Termination of the given JBC could be proven:

0 JBC
↳1 JBC2FIG (⇐)
↳2 FIGraph
↳3 FIGtoITRSProof (⇐)
↳4 ITRS
↳5 DuplicateArgsRemoverProof (⇔)
↳6 ITRS
↳7 ITRStoIDPProof (⇔)
↳8 IDP
↳9 UsableRulesProof (⇔)
↳10 IDP
↳11 IDPNonInfProof (⇐)
↳12 AND
    ↳13 IDP
        ↳14 IDependencyGraphProof (⇔)
        ↳15 TRUE
    ↳16 IDP
        ↳17 IDependencyGraphProof (⇔)
        ↳18 TRUE

(0) Obligation:

JBC Problem based on JBC Program:
Manifest-Version: 1.0 Created-By: 1.6.0_16 (Sun Microsystems Inc.) Main-Class: LogIterative 
public class LogIterative {
  public static int log(int x, int y) {
    int res = 0;
    while (x >= y && y > 1) {
      res++;
      x = x/y;
    }
    return res;
  } 

  public static void main(String[] args) {
    Random.args = args;
    int x = Random.random();
    int y = Random.random();
    log(x, y);
  }
}


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 206 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:
Load1335(i349, i315, i349, i317) → Cond_Load1335(i349 > 1 && i315 >= i349 && i317 + 1 > 0, i349, i315, i349, i317)
Cond_Load1335(TRUE, i349, i315, i349, i317) → Load1335(i349, i315 / i349, i349, i317 + 1)
The set Q consists of the following terms:
Load1335(x0, x1, x0, x2)
Cond_Load1335(TRUE, x0, x1, x0, x2)

(5) DuplicateArgsRemoverProof (EQUIVALENT transformation)

Some arguments are removed because they only appear as duplicates.
We removed arguments according to the following replacements:

Load1335(x1, x2, x3, x4) → Load1335(x2, x3, x4)
Cond_Load1335(x1, x2, x3, x4, x5) → Cond_Load1335(x1, x3, x4, x5)

(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:
Load1335(i315, i349, i317) → Cond_Load1335(i349 > 1 && i315 >= i349 && i317 + 1 > 0, i315, i349, i317)
Cond_Load1335(TRUE, i315, i349, i317) → Load1335(i315 / i349, i349, i317 + 1)
The set Q consists of the following terms:
Load1335(x0, x1, x2)
Cond_Load1335(TRUE, x0, x1, x2)

(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:
Load1335(i315, i349, i317) → Cond_Load1335(i349 > 1 && i315 >= i349 && i317 + 1 > 0, i315, i349, i317)
Cond_Load1335(TRUE, i315, i349, i317) → Load1335(i315 / i349, i349, i317 + 1)

The integer pair graph contains the following rules and edges:
(0): LOAD1335(i315[0], i349[0], i317[0]) → COND_LOAD1335(i349[0] > 1 && i315[0] >= i349[0] && i317[0] + 1 > 0, i315[0], i349[0], i317[0])
(1): COND_LOAD1335(TRUE, i315[1], i349[1], i317[1]) → LOAD1335(i315[1] / i349[1], i349[1], i317[1] + 1)

(0) -> (1), if ((i315[0] →* i315[1])∧(i349[0] > 1 && i315[0] >= i349[0] && i317[0] + 1 > 0 →* TRUE)∧(i317[0] →* i317[1])∧(i349[0] →* i349[1]))


(1) -> (0), if ((i315[1] / i349[1] →* i315[0])∧(i317[1] + 1 →* i317[0])∧(i349[1] →* i349[0]))



The set Q consists of the following terms:
Load1335(x0, x1, x2)
Cond_Load1335(TRUE, x0, x1, x2)

(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): LOAD1335(i315[0], i349[0], i317[0]) → COND_LOAD1335(i349[0] > 1 && i315[0] >= i349[0] && i317[0] + 1 > 0, i315[0], i349[0], i317[0])
(1): COND_LOAD1335(TRUE, i315[1], i349[1], i317[1]) → LOAD1335(i315[1] / i349[1], i349[1], i317[1] + 1)

(0) -> (1), if ((i315[0] →* i315[1])∧(i349[0] > 1 && i315[0] >= i349[0] && i317[0] + 1 > 0 →* TRUE)∧(i317[0] →* i317[1])∧(i349[0] →* i349[1]))


(1) -> (0), if ((i315[1] / i349[1] →* i315[0])∧(i317[1] + 1 →* i317[0])∧(i349[1] →* i349[0]))



The set Q consists of the following terms:
Load1335(x0, x1, x2)
Cond_Load1335(TRUE, x0, x1, x2)

(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 LOAD1335(i315, i349, i317) → COND_LOAD1335(&&(&&(>(i349, 1), >=(i315, i349)), >(+(i317, 1), 0)), i315, i349, i317) the following chains were created:
  • We consider the chain LOAD1335(i315[0], i349[0], i317[0]) → COND_LOAD1335(&&(&&(>(i349[0], 1), >=(i315[0], i349[0])), >(+(i317[0], 1), 0)), i315[0], i349[0], i317[0]), COND_LOAD1335(TRUE, i315[1], i349[1], i317[1]) → LOAD1335(/(i315[1], i349[1]), i349[1], +(i317[1], 1)) which results in the following constraint:

    (1)    (i315[0]=i315[1]∧&&(&&(>(i349[0], 1), >=(i315[0], i349[0])), >(+(i317[0], 1), 0))=TRUE∧i317[0]=i317[1]∧i349[0]=i349[1] ⇒ LOAD1335(i315[0], i349[0], i317[0])≥NonInfC∧LOAD1335(i315[0], i349[0], i317[0])≥COND_LOAD1335(&&(&&(>(i349[0], 1), >=(i315[0], i349[0])), >(+(i317[0], 1), 0)), i315[0], i349[0], i317[0])∧(UIncreasing(COND_LOAD1335(&&(&&(>(i349[0], 1), >=(i315[0], i349[0])), >(+(i317[0], 1), 0)), i315[0], i349[0], i317[0])), ≥))



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

    (2)    (>(+(i317[0], 1), 0)=TRUE∧>(i349[0], 1)=TRUE∧>=(i315[0], i349[0])=TRUE ⇒ LOAD1335(i315[0], i349[0], i317[0])≥NonInfC∧LOAD1335(i315[0], i349[0], i317[0])≥COND_LOAD1335(&&(&&(>(i349[0], 1), >=(i315[0], i349[0])), >(+(i317[0], 1), 0)), i315[0], i349[0], i317[0])∧(UIncreasing(COND_LOAD1335(&&(&&(>(i349[0], 1), >=(i315[0], i349[0])), >(+(i317[0], 1), 0)), i315[0], i349[0], i317[0])), ≥))



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

    (3)    (i317[0] ≥ 0∧i349[0] + [-2] ≥ 0∧i315[0] + [-1]i349[0] ≥ 0 ⇒ (UIncreasing(COND_LOAD1335(&&(&&(>(i349[0], 1), >=(i315[0], i349[0])), >(+(i317[0], 1), 0)), i315[0], i349[0], i317[0])), ≥)∧[(-1)bni_15 + (-1)Bound*bni_15] + [bni_15]i315[0] ≥ 0∧[(-1)bso_16] ≥ 0)



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

    (4)    (i317[0] ≥ 0∧i349[0] + [-2] ≥ 0∧i315[0] + [-1]i349[0] ≥ 0 ⇒ (UIncreasing(COND_LOAD1335(&&(&&(>(i349[0], 1), >=(i315[0], i349[0])), >(+(i317[0], 1), 0)), i315[0], i349[0], i317[0])), ≥)∧[(-1)bni_15 + (-1)Bound*bni_15] + [bni_15]i315[0] ≥ 0∧[(-1)bso_16] ≥ 0)



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

    (5)    (i317[0] ≥ 0∧i349[0] + [-2] ≥ 0∧i315[0] + [-1]i349[0] ≥ 0 ⇒ (UIncreasing(COND_LOAD1335(&&(&&(>(i349[0], 1), >=(i315[0], i349[0])), >(+(i317[0], 1), 0)), i315[0], i349[0], i317[0])), ≥)∧[(-1)bni_15 + (-1)Bound*bni_15] + [bni_15]i315[0] ≥ 0∧[(-1)bso_16] ≥ 0)



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

    (6)    (i317[0] ≥ 0∧i349[0] ≥ 0∧i315[0] + [-2] + [-1]i349[0] ≥ 0 ⇒ (UIncreasing(COND_LOAD1335(&&(&&(>(i349[0], 1), >=(i315[0], i349[0])), >(+(i317[0], 1), 0)), i315[0], i349[0], i317[0])), ≥)∧[(-1)bni_15 + (-1)Bound*bni_15] + [bni_15]i315[0] ≥ 0∧[(-1)bso_16] ≥ 0)



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

    (7)    (i317[0] ≥ 0∧i349[0] ≥ 0∧i315[0] ≥ 0 ⇒ (UIncreasing(COND_LOAD1335(&&(&&(>(i349[0], 1), >=(i315[0], i349[0])), >(+(i317[0], 1), 0)), i315[0], i349[0], i317[0])), ≥)∧[bni_15 + (-1)Bound*bni_15] + [bni_15]i349[0] + [bni_15]i315[0] ≥ 0∧[(-1)bso_16] ≥ 0)







For Pair COND_LOAD1335(TRUE, i315, i349, i317) → LOAD1335(/(i315, i349), i349, +(i317, 1)) the following chains were created:
  • We consider the chain LOAD1335(i315[0], i349[0], i317[0]) → COND_LOAD1335(&&(&&(>(i349[0], 1), >=(i315[0], i349[0])), >(+(i317[0], 1), 0)), i315[0], i349[0], i317[0]), COND_LOAD1335(TRUE, i315[1], i349[1], i317[1]) → LOAD1335(/(i315[1], i349[1]), i349[1], +(i317[1], 1)), LOAD1335(i315[0], i349[0], i317[0]) → COND_LOAD1335(&&(&&(>(i349[0], 1), >=(i315[0], i349[0])), >(+(i317[0], 1), 0)), i315[0], i349[0], i317[0]) which results in the following constraint:

    (8)    (i315[0]=i315[1]∧&&(&&(>(i349[0], 1), >=(i315[0], i349[0])), >(+(i317[0], 1), 0))=TRUE∧i317[0]=i317[1]∧i349[0]=i349[1]∧/(i315[1], i349[1])=i315[0]1∧+(i317[1], 1)=i317[0]1∧i349[1]=i349[0]1 ⇒ COND_LOAD1335(TRUE, i315[1], i349[1], i317[1])≥NonInfC∧COND_LOAD1335(TRUE, i315[1], i349[1], i317[1])≥LOAD1335(/(i315[1], i349[1]), i349[1], +(i317[1], 1))∧(UIncreasing(LOAD1335(/(i315[1], i349[1]), i349[1], +(i317[1], 1))), ≥))



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

    (9)    (>(+(i317[0], 1), 0)=TRUE∧>(i349[0], 1)=TRUE∧>=(i315[0], i349[0])=TRUE ⇒ COND_LOAD1335(TRUE, i315[0], i349[0], i317[0])≥NonInfC∧COND_LOAD1335(TRUE, i315[0], i349[0], i317[0])≥LOAD1335(/(i315[0], i349[0]), i349[0], +(i317[0], 1))∧(UIncreasing(LOAD1335(/(i315[1], i349[1]), i349[1], +(i317[1], 1))), ≥))



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

    (10)    (i317[0] ≥ 0∧i349[0] + [-2] ≥ 0∧i315[0] + [-1]i349[0] ≥ 0 ⇒ (UIncreasing(LOAD1335(/(i315[1], i349[1]), i349[1], +(i317[1], 1))), ≥)∧[(-1)bni_17 + (-1)Bound*bni_17] + [bni_17]i315[0] ≥ 0∧[(-1)bso_21] + i315[0] + [-1]max{i315[0], [-1]i315[0]} + min{max{i349[0], [-1]i349[0]} + [-1], max{i315[0], [-1]i315[0]}} ≥ 0)



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

    (11)    (i317[0] ≥ 0∧i349[0] + [-2] ≥ 0∧i315[0] + [-1]i349[0] ≥ 0 ⇒ (UIncreasing(LOAD1335(/(i315[1], i349[1]), i349[1], +(i317[1], 1))), ≥)∧[(-1)bni_17 + (-1)Bound*bni_17] + [bni_17]i315[0] ≥ 0∧[(-1)bso_21] + i315[0] + [-1]max{i315[0], [-1]i315[0]} + min{max{i349[0], [-1]i349[0]} + [-1], max{i315[0], [-1]i315[0]}} ≥ 0)



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

    (12)    (i317[0] ≥ 0∧i349[0] + [-2] ≥ 0∧i315[0] + [-1]i349[0] ≥ 0∧[2]i315[0] ≥ 0∧[2]i349[0] ≥ 0 ⇒ (UIncreasing(LOAD1335(/(i315[1], i349[1]), i349[1], +(i317[1], 1))), ≥)∧[(-1)bni_17 + (-1)Bound*bni_17] + [bni_17]i315[0] ≥ 0∧[-1 + (-1)bso_21] + i349[0] ≥ 0)



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

    (13)    (i317[0] ≥ 0∧i349[0] ≥ 0∧i315[0] + [-2] + [-1]i349[0] ≥ 0∧[2]i315[0] ≥ 0∧[4] + [2]i349[0] ≥ 0 ⇒ (UIncreasing(LOAD1335(/(i315[1], i349[1]), i349[1], +(i317[1], 1))), ≥)∧[(-1)bni_17 + (-1)Bound*bni_17] + [bni_17]i315[0] ≥ 0∧[1 + (-1)bso_21] + i349[0] ≥ 0)



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

    (14)    (i317[0] ≥ 0∧i349[0] ≥ 0∧i315[0] ≥ 0∧[4] + [2]i349[0] + [2]i315[0] ≥ 0∧[4] + [2]i349[0] ≥ 0 ⇒ (UIncreasing(LOAD1335(/(i315[1], i349[1]), i349[1], +(i317[1], 1))), ≥)∧[bni_17 + (-1)Bound*bni_17] + [bni_17]i349[0] + [bni_17]i315[0] ≥ 0∧[1 + (-1)bso_21] + i349[0] ≥ 0)



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

    (15)    (i317[0] ≥ 0∧i349[0] ≥ 0∧i315[0] ≥ 0∧[2] + i349[0] + i315[0] ≥ 0∧[2] + i349[0] ≥ 0 ⇒ (UIncreasing(LOAD1335(/(i315[1], i349[1]), i349[1], +(i317[1], 1))), ≥)∧[bni_17 + (-1)Bound*bni_17] + [bni_17]i349[0] + [bni_17]i315[0] ≥ 0∧[1 + (-1)bso_21] + i349[0] ≥ 0)







To summarize, we get the following constraints P≥ for the following pairs.
  • LOAD1335(i315, i349, i317) → COND_LOAD1335(&&(&&(>(i349, 1), >=(i315, i349)), >(+(i317, 1), 0)), i315, i349, i317)
    • (i317[0] ≥ 0∧i349[0] ≥ 0∧i315[0] ≥ 0 ⇒ (UIncreasing(COND_LOAD1335(&&(&&(>(i349[0], 1), >=(i315[0], i349[0])), >(+(i317[0], 1), 0)), i315[0], i349[0], i317[0])), ≥)∧[bni_15 + (-1)Bound*bni_15] + [bni_15]i349[0] + [bni_15]i315[0] ≥ 0∧[(-1)bso_16] ≥ 0)

  • COND_LOAD1335(TRUE, i315, i349, i317) → LOAD1335(/(i315, i349), i349, +(i317, 1))
    • (i317[0] ≥ 0∧i349[0] ≥ 0∧i315[0] ≥ 0∧[2] + i349[0] + i315[0] ≥ 0∧[2] + i349[0] ≥ 0 ⇒ (UIncreasing(LOAD1335(/(i315[1], i349[1]), i349[1], +(i317[1], 1))), ≥)∧[bni_17 + (-1)Bound*bni_17] + [bni_17]i349[0] + [bni_17]i315[0] ≥ 0∧[1 + (-1)bso_21] + i349[0] ≥ 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(LOAD1335(x1, x2, x3)) = [-1] + x1   
POL(COND_LOAD1335(x1, x2, x3, x4)) = [-1] + x2 + [-1]x1   
POL(&&(x1, x2)) = 0   
POL(>(x1, x2)) = [-1]   
POL(1) = [1]   
POL(>=(x1, x2)) = [-1]   
POL(+(x1, x2)) = x1 + x2   
POL(0) = 0   

Polynomial Interpretations with Context Sensitive Arithemetic Replacement
POL(TermCSAR-Mode @ Context)

POL(/(x1, i349[0])1 @ {LOAD1335_3/0}) = max{x1, [-1]x1} + [-1]min{max{x2, [-1]x2} + [-1], max{x1, [-1]x1}}   

The following pairs are in P>:

COND_LOAD1335(TRUE, i315[1], i349[1], i317[1]) → LOAD1335(/(i315[1], i349[1]), i349[1], +(i317[1], 1))

The following pairs are in Pbound:

LOAD1335(i315[0], i349[0], i317[0]) → COND_LOAD1335(&&(&&(>(i349[0], 1), >=(i315[0], i349[0])), >(+(i317[0], 1), 0)), i315[0], i349[0], i317[0])
COND_LOAD1335(TRUE, i315[1], i349[1], i317[1]) → LOAD1335(/(i315[1], i349[1]), i349[1], +(i317[1], 1))

The following pairs are in P≥:

LOAD1335(i315[0], i349[0], i317[0]) → COND_LOAD1335(&&(&&(>(i349[0], 1), >=(i315[0], i349[0])), >(+(i317[0], 1), 0)), i315[0], i349[0], i317[0])

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

&&(TRUE, TRUE)1 ↔ TRUE1
&&(TRUE, FALSE)1 ↔ FALSE1
&&(FALSE, TRUE)1 ↔ FALSE1
&&(FALSE, FALSE)1 ↔ FALSE1
/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): LOAD1335(i315[0], i349[0], i317[0]) → COND_LOAD1335(i349[0] > 1 && i315[0] >= i349[0] && i317[0] + 1 > 0, i315[0], i349[0], i317[0])


The set Q consists of the following terms:
Load1335(x0, x1, x2)
Cond_Load1335(TRUE, x0, x1, x2)

(14) IDependencyGraphProof (EQUIVALENT transformation)

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

(15) TRUE

(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:
none


R is empty.

The integer pair graph is empty.

The set Q consists of the following terms:
Load1335(x0, x1, x2)
Cond_Load1335(TRUE, x0, x1, x2)

(17) IDependencyGraphProof (EQUIVALENT transformation)

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

(18) TRUE