2] + [-1] ≥ 0∧i33[2] + [-1]i49[2] ≥ 0 ⇒ (UIncreasing(COND_LOAD680(&&(>(i49[2], 0), >=(i33[2], i49[2])), i33[2], i49[2])), ≥)∧[(-1)bni_27 + (-1)Bound*bni_27] + [bni_27]i33[2] ≥ 0∧[(-1)bso_28] ≥ 0)



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

    (20)    (i49[2] + [-1] ≥ 0∧i33[2] + [-1]i49[2] ≥ 0 ⇒ (UIncreasing(COND_LOAD680(&&(>(i49[2], 0), >=(i33[2], i49[2])), i33[2], i49[2])), ≥)∧[(-1)bni_27 + (-1)Bound*bni_27] + [bni_27]i33[2] ≥ 0∧[(-1)bso_28] ≥ 0)



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

    (21)    (i49[2] ≥ 0∧i33[2] + [-1] + [-1]i49[2] ≥ 0 ⇒ (UIncreasing(COND_LOAD680(&&(>(i49[2], 0), >=(i33[2], i49[2])), i33[2], i49[2])), ≥)∧[(-1)bni_27 + (-1)Bound*bni_27] + [bni_27]i33[2] ≥ 0∧[(-1)bso_28] ≥ 0)



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

    (22)    (i49[2] ≥ 0∧i33[2] ≥ 0 ⇒ (UIncreasing(COND_LOAD680(&&(>(i49[2], 0), >=(i33[2], i49[2])), i33[2], i49[2])), ≥)∧[(-1)Bound*bni_27] + [bni_27]i49[2] + [bni_27]i33[2] ≥ 0∧[(-1)bso_28] ≥ 0)







For Pair COND_LOAD680(TRUE, i33, i49) → LOAD680(i33, +(i49, 1)) the following chains were created:
  • We consider the chain LOAD680(i33[2], i49[2]) → COND_LOAD680(&&(>(i49[2], 0), >=(i33[2], i49[2])), i33[2], i49[2]), COND_LOAD680(TRUE, i33[3], i49[3]) → LOAD680(i33[3], +(i49[3], 1)), LOAD680(i33[2], i49[2]) → COND_LOAD680(&&(>(i49[2], 0), >=(i33[2], i49[2])), i33[2], i49[2]) which results in the following constraint:

    (23)    (i49[2]=i49[3]∧i33[2]=i33[3]∧&&(>(i49[2], 0), >=(i33[2], i49[2]))=TRUE∧+(i49[3], 1)=i49[2]1∧i33[3]=i33[2]1 ⇒ COND_LOAD680(TRUE, i33[3], i49[3])≥NonInfC∧COND_LOAD680(TRUE, i33[3], i49[3])≥LOAD680(i33[3], +(i49[3], 1))∧(UIncreasing(LOAD680(i33[3], +(i49[3], 1))), ≥))



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

    (24)    (>(i49[2], 0)=TRUE∧>=(i33[2], i49[2])=TRUE ⇒ COND_LOAD680(TRUE, i33[2], i49[2])≥NonInfC∧COND_LOAD680(TRUE, i33[2], i49[2])≥LOAD680(i33[2], +(i49[2], 1))∧(UIncreasing(LOAD680(i33[3], +(i49[3], 1))), ≥))



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

    (25)    (i49[2] + [-1] ≥ 0∧i33[2] + [-1]i49[2] ≥ 0 ⇒ (UIncreasing(LOAD680(i33[3], +(i49[3], 1))), ≥)∧[(-1)bni_29 + (-1)Bound*bni_29] + [bni_29]i33[2] ≥ 0∧[(-1)bso_30] ≥ 0)



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

    (26)    (i49[2] + [-1] ≥ 0∧i33[2] + [-1]i49[2] ≥ 0 ⇒ (UIncreasing(LOAD680(i33[3], +(i49[3], 1))), ≥)∧[(-1)bni_29 + (-1)Bound*bni_29] + [bni_29]i33[2] ≥ 0∧[(-1)bso_30] ≥ 0)



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

    (27)    (i49[2] + [-1] ≥ 0∧i33[2] + [-1]i49[2] ≥ 0 ⇒ (UIncreasing(LOAD680(i33[3], +(i49[3], 1))), ≥)∧[(-1)bni_29 + (-1)Bound*bni_29] + [bni_29]i33[2] ≥ 0∧[(-1)bso_30] ≥ 0)



