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: PastaB14 
/**
 * 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 PastaB14 {
    public static void main(String[] args) {
        Random.args = args;
        int x = Random.random();
        int y = Random.random();

        while (x == y && x > 0) {
            while (y > 0) {
                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 197 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:
Load804(i108) → Cond_Load804(i108 > 0, i108)
Cond_Load804(TRUE, i108) → Load1161(i108, i108)
Load1161(i139, i141) → Cond_Load1161(i141 > 0, i139, i141)
Cond_Load1161(TRUE, i139, i141) → Load1161(i139 + -1, i141 + -1)
Load1161(i139, 0) → Load804(i139)
The set Q consists of the following terms:
Load804(x0)
Cond_Load804(TRUE, x0)
Load1161(x0, x1)
Cond_Load1161(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


The ITRS R consists of the following rules:
Load804(i108) → Cond_Load804(i108 > 0, i108)
Cond_Load804(TRUE, i108) → Load1161(i108, i108)
Load1161(i139, i141) → Cond_Load1161(i141 > 0, i139, i141)
Cond_Load1161(TRUE, i139, i141) → Load1161(i139 + -1, i141 + -1)
Load1161(i139, 0) → Load804(i139)

The integer pair graph contains the following rules and edges:
(0): LOAD804(i108[0]) → COND_LOAD804(i108[0] > 0, i108[0])
(1): COND_LOAD804(TRUE, i108[1]) → LOAD1161(i108[1], i108[1])
(2): LOAD1161(i139[2], i141[2]) → COND_LOAD1161(i141[2] > 0, i139[2], i141[2])
(3): COND_LOAD1161(TRUE, i139[3], i141[3]) → LOAD1161(i139[3] + -1, i141[3] + -1)
(4): LOAD1161(i139[4], 0) → LOAD804(i139[4])

(0) -> (1), if ((i108[0] > 0 →* TRUE)∧(i108[0] →* i108[1]))


(1) -> (2), if ((i108[1] →* i139[2])∧(i108[1] →* i141[2]))


(1) -> (4), if ((i108[1] →* i139[4])∧(i108[1] →* 0))


(2) -> (3), if ((i139[2] →* i139[3])∧(i141[2] > 0 →* TRUE)∧(i141[2] →* i141[3]))


(3) -> (2), if ((i141[3] + -1 →* i141[2])∧(i139[3] + -1 →* i139[2]))


(3) -> (4), if ((i139[3] + -1 →* i139[4])∧(i141[3] + -1 →* 0))


(4) -> (0), if ((i139[4] →* i108[0]))



The set Q consists of the following terms:
Load804(x0)
Cond_Load804(TRUE, x0)
Load1161(x0, x1)
Cond_Load1161(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


R is empty.

The integer pair graph contains the following rules and edges:
(0): LOAD804(i108[0]) → COND_LOAD804(i108[0] > 0, i108[0])
(1): COND_LOAD804(TRUE, i108[1]) → LOAD1161(i108[1], i108[1])
(2): LOAD1161(i139[2], i141[2]) → COND_LOAD1161(i141[2] > 0, i139[2], i141[2])
(3): COND_LOAD1161(TRUE, i139[3], i141[3]) → LOAD1161(i139[3] + -1, i141[3] + -1)
(4): LOAD1161(i139[4], 0) → LOAD804(i139[4])

(0) -> (1), if ((i108[0] > 0 →* TRUE)∧(i108[0] →* i108[1]))


(1) -> (2), if ((i108[1] →* i139[2])∧(i108[1] →* i141[2]))


(1) -> (4), if ((i108[1] →* i139[4])∧(i108[1] →* 0))


(2) -> (3), if ((i139[2] →* i139[3])∧(i141[2] > 0 →* TRUE)∧(i141[2] →* i141[3]))


(3) -> (2), if ((i141[3] + -1 →* i141[2])∧(i139[3] + -1 →* i139[2]))


(3) -> (4), if ((i139[3] + -1 →* i139[4])∧(i141[3] + -1 →* 0))


(4) -> (0), if ((i139[4] →* i108[0]))



The set Q consists of the following terms:
Load804(x0)
Cond_Load804(TRUE, x0)
Load1161(x0, x1)
Cond_Load1161(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 LOAD804(i108) → COND_LOAD804(>(i108, 0), i108) the following chains were created:
  • We consider the chain LOAD804(i108[0]) → COND_LOAD804(>(i108[0], 0), i108[0]), COND_LOAD804(TRUE, i108[1]) → LOAD1161(i108[1], i108[1]) which results in the following constraint:

    (1)    (>(i108[0], 0)=TRUE∧i108[0]=i108[1] ⇒ LOAD804(i108[0])≥NonInfC∧LOAD804(i108[0])≥COND_LOAD804(>(i108[0], 0), i108[0])∧(UIncreasing(COND_LOAD804(>(i108[0], 0), i108[0])), ≥))



