Termination of the given JBC could be proven:

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

(0) Obligation:

JBC Problem based on JBC Program:
Manifest-Version: 1.0 Created-By: 1.6.0_16 (Sun Microsystems Inc.) Main-Class: CountUpRound 
public class CountUpRound{
  public static int round (int x) {

    if (x % 2 == 0) return x;
    else return x+1;
  }


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



    while (x > y) {

      y = round(y+1);

    }


  }

}


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 201 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:
Load761(i11, i42) → Cond_Load761(!(i42 + 1 % 2 = 0) && i11 > i42, i11, i42)
Cond_Load761(TRUE, i11, i42) → Store1059(i11, i42 + 1 + 1)
Store1059(i11, i70) → Load761(i11, i70)
Load761(i11, i42) → Cond_Load7611(i11 > i42 && 0 = i42 + 1 % 2, i11, i42)
Cond_Load7611(TRUE, i11, i42) → Load761(i11, i42 + 1)
The set Q consists of the following terms:
Load761(x0, x1)
Cond_Load761(TRUE, x0, x1)
Store1059(x0, x1)
Cond_Load7611(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:

Boolean, Integer


The ITRS R consists of the following rules:
Load761(i11, i42) → Cond_Load761(!(i42 + 1 % 2 = 0) && i11 > i42, i11, i42)
Cond_Load761(TRUE, i11, i42) → Store1059(i11, i42 + 1 + 1)
Store1059(i11, i70) → Load761(i11, i70)
Load761(i11, i42) → Cond_Load7611(i11 > i42 && 0 = i42 + 1 % 2, i11, i42)
Cond_Load7611(TRUE, i11, i42) → Load761(i11, i42 + 1)

The integer pair graph contains the following rules and edges:
(0): LOAD761(i11[0], i42[0]) → COND_LOAD761(!(i42[0] + 1 % 2 = 0) && i11[0] > i42[0], i11[0], i42[0])
(1): COND_LOAD761(TRUE, i11[1], i42[1]) → STORE1059(i11[1], i42[1] + 1 + 1)
(2): STORE1059(i11[2], i70[2]) → LOAD761(i11[2], i70[2])
(3): LOAD761(i11[3], i42[3]) → COND_LOAD7611(i11[3] > i42[3] && 0 = i42[3] + 1 % 2, i11[3], i42[3])
(4): COND_LOAD7611(TRUE, i11[4], i42[4]) → LOAD761(i11[4], i42[4] + 1)

(0) -> (1), if ((i42[0] →* i42[1])∧(!(i42[0] + 1 % 2 = 0) && i11[0] > i42[0] →* TRUE)∧(i11[0] →* i11[1]))


(1) -> (2), if ((i42[1] + 1 + 1 →* i70[2])∧(i11[1] →* i11[2]))


(2) -> (0), if ((i70[2] →* i42[0])∧(i11[2] →* i11[0]))


(2) -> (3), if ((i70[2] →* i42[3])∧(i11[2] →* i11[3]))


(3) -> (4), if ((i11[3] →* i11[4])∧(i42[3] →* i42[4])∧(i11[3] > i42[3] && 0 = i42[3] + 1 % 2 →* TRUE))


(4) -> (0), if ((i11[4] →* i11[0])∧(i42[4] + 1 →* i42[0]))


(4) -> (3), if ((i42[4] + 1 →* i42[3])∧(i11[4] →* i11[3]))



The set Q consists of the following terms:
Load761(x0, x1)
Cond_Load761(TRUE, x0, x1)
Store1059(x0, x1)
Cond_Load7611(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:

Boolean, Integer


R is empty.

The integer pair graph contains the following rules and edges:
(0): LOAD761(i11[0], i42[0]) → COND_LOAD761(!(i42[0] + 1 % 2 = 0) && i11[0] > i42[0], i11[0], i42[0])
(1): COND_LOAD761(TRUE, i11[1], i42[1]) → STORE1059(i11[1], i42[1] + 1 + 1)
(2): STORE1059(i11[2], i70[2]) → LOAD761(i11[2], i70[2])
(3): LOAD761(i11[3], i42[3]) → COND_LOAD7611(i11[3] > i42[3] && 0 = i42[3] + 1 % 2, i11[3], i42[3])
(4): COND_LOAD7611(TRUE, i11[4], i42[4]) → LOAD761(i11[4], i42[4] + 1)

(0) -> (1), if ((i42[0] →* i42[1])∧(!(i42[0] + 1 % 2 = 0) && i11[0] > i42[0] →* TRUE)∧(i11[0] →* i11[1]))


(1) -> (2), if ((i42[1] + 1 + 1 →* i70[2])∧(i11[1] →* i11[2]))


(2) -> (0), if ((i70[2] →* i42[0])∧(i11[2] →* i11[0]))


(2) -> (3), if ((i70[2] →* i42[3])∧(i11[2] →* i11[3]))


(3) -> (4), if ((i11[3] →* i11[4])∧(i42[3] →* i42[4])∧(i11[3] > i42[3] && 0 = i42[3] + 1 % 2 →* TRUE))


(4) -> (0), if ((i11[4] →* i11[0])∧(i42[4] + 1 →* i42[0]))


(4) -> (3), if ((i42[4] + 1 →* i42[3])∧(i11[4] →* i11[3]))



The set Q consists of the following terms:
Load761(x0, x1)
Cond_Load761(TRUE, x0, x1)
Store1059(x0, x1)
Cond_Load7611(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 LOAD761(i11, i42) → COND_LOAD761(&&(!(=(%(+(i42, 1), 2), 0)), >(i11, i42)), i11, i42) the following chains were created:
  • We consider the chain LOAD761(i11[0], i42[0]) → COND_LOAD761(&&(!(=(%(+(i42[0], 1), 2), 0)), >(i11[0], i42[0])), i11[0], i42[0]), COND_LOAD761(TRUE, i11[1], i42[1]) → STORE1059(i11[1], +(+(i42[1], 1), 1)) which results in the following constraint:

    (1)    (i42[0]=i42[1]∧&&(!(=(%(+(i42[0], 1), 2), 0)), >(i11[0], i42[0]))=TRUE∧i11[0]=i11[1] ⇒ LOAD761(i11[0], i42[0])≥NonInfC∧LOAD761(i11[0], i42[0])≥COND_LOAD761(&&(!(=(%(+(i42[0], 1), 2), 0)), >(i11[0], i42[0])), i11[0], i42[0])∧(UIncreasing(COND_LOAD761(&&(!(=(%(+(i42[0], 1), 2), 0)), >(i11[0], i42[0])), i11[0], i42[0])), ≥))



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

    (2)    (>(i11[0], i42[0])=TRUE∧<(%(+(i42[0], 1), 2), 0)=TRUE ⇒ LOAD761(i11[0], i42[0])≥NonInfC∧LOAD761(i11[0], i42[0])≥COND_LOAD761(&&(!(=(%(+(i42[0], 1), 2), 0)), >(i11[0], i42[0])), i11[0], i42[0])∧(UIncreasing(COND_LOAD761(&&(!(=(%(+(i42[0], 1), 2), 0)), >(i11[0], i42[0])), i11[0], i42[0])), ≥))


    (3)    (>(i11[0], i42[0])=TRUE∧>(%(+(i42[0], 1), 2), 0)=TRUE ⇒ LOAD761(i11[0], i42[0])≥NonInfC∧LOAD761(i11[0], i42[0])≥COND_LOAD761(&&(!(=(%(+(i42[0], 1), 2), 0)), >(i11[0], i42[0])), i11[0], i42[0])∧(UIncreasing(COND_LOAD761(&&(!(=(%(+(i42[0], 1), 2), 0)), >(i11[0], i42[0])), i11[0], i42[0])), ≥))



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

    (4)    (i11[0] + [-1] + [-1]i42[0] ≥ 0∧[-1] + [-1]min{[2], [-2]} ≥ 0 ⇒ (UIncreasing(COND_LOAD761(&&(!(=(%(+(i42[0], 1), 2), 0)), >(i11[0], i42[0])), i11[0], i42[0])), ≥)∧[(-1)Bound*bni_14] + [bni_14]i11[0] + [(-1)bni_14]i42[0] ≥ 0∧[1 + (-1)bso_15] ≥ 0)



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

    (5)    (i11[0] + [-1] + [-1]i42[0] ≥ 0∧max{[2], [-2]} + [-1] ≥ 0 ⇒ (UIncreasing(COND_LOAD761(&&(!(=(%(+(i42[0], 1), 2), 0)), >(i11[0], i42[0])), i11[0], i42[0])), ≥)∧[(-1)Bound*bni_14] + [bni_14]i11[0] + [(-1)bni_14]i42[0] ≥ 0∧[1 + (-1)bso_15] ≥ 0)



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

    (6)    (i11[0] + [-1] + [-1]i42[0] ≥ 0∧[-1] + [-1]min{[2], [-2]} ≥ 0 ⇒ (UIncreasing(COND_LOAD761(&&(!(=(%(+(i42[0], 1), 2), 0)), >(i11[0], i42[0])), i11[0], i42[0])), ≥)∧[(-1)Bound*bni_14] + [bni_14]i11[0] + [(-1)bni_14]i42[0] ≥ 0∧[1 + (-1)bso_15] ≥ 0)



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

    (7)    (i11[0] + [-1] + [-1]i42[0] ≥ 0∧max{[2], [-2]} + [-1] ≥ 0 ⇒ (UIncreasing(COND_LOAD761(&&(!(=(%(+(i42[0], 1), 2), 0)), >(i11[0], i42[0])), i11[0], i42[0])), ≥)∧[(-1)Bound*bni_14] + [bni_14]i11[0] + [(-1)bni_14]i42[0] ≥ 0∧[1 + (-1)bso_15] ≥ 0)



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

    (8)    (i11[0] + [-1] + [-1]i42[0] ≥ 0∧[4] ≥ 0∧[1] ≥ 0 ⇒ (UIncreasing(COND_LOAD761(&&(!(=(%(+(i42[0], 1), 2), 0)), >(i11[0], i42[0])), i11[0], i42[0])), ≥)∧[(-1)Bound*bni_14] + [bni_14]i11[0] + [(-1)bni_14]i42[0] ≥ 0∧[1 + (-1)bso_15] ≥ 0)



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

    (9)    (i11[0] + [-1] + [-1]i42[0] ≥ 0∧[4] ≥ 0∧[1] ≥ 0 ⇒ (UIncreasing(COND_LOAD761(&&(!(=(%(+(i42[0], 1), 2), 0)), >(i11[0], i42[0])), i11[0], i42[0])), ≥)∧[(-1)Bound*bni_14] + [bni_14]i11[0] + [(-1)bni_14]i42[0] ≥ 0∧[1 + (-1)bso_15] ≥ 0)



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

    (10)    (i11[0] ≥ 0∧[4] ≥ 0∧[1] ≥ 0 ⇒ (UIncreasing(COND_LOAD761(&&(!(=(%(+(i42[0], 1), 2), 0)), >(i11[0], i42[0])), i11[0], i42[0])), ≥)∧[(-1)Bound*bni_14 + bni_14] + [bni_14]i11[0] ≥ 0∧[1 + (-1)bso_15] ≥ 0)



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

    (11)    (i11[0] ≥ 0∧[4] ≥ 0∧[1] ≥ 0 ⇒ (UIncreasing(COND_LOAD761(&&(!(=(%(+(i42[0], 1), 2), 0)), >(i11[0], i42[0])), i11[0], i42[0])), ≥)∧[(-1)Bound*bni_14 + bni_14] + [bni_14]i11[0] ≥ 0∧[1 + (-1)bso_15] ≥ 0)



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

    (12)    (i11[0] ≥ 0∧[4] ≥ 0∧[1] ≥ 0∧i42[0] ≥ 0 ⇒ (UIncreasing(COND_LOAD761(&&(!(=(%(+(i42[0], 1), 2), 0)), >(i11[0], i42[0])), i11[0], i42[0])), ≥)∧[(-1)Bound*bni_14 + bni_14] + [bni_14]i11[0] ≥ 0∧[1 + (-1)bso_15] ≥ 0)


    (13)    (i11[0] ≥ 0∧[4] ≥ 0∧[1] ≥ 0∧i42[0] ≥ 0 ⇒ (UIncreasing(COND_LOAD761(&&(!(=(%(+(i42[0], 1), 2), 0)), >(i11[0], i42[0])), i11[0], i42[0])), ≥)∧[(-1)Bound*bni_14 + bni_14] + [bni_14]i11[0] ≥ 0∧[1 + (-1)bso_15] ≥ 0)



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

    (14)    (i11[0] ≥ 0∧[1] ≥ 0∧i42[0] ≥ 0∧[1] ≥ 0 ⇒ (UIncreasing(COND_LOAD761(&&(!(=(%(+(i42[0], 1), 2), 0)), >(i11[0], i42[0])), i11[0], i42[0])), ≥)∧[(-1)Bound*bni_14 + bni_14] + [bni_14]i11[0] ≥ 0∧[1 + (-1)bso_15] ≥ 0)



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

    (15)    (i11[0] ≥ 0∧[1] ≥ 0∧i42[0] ≥ 0∧[1] ≥ 0 ⇒ (UIncreasing(COND_LOAD761(&&(!(=(%(+(i42[0], 1), 2), 0)), >(i11[0], i42[0])), i11[0], i42[0])), ≥)∧[(-1)Bound*bni_14 + bni_14] + [bni_14]i11[0] ≥ 0∧[1 + (-1)bso_15] ≥ 0)



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

    (16)    (i11[0] ≥ 0∧[4] ≥ 0∧[1] ≥ 0∧i42[0] ≥ 0 ⇒ (UIncreasing(COND_LOAD761(&&(!(=(%(+(i42[0], 1), 2), 0)), >(i11[0], i42[0])), i11[0], i42[0])), ≥)∧[(-1)Bound*bni_14 + bni_14] + [bni_14]i11[0] ≥ 0∧[1 + (-1)bso_15] ≥ 0)


    (17)    (i11[0] ≥ 0∧[4] ≥ 0∧[1] ≥ 0∧i42[0] ≥ 0 ⇒ (UIncreasing(COND_LOAD761(&&(!(=(%(+(i42[0], 1), 2), 0)), >(i11[0], i42[0])), i11[0], i42[0])), ≥)∧[(-1)Bound*bni_14 + bni_14] + [bni_14]i11[0] ≥ 0∧[1 + (-1)bso_15] ≥ 0)



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

    (18)    (i11[0] ≥ 0∧[1] ≥ 0∧i42[0] ≥ 0∧[1] ≥ 0 ⇒ (UIncreasing(COND_LOAD761(&&(!(=(%(+(i42[0], 1), 2), 0)), >(i11[0], i42[0])), i11[0], i42[0])), ≥)∧[(-1)Bound*bni_14 + bni_14] + [bni_14]i11[0] ≥ 0∧[1 + (-1)bso_15] ≥ 0)



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

    (19)    (i11[0] ≥ 0∧[1] ≥ 0∧i42[0] ≥ 0∧[1] ≥ 0 ⇒ (UIncreasing(COND_LOAD761(&&(!(=(%(+(i42[0], 1), 2), 0)), >(i11[0], i42[0])), i11[0], i42[0])), ≥)∧[(-1)Bound*bni_14 + bni_14] + [bni_14]i11[0] ≥ 0∧[1 + (-1)bso_15] ≥ 0)







For Pair COND_LOAD761(TRUE, i11, i42) → STORE1059(i11, +(+(i42, 1), 1)) the following chains were created:
  • We consider the chain COND_LOAD761(TRUE, i11[1], i42[1]) → STORE1059(i11[1], +(+(i42[1], 1), 1)) which results in the following constraint:

    (20)    (COND_LOAD761(TRUE, i11[1], i42[1])≥NonInfC∧COND_LOAD761(TRUE, i11[1], i42[1])≥STORE1059(i11[1], +(+(i42[1], 1), 1))∧(UIncreasing(STORE1059(i11[1], +(+(i42[1], 1), 1))), ≥))



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

    (21)    ((UIncreasing(STORE1059(i11[1], +(+(i42[1], 1), 1))), ≥)∧[(-1)bso_17] ≥ 0)



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

    (22)    ((UIncreasing(STORE1059(i11[1], +(+(i42[1], 1), 1))), ≥)∧[(-1)bso_17] ≥ 0)



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

    (23)    ((UIncreasing(STORE1059(i11[1], +(+(i42[1], 1), 1))), ≥)∧[(-1)bso_17] ≥ 0)



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

    (24)    ((UIncreasing(STORE1059(i11[1], +(+(i42[1], 1), 1))), ≥)∧0 = 0∧0 = 0∧[(-1)bso_17] ≥ 0)







For Pair STORE1059(i11, i70) → LOAD761(i11, i70) the following chains were created:
  • We consider the chain STORE1059(i11[2], i70[2]) → LOAD761(i11[2], i70[2]), LOAD761(i11[0], i42[0]) → COND_LOAD761(&&(!(=(%(+(i42[0], 1), 2), 0)), >(i11[0], i42[0])), i11[0], i42[0]) which results in the following constraint:

    (25)    (i70[2]=i42[0]∧i11[2]=i11[0] ⇒ STORE1059(i11[2], i70[2])≥NonInfC∧STORE1059(i11[2], i70[2])≥LOAD761(i11[2], i70[2])∧(UIncreasing(LOAD761(i11[2], i70[2])), ≥))



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

    (26)    (STORE1059(i11[2], i70[2])≥NonInfC∧STORE1059(i11[2], i70[2])≥LOAD761(i11[2], i70[2])∧(UIncreasing(LOAD761(i11[2], i70[2])), ≥))



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

    (27)    ((UIncreasing(LOAD761(i11[2], i70[2])), ≥)∧[1 + (-1)bso_19] ≥ 0)



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

    (28)    ((UIncreasing(LOAD761(i11[2], i70[2])), ≥)∧[1 + (-1)bso_19] ≥ 0)



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

    (29)    ((UIncreasing(LOAD761(i11[2], i70[2])), ≥)∧[1 + (-1)bso_19] ≥ 0)



    We simplified constraint (29) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint:

    (30)    ((UIncreasing(LOAD761(i11[2], i70[2])), ≥)∧0 = 0∧0 = 0∧[1 + (-1)bso_19] ≥ 0)



  • We consider the chain STORE1059(i11[2], i70[2]) → LOAD761(i11[2], i70[2]), LOAD761(i11[3], i42[3]) → COND_LOAD7611(&&(>(i11[3], i42[3]), =(0, %(+(i42[3], 1), 2))), i11[3], i42[3]) which results in the following constraint:

    (31)    (i70[2]=i42[3]∧i11[2]=i11[3] ⇒ STORE1059(i11[2], i70[2])≥NonInfC∧STORE1059(i11[2], i70[2])≥LOAD761(i11[2], i70[2])∧(UIncreasing(LOAD761(i11[2], i70[2])), ≥))



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

    (32)    (STORE1059(i11[2], i70[2])≥NonInfC∧STORE1059(i11[2], i70[2])≥LOAD761(i11[2], i70[2])∧(UIncreasing(LOAD761(i11[2], i70[2])), ≥))



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

    (33)    ((UIncreasing(LOAD761(i11[2], i70[2])), ≥)∧[1 + (-1)bso_19] ≥ 0)



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

    (34)    ((UIncreasing(LOAD761(i11[2], i70[2])), ≥)∧[1 + (-1)bso_19] ≥ 0)



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

    (35)    ((UIncreasing(LOAD761(i11[2], i70[2])), ≥)∧[1 + (-1)bso_19] ≥ 0)



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

    (36)    ((UIncreasing(LOAD761(i11[2], i70[2])), ≥)∧0 = 0∧0 = 0∧[1 + (-1)bso_19] ≥ 0)







For Pair LOAD761(i11, i42) → COND_LOAD7611(&&(>(i11, i42), =(0, %(+(i42, 1), 2))), i11, i42) the following chains were created:
  • We consider the chain LOAD761(i11[3], i42[3]) → COND_LOAD7611(&&(>(i11[3], i42[3]), =(0, %(+(i42[3], 1), 2))), i11[3], i42[3]), COND_LOAD7611(TRUE, i11[4], i42[4]) → LOAD761(i11[4], +(i42[4], 1)) which results in the following constraint:

    (37)    (i11[3]=i11[4]∧i42[3]=i42[4]∧&&(>(i11[3], i42[3]), =(0, %(+(i42[3], 1), 2)))=TRUE ⇒ LOAD761(i11[3], i42[3])≥NonInfC∧LOAD761(i11[3], i42[3])≥COND_LOAD7611(&&(>(i11[3], i42[3]), =(0, %(+(i42[3], 1), 2))), i11[3], i42[3])∧(UIncreasing(COND_LOAD7611(&&(>(i11[3], i42[3]), =(0, %(+(i42[3], 1), 2))), i11[3], i42[3])), ≥))



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

    (38)    (>(i11[3], i42[3])=TRUE∧>=(0, %(+(i42[3], 1), 2))=TRUE∧<=(0, %(+(i42[3], 1), 2))=TRUE ⇒ LOAD761(i11[3], i42[3])≥NonInfC∧LOAD761(i11[3], i42[3])≥COND_LOAD7611(&&(>(i11[3], i42[3]), =(0, %(+(i42[3], 1), 2))), i11[3], i42[3])∧(UIncreasing(COND_LOAD7611(&&(>(i11[3], i42[3]), =(0, %(+(i42[3], 1), 2))), i11[3], i42[3])), ≥))



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

    (39)    (i11[3] + [-1] + [-1]i42[3] ≥ 0∧[-1]min{[2], [-2]} ≥ 0∧max{[2], [-2]} ≥ 0 ⇒ (UIncreasing(COND_LOAD7611(&&(>(i11[3], i42[3]), =(0, %(+(i42[3], 1), 2))), i11[3], i42[3])), ≥)∧[(-1)Bound*bni_20] + [bni_20]i11[3] + [(-1)bni_20]i42[3] ≥ 0∧[1 + (-1)bso_21] ≥ 0)



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

    (40)    (i11[3] + [-1] + [-1]i42[3] ≥ 0∧[-1]min{[2], [-2]} ≥ 0∧max{[2], [-2]} ≥ 0 ⇒ (UIncreasing(COND_LOAD7611(&&(>(i11[3], i42[3]), =(0, %(+(i42[3], 1), 2))), i11[3], i42[3])), ≥)∧[(-1)Bound*bni_20] + [bni_20]i11[3] + [(-1)bni_20]i42[3] ≥ 0∧[1 + (-1)bso_21] ≥ 0)



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

    (41)    (i11[3] + [-1] + [-1]i42[3] ≥ 0∧[4] ≥ 0∧[2] ≥ 0∧[2] ≥ 0 ⇒ (UIncreasing(COND_LOAD7611(&&(>(i11[3], i42[3]), =(0, %(+(i42[3], 1), 2))), i11[3], i42[3])), ≥)∧[(-1)Bound*bni_20] + [bni_20]i11[3] + [(-1)bni_20]i42[3] ≥ 0∧[1 + (-1)bso_21] ≥ 0)



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

    (42)    (i11[3] ≥ 0∧[4] ≥ 0∧[2] ≥ 0∧[2] ≥ 0 ⇒ (UIncreasing(COND_LOAD7611(&&(>(i11[3], i42[3]), =(0, %(+(i42[3], 1), 2))), i11[3], i42[3])), ≥)∧[(-1)Bound*bni_20 + bni_20] + [bni_20]i11[3] ≥ 0∧[1 + (-1)bso_21] ≥ 0)



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

    (43)    (i11[3] ≥ 0∧[4] ≥ 0∧[2] ≥ 0∧[2] ≥ 0∧i42[3] ≥ 0 ⇒ (UIncreasing(COND_LOAD7611(&&(>(i11[3], i42[3]), =(0, %(+(i42[3], 1), 2))), i11[3], i42[3])), ≥)∧[(-1)Bound*bni_20 + bni_20] + [bni_20]i11[3] ≥ 0∧[1 + (-1)bso_21] ≥ 0)


    (44)    (i11[3] ≥ 0∧[4] ≥ 0∧[2] ≥ 0∧[2] ≥ 0∧i42[3] ≥ 0 ⇒ (UIncreasing(COND_LOAD7611(&&(>(i11[3], i42[3]), =(0, %(+(i42[3], 1), 2))), i11[3], i42[3])), ≥)∧[(-1)Bound*bni_20 + bni_20] + [bni_20]i11[3] ≥ 0∧[1 + (-1)bso_21] ≥ 0)



    We simplified constraint (43) using rule (IDP_POLY_GCD) which results in the following new constraint:

    (45)    (i11[3] ≥ 0∧i42[3] ≥ 0∧[1] ≥ 0∧[1] ≥ 0∧[1] ≥ 0 ⇒ (UIncreasing(COND_LOAD7611(&&(>(i11[3], i42[3]), =(0, %(+(i42[3], 1), 2))), i11[3], i42[3])), ≥)∧[(-1)Bound*bni_20 + bni_20] + [bni_20]i11[3] ≥ 0∧[1 + (-1)bso_21] ≥ 0)



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

    (46)    (i11[3] ≥ 0∧i42[3] ≥ 0∧[1] ≥ 0∧[1] ≥ 0∧[1] ≥ 0 ⇒ (UIncreasing(COND_LOAD7611(&&(>(i11[3], i42[3]), =(0, %(+(i42[3], 1), 2))), i11[3], i42[3])), ≥)∧[(-1)Bound*bni_20 + bni_20] + [bni_20]i11[3] ≥ 0∧[1 + (-1)bso_21] ≥ 0)







For Pair COND_LOAD7611(TRUE, i11, i42) → LOAD761(i11, +(i42, 1)) the following chains were created:
  • We consider the chain COND_LOAD7611(TRUE, i11[4], i42[4]) → LOAD761(i11[4], +(i42[4], 1)) which results in the following constraint:

    (47)    (COND_LOAD7611(TRUE, i11[4], i42[4])≥NonInfC∧COND_LOAD7611(TRUE, i11[4], i42[4])≥LOAD761(i11[4], +(i42[4], 1))∧(UIncreasing(LOAD761(i11[4], +(i42[4], 1))), ≥))



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

    (48)    ((UIncreasing(LOAD761(i11[4], +(i42[4], 1))), ≥)∧[(-1)bso_23] ≥ 0)



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

    (49)    ((UIncreasing(LOAD761(i11[4], +(i42[4], 1))), ≥)∧[(-1)bso_23] ≥ 0)



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

    (50)    ((UIncreasing(LOAD761(i11[4], +(i42[4], 1))), ≥)∧[(-1)bso_23] ≥ 0)



    We simplified constraint (50) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint:

    (51)    ((UIncreasing(LOAD761(i11[4], +(i42[4], 1))), ≥)∧0 = 0∧0 = 0∧[(-1)bso_23] ≥ 0)







To summarize, we get the following constraints P≥ for the following pairs.
  • LOAD761(i11, i42) → COND_LOAD761(&&(!(=(%(+(i42, 1), 2), 0)), >(i11, i42)), i11, i42)
    • (i11[0] ≥ 0∧[1] ≥ 0∧i42[0] ≥ 0∧[1] ≥ 0 ⇒ (UIncreasing(COND_LOAD761(&&(!(=(%(+(i42[0], 1), 2), 0)), >(i11[0], i42[0])), i11[0], i42[0])), ≥)∧[(-1)Bound*bni_14 + bni_14] + [bni_14]i11[0] ≥ 0∧[1 + (-1)bso_15] ≥ 0)
    • (i11[0] ≥ 0∧[1] ≥ 0∧i42[0] ≥ 0∧[1] ≥ 0 ⇒ (UIncreasing(COND_LOAD761(&&(!(=(%(+(i42[0], 1), 2), 0)), >(i11[0], i42[0])), i11[0], i42[0])), ≥)∧[(-1)Bound*bni_14 + bni_14] + [bni_14]i11[0] ≥ 0∧[1 + (-1)bso_15] ≥ 0)
    • (i11[0] ≥ 0∧[1] ≥ 0∧i42[0] ≥ 0∧[1] ≥ 0 ⇒ (UIncreasing(COND_LOAD761(&&(!(=(%(+(i42[0], 1), 2), 0)), >(i11[0], i42[0])), i11[0], i42[0])), ≥)∧[(-1)Bound*bni_14 + bni_14] + [bni_14]i11[0] ≥ 0∧[1 + (-1)bso_15] ≥ 0)
    • (i11[0] ≥ 0∧[1] ≥ 0∧i42[0] ≥ 0∧[1] ≥ 0 ⇒ (UIncreasing(COND_LOAD761(&&(!(=(%(+(i42[0], 1), 2), 0)), >(i11[0], i42[0])), i11[0], i42[0])), ≥)∧[(-1)Bound*bni_14 + bni_14] + [bni_14]i11[0] ≥ 0∧[1 + (-1)bso_15] ≥ 0)

  • COND_LOAD761(TRUE, i11, i42) → STORE1059(i11, +(+(i42, 1), 1))
    • ((UIncreasing(STORE1059(i11[1], +(+(i42[1], 1), 1))), ≥)∧0 = 0∧0 = 0∧[(-1)bso_17] ≥ 0)

  • STORE1059(i11, i70) → LOAD761(i11, i70)
    • ((UIncreasing(LOAD761(i11[2], i70[2])), ≥)∧0 = 0∧0 = 0∧[1 + (-1)bso_19] ≥ 0)
    • ((UIncreasing(LOAD761(i11[2], i70[2])), ≥)∧0 = 0∧0 = 0∧[1 + (-1)bso_19] ≥ 0)

  • LOAD761(i11, i42) → COND_LOAD7611(&&(>(i11, i42), =(0, %(+(i42, 1), 2))), i11, i42)
    • (i11[3] ≥ 0∧i42[3] ≥ 0∧[1] ≥ 0∧[1] ≥ 0∧[1] ≥ 0 ⇒ (UIncreasing(COND_LOAD7611(&&(>(i11[3], i42[3]), =(0, %(+(i42[3], 1), 2))), i11[3], i42[3])), ≥)∧[(-1)Bound*bni_20 + bni_20] + [bni_20]i11[3] ≥ 0∧[1 + (-1)bso_21] ≥ 0)
    • (i11[3] ≥ 0∧i42[3] ≥ 0∧[1] ≥ 0∧[1] ≥ 0∧[1] ≥ 0 ⇒ (UIncreasing(COND_LOAD7611(&&(>(i11[3], i42[3]), =(0, %(+(i42[3], 1), 2))), i11[3], i42[3])), ≥)∧[(-1)Bound*bni_20 + bni_20] + [bni_20]i11[3] ≥ 0∧[1 + (-1)bso_21] ≥ 0)

  • COND_LOAD7611(TRUE, i11, i42) → LOAD761(i11, +(i42, 1))
    • ((UIncreasing(LOAD761(i11[4], +(i42[4], 1))), ≥)∧0 = 0∧0 = 0∧[(-1)bso_23] ≥ 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(LOAD761(x1, x2)) = x1 + [-1]x2   
POL(COND_LOAD761(x1, x2, x3)) = [-1] + x2 + [-1]x3   
POL(&&(x1, x2)) = [-1]   
POL(!(x1)) = [-1]   
POL(=(x1, x2)) = [-1]   
POL(+(x1, x2)) = x1 + x2   
POL(1) = [1]   
POL(2) = [2]   
POL(0) = 0   
POL(>(x1, x2)) = [-1]   
POL(STORE1059(x1, x2)) = [1] + [-1]x2 + x1   
POL(COND_LOAD7611(x1, x2, x3)) = [-1] + [-1]x3 + x2   

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

POL(%(x1, 2)-1 @ {}) = min{x2, [-1]x2}   
POL(%(x1, 2)1 @ {}) = max{x2, [-1]x2}   

The following pairs are in P>:

LOAD761(i11[0], i42[0]) → COND_LOAD761(&&(!(=(%(+(i42[0], 1), 2), 0)), >(i11[0], i42[0])), i11[0], i42[0])
STORE1059(i11[2], i70[2]) → LOAD761(i11[2], i70[2])
LOAD761(i11[3], i42[3]) → COND_LOAD7611(&&(>(i11[3], i42[3]), =(0, %(+(i42[3], 1), 2))), i11[3], i42[3])

The following pairs are in Pbound:

LOAD761(i11[0], i42[0]) → COND_LOAD761(&&(!(=(%(+(i42[0], 1), 2), 0)), >(i11[0], i42[0])), i11[0], i42[0])
LOAD761(i11[3], i42[3]) → COND_LOAD7611(&&(>(i11[3], i42[3]), =(0, %(+(i42[3], 1), 2))), i11[3], i42[3])

The following pairs are in P≥:

COND_LOAD761(TRUE, i11[1], i42[1]) → STORE1059(i11[1], +(+(i42[1], 1), 1))
COND_LOAD7611(TRUE, i11[4], i42[4]) → LOAD761(i11[4], +(i42[4], 1))

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


R is empty.

The integer pair graph contains the following rules and edges:
(1): COND_LOAD761(TRUE, i11[1], i42[1]) → STORE1059(i11[1], i42[1] + 1 + 1)
(4): COND_LOAD7611(TRUE, i11[4], i42[4]) → LOAD761(i11[4], i42[4] + 1)


The set Q consists of the following terms:
Load761(x0, x1)
Cond_Load761(TRUE, x0, x1)
Store1059(x0, x1)
Cond_Load7611(TRUE, x0, x1)

(12) IDependencyGraphProof (EQUIVALENT transformation)

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

(13) TRUE

(14) Obligation:

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


The following domains are used:

Integer


R is empty.

The integer pair graph contains the following rules and edges:
(1): COND_LOAD761(TRUE, i11[1], i42[1]) → STORE1059(i11[1], i42[1] + 1 + 1)
(2): STORE1059(i11[2], i70[2]) → LOAD761(i11[2], i70[2])
(4): COND_LOAD7611(TRUE, i11[4], i42[4]) → LOAD761(i11[4], i42[4] + 1)

(1) -> (2), if ((i42[1] + 1 + 1 →* i70[2])∧(i11[1] →* i11[2]))



The set Q consists of the following terms:
Load761(x0, x1)
Cond_Load761(TRUE, x0, x1)
Store1059(x0, x1)
Cond_Load7611(TRUE, x0, x1)

(15) IDependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 3 less nodes.

(16) TRUE