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

    (28)    (i49[2] ≥ 0∧i33[2] + [-1] + [-1]i49[2] ≥ 0 ⇒ (UIncreasing(LOAD680(i33[3], +(i49[3], 1))), ≥)∧[(-1)bni_29 + (-1)Bound*bni_29] + [bni_29]i33[2] ≥ 0∧[(-1)bso_30] ≥ 0)



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

    (29)    (i49[2] ≥ 0∧i33[2] ≥ 0 ⇒ (UIncreasing(LOAD680(i33[3], +(i49[3], 1))), ≥)∧[(-1)Bound*bni_29] + [bni_29]i49[2] + [bni_29]i33[2] ≥ 0∧[(-1)bso_30] ≥ 0)



  • We consider the chain LOAD680(i33[2], i49[2]) → COND_LOAD680(&&(>(i49[2], 0), >=(i33[2], i49[2])), i33[2], i49[2]), COND_LOAD680(TRUE, i33[3], i49[3]) → LOAD680(i33[3], +(i49[3], 1)), LOAD680(i33[4], i49[4]) → COND_LOAD6801(&&(>(i33[4], 0), <(i33[4], i49[4])), i33[4], i49[4]) which results in the following constraint:

    (30)    (i49[2]=i49[3]∧i33[2]=i33[3]∧&&(>(i49[2], 0), >=(i33[2], i49[2]))=TRUE∧+(i49[3], 1)=i49[4]∧i33[3]=i33[4] ⇒ COND_LOAD680(TRUE, i33[3], i49[3])≥NonInfC∧COND_LOAD680(TRUE, i33[3], i49[3])≥LOAD680(i33[3], +(i49[3], 1))∧(UIncreasing(LOAD680(i33[3], +(i49[3], 1))), ≥))



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

    (31)    (>(i49[2], 0)=TRUE∧>=(i33[2], i49[2])=TRUE ⇒ COND_LOAD680(TRUE, i33[2], i49[2])≥NonInfC∧COND_LOAD680(TRUE, i33[2], i49[2])≥LOAD680(i33[2], +(i49[2], 1))∧(UIncreasing(LOAD680(i33[3], +(i49[3], 1))), ≥))



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

    (32)    (i49[2] + [-1] ≥ 0∧i33[2] + [-1]i49[2] ≥ 0 ⇒ (UIncreasing(LOAD680(i33[3], +(i49[3], 1))), ≥)∧[(-1)bni_29 + (-1)Bound*bni_29] + [bni_29]i33[2] ≥ 0∧[(-1)bso_30] ≥ 0)



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

    (33)    (i49[2] + [-1] ≥ 0∧i33[2] + [-1]i49[2] ≥ 0 ⇒ (UIncreasing(LOAD680(i33[3], +(i49[3], 1))), ≥)∧[(-1)bni_29 + (-1)Bound*bni_29] + [bni_29]i33[2] ≥ 0∧[(-1)bso_30] ≥ 0)



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

    (34)    (i49[2] + [-1] ≥ 0∧i33[2] + [-1]i49[2] ≥ 0 ⇒ (UIncreasing(LOAD680(i33[3], +(i49[3], 1))), ≥)∧[(-1)bni_29 + (-1)Bound*bni_29] + [bni_29]i33[2] ≥ 0∧[(-1)bso_30] ≥ 0)



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

    (35)    (i49[2] ≥ 0∧i33[2] + [-1] + [-1]i49[2] ≥ 0 ⇒ (UIncreasing(LOAD680(i33[3], +(i49[3], 1))), ≥)∧[(-1)bni_29 + (-1)Bound*bni_29] + [bni_29]i33[2] ≥ 0∧[(-1)bso_30] ≥ 0)



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

    (36)    (i49[2] ≥ 0∧i33[2] ≥ 0 ⇒ (UIncreasing(LOAD680(i33[3], +(i49[3], 1))), ≥)∧[(-1)Bound*bni_29] + [bni_29]i49[2] + [bni_29]i33[2] ≥ 0∧[(-1)bso_30] ≥ 0)







For Pair LOAD680(i33, i49) → COND_LOAD6801(&&(>(i33, 0), <(i33, i49)), i33, i49) the following chains were created:
  • We consider the chain LOAD680(i33[4], i49[4]) → COND_LOAD6801(&&(>(i33[4], 0), <(i33[4], i49[4])), i33[4], i49[4]), COND_LOAD6801(TRUE, i33[5], i49[5]) → LOAD291(-(i33[5], 2)) which results in the following constraint:

    (37)    (&&(>(i33[4], 0), <(i33[4], i49[4]))=TRUE∧i49[4]=i49[5]∧i33[4]=i33[5] ⇒ LOAD680(i33[4], i49[4])≥NonInfC∧LOAD680(i33[4], i49[4])≥COND_LOAD6801(&&(>(i33[4], 0), <(i33[4], i49[4])), i33[4], i49[4])∧(UIncreasing(COND_LOAD6801(&&(>(i33[4], 0), <(i33[4], i49[4])), i33[4], i49[4])), ≥))



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

    (38)    (>(i33[4], 0)=TRUE∧<(i33[4], i49[4])=TRUE ⇒ LOAD680(i33[4], i49[4])≥NonInfC∧LOAD680(i33[4], i49[4])≥COND_LOAD6801(&&(>(i33[4], 0), <(i33[4], i49[4])), i33[4], i49[4])∧(UIncreasing(COND_LOAD6801(&&(>(i33[4], 0), <(i33[4], i49[4])), i33[4], i49[4])), ≥))



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

    (39)    (i33[4] + [-1] ≥ 0∧i49[4] + [-1] + [-1]i33[4] ≥ 0 ⇒ (UIncreasing(COND_LOAD6801(&&(>(i33[4], 0), <(i33[4], i49[4])), i33[4], i49[4])), ≥)∧[(-1)bni_31 + (-1)Bound*bni_31] + [bni_31]i33[4] ≥ 0∧[(-1)bso_32] ≥ 0)



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

    (40)    (i33[4] + [-1] ≥ 0∧i49[4] + [-1] + [-1]i33[4] ≥ 0 ⇒ (UIncreasing(COND_LOAD6801(&&(>(i33[4], 0), <(i33[4], i49[4])), i33[4], i49[4])), ≥)∧[(-1)bni_31 + (-1)Bound*bni_31] + [bni_31]i33[4] ≥ 0∧[(-1)bso_32] ≥ 0)



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

    (41)    (i33[4] + [-1] ≥ 0∧i49[4] + [-1] + [-1]i33[4] ≥ 0 ⇒ (UIncreasing(COND_LOAD6801(&&(>(i33[4], 0), <(i33[4], i49[4])), i33[4], i49[4])), ≥)∧[(-1)bni_31 + (-1)Bound*bni_31] + [bni_31]i33[4] ≥ 0∧[(-1)bso_32] ≥ 0)



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

    (42)    (i33[4] ≥ 0∧i49[4] + [-2] + [-1]i33[4] ≥ 0 ⇒ (UIncreasing(COND_LOAD6801(&&(>(i33[4], 0), <(i33[4], i49[4])), i33[4], i49[4])), ≥)∧[(-1)Bound*bni_31] + [bni_31]i33[4] ≥ 0∧[(-1)bso_32] ≥ 0)



    We simplified constraint (42) using rule (IDP_SMT_SPLIT) which results in the following new constraint:

    (43)    (i33[4] ≥ 0∧i49[4] ≥ 0 ⇒ (UIncreasing(COND_LOAD6801(&&(>(i33[4], 0), <(i33[4], i49[4])), i33[4], i49[4])), ≥)∧[(-1)Bound*bni_31] + [bni_31]i33[4] ≥ 0∧[(-1)bso_32] ≥ 0)







For Pair COND_LOAD6801(TRUE, i33, i49) → LOAD291(-(i33, 2)) the following chains were created:
  • We consider the chain LOAD680(i33[4], i49[4]) → COND_LOAD6801(&&(>(i33[4], 0), <(i33[4], i49[4])), i33[4], i49[4]), COND_LOAD6801(TRUE, i33[5], i49[5]) → LOAD291(-(i33[5], 2)), LOAD291(i21[0]) → COND_LOAD291(&&(>=(i21[0], 0), >(+(i21[0], 1), 0)), i21[0]) which results in the following constraint:

    (44)    (&&(>(i33[4], 0), <(i33[4], i49[4]))=TRUE∧i49[4]=i49[5]∧i33[4]=i33[5]∧-(i33[5], 2)=i21[0] ⇒ COND_LOAD6801(TRUE, i33[5], i49[5])≥NonInfC∧COND_LOAD6801(TRUE, i33[5], i49[5])≥LOAD291(-(i33[5], 2))∧(UIncreasing(LOAD291(-(i33[5], 2))), ≥))



    We simplified constraint (44) using rules (III), (IV), (IDP_BOOLEAN) which results in the following new constraint:

    (45)    (>(i33[4], 0)=TRUE∧<(i33[4], i49[4])=TRUE ⇒ COND_LOAD6801(TRUE, i33[4], i49[4])≥NonInfC∧COND_LOAD6801(TRUE, i33[4], i49[4])≥LOAD291(-(i33[4], 2))∧(UIncreasing(LOAD291(-(i33[5], 2))), ≥))



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

    (46)    (i33[4] + [-1] ≥ 0∧i49[4] + [-1] + [-1]i33[4] ≥ 0 ⇒ (UIncreasing(LOAD291(-(i33[5], 2))), ≥)∧[(-1)bni_33 + (-1)Bound*bni_33] + [bni_33]i33[4] ≥ 0∧[1 + (-1)bso_34] ≥ 0)



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

    (47)    (i33[4] + [-1] ≥ 0∧i49[4] + [-1] + [-1]i33[4] ≥ 0 ⇒ (UIncreasing(LOAD291(-(i33[5], 2))), ≥)∧[(-1)bni_33 + (-1)Bound*bni_33] + [bni_33]i33[4] ≥ 0∧[1 + (-1)bso_34] ≥ 0)



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

    (48)    (i33[4] + [-1] ≥ 0∧i49[4] + [-1] + [-1]i33[4] ≥ 0 ⇒ (UIncreasing(LOAD291(-(i33[5], 2))), ≥)∧[(-1)bni_33 + (-1)Bound*bni_33] + [bni_33]i33[4] ≥ 0∧[1 + (-1)bso_34] ≥ 0)



    We simplified constraint (48) using rule (IDP_SMT_SPLIT) which results in the following new constraint:

    (49)    (i33[4] ≥ 0∧i49[4] + [-2] + [-1]i33[4] ≥ 0 ⇒ (UIncreasing(LOAD291(-(i33[5], 2))), ≥)∧[(-1)Bound*bni_33] + [bni_33]i33[4] ≥ 0∧[1 + (-1)bso_34] ≥ 0)



    We simplified constraint (49) using rule (IDP_SMT_SPLIT) which results in the following new constraint:

    (50)    (i33[4] ≥ 0∧i49[4] ≥ 0 ⇒ (UIncreasing(LOAD291(-(i33[5], 2))), ≥)∧[(-1)Bound*bni_33] + [bni_33]i33[4] ≥ 0∧[1 + (-1)bso_34] ≥ 0)







To summarize, we get the following constraints P≥ for the following pairs.
  • LOAD291(i21) → COND_LOAD291(&&(>=(i21, 0), >(+(i21, 1), 0)), i21)
    • (i21[0] ≥ 0∧i21[0] ≥ 0 ⇒ (UIncreasing(COND_LOAD291(&&(>=(i21[0], 0), >(+(i21[0], 1), 0)), i21[0])), ≥)∧[(-1)Bound*bni_23] + [bni_23]i21[0] ≥ 0∧[(-1)bso_24] ≥ 0)

  • COND_LOAD291(TRUE, i21) → LOAD680(+(i21, 1), 1)
    • (i21[0] ≥ 0∧i21[0] ≥ 0 ⇒ (UIncreasing(LOAD680(+(i21[1], 1), 1)), ≥)∧[(-1)Bound*bni_25] + [bni_25]i21[0] ≥ 0∧[(-1)bso_26] ≥ 0)
    • (i21[0] ≥ 0∧i21[0] ≥ 0 ⇒ (UIncreasing(LOAD680(+(i21[1], 1), 1)), ≥)∧[(-1)Bound*bni_25] + [bni_25]i21[0] ≥ 0∧[(-1)bso_26] ≥ 0)

  • LOAD680(i33, i49) → COND_LOAD680(&&(>(i49, 0), >=(i33, i49)), i33, i49)
    • (i49[2] ≥ 0∧i33[2] ≥ 0 ⇒ (UIncreasing(COND_LOAD680(&&(>(i49[2], 0), >=(i33[2], i49[2])), i33[2], i49[2])), ≥)∧[(-1)Bound*bni_27] + [bni_27]i49[2] + [bni_27]i33[2] ≥ 0∧[(-1)bso_28] ≥ 0)

  • COND_LOAD680(TRUE, i33, i49) → LOAD680(i33, +(i49, 1))
    • (i49[2] ≥ 0∧i33[2] ≥ 0 ⇒ (UIncreasing(LOAD680(i33[3], +(i49[3], 1))), ≥)∧[(-1)Bound*bni_29] + [bni_29]i49[2] + [bni_29]i33[2] ≥ 0∧[(-1)bso_30] ≥ 0)
    • (i49[2] ≥ 0∧i33[2] ≥ 0 ⇒ (UIncreasing(LOAD680(i33[3], +(i49[3], 1))), ≥)∧[(-1)Bound*bni_29] + [bni_29]i49[2] + [bni_29]i33[2] ≥ 0∧[(-1)bso_30] ≥ 0)

  • LOAD680(i33, i49) → COND_LOAD6801(&&(>(i33, 0), <(i33, i49)), i33, i49)
    • (i33[4] ≥ 0∧i49[4] ≥ 0 ⇒ (UIncreasing(COND_LOAD6801(&&(>(i33[4], 0), <(i33[4], i49[4])), i33[4], i49[4])), ≥)∧[(-1)Bound*bni_31] + [bni_31]i33[4] ≥ 0∧[(-1)bso_32] ≥ 0)

  • COND_LOAD6801(TRUE, i33, i49) → LOAD291(-(i33, 2))
    • (i33[4] ≥ 0∧i49[4] ≥ 0 ⇒ (UIncreasing(LOAD291(-(i33[5], 2))), ≥)∧[(-1)Bound*bni_33] + [bni_33]i33[4] ≥ 0∧[1 + (-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) = 0   
POL(LOAD291(x1)) = x1   
POL(COND_LOAD291(x1, x2)) = 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(LOAD680(x1, x2)) = [-1] + x1   
POL(COND_LOAD680(x1, x2, x3)) = [-1] + x2   
POL(COND_LOAD6801(x1, x2, x3)) = [-1] + x2   
POL(<(x1, x2)) = [-1]   
POL(-(x1, x2)) = x1 + [-1]x2   
POL(2) = [2]   

The following pairs are in P>:

COND_LOAD6801(TRUE, i33[5], i49[5]) → LOAD291(-(i33[5], 2))

The following pairs are in Pbound:

LOAD291(i21[0]) → COND_LOAD291(&&(>=(i21[0], 0), >(+(i21[0], 1), 0)), i21[0])
COND_LOAD291(TRUE, i21[1]) → LOAD680(+(i21[1], 1), 1)
LOAD680(i33[2], i49[2]) → COND_LOAD680(&&(>(i49[2], 0), >=(i33[2], i49[2])), i33[2], i49[2])
COND_LOAD680(TRUE, i33[3], i49[3]) → LOAD680(i33[3], +(i49[3], 1))
LOAD680(i33[4], i49[4]) → COND_LOAD6801(&&(>(i33[4], 0), <(i33[4], i49[4])), i33[4], i49[4])
COND_LOAD6801(TRUE, i33[5], i49[5]) → LOAD291(-(i33[5], 2))

The following pairs are in P≥:

LOAD291(i21[0]) → COND_LOAD291(&&(>=(i21[0], 0), >(+(i21[0], 1), 0)), i21[0])
COND_LOAD291(TRUE, i21[1]) → LOAD680(+(i21[1], 1), 1)
LOAD680(i33[2], i49[2]) → COND_LOAD680(&&(>(i49[2], 0), >=(i33[2], i49[2])), i33[2], i49[2])
COND_LOAD680(TRUE, i33[3], i49[3]) → LOAD680(i33[3], +(i49[3], 1))
LOAD680(i33[4], i49[4]) → COND_LOAD6801(&&(>(i33[4], 0), <(i33[4], i49[4])), i33[4], i49[4])

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:
(0): LOAD291(i21[0]) → COND_LOAD291(i21[0] >= 0 && i21[0] + 1 > 0, i21[0])
(1): COND_LOAD291(TRUE, i21[1]) → LOAD680(i21[1] + 1, 1)
(2): LOAD680(i33[2], i49[2]) → COND_LOAD680(i49[2] > 0 && i33[2] >= i49[2], i33[2], i49[2])
(3): COND_LOAD680(TRUE, i33[3], i49[3]) → LOAD680(i33[3], i49[3] + 1)
(4): LOAD680(i33[4], i49[4]) → COND_LOAD6801(i33[4] > 0 && i33[4] < i49[4], i33[4], i49[4])

(0) -> (1), if ((i21[0] >= 0 && i21[0] + 1 > 0 →* TRUE)∧(i21[0] →* i21[1]))


(1) -> (2), if ((1 →* i49[2])∧(i21[1] + 1 →* i33[2]))


(3) -> (2), if ((i49[3] + 1 →* i49[2])∧(i33[3] →* i33[2]))


(2) -> (3), if ((i49[2] →* i49[3])∧(i33[2] →* i33[3])∧(i49[2] > 0 && i33[2] >= i49[2] →* TRUE))


(1) -> (4), if ((1 →* i49[4])∧(i21[1] + 1 →* i33[4]))


(3) -> (4), if ((i49[3] + 1 →* i49[4])∧(i33[3] →* i33[4]))



The set Q consists of the following terms:
Load291(x0)
Cond_Load291(TRUE, x0)
Load680(x0, x1)
Cond_Load680(TRUE, x0, x1)
Cond_Load6801(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_LOAD680(TRUE, i33[3], i49[3]) → LOAD680(i33[3], i49[3] + 1)
(2): LOAD680(i33[2], i49[2]) → COND_LOAD680(i49[2] > 0 && i33[2] >= i49[2], i33[2], i49[2])

(3) -> (2), if ((i49[3] + 1 →* i49[2])∧(i33[3] →* i33[2]))


(2) -> (3), if ((i49[2] →* i49[3])∧(i33[2] →* i33[3])∧(i49[2] > 0 && i33[2] >= i49[2] →* TRUE))



The set Q consists of the following terms:
Load291(x0)
Cond_Load291(TRUE, x0)
Load680(x0, x1)
Cond_Load680(TRUE, x0, x1)
Cond_Load6801(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_LOAD680(TRUE, i33[3], i49[3]) → LOAD680(i33[3], +(i49[3], 1)) the following chains were created:
  • We consider the chain LOAD680(i33[2], i49[2]) → COND_LOAD680(&&(>(i49[2], 0), >=(i33[2], i49[2])), i33[2], i49[2]), COND_LOAD680(TRUE, i33[3], i49[3]) → LOAD680(i33[3], +(i49[3], 1)), LOAD680(i33[2], i49[2]) → COND_LOAD680(&&(>(i49[2], 0), >=(i33[2], i49[2])), i33[2], i49[2]) which results in the following constraint:

    (1)    (i49[2]=i49[3]∧i33[2]=i33[3]∧&&(>(i49[2], 0), >=(i33[2], i49[2]))=TRUE∧+(i49[3], 1)=i49[2]1∧i33[3]=i33[2]1 ⇒ COND_LOAD680(TRUE, i33[3], i49[3])≥NonInfC∧COND_LOAD680(TRUE, i33[3], i49[3])≥LOAD680(i33[3], +(i49[3], 1))∧(UIncreasing(LOAD680(i33[3], +(i49[3], 1))), ≥))



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

    (2)    (>(i49[2], 0)=TRUE∧>=(i33[2], i49[2])=TRUE ⇒ COND_LOAD680(TRUE, i33[2], i49[2])≥NonInfC∧COND_LOAD680(TRUE, i33[2], i49[2])≥LOAD680(i33[2], +(i49[2], 1))∧(UIncreasing(LOAD680(i33[3], +(i49[3], 1))), ≥))



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

    (3)    (i49[2] + [-1] ≥ 0∧i33[2] + [-1]i49[2] ≥ 0 ⇒ (UIncreasing(LOAD680(i33[3], +(i49[3], 1))), ≥)∧[(-1)bni_13 + (-1)Bound*bni_13] + [(-1)bni_13]i49[2] + [bni_13]i33[2] ≥ 0∧[(-1)bso_14] ≥ 0)



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

    (4)    (i49[2] + [-1] ≥ 0∧i33[2] + [-1]i49[2] ≥ 0 ⇒ (UIncreasing(LOAD680(i33[3], +(i49[3], 1))), ≥)∧[(-1)bni_13 + (-1)Bound*bni_13] + [(-1)bni_13]i49[2] + [bni_13]i33[2] ≥ 0∧[(-1)bso_14] ≥ 0)



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

    (5)    (i49[2] + [-1] ≥ 0∧i33[2] + [-1]i49[2] ≥ 0 ⇒ (UIncreasing(LOAD680(i33[3], +(i49[3], 1))), ≥)∧[(-1)bni_13 + (-1)Bound*bni_13] + [(-1)bni_13]i49[2] + [bni_13]i33[2] ≥ 0∧[(-1)bso_14] ≥ 0)



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

    (6)    (i49[2] ≥ 0∧i33[2] + [-1] + [-1]i49[2] ≥ 0 ⇒ (UIncreasing(LOAD680(i33[3], +(i49[3], 1))), ≥)∧[(-2)bni_13 + (-1)Bound*bni_13] + [(-1)bni_13]i49[2] + [bni_13]i33[2] ≥ 0∧[(-1)bso_14] ≥ 0)



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

    (7)    (i49[2] ≥ 0∧i33[2] ≥ 0 ⇒ (UIncreasing(LOAD680(i33[3], +(i49[3], 1))), ≥)∧[(-1)bni_13 + (-1)Bound*bni_13] + [bni_13]i33[2] ≥ 0∧[(-1)bso_14] ≥ 0)







For Pair LOAD680(i33[2], i49[2]) → COND_LOAD680(&&(>(i49[2], 0), >=(i33[2], i49[2])), i33[2], i49[2]) the following chains were created:
  • We consider the chain LOAD680(i33[2], i49[2]) → COND_LOAD680(&&(>(i49[2], 0), >=(i33[2], i49[2])), i33[2], i49[2]), COND_LOAD680(TRUE, i33[3], i49[3]) → LOAD680(i33[3], +(i49[3], 1)) which results in the following constraint:

    (8)    (i49[2]=i49[3]∧i33[2]=i33[3]∧&&(>(i49[2], 0), >=(i33[2], i49[2]))=TRUE ⇒ LOAD680(i33[2], i49[2])≥NonInfC∧LOAD680(i33[2], i49[2])≥COND_LOAD680(&&(>(i49[2], 0), >=(i33[2], i49[2])), i33[2], i49[2])∧(UIncreasing(COND_LOAD680(&&(>(i49[2], 0), >=(i33[2], i49[2])), i33[2], i49[2])), ≥))



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

    (9)    (>(i49[2], 0)=TRUE∧>=(i33[2], i49[2])=TRUE ⇒ LOAD680(i33[2], i49[2])≥NonInfC∧LOAD680(i33[2], i49[2])≥COND_LOAD680(&&(>(i49[2], 0), >=(i33[2], i49[2])), i33[2], i49[2])∧(UIncreasing(COND_LOAD680(&&(>(i49[2], 0), >=(i33[2], i49[2])), i33[2], i49[2])), ≥))



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

    (10)    (i49[2] + [-1] ≥ 0∧i33[2] + [-1]i49[2] ≥ 0 ⇒ (UIncreasing(COND_LOAD680(&&(>(i49[2], 0), >=(i33[2], i49[2])), i33[2], i49[2])), ≥)∧[(-1)Bound*bni_15] + [(-1)bni_15]i49[2] + [bni_15]i33[2] ≥ 0∧[1 + (-1)bso_16] ≥ 0)



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

    (11)    (i49[2] + [-1] ≥ 0∧i33[2] + [-1]i49[2] ≥ 0 ⇒ (UIncreasing(COND_LOAD680(&&(>(i49[2], 0), >=(i33[2], i49[2])), i33[2], i49[2])), ≥)∧[(-1)Bound*bni_15] + [(-1)bni_15]i49[2] + [bni_15]i33[2] ≥ 0∧[1 + (-1)bso_16] ≥ 0)



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

    (12)    (i49[2] + [-1] ≥ 0∧i33[2] + [-1]i49[2] ≥ 0 ⇒ (UIncreasing(COND_LOAD680(&&(>(i49[2], 0), >=(i33[2], i49[2])), i33[2], i49[2])), ≥)∧[(-1)Bound*bni_15] + [(-1)bni_15]i49[2] + [bni_15]i33[2] ≥ 0∧[1 + (-1)bso_16] ≥ 0)



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

    (13)    (i49[2] ≥ 0∧i33[2] + [-1] + [-1]i49[2] ≥ 0 ⇒ (UIncreasing(COND_LOAD680(&&(>(i49[2], 0), >=(i33[2], i49[2])), i33[2], i49[2])), ≥)∧[(-1)Bound*bni_15 + (-1)bni_15] + [(-1)bni_15]i49[2] + [bni_15]i33[2] ≥ 0∧[1 + (-1)bso_16] ≥ 0)



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

    (14)    (i49[2] ≥ 0∧i33[2] ≥ 0 ⇒ (UIncreasing(COND_LOAD680(&&(>(i49[2], 0), >=(i33[2], i49[2])), i33[2], i49[2])), ≥)∧[(-1)Bound*bni_15] + [bni_15]i33[2] ≥ 0∧[1 + (-1)bso_16] ≥ 0)







To summarize, we get the following constraints P≥ for the following pairs.
  • COND_LOAD680(TRUE, i33[3], i49[3]) → LOAD680(i33[3], +(i49[3], 1))
    • (i49[2] ≥ 0∧i33[2] ≥ 0 ⇒ (UIncreasing(LOAD680(i33[3], +(i49[3], 1))), ≥)∧[(-1)bni_13 + (-1)Bound*bni_13] + [bni_13]i33[2] ≥ 0∧[(-1)bso_14] ≥ 0)

  • LOAD680(i33[2], i49[2]) → COND_LOAD680(&&(>(i49[2], 0), >=(i33[2], i49[2])), i33[2], i49[2])
    • (i49[2] ≥ 0∧i33[2] ≥ 0 ⇒ (UIncreasing(COND_LOAD680(&&(>(i49[2], 0), >=(i33[2], i49[2])), i33[2], i49[2])), ≥)∧[(-1)Bound*bni_15] + [bni_15]i33[2] ≥ 0∧[1 + (-1)bso_16] ≥ 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(COND_LOAD680(x1, x2, x3)) = [-1] + [-1]x3 + x2   
POL(LOAD680(x1, x2)) = [-1]x2 + x1   
POL(+(x1, x2)) = x1 + x2   
POL(1) = [1]   
POL(&&(x1, x2)) = 0   
POL(>(x1, x2)) = [-1]   
POL(0) = 0   
POL(>=(x1, x2)) = [-1]   

The following pairs are in P>:

LOAD680(i33[2], i49[2]) → COND_LOAD680(&&(>(i49[2], 0), >=(i33[2], i49[2])), i33[2], i49[2])

The following pairs are in Pbound:

COND_LOAD680(TRUE, i33[3], i49[3]) → LOAD680(i33[3], +(i49[3], 1))
LOAD680(i33[2], i49[2]) → COND_LOAD680(&&(>(i49[2], 0), >=(i33[2], i49[2])), i33[2], i49[2])

The following pairs are in P≥:

COND_LOAD680(TRUE, i33[3], i49[3]) → LOAD680(i33[3], +(i49[3], 1))

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

TRUE1 → &&(TRUE, TRUE)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:

Integer


R is empty.

The integer pair graph contains the following rules and edges:
(3): COND_LOAD680(TRUE, i33[3], i49[3]) → LOAD680(i33[3], i49[3] + 1)


The set Q consists of the following terms:
Load291(x0)
Cond_Load291(TRUE, x0)
Load680(x0, x1)
Cond_Load680(TRUE, x0, x1)
Cond_Load6801(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:
Load291(x0)
Cond_Load291(TRUE, x0)
Load680(x0, x1)
Cond_Load680(TRUE, x0, x1)
Cond_Load6801(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:
Load291(x0)
Cond_Load291(TRUE, x0)
Load680(x0, x1)
Cond_Load680(TRUE, x0, x1)
Cond_Load6801(TRUE, x0, x1)

(25) IDependencyGraphProof (EQUIVALENT transformation)

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

(26) TRUE