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

    (2)    (>(i108[0], 0)=TRUE ⇒ LOAD804(i108[0])≥NonInfC∧LOAD804(i108[0])≥COND_LOAD804(>(i108[0], 0), i108[0])∧(UIncreasing(COND_LOAD804(>(i108[0], 0), i108[0])), ≥))



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

    (3)    (i108[0] + [-1] ≥ 0 ⇒ (UIncreasing(COND_LOAD804(>(i108[0], 0), i108[0])), ≥)∧[(-1)bni_16 + (-1)Bound*bni_16] + [bni_16]i108[0] ≥ 0∧[-1 + (-1)bso_17] + i108[0] ≥ 0)



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

    (4)    (i108[0] + [-1] ≥ 0 ⇒ (UIncreasing(COND_LOAD804(>(i108[0], 0), i108[0])), ≥)∧[(-1)bni_16 + (-1)Bound*bni_16] + [bni_16]i108[0] ≥ 0∧[-1 + (-1)bso_17] + i108[0] ≥ 0)



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

    (5)    (i108[0] + [-1] ≥ 0 ⇒ (UIncreasing(COND_LOAD804(>(i108[0], 0), i108[0])), ≥)∧[(-1)bni_16 + (-1)Bound*bni_16] + [bni_16]i108[0] ≥ 0∧[-1 + (-1)bso_17] + i108[0] ≥ 0)



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

    (6)    (i108[0] ≥ 0 ⇒ (UIncreasing(COND_LOAD804(>(i108[0], 0), i108[0])), ≥)∧[(-1)Bound*bni_16] + [bni_16]i108[0] ≥ 0∧[(-1)bso_17] + i108[0] ≥ 0)







For Pair COND_LOAD804(TRUE, i108) → LOAD1161(i108, i108) the following chains were created:
  • We consider the chain COND_LOAD804(TRUE, i108[1]) → LOAD1161(i108[1], i108[1]), LOAD1161(i139[2], i141[2]) → COND_LOAD1161(>(i141[2], 0), i139[2], i141[2]) which results in the following constraint:

    (7)    (i108[1]=i139[2]∧i108[1]=i141[2] ⇒ COND_LOAD804(TRUE, i108[1])≥NonInfC∧COND_LOAD804(TRUE, i108[1])≥LOAD1161(i108[1], i108[1])∧(UIncreasing(LOAD1161(i108[1], i108[1])), ≥))



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

    (8)    (COND_LOAD804(TRUE, i108[1])≥NonInfC∧COND_LOAD804(TRUE, i108[1])≥LOAD1161(i108[1], i108[1])∧(UIncreasing(LOAD1161(i108[1], i108[1])), ≥))



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

    (9)    ((UIncreasing(LOAD1161(i108[1], i108[1])), ≥)∧[1 + (-1)bso_19] ≥ 0)



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

    (10)    ((UIncreasing(LOAD1161(i108[1], i108[1])), ≥)∧[1 + (-1)bso_19] ≥ 0)



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

    (11)    ((UIncreasing(LOAD1161(i108[1], i108[1])), ≥)∧[1 + (-1)bso_19] ≥ 0)



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

    (12)    ((UIncreasing(LOAD1161(i108[1], i108[1])), ≥)∧0 = 0∧[1 + (-1)bso_19] ≥ 0)



  • We consider the chain COND_LOAD804(TRUE, i108[1]) → LOAD1161(i108[1], i108[1]), LOAD1161(i139[4], 0) → LOAD804(i139[4]) which results in the following constraint:

    (13)    (i108[1]=i139[4]∧i108[1]=0 ⇒ COND_LOAD804(TRUE, i108[1])≥NonInfC∧COND_LOAD804(TRUE, i108[1])≥LOAD1161(i108[1], i108[1])∧(UIncreasing(LOAD1161(i108[1], i108[1])), ≥))



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

    (14)    (COND_LOAD804(TRUE, 0)≥NonInfC∧COND_LOAD804(TRUE, 0)≥LOAD1161(0, 0)∧(UIncreasing(LOAD1161(i108[1], i108[1])), ≥))



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

    (15)    ((UIncreasing(LOAD1161(i108[1], i108[1])), ≥)∧[1 + (-1)bso_19] ≥ 0)



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

    (16)    ((UIncreasing(LOAD1161(i108[1], i108[1])), ≥)∧[1 + (-1)bso_19] ≥ 0)



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

    (17)    ((UIncreasing(LOAD1161(i108[1], i108[1])), ≥)∧[1 + (-1)bso_19] ≥ 0)







For Pair LOAD1161(i139, i141) → COND_LOAD1161(>(i141, 0), i139, i141) the following chains were created:
  • We consider the chain LOAD1161(i139[2], i141[2]) → COND_LOAD1161(>(i141[2], 0), i139[2], i141[2]), COND_LOAD1161(TRUE, i139[3], i141[3]) → LOAD1161(+(i139[3], -1), +(i141[3], -1)) which results in the following constraint:

