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 IDP
        ↳14 IDPNonInfProof (⇐)
        ↳15 AND
            ↳16 IDP
                ↳17 IDependencyGraphProof (⇔)
                ↳18 TRUE
            ↳19 IDP
                ↳20 IDependencyGraphProof (⇔)
                ↳21 TRUE
    ↳22 IDP
        ↳23 IDependencyGraphProof (⇔)
        ↳24 IDP
        ↳25 IDPNonInfProof (⇐)
        ↳26 AND
            ↳27 IDP
                ↳28 IDependencyGraphProof (⇔)
                ↳29 TRUE
            ↳30 IDP
                ↳31 IDependencyGraphProof (⇔)
                ↳32 TRUE

(0) Obligation:

JBC Problem based on JBC Program:
Manifest-Version: 1.0 Created-By: 1.6.0_16 (Sun Microsystems Inc.) Main-Class: PastaB12 
/**
 * 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 PastaB12 {
    public static void main(String[] args) {
        Random.args = args;
        int x = Random.random();
        int y = Random.random();

        while (x > 0 || y > 0) {
            if (x > 0) {
                x--;
            } else if (y > 0) {
                y--;
            } else {
                continue;
            }
        }
    }
}


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 196 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:
Load635(i49, i36) → Cond_Load635(i49 > 0, i49, i36)
Cond_Load635(TRUE, i49, i36) → Load635(i49 + -1, i36)
Load635(i50, i57) → Cond_Load6351(i57 > 0 && i50 <= 0, i50, i57)
Cond_Load6351(TRUE, i50, i57) → Load635(i50, i57 + -1)
The set Q consists of the following terms:
Load635(x0, x1)
Cond_Load635(TRUE, x0, x1)
Cond_Load6351(TRUE, x0, x1)

(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:

Integer, Boolean


The ITRS R consists of the following rules:
Load635(i49, i36) → Cond_Load635(i49 > 0, i49, i36)
Cond_Load635(TRUE, i49, i36) → Load635(i49 + -1, i36)
Load635(i50, i57) → Cond_Load6351(i57 > 0 && i50 <= 0, i50, i57)
Cond_Load6351(TRUE, i50, i57) → Load635(i50, i57 + -1)

The integer pair graph contains the following rules and edges:
(0): LOAD635(i49[0], i36[0]) → COND_LOAD635(i49[0] > 0, i49[0], i36[0])
(1): COND_LOAD635(TRUE, i49[1], i36[1]) → LOAD635(i49[1] + -1, i36[1])
(2): LOAD635(i50[2], i57[2]) → COND_LOAD6351(i57[2] > 0 && i50[2] <= 0, i50[2], i57[2])
(3): COND_LOAD6351(TRUE, i50[3], i57[3]) → LOAD635(i50[3], i57[3] + -1)

(0) -> (1), if ((i36[0] →* i36[1])∧(i49[0] →* i49[1])∧(i49[0] > 0 →* TRUE))


(1) -> (0), if ((i49[1] + -1 →* i49[0])∧(i36[1] →* i36[0]))


(1) -> (2), if ((i49[1] + -1 →* i50[2])∧(i36[1] →* i57[2]))


(2) -> (3), if ((i57[2] > 0 && i50[2] <= 0 →* TRUE)∧(i50[2] →* i50[3])∧(i57[2] →* i57[3]))


(3) -> (0), if ((i57[3] + -1 →* i36[0])∧(i50[3] →* i49[0]))


(3) -> (2), if ((i50[3] →* i50[2])∧(i57[3] + -1 →* i57[2]))



The set Q consists of the following terms:
Load635(x0, x1)
Cond_Load635(TRUE, x0, x1)
Cond_Load6351(TRUE, x0, x1)

(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:

Integer, Boolean


R is empty.

The integer pair graph contains the following rules and edges:
(0): LOAD635(i49[0], i36[0]) → COND_LOAD635(i49[0] > 0, i49[0], i36[0])
(1): COND_LOAD635(TRUE, i49[1], i36[1]) → LOAD635(i49[1] + -1, i36[1])
(2): LOAD635(i50[2], i57[2]) → COND_LOAD6351(i57[2] > 0 && i50[2] <= 0, i50[2], i57[2])
(3): COND_LOAD6351(TRUE, i50[3], i57[3]) → LOAD635(i50[3], i57[3] + -1)

(0) -> (1), if ((i36[0] →* i36[1])∧(i49[0] →* i49[1])∧(i49[0] > 0 →* TRUE))


(1) -> (0), if ((i49[1] + -1 →* i49[0])∧(i36[1] →* i36[0]))


(1) -> (2), if ((i49[1] + -1 →* i50[2])∧(i36[1] →* i57[2]))


(2) -> (3), if ((i57[2] > 0 && i50[2] <= 0 →* TRUE)∧(i50[2] →* i50[3])∧(i57[2] →* i57[3]))


(3) -> (0), if ((i57[3] + -1 →* i36[0])∧(i50[3] →* i49[0]))


(3) -> (2), if ((i50[3] →* i50[2])∧(i57[3] + -1 →* i57[2]))



The set Q consists of the following terms:
Load635(x0, x1)
Cond_Load635(TRUE, x0, x1)
Cond_Load6351(TRUE, x0, x1)

(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 LOAD635(i49, i36) → COND_LOAD635(>(i49, 0), i49, i36) the following chains were created:
  • We consider the chain LOAD635(i49[0], i36[0]) → COND_LOAD635(>(i49[0], 0), i49[0], i36[0]), COND_LOAD635(TRUE, i49[1], i36[1]) → LOAD635(+(i49[1], -1), i36[1]) which results in the following constraint:

    (1)    (i36[0]=i36[1]∧i49[0]=i49[1]∧>(i49[0], 0)=TRUE ⇒ LOAD635(i49[0], i36[0])≥NonInfC∧LOAD635(i49[0], i36[0])≥COND_LOAD635(>(i49[0], 0), i49[0], i36[0])∧(UIncreasing(COND_LOAD635(>(i49[0], 0), i49[0], i36[0])), ≥))



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

    (2)    (>(i49[0], 0)=TRUE ⇒ LOAD635(i49[0], i36[0])≥NonInfC∧LOAD635(i49[0], i36[0])≥COND_LOAD635(>(i49[0], 0), i49[0], i36[0])∧(UIncreasing(COND_LOAD635(>(i49[0], 0), i49[0], i36[0])), ≥))



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

    (3)    (i49[0] + [-1] ≥ 0 ⇒ (UIncreasing(COND_LOAD635(>(i49[0], 0), i49[0], i36[0])), ≥)∧[(-1)bni_11 + (-1)Bound*bni_11] + [bni_11]i36[0] ≥ 0∧[(-1)bso_12] ≥ 0)



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

    (4)    (i49[0] + [-1] ≥ 0 ⇒ (UIncreasing(COND_LOAD635(>(i49[0], 0), i49[0], i36[0])), ≥)∧[(-1)bni_11 + (-1)Bound*bni_11] + [bni_11]i36[0] ≥ 0∧[(-1)bso_12] ≥ 0)



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

    (5)    (i49[0] + [-1] ≥ 0 ⇒ (UIncreasing(COND_LOAD635(>(i49[0], 0), i49[0], i36[0])), ≥)∧[(-1)bni_11 + (-1)Bound*bni_11] + [bni_11]i36[0] ≥ 0∧[(-1)bso_12] ≥ 0)



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

    (6)    (i49[0] + [-1] ≥ 0 ⇒ (UIncreasing(COND_LOAD635(>(i49[0], 0), i49[0], i36[0])), ≥)∧[bni_11] = 0∧[(-1)bni_11 + (-1)Bound*bni_11] ≥ 0∧0 = 0∧[(-1)bso_12] ≥ 0)



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

    (7)    (i49[0] ≥ 0 ⇒ (UIncreasing(COND_LOAD635(>(i49[0], 0), i49[0], i36[0])), ≥)∧[bni_11] = 0∧[(-1)bni_11 + (-1)Bound*bni_11] ≥ 0∧0 = 0∧[(-1)bso_12] ≥ 0)







For Pair COND_LOAD635(TRUE, i49, i36) → LOAD635(+(i49, -1), i36) the following chains were created:
  • We consider the chain COND_LOAD635(TRUE, i49[1], i36[1]) → LOAD635(+(i49[1], -1), i36[1]) which results in the following constraint:

    (8)    (COND_LOAD635(TRUE, i49[1], i36[1])≥NonInfC∧COND_LOAD635(TRUE, i49[1], i36[1])≥LOAD635(+(i49[1], -1), i36[1])∧(UIncreasing(LOAD635(+(i49[1], -1), i36[1])), ≥))



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

    (9)    ((UIncreasing(LOAD635(+(i49[1], -1), i36[1])), ≥)∧[(-1)bso_14] ≥ 0)



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

    (10)    ((UIncreasing(LOAD635(+(i49[1], -1), i36[1])), ≥)∧[(-1)bso_14] ≥ 0)



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

    (11)    ((UIncreasing(LOAD635(+(i49[1], -1), i36[1])), ≥)∧[(-1)bso_14] ≥ 0)



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

    (12)    ((UIncreasing(LOAD635(+(i49[1], -1), i36[1])), ≥)∧0 = 0∧0 = 0∧[(-1)bso_14] ≥ 0)







For Pair LOAD635(i50, i57) → COND_LOAD6351(&&(>(i57, 0), <=(i50, 0)), i50, i57) the following chains were created:
  • We consider the chain LOAD635(i50[2], i57[2]) → COND_LOAD6351(&&(>(i57[2], 0), <=(i50[2], 0)), i50[2], i57[2]), COND_LOAD6351(TRUE, i50[3], i57[3]) → LOAD635(i50[3], +(i57[3], -1)) which results in the following constraint:

    (13)    (&&(>(i57[2], 0), <=(i50[2], 0))=TRUE∧i50[2]=i50[3]∧i57[2]=i57[3] ⇒ LOAD635(i50[2], i57[2])≥NonInfC∧LOAD635(i50[2], i57[2])≥COND_LOAD6351(&&(>(i57[2], 0), <=(i50[2], 0)), i50[2], i57[2])∧(UIncreasing(COND_LOAD6351(&&(>(i57[2], 0), <=(i50[2], 0)), i50[2], i57[2])), ≥))



    We simplified constraint (13) using rules (IV), (IDP_BOOLEAN) which results in the following new constraint:

    (14)    (>(i57[2], 0)=TRUE∧<=(i50[2], 0)=TRUE ⇒ LOAD635(i50[2], i57[2])≥NonInfC∧LOAD635(i50[2], i57[2])≥COND_LOAD6351(&&(>(i57[2], 0), <=(i50[2], 0)), i50[2], i57[2])∧(UIncreasing(COND_LOAD6351(&&(>(i57[2], 0), <=(i50[2], 0)), i50[2], i57[2])), ≥))



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

    (15)    (i57[2] + [-1] ≥ 0∧[-1]i50[2] ≥ 0 ⇒ (UIncreasing(COND_LOAD6351(&&(>(i57[2], 0), <=(i50[2], 0)), i50[2], i57[2])), ≥)∧[(-1)bni_15 + (-1)Bound*bni_15] + [bni_15]i57[2] ≥ 0∧[(-1)bso_16] ≥ 0)



    We simplified constraint (15) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:

    (16)    (i57[2] + [-1] ≥ 0∧[-1]i50[2] ≥ 0 ⇒ (UIncreasing(COND_LOAD6351(&&(>(i57[2], 0), <=(i50[2], 0)), i50[2], i57[2])), ≥)∧[(-1)bni_15 + (-1)Bound*bni_15] + [bni_15]i57[2] ≥ 0∧[(-1)bso_16] ≥ 0)



    We simplified constraint (16) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:

    (17)    (i57[2] + [-1] ≥ 0∧[-1]i50[2] ≥ 0 ⇒ (UIncreasing(COND_LOAD6351(&&(>(i57[2], 0), <=(i50[2], 0)), i50[2], i57[2])), ≥)∧[(-1)bni_15 + (-1)Bound*bni_15] + [bni_15]i57[2] ≥ 0∧[(-1)bso_16] ≥ 0)



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

    (18)    (i57[2] ≥ 0∧[-1]i50[2] ≥ 0 ⇒ (UIncreasing(COND_LOAD6351(&&(>(i57[2], 0), <=(i50[2], 0)), i50[2], i57[2])), ≥)∧[(-1)Bound*bni_15] + [bni_15]i57[2] ≥ 0∧[(-1)bso_16] ≥ 0)



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

    (19)    (i57[2] ≥ 0∧i50[2] ≥ 0 ⇒ (UIncreasing(COND_LOAD6351(&&(>(i57[2], 0), <=(i50[2], 0)), i50[2], i57[2])), ≥)∧[(-1)Bound*bni_15] + [bni_15]i57[2] ≥ 0∧[(-1)bso_16] ≥ 0)







For Pair COND_LOAD6351(TRUE, i50, i57) → LOAD635(i50, +(i57, -1)) the following chains were created:
  • We consider the chain COND_LOAD6351(TRUE, i50[3], i57[3]) → LOAD635(i50[3], +(i57[3], -1)) which results in the following constraint:

    (20)    (COND_LOAD6351(TRUE, i50[3], i57[3])≥NonInfC∧COND_LOAD6351(TRUE, i50[3], i57[3])≥LOAD635(i50[3], +(i57[3], -1))∧(UIncreasing(LOAD635(i50[3], +(i57[3], -1))), ≥))



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

    (21)    ((UIncreasing(LOAD635(i50[3], +(i57[3], -1))), ≥)∧[1 + (-1)bso_18] ≥ 0)



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

    (22)    ((UIncreasing(LOAD635(i50[3], +(i57[3], -1))), ≥)∧[1 + (-1)bso_18] ≥ 0)



    We simplified constraint (22) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:

    (23)    ((UIncreasing(LOAD635(i50[3], +(i57[3], -1))), ≥)∧[1 + (-1)bso_18] ≥ 0)



    We simplified constraint (23) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint:

    (24)    ((UIncreasing(LOAD635(i50[3], +(i57[3], -1))), ≥)∧0 = 0∧0 = 0∧[1 + (-1)bso_18] ≥ 0)







To summarize, we get the following constraints P≥ for the following pairs.
  • LOAD635(i49, i36) → COND_LOAD635(>(i49, 0), i49, i36)
    • (i49[0] ≥ 0 ⇒ (UIncreasing(COND_LOAD635(>(i49[0], 0), i49[0], i36[0])), ≥)∧[bni_11] = 0∧[(-1)bni_11 + (-1)Bound*bni_11] ≥ 0∧0 = 0∧[(-1)bso_12] ≥ 0)

  • COND_LOAD635(TRUE, i49, i36) → LOAD635(+(i49, -1), i36)
    • ((UIncreasing(LOAD635(+(i49[1], -1), i36[1])), ≥)∧0 = 0∧0 = 0∧[(-1)bso_14] ≥ 0)

  • LOAD635(i50, i57) → COND_LOAD6351(&&(>(i57, 0), <=(i50, 0)), i50, i57)
    • (i57[2] ≥ 0∧i50[2] ≥ 0 ⇒ (UIncreasing(COND_LOAD6351(&&(>(i57[2], 0), <=(i50[2], 0)), i50[2], i57[2])), ≥)∧[(-1)Bound*bni_15] + [bni_15]i57[2] ≥ 0∧[(-1)bso_16] ≥ 0)

  • COND_LOAD6351(TRUE, i50, i57) → LOAD635(i50, +(i57, -1))
    • ((UIncreasing(LOAD635(i50[3], +(i57[3], -1))), ≥)∧0 = 0∧0 = 0∧[1 + (-1)bso_18] ≥ 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(LOAD635(x1, x2)) = [-1] + x2   
POL(COND_LOAD635(x1, x2, x3)) = [-1] + x3   
POL(>(x1, x2)) = [-1]   
POL(0) = 0   
POL(+(x1, x2)) = x1 + x2   
POL(-1) = [-1]   
POL(COND_LOAD6351(x1, x2, x3)) = [-1] + x3   
POL(&&(x1, x2)) = [-1]   
POL(<=(x1, x2)) = [-1]   

The following pairs are in P>:

COND_LOAD6351(TRUE, i50[3], i57[3]) → LOAD635(i50[3], +(i57[3], -1))

The following pairs are in Pbound:

LOAD635(i50[2], i57[2]) → COND_LOAD6351(&&(>(i57[2], 0), <=(i50[2], 0)), i50[2], i57[2])

The following pairs are in P≥:

LOAD635(i49[0], i36[0]) → COND_LOAD635(>(i49[0], 0), i49[0], i36[0])
COND_LOAD635(TRUE, i49[1], i36[1]) → LOAD635(+(i49[1], -1), i36[1])
LOAD635(i50[2], i57[2]) → COND_LOAD6351(&&(>(i57[2], 0), <=(i50[2], 0)), i50[2], i57[2])

There are no usable rules.

(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:

Integer, Boolean


R is empty.

The integer pair graph contains the following rules and edges:
(0): LOAD635(i49[0], i36[0]) → COND_LOAD635(i49[0] > 0, i49[0], i36[0])
(1): COND_LOAD635(TRUE, i49[1], i36[1]) → LOAD635(i49[1] + -1, i36[1])
(2): LOAD635(i50[2], i57[2]) → COND_LOAD6351(i57[2] > 0 && i50[2] <= 0, i50[2], i57[2])

(1) -> (0), if ((i49[1] + -1 →* i49[0])∧(i36[1] →* i36[0]))


(0) -> (1), if ((i36[0] →* i36[1])∧(i49[0] →* i49[1])∧(i49[0] > 0 →* TRUE))


(1) -> (2), if ((i49[1] + -1 →* i50[2])∧(i36[1] →* i57[2]))



The set Q consists of the following terms:
Load635(x0, x1)
Cond_Load635(TRUE, x0, x1)
Cond_Load6351(TRUE, x0, x1)

(12) IDependencyGraphProof (EQUIVALENT transformation)

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

(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:

Integer


R is empty.

The integer pair graph contains the following rules and edges:
(1): COND_LOAD635(TRUE, i49[1], i36[1]) → LOAD635(i49[1] + -1, i36[1])
(0): LOAD635(i49[0], i36[0]) → COND_LOAD635(i49[0] > 0, i49[0], i36[0])

(1) -> (0), if ((i49[1] + -1 →* i49[0])∧(i36[1] →* i36[0]))


(0) -> (1), if ((i36[0] →* i36[1])∧(i49[0] →* i49[1])∧(i49[0] > 0 →* TRUE))



The set Q consists of the following terms:
Load635(x0, x1)
Cond_Load635(TRUE, x0, x1)
Cond_Load6351(TRUE, x0, x1)

(14) 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_LOAD635(TRUE, i49[1], i36[1]) → LOAD635(+(i49[1], -1), i36[1]) the following chains were created:
  • We consider the chain COND_LOAD635(TRUE, i49[1], i36[1]) → LOAD635(+(i49[1], -1), i36[1]) which results in the following constraint:

    (1)    (COND_LOAD635(TRUE, i49[1], i36[1])≥NonInfC∧COND_LOAD635(TRUE, i49[1], i36[1])≥LOAD635(+(i49[1], -1), i36[1])∧(UIncreasing(LOAD635(+(i49[1], -1), i36[1])), ≥))



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

    (2)    ((UIncreasing(LOAD635(+(i49[1], -1), i36[1])), ≥)∧[1 + (-1)bso_7] ≥ 0)



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

    (3)    ((UIncreasing(LOAD635(+(i49[1], -1), i36[1])), ≥)∧[1 + (-1)bso_7] ≥ 0)



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

    (4)    ((UIncreasing(LOAD635(+(i49[1], -1), i36[1])), ≥)∧[1 + (-1)bso_7] ≥ 0)



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

    (5)    ((UIncreasing(LOAD635(+(i49[1], -1), i36[1])), ≥)∧0 = 0∧[1 + (-1)bso_7] ≥ 0)







For Pair LOAD635(i49[0], i36[0]) → COND_LOAD635(>(i49[0], 0), i49[0], i36[0]) the following chains were created:
  • We consider the chain LOAD635(i49[0], i36[0]) → COND_LOAD635(>(i49[0], 0), i49[0], i36[0]), COND_LOAD635(TRUE, i49[1], i36[1]) → LOAD635(+(i49[1], -1), i36[1]) which results in the following constraint:

    (6)    (i36[0]=i36[1]∧i49[0]=i49[1]∧>(i49[0], 0)=TRUE ⇒ LOAD635(i49[0], i36[0])≥NonInfC∧LOAD635(i49[0], i36[0])≥COND_LOAD635(>(i49[0], 0), i49[0], i36[0])∧(UIncreasing(COND_LOAD635(>(i49[0], 0), i49[0], i36[0])), ≥))



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

    (7)    (>(i49[0], 0)=TRUE ⇒ LOAD635(i49[0], i36[0])≥NonInfC∧LOAD635(i49[0], i36[0])≥COND_LOAD635(>(i49[0], 0), i49[0], i36[0])∧(UIncreasing(COND_LOAD635(>(i49[0], 0), i49[0], i36[0])), ≥))



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

    (8)    (i49[0] + [-1] ≥ 0 ⇒ (UIncreasing(COND_LOAD635(>(i49[0], 0), i49[0], i36[0])), ≥)∧[bni_8 + (-1)Bound*bni_8] + [bni_8]i49[0] ≥ 0∧[(-1)bso_9] ≥ 0)



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

    (9)    (i49[0] + [-1] ≥ 0 ⇒ (UIncreasing(COND_LOAD635(>(i49[0], 0), i49[0], i36[0])), ≥)∧[bni_8 + (-1)Bound*bni_8] + [bni_8]i49[0] ≥ 0∧[(-1)bso_9] ≥ 0)



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

    (10)    (i49[0] + [-1] ≥ 0 ⇒ (UIncreasing(COND_LOAD635(>(i49[0], 0), i49[0], i36[0])), ≥)∧[bni_8 + (-1)Bound*bni_8] + [bni_8]i49[0] ≥ 0∧[(-1)bso_9] ≥ 0)



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

    (11)    (i49[0] ≥ 0 ⇒ (UIncreasing(COND_LOAD635(>(i49[0], 0), i49[0], i36[0])), ≥)∧[(2)bni_8 + (-1)Bound*bni_8] + [bni_8]i49[0] ≥ 0∧[(-1)bso_9] ≥ 0)







To summarize, we get the following constraints P≥ for the following pairs.
  • COND_LOAD635(TRUE, i49[1], i36[1]) → LOAD635(+(i49[1], -1), i36[1])
    • ((UIncreasing(LOAD635(+(i49[1], -1), i36[1])), ≥)∧0 = 0∧[1 + (-1)bso_7] ≥ 0)

  • LOAD635(i49[0], i36[0]) → COND_LOAD635(>(i49[0], 0), i49[0], i36[0])
    • (i49[0] ≥ 0 ⇒ (UIncreasing(COND_LOAD635(>(i49[0], 0), i49[0], i36[0])), ≥)∧[(2)bni_8 + (-1)Bound*bni_8] + [bni_8]i49[0] ≥ 0∧[(-1)bso_9] ≥ 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(COND_LOAD635(x1, x2, x3)) = [1] + x2   
POL(LOAD635(x1, x2)) = [1] + x1   
POL(+(x1, x2)) = x1 + x2   
POL(-1) = [-1]   
POL(>(x1, x2)) = [1]   
POL(0) = 0   

The following pairs are in P>:

COND_LOAD635(TRUE, i49[1], i36[1]) → LOAD635(+(i49[1], -1), i36[1])

The following pairs are in Pbound:

LOAD635(i49[0], i36[0]) → COND_LOAD635(>(i49[0], 0), i49[0], i36[0])

The following pairs are in P≥:

LOAD635(i49[0], i36[0]) → COND_LOAD635(>(i49[0], 0), i49[0], i36[0])

There are no usable rules.

(15) Complex Obligation (AND)

(16) 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:
(0): LOAD635(i49[0], i36[0]) → COND_LOAD635(i49[0] > 0, i49[0], i36[0])


The set Q consists of the following terms:
Load635(x0, x1)
Cond_Load635(TRUE, x0, x1)
Cond_Load6351(TRUE, x0, x1)

(17) IDependencyGraphProof (EQUIVALENT transformation)

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

(18) TRUE

(19) Obligation:

IDP problem:
The following function symbols are pre-defined:
!=~Neq: (Integer, Integer) -> Boolean
*~Mul: (Integer, Integer) -> Integer
>=~Ge: (Integer, Integer) -> Boolean
-1~UnaryMinus: (Integer) -> Integer
|~Bwor: (Integer, Integer) -> Integer
/~Div: (Integer, Integer) -> Integer
=~Eq: (Integer, Integer) -> Boolean
~Bwxor: (Integer, Integer) -> Integer
||~Lor: (Boolean, Boolean) -> Boolean
!~Lnot: (Boolean) -> Boolean
<~Lt: (Integer, Integer) -> Boolean
-~Sub: (Integer, Integer) -> Integer
<=~Le: (Integer, Integer) -> Boolean
>~Gt: (Integer, Integer) -> Boolean
~~Bwnot: (Integer) -> Integer
%~Mod: (Integer, Integer) -> Integer
&~Bwand: (Integer, Integer) -> Integer
+~Add: (Integer, Integer) -> Integer
&&~Land: (Boolean, Boolean) -> Boolean


The following domains are used:

Integer


R is empty.

The integer pair graph contains the following rules and edges:
(1): COND_LOAD635(TRUE, i49[1], i36[1]) → LOAD635(i49[1] + -1, i36[1])


The set Q consists of the following terms:
Load635(x0, x1)
Cond_Load635(TRUE, x0, x1)
Cond_Load6351(TRUE, x0, x1)

(20) IDependencyGraphProof (EQUIVALENT transformation)

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

(21) TRUE

(22) 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:
(0): LOAD635(i49[0], i36[0]) → COND_LOAD635(i49[0] > 0, i49[0], i36[0])
(1): COND_LOAD635(TRUE, i49[1], i36[1]) → LOAD635(i49[1] + -1, i36[1])
(3): COND_LOAD6351(TRUE, i50[3], i57[3]) → LOAD635(i50[3], i57[3] + -1)

(1) -> (0), if ((i49[1] + -1 →* i49[0])∧(i36[1] →* i36[0]))


(3) -> (0), if ((i57[3] + -1 →* i36[0])∧(i50[3] →* i49[0]))


(0) -> (1), if ((i36[0] →* i36[1])∧(i49[0] →* i49[1])∧(i49[0] > 0 →* TRUE))



The set Q consists of the following terms:
Load635(x0, x1)
Cond_Load635(TRUE, x0, x1)
Cond_Load6351(TRUE, x0, x1)

(23) IDependencyGraphProof (EQUIVALENT transformation)

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

(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:

Integer


R is empty.

The integer pair graph contains the following rules and edges:
(1): COND_LOAD635(TRUE, i49[1], i36[1]) → LOAD635(i49[1] + -1, i36[1])
(0): LOAD635(i49[0], i36[0]) → COND_LOAD635(i49[0] > 0, i49[0], i36[0])

(1) -> (0), if ((i49[1] + -1 →* i49[0])∧(i36[1] →* i36[0]))


(0) -> (1), if ((i36[0] →* i36[1])∧(i49[0] →* i49[1])∧(i49[0] > 0 →* TRUE))



The set Q consists of the following terms:
Load635(x0, x1)
Cond_Load635(TRUE, x0, x1)
Cond_Load6351(TRUE, x0, x1)

(25) 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_LOAD635(TRUE, i49[1], i36[1]) → LOAD635(+(i49[1], -1), i36[1]) the following chains were created:
  • We consider the chain COND_LOAD635(TRUE, i49[1], i36[1]) → LOAD635(+(i49[1], -1), i36[1]) which results in the following constraint:

    (1)    (COND_LOAD635(TRUE, i49[1], i36[1])≥NonInfC∧COND_LOAD635(TRUE, i49[1], i36[1])≥LOAD635(+(i49[1], -1), i36[1])∧(UIncreasing(LOAD635(+(i49[1], -1), i36[1])), ≥))



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

    (2)    ((UIncreasing(LOAD635(+(i49[1], -1), i36[1])), ≥)∧[1 + (-1)bso_8] ≥ 0)



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

    (3)    ((UIncreasing(LOAD635(+(i49[1], -1), i36[1])), ≥)∧[1 + (-1)bso_8] ≥ 0)



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

    (4)    ((UIncreasing(LOAD635(+(i49[1], -1), i36[1])), ≥)∧[1 + (-1)bso_8] ≥ 0)



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

    (5)    ((UIncreasing(LOAD635(+(i49[1], -1), i36[1])), ≥)∧0 = 0∧[1 + (-1)bso_8] ≥ 0)







For Pair LOAD635(i49[0], i36[0]) → COND_LOAD635(>(i49[0], 0), i49[0], i36[0]) the following chains were created:
  • We consider the chain LOAD635(i49[0], i36[0]) → COND_LOAD635(>(i49[0], 0), i49[0], i36[0]), COND_LOAD635(TRUE, i49[1], i36[1]) → LOAD635(+(i49[1], -1), i36[1]) which results in the following constraint:

    (6)    (i36[0]=i36[1]∧i49[0]=i49[1]∧>(i49[0], 0)=TRUE ⇒ LOAD635(i49[0], i36[0])≥NonInfC∧LOAD635(i49[0], i36[0])≥COND_LOAD635(>(i49[0], 0), i49[0], i36[0])∧(UIncreasing(COND_LOAD635(>(i49[0], 0), i49[0], i36[0])), ≥))



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

    (7)    (>(i49[0], 0)=TRUE ⇒ LOAD635(i49[0], i36[0])≥NonInfC∧LOAD635(i49[0], i36[0])≥COND_LOAD635(>(i49[0], 0), i49[0], i36[0])∧(UIncreasing(COND_LOAD635(>(i49[0], 0), i49[0], i36[0])), ≥))



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

    (8)    (i49[0] + [-1] ≥ 0 ⇒ (UIncreasing(COND_LOAD635(>(i49[0], 0), i49[0], i36[0])), ≥)∧[bni_9 + (-1)Bound*bni_9] + [bni_9]i49[0] ≥ 0∧[(-1)bso_10] ≥ 0)



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

    (9)    (i49[0] + [-1] ≥ 0 ⇒ (UIncreasing(COND_LOAD635(>(i49[0], 0), i49[0], i36[0])), ≥)∧[bni_9 + (-1)Bound*bni_9] + [bni_9]i49[0] ≥ 0∧[(-1)bso_10] ≥ 0)



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

    (10)    (i49[0] + [-1] ≥ 0 ⇒ (UIncreasing(COND_LOAD635(>(i49[0], 0), i49[0], i36[0])), ≥)∧[bni_9 + (-1)Bound*bni_9] + [bni_9]i49[0] ≥ 0∧[(-1)bso_10] ≥ 0)



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

    (11)    (i49[0] ≥ 0 ⇒ (UIncreasing(COND_LOAD635(>(i49[0], 0), i49[0], i36[0])), ≥)∧[(2)bni_9 + (-1)Bound*bni_9] + [bni_9]i49[0] ≥ 0∧[(-1)bso_10] ≥ 0)







To summarize, we get the following constraints P≥ for the following pairs.
  • COND_LOAD635(TRUE, i49[1], i36[1]) → LOAD635(+(i49[1], -1), i36[1])
    • ((UIncreasing(LOAD635(+(i49[1], -1), i36[1])), ≥)∧0 = 0∧[1 + (-1)bso_8] ≥ 0)

  • LOAD635(i49[0], i36[0]) → COND_LOAD635(>(i49[0], 0), i49[0], i36[0])
    • (i49[0] ≥ 0 ⇒ (UIncreasing(COND_LOAD635(>(i49[0], 0), i49[0], i36[0])), ≥)∧[(2)bni_9 + (-1)Bound*bni_9] + [bni_9]i49[0] ≥ 0∧[(-1)bso_10] ≥ 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(COND_LOAD635(x1, x2, x3)) = [1] + x2   
POL(LOAD635(x1, x2)) = [1] + x1   
POL(+(x1, x2)) = x1 + x2   
POL(-1) = [-1]   
POL(>(x1, x2)) = [-1]   
POL(0) = 0   

The following pairs are in P>:

COND_LOAD635(TRUE, i49[1], i36[1]) → LOAD635(+(i49[1], -1), i36[1])

The following pairs are in Pbound:

LOAD635(i49[0], i36[0]) → COND_LOAD635(>(i49[0], 0), i49[0], i36[0])

The following pairs are in P≥:

LOAD635(i49[0], i36[0]) → COND_LOAD635(>(i49[0], 0), i49[0], i36[0])

There are no usable rules.

(26) Complex Obligation (AND)

(27) 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:
(0): LOAD635(i49[0], i36[0]) → COND_LOAD635(i49[0] > 0, i49[0], i36[0])


The set Q consists of the following terms:
Load635(x0, x1)
Cond_Load635(TRUE, x0, x1)
Cond_Load6351(TRUE, x0, x1)

(28) IDependencyGraphProof (EQUIVALENT transformation)

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

(29) TRUE

(30) Obligation:

IDP problem:
The following function symbols are pre-defined:
!=~Neq: (Integer, Integer) -> Boolean
*~Mul: (Integer, Integer) -> Integer
>=~Ge: (Integer, Integer) -> Boolean
-1~UnaryMinus: (Integer) -> Integer
|~Bwor: (Integer, Integer) -> Integer
/~Div: (Integer, Integer) -> Integer
=~Eq: (Integer, Integer) -> Boolean
~Bwxor: (Integer, Integer) -> Integer
||~Lor: (Boolean, Boolean) -> Boolean
!~Lnot: (Boolean) -> Boolean
<~Lt: (Integer, Integer) -> Boolean
-~Sub: (Integer, Integer) -> Integer
<=~Le: (Integer, Integer) -> Boolean
>~Gt: (Integer, Integer) -> Boolean
~~Bwnot: (Integer) -> Integer
%~Mod: (Integer, Integer) -> Integer
&~Bwand: (Integer, Integer) -> Integer
+~Add: (Integer, Integer) -> Integer
&&~Land: (Boolean, Boolean) -> Boolean


The following domains are used:

Integer


R is empty.

The integer pair graph contains the following rules and edges:
(1): COND_LOAD635(TRUE, i49[1], i36[1]) → LOAD635(i49[1] + -1, i36[1])


The set Q consists of the following terms:
Load635(x0, x1)
Cond_Load635(TRUE, x0, x1)
Cond_Load6351(TRUE, x0, x1)

(31) IDependencyGraphProof (EQUIVALENT transformation)

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

(32) TRUE