    (18)    (i139[2]=i139[3]∧>(i141[2], 0)=TRUE∧i141[2]=i141[3] ⇒ LOAD1161(i139[2], i141[2])≥NonInfC∧LOAD1161(i139[2], i141[2])≥COND_LOAD1161(>(i141[2], 0), i139[2], i141[2])∧(UIncreasing(COND_LOAD1161(>(i141[2], 0), i139[2], i141[2])), ≥))



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

    (19)    (>(i141[2], 0)=TRUE ⇒ LOAD1161(i139[2], i141[2])≥NonInfC∧LOAD1161(i139[2], i141[2])≥COND_LOAD1161(>(i141[2], 0), i139[2], i141[2])∧(UIncreasing(COND_LOAD1161(>(i141[2], 0), i139[2], i141[2])), ≥))



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

    (20)    (i141[2] + [-1] ≥ 0 ⇒ (UIncreasing(COND_LOAD1161(>(i141[2], 0), i139[2], i141[2])), ≥)∧[(-1)bni_20 + (-1)Bound*bni_20] + [(-1)bni_20]i141[2] + [bni_20]i139[2] ≥ 0∧[(-1)bso_21] ≥ 0)



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

    (21)    (i141[2] + [-1] ≥ 0 ⇒ (UIncreasing(COND_LOAD1161(>(i141[2], 0), i139[2], i141[2])), ≥)∧[(-1)bni_20 + (-1)Bound*bni_20] + [(-1)bni_20]i141[2] + [bni_20]i139[2] ≥ 0∧[(-1)bso_21] ≥ 0)



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

    (22)    (i141[2] + [-1] ≥ 0 ⇒ (UIncreasing(COND_LOAD1161(>(i141[2], 0), i139[2], i141[2])), ≥)∧[(-1)bni_20 + (-1)Bound*bni_20] + [(-1)bni_20]i141[2] + [bni_20]i139[2] ≥ 0∧[(-1)bso_21] ≥ 0)



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

    (23)    (i141[2] + [-1] ≥ 0 ⇒ (UIncreasing(COND_LOAD1161(>(i141[2], 0), i139[2], i141[2])), ≥)∧[bni_20] = 0∧[(-1)bni_20 + (-1)Bound*bni_20] + [(-1)bni_20]i141[2] ≥ 0∧0 = 0∧[(-1)bso_21] ≥ 0)



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

    (24)    (i141[2] ≥ 0 ⇒ (UIncreasing(COND_LOAD1161(>(i141[2], 0), i139[2], i141[2])), ≥)∧[bni_20] = 0∧[(-2)bni_20 + (-1)Bound*bni_20] + [(-1)bni_20]i141[2] ≥ 0∧0 = 0∧[(-1)bso_21] ≥ 0)







For Pair COND_LOAD1161(TRUE, i139, i141) → LOAD1161(+(i139, -1), +(i141, -1)) the following chains were created:
  • We consider the chain LOAD1161(i139[2], i141[2]) → COND_LOAD1161(>(i141[2], 0), i139[2], i141[2]), COND_LOAD1161(TRUE, i139[3], i141[3]) → LOAD1161(+(i139[3], -1), +(i141[3], -1)), LOAD1161(i139[2], i141[2]) → COND_LOAD1161(>(i141[2], 0), i139[2], i141[2]) which results in the following constraint:

    (25)    (i139[2]=i139[3]∧>(i141[2], 0)=TRUE∧i141[2]=i141[3]∧+(i141[3], -1)=i141[2]1∧+(i139[3], -1)=i139[2]1 ⇒ COND_LOAD1161(TRUE, i139[3], i141[3])≥NonInfC∧COND_LOAD1161(TRUE, i139[3], i141[3])≥LOAD1161(+(i139[3], -1), +(i141[3], -1))∧(UIncreasing(LOAD1161(+(i139[3], -1), +(i141[3], -1))), ≥))



    We simplified constraint (25) using rules (III), (IV) which results in the following new constraint:

    (26)    (>(i141[2], 0)=TRUE ⇒ COND_LOAD1161(TRUE, i139[2], i141[2])≥NonInfC∧COND_LOAD1161(TRUE, i139[2], i141[2])≥LOAD1161(+(i139[2], -1), +(i141[2], -1))∧(UIncreasing(LOAD1161(+(i139[3], -1), +(i141[3], -1))), ≥))



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

    (27)    (i141[2] + [-1] ≥ 0 ⇒ (UIncreasing(LOAD1161(+(i139[3], -1), +(i141[3], -1))), ≥)∧[(-1)bni_22 + (-1)Bound*bni_22] + [(-1)bni_22]i141[2] + [bni_22]i139[2] ≥ 0∧[(-1)bso_23] ≥ 0)



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

    (28)    (i141[2] + [-1] ≥ 0 ⇒ (UIncreasing(LOAD1161(+(i139[3], -1), +(i141[3], -1))), ≥)∧[(-1)bni_22 + (-1)Bound*bni_22] + [(-1)bni_22]i141[2] + [bni_22]i139[2] ≥ 0∧[(-1)bso_23] ≥ 0)



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

    (29)    (i141[2] + [-1] ≥ 0 ⇒ (UIncreasing(LOAD1161(+(i139[3], -1), +(i141[3], -1))), ≥)∧[(-1)bni_22 + (-1)Bound*bni_22] + [(-1)bni_22]i141[2] + [bni_22]i139[2] ≥ 0∧[(-1)bso_23] ≥ 0)



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

    (30)    (i141[2] + [-1] ≥ 0 ⇒ (UIncreasing(LOAD1161(+(i139[3], -1), +(i141[3], -1))), ≥)∧[bni_22] = 0∧[(-1)bni_22 + (-1)Bound*bni_22] + [(-1)bni_22]i141[2] ≥ 0∧0 = 0∧[(-1)bso_23] ≥ 0)



    We simplified constraint (30) using rule (IDP_SMT_SPLIT) which results in the following new constraint:

    (31)    (i141[2] ≥ 0 ⇒ (UIncreasing(LOAD1161(+(i139[3], -1), +(i141[3], -1))), ≥)∧[bni_22] = 0∧[(-2)bni_22 + (-1)Bound*bni_22] + [(-1)bni_22]i141[2] ≥ 0∧0 = 0∧[(-1)bso_23] ≥ 0)



  • We consider the chain LOAD1161(i139[2], i141[2]) → COND_LOAD1161(>(i141[2], 0), i139[2], i141[2]), COND_LOAD1161(TRUE, i139[3], i141[3]) → LOAD1161(+(i139[3], -1), +(i141[3], -1)), LOAD1161(i139[4], 0) → LOAD804(i139[4]) which results in the following constraint:

    (32)    (i139[2]=i139[3]∧>(i141[2], 0)=TRUE∧i141[2]=i141[3]∧+(i139[3], -1)=i139[4]∧+(i141[3], -1)=0 ⇒ COND_LOAD1161(TRUE, i139[3], i141[3])≥NonInfC∧COND_LOAD1161(TRUE, i139[3], i141[3])≥LOAD1161(+(i139[3], -1), +(i141[3], -1))∧(UIncreasing(LOAD1161(+(i139[3], -1), +(i141[3], -1))), ≥))



    We simplified constraint (32) using rules (III), (IV) which results in the following new constraint:

    (33)    (>(i141[2], 0)=TRUE∧+(i141[2], -1)=0 ⇒ COND_LOAD1161(TRUE, i139[2], i141[2])≥NonInfC∧COND_LOAD1161(TRUE, i139[2], i141[2])≥LOAD1161(+(i139[2], -1), +(i141[2], -1))∧(UIncreasing(LOAD1161(+(i139[3], -1), +(i141[3], -1))), ≥))



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

    (34)    (i141[2] + [-1] ≥ 0∧i141[2] + [-1] ≥ 0 ⇒ (UIncreasing(LOAD1161(+(i139[3], -1), +(i141[3], -1))), ≥)∧[(-1)bni_22 + (-1)Bound*bni_22] + [(-1)bni_22]i141[2] + [bni_22]i139[2] ≥ 0∧[(-1)bso_23] ≥ 0)



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

    (35)    (i141[2] + [-1] ≥ 0∧i141[2] + [-1] ≥ 0 ⇒ (UIncreasing(LOAD1161(+(i139[3], -1), +(i141[3], -1))), ≥)∧[(-1)bni_22 + (-1)Bound*bni_22] + [(-1)bni_22]i141[2] + [bni_22]i139[2] ≥ 0∧[(-1)bso_23] ≥ 0)



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

    (36)    (i141[2] + [-1] ≥ 0∧i141[2] + [-1] ≥ 0 ⇒ (UIncreasing(LOAD1161(+(i139[3], -1), +(i141[3], -1))), ≥)∧[(-1)bni_22 + (-1)Bound*bni_22] + [(-1)bni_22]i141[2] + [bni_22]i139[2] ≥ 0∧[(-1)bso_23] ≥ 0)



    We simplified constraint (36) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint:

    (37)    (i141[2] + [-1] ≥ 0∧i141[2] + [-1] ≥ 0 ⇒ (UIncreasing(LOAD1161(+(i139[3], -1), +(i141[3], -1))), ≥)∧[bni_22] = 0∧[(-1)bni_22 + (-1)Bound*bni_22] + [(-1)bni_22]i141[2] ≥ 0∧0 = 0∧[(-1)bso_23] ≥ 0)



    We simplified constraint (37) using rule (IDP_SMT_SPLIT) which results in the following new constraint:

    (38)    (i141[2] ≥ 0∧i141[2] ≥ 0 ⇒ (UIncreasing(LOAD1161(+(i139[3], -1), +(i141[3], -1))), ≥)∧[bni_22] = 0∧[(-2)bni_22 + (-1)Bound*bni_22] + [(-1)bni_22]i141[2] ≥ 0∧0 = 0∧[(-1)bso_23] ≥ 0)







For Pair LOAD1161(i139, 0) → LOAD804(i139) the following chains were created:
  • We consider the chain LOAD1161(i139[4], 0) → LOAD804(i139[4]), LOAD804(i108[0]) → COND_LOAD804(>(i108[0], 0), i108[0]) which results in the following constraint:

    (39)    (i139[4]=i108[0] ⇒ LOAD1161(i139[4], 0)≥NonInfC∧LOAD1161(i139[4], 0)≥LOAD804(i139[4])∧(UIncreasing(LOAD804(i139[4])), ≥))



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

    (40)    (LOAD1161(i139[4], 0)≥NonInfC∧LOAD1161(i139[4], 0)≥LOAD804(i139[4])∧(UIncreasing(LOAD804(i139[4])), ≥))



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

    (41)    ((UIncreasing(LOAD804(i139[4])), ≥)∧[(-1)bso_25] ≥ 0)



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

    (42)    ((UIncreasing(LOAD804(i139[4])), ≥)∧[(-1)bso_25] ≥ 0)



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

    (43)    ((UIncreasing(LOAD804(i139[4])), ≥)∧[(-1)bso_25] ≥ 0)



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

    (44)    ((UIncreasing(LOAD804(i139[4])), ≥)∧0 = 0∧[(-1)bso_25] ≥ 0)







To summarize, we get the following constraints P≥ for the following pairs.
  • LOAD804(i108) → COND_LOAD804(>(i108, 0), i108)
    • (i108[0] ≥ 0 ⇒ (UIncreasing(COND_LOAD804(>(i108[0], 0), i108[0])), ≥)∧[(-1)Bound*bni_16] + [bni_16]i108[0] ≥ 0∧[(-1)bso_17] + i108[0] ≥ 0)

  • COND_LOAD804(TRUE, i108) → LOAD1161(i108, i108)
    • ((UIncreasing(LOAD1161(i108[1], i108[1])), ≥)∧0 = 0∧[1 + (-1)bso_19] ≥ 0)
    • ((UIncreasing(LOAD1161(i108[1], i108[1])), ≥)∧[1 + (-1)bso_19] ≥ 0)

  • LOAD1161(i139, i141) → COND_LOAD1161(>(i141, 0), i139, i141)
    • (i141[2] ≥ 0 ⇒ (UIncreasing(COND_LOAD1161(>(i141[2], 0), i139[2], i141[2])), ≥)∧[bni_20] = 0∧[(-2)bni_20 + (-1)Bound*bni_20] + [(-1)bni_20]i141[2] ≥ 0∧0 = 0∧[(-1)bso_21] ≥ 0)

  • COND_LOAD1161(TRUE, i139, i141) → LOAD1161(+(i139, -1), +(i141, -1))
    • (i141[2] ≥ 0 ⇒ (UIncreasing(LOAD1161(+(i139[3], -1), +(i141[3], -1))), ≥)∧[bni_22] = 0∧[(-2)bni_22 + (-1)Bound*bni_22] + [(-1)bni_22]i141[2] ≥ 0∧0 = 0∧[(-1)bso_23] ≥ 0)
    • (i141[2] ≥ 0∧i141[2] ≥ 0 ⇒ (UIncreasing(LOAD1161(+(i139[3], -1), +(i141[3], -1))), ≥)∧[bni_22] = 0∧[(-2)bni_22 + (-1)Bound*bni_22] + [(-1)bni_22]i141[2] ≥ 0∧0 = 0∧[(-1)bso_23] ≥ 0)

  • LOAD1161(i139, 0) → LOAD804(i139)
    • ((UIncreasing(LOAD804(i139[4])), ≥)∧0 = 0∧[(-1)bso_25] ≥ 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(LOAD804(x1)) = [-1] + x1   
POL(COND_LOAD804(x1, x2)) = 0   
POL(>(x1, x2)) = [1]   
POL(0) = 0   
POL(LOAD1161(x1, x2)) = [-1] + [-1]x2 + x1   
POL(COND_LOAD1161(x1, x2, x3)) = [-1] + [-1]x3 + x2   
POL(+(x1, x2)) = x1 + x2   
POL(-1) = [-1]   

The following pairs are in P>:

COND_LOAD804(TRUE, i108[1]) → LOAD1161(i108[1], i108[1])

The following pairs are in Pbound:

LOAD804(i108[0]) → COND_LOAD804(>(i108[0], 0), i108[0])

The following pairs are in P≥:

LOAD804(i108[0]) → COND_LOAD804(>(i108[0], 0), i108[0])
LOAD1161(i139[2], i141[2]) → COND_LOAD1161(>(i141[2], 0), i139[2], i141[2])
COND_LOAD1161(TRUE, i139[3], i141[3]) → LOAD1161(+(i139[3], -1), +(i141[3], -1))
LOAD1161(i139[4], 0) → LOAD804(i139[4])

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:
(0): LOAD804(i108[0]) → COND_LOAD804(i108[0] > 0, i108[0])
(2): LOAD1161(i139[2], i141[2]) → COND_LOAD1161(i141[2] > 0, i139[2], i141[2])
(3): COND_LOAD1161(TRUE, i139[3], i141[3]) → LOAD1161(i139[3] + -1, i141[3] + -1)
(4): LOAD1161(i139[4], 0) → LOAD804(i139[4])

(4) -> (0), if ((i139[4] →* i108[0]))


(3) -> (2), if ((i141[3] + -1 →* i141[2])∧(i139[3] + -1 →* i139[2]))


(2) -> (3), if ((i139[2] →* i139[3])∧(i141[2] > 0 →* TRUE)∧(i141[2] →* i141[3]))


(3) -> (4), if ((i139[3] + -1 →* i139[4])∧(i141[3] + -1 →* 0))



The set Q consists of the following terms:
Load804(x0)
Cond_Load804(TRUE, x0)
Load1161(x0, x1)
Cond_Load1161(TRUE, x0, x1)

(12) IDependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 2 less nodes.

(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:
(3): COND_LOAD1161(TRUE, i139[3], i141[3]) → LOAD1161(i139[3] + -1, i141[3] + -1)
(2): LOAD1161(i139[2], i141[2]) → COND_LOAD1161(i141[2] > 0, i139[2], i141[2])

(3) -> (2), if ((i141[3] + -1 →* i141[2])∧(i139[3] + -1 →* i139[2]))


(2) -> (3), if ((i139[2] →* i139[3])∧(i141[2] > 0 →* TRUE)∧(i141[2] →* i141[3]))



The set Q consists of the following terms:
Load804(x0)
Cond_Load804(TRUE, x0)
Load1161(x0, x1)
Cond_Load1161(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_LOAD1161(TRUE, i139[3], i141[3]) → LOAD1161(+(i139[3], -1), +(i141[3], -1)) the following chains were created:
  • We consider the chain LOAD1161(i139[2], i141[2]) → COND_LOAD1161(>(i141[2], 0), i139[2], i141[2]), COND_LOAD1161(TRUE, i139[3], i141[3]) → LOAD1161(+(i139[3], -1), +(i141[3], -1)), LOAD1161(i139[2], i141[2]) → COND_LOAD1161(>(i141[2], 0), i139[2], i141[2]) which results in the following constraint:

    (1)    (i139[2]=i139[3]∧>(i141[2], 0)=TRUE∧i141[2]=i141[3]∧+(i141[3], -1)=i141[2]1∧+(i139[3], -1)=i139[2]1 ⇒ COND_LOAD1161(TRUE, i139[3], i141[3])≥NonInfC∧COND_LOAD1161(TRUE, i139[3], i141[3])≥LOAD1161(+(i139[3], -1), +(i141[3], -1))∧(UIncreasing(LOAD1161(+(i139[3], -1), +(i141[3], -1))), ≥))



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

    (2)    (>(i141[2], 0)=TRUE ⇒ COND_LOAD1161(TRUE, i139[2], i141[2])≥NonInfC∧COND_LOAD1161(TRUE, i139[2], i141[2])≥LOAD1161(+(i139[2], -1), +(i141[2], -1))∧(UIncreasing(LOAD1161(+(i139[3], -1), +(i141[3], -1))), ≥))



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

    (3)    (i141[2] + [-1] ≥ 0 ⇒ (UIncreasing(LOAD1161(+(i139[3], -1), +(i141[3], -1))), ≥)∧[(-1)bni_11 + (-1)Bound*bni_11] + [bni_11]i141[2] ≥ 0∧[1 + (-1)bso_12] ≥ 0)



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

    (4)    (i141[2] + [-1] ≥ 0 ⇒ (UIncreasing(LOAD1161(+(i139[3], -1), +(i141[3], -1))), ≥)∧[(-1)bni_11 + (-1)Bound*bni_11] + [bni_11]i141[2] ≥ 0∧[1 + (-1)bso_12] ≥ 0)



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

    (5)    (i141[2] + [-1] ≥ 0 ⇒ (UIncreasing(LOAD1161(+(i139[3], -1), +(i141[3], -1))), ≥)∧[(-1)bni_11 + (-1)Bound*bni_11] + [bni_11]i141[2] ≥ 0∧[1 + (-1)bso_12] ≥ 0)



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

    (6)    (i141[2] + [-1] ≥ 0 ⇒ (UIncreasing(LOAD1161(+(i139[3], -1), +(i141[3], -1))), ≥)∧0 = 0∧[(-1)bni_11 + (-1)Bound*bni_11] + [bni_11]i141[2] ≥ 0∧0 = 0∧[1 + (-1)bso_12] ≥ 0)



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

    (7)    (i141[2] ≥ 0 ⇒ (UIncreasing(LOAD1161(+(i139[3], -1), +(i141[3], -1))), ≥)∧0 = 0∧[(-1)Bound*bni_11] + [bni_11]i141[2] ≥ 0∧0 = 0∧[1 + (-1)bso_12] ≥ 0)







For Pair LOAD1161(i139[2], i141[2]) → COND_LOAD1161(>(i141[2], 0), i139[2], i141[2]) the following chains were created:
  • We consider the chain LOAD1161(i139[2], i141[2]) → COND_LOAD1161(>(i141[2], 0), i139[2], i141[2]), COND_LOAD1161(TRUE, i139[3], i141[3]) → LOAD1161(+(i139[3], -1), +(i141[3], -1)) which results in the following constraint:

    (8)    (i139[2]=i139[3]∧>(i141[2], 0)=TRUE∧i141[2]=i141[3] ⇒ LOAD1161(i139[2], i141[2])≥NonInfC∧LOAD1161(i139[2], i141[2])≥COND_LOAD1161(>(i141[2], 0), i139[2], i141[2])∧(UIncreasing(COND_LOAD1161(>(i141[2], 0), i139[2], i141[2])), ≥))



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

    (9)    (>(i141[2], 0)=TRUE ⇒ LOAD1161(i139[2], i141[2])≥NonInfC∧LOAD1161(i139[2], i141[2])≥COND_LOAD1161(>(i141[2], 0), i139[2], i141[2])∧(UIncreasing(COND_LOAD1161(>(i141[2], 0), i139[2], i141[2])), ≥))



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

    (10)    (i141[2] + [-1] ≥ 0 ⇒ (UIncreasing(COND_LOAD1161(>(i141[2], 0), i139[2], i141[2])), ≥)∧[(-1)bni_13 + (-1)Bound*bni_13] + [bni_13]i141[2] ≥ 0∧[(-1)bso_14] ≥ 0)



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

    (11)    (i141[2] + [-1] ≥ 0 ⇒ (UIncreasing(COND_LOAD1161(>(i141[2], 0), i139[2], i141[2])), ≥)∧[(-1)bni_13 + (-1)Bound*bni_13] + [bni_13]i141[2] ≥ 0∧[(-1)bso_14] ≥ 0)



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

    (12)    (i141[2] + [-1] ≥ 0 ⇒ (UIncreasing(COND_LOAD1161(>(i141[2], 0), i139[2], i141[2])), ≥)∧[(-1)bni_13 + (-1)Bound*bni_13] + [bni_13]i141[2] ≥ 0∧[(-1)bso_14] ≥ 0)



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

    (13)    (i141[2] + [-1] ≥ 0 ⇒ (UIncreasing(COND_LOAD1161(>(i141[2], 0), i139[2], i141[2])), ≥)∧0 = 0∧[(-1)bni_13 + (-1)Bound*bni_13] + [bni_13]i141[2] ≥ 0∧0 = 0∧[(-1)bso_14] ≥ 0)



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

    (14)    (i141[2] ≥ 0 ⇒ (UIncreasing(COND_LOAD1161(>(i141[2], 0), i139[2], i141[2])), ≥)∧0 = 0∧[(-1)Bound*bni_13] + [bni_13]i141[2] ≥ 0∧0 = 0∧[(-1)bso_14] ≥ 0)







To summarize, we get the following constraints P≥ for the following pairs.
  • COND_LOAD1161(TRUE, i139[3], i141[3]) → LOAD1161(+(i139[3], -1), +(i141[3], -1))
    • (i141[2] ≥ 0 ⇒ (UIncreasing(LOAD1161(+(i139[3], -1), +(i141[3], -1))), ≥)∧0 = 0∧[(-1)Bound*bni_11] + [bni_11]i141[2] ≥ 0∧0 = 0∧[1 + (-1)bso_12] ≥ 0)

  • LOAD1161(i139[2], i141[2]) → COND_LOAD1161(>(i141[2], 0), i139[2], i141[2])
    • (i141[2] ≥ 0 ⇒ (UIncreasing(COND_LOAD1161(>(i141[2], 0), i139[2], i141[2])), ≥)∧0 = 0∧[(-1)Bound*bni_13] + [bni_13]i141[2] ≥ 0∧0 = 0∧[(-1)bso_14] ≥ 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_LOAD1161(x1, x2, x3)) = [-1] + x3   
POL(LOAD1161(x1, x2)) = [-1] + x2   
POL(+(x1, x2)) = x1 + x2   
POL(-1) = [-1]   
POL(>(x1, x2)) = [2]   
POL(0) = 0   

The following pairs are in P>:

COND_LOAD1161(TRUE, i139[3], i141[3]) → LOAD1161(+(i139[3], -1), +(i141[3], -1))

The following pairs are in Pbound:

COND_LOAD1161(TRUE, i139[3], i141[3]) → LOAD1161(+(i139[3], -1), +(i141[3], -1))
LOAD1161(i139[2], i141[2]) → COND_LOAD1161(>(i141[2], 0), i139[2], i141[2])

The following pairs are in P≥:

LOAD1161(i139[2], i141[2]) → COND_LOAD1161(>(i141[2], 0), i139[2], i141[2])

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:
(2): LOAD1161(i139[2], i141[2]) → COND_LOAD1161(i141[2] > 0, i139[2], i141[2])


The set Q consists of the following terms:
Load804(x0)
Cond_Load804(TRUE, x0)
Load1161(x0, x1)
Cond_Load1161(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:
none


R is empty.

The integer pair graph is empty.

The set Q consists of the following terms:
Load804(x0)
Cond_Load804(TRUE, x0)
Load1161(x0, x1)
Cond_Load1161(TRUE, x0, x1)

(20) IDependencyGraphProof (EQUIVALENT transformation)

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

(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:
(1): COND_LOAD804(TRUE, i108[1]) → LOAD1161(i108[1], i108[1])
(2): LOAD1161(i139[2], i141[2]) → COND_LOAD1161(i141[2] > 0, i139[2], i141[2])
(3): COND_LOAD1161(TRUE, i139[3], i141[3]) → LOAD1161(i139[3] + -1, i141[3] + -1)
(4): LOAD1161(i139[4], 0) → LOAD804(i139[4])

(1) -> (2), if ((i108[1] →* i139[2])∧(i108[1] →* i141[2]))


(3) -> (2), if ((i141[3] + -1 →* i141[2])∧(i139[3] + -1 →* i139[2]))


(2) -> (3), if ((i139[2] →* i139[3])∧(i141[2] > 0 →* TRUE)∧(i141[2] →* i141[3]))


(1) -> (4), if ((i108[1] →* i139[4])∧(i108[1] →* 0))


(3) -> (4), if ((i139[3] + -1 →* i139[4])∧(i141[3] + -1 →* 0))



The set Q consists of the following terms:
Load804(x0)
Cond_Load804(TRUE, x0)
Load1161(x0, x1)
Cond_Load1161(TRUE, x0, x1)

(23) IDependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 2 less nodes.

(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:
(3): COND_LOAD1161(TRUE, i139[3], i141[3]) → LOAD1161(i139[3] + -1, i141[3] + -1)
(2): LOAD1161(i139[2], i141[2]) → COND_LOAD1161(i141[2] > 0, i139[2], i141[2])

(3) -> (2), if ((i141[3] + -1 →* i141[2])∧(i139[3] + -1 →* i139[2]))


(2) -> (3), if ((i139[2] →* i139[3])∧(i141[2] > 0 →* TRUE)∧(i141[2] →* i141[3]))



The set Q consists of the following terms:
Load804(x0)
Cond_Load804(TRUE, x0)
Load1161(x0, x1)
Cond_Load1161(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_LOAD1161(TRUE, i139[3], i141[3]) → LOAD1161(+(i139[3], -1), +(i141[3], -1)) the following chains were created:
  • We consider the chain LOAD1161(i139[2], i141[2]) → COND_LOAD1161(>(i141[2], 0), i139[2], i141[2]), COND_LOAD1161(TRUE, i139[3], i141[3]) → LOAD1161(+(i139[3], -1), +(i141[3], -1)), LOAD1161(i139[2], i141[2]) → COND_LOAD1161(>(i141[2], 0), i139[2], i141[2]) which results in the following constraint:

    (1)    (i139[2]=i139[3]∧>(i141[2], 0)=TRUE∧i141[2]=i141[3]∧+(i141[3], -1)=i141[2]1∧+(i139[3], -1)=i139[2]1 ⇒ COND_LOAD1161(TRUE, i139[3], i141[3])≥NonInfC∧COND_LOAD1161(TRUE, i139[3], i141[3])≥LOAD1161(+(i139[3], -1), +(i141[3], -1))∧(UIncreasing(LOAD1161(+(i139[3], -1), +(i141[3], -1))), ≥))



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

    (2)    (>(i141[2], 0)=TRUE ⇒ COND_LOAD1161(TRUE, i139[2], i141[2])≥NonInfC∧COND_LOAD1161(TRUE, i139[2], i141[2])≥LOAD1161(+(i139[2], -1), +(i141[2], -1))∧(UIncreasing(LOAD1161(+(i139[3], -1), +(i141[3], -1))), ≥))



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

    (3)    (i141[2] + [-1] ≥ 0 ⇒ (UIncreasing(LOAD1161(+(i139[3], -1), +(i141[3], -1))), ≥)∧[(-1)bni_11 + (-1)Bound*bni_11] + [bni_11]i141[2] ≥ 0∧[1 + (-1)bso_12] ≥ 0)



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

    (4)    (i141[2] + [-1] ≥ 0 ⇒ (UIncreasing(LOAD1161(+(i139[3], -1), +(i141[3], -1))), ≥)∧[(-1)bni_11 + (-1)Bound*bni_11] + [bni_11]i141[2] ≥ 0∧[1 + (-1)bso_12] ≥ 0)



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

    (5)    (i141[2] + [-1] ≥ 0 ⇒ (UIncreasing(LOAD1161(+(i139[3], -1), +(i141[3], -1))), ≥)∧[(-1)bni_11 + (-1)Bound*bni_11] + [bni_11]i141[2] ≥ 0∧[1 + (-1)bso_12] ≥ 0)



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

    (6)    (i141[2] + [-1] ≥ 0 ⇒ (UIncreasing(LOAD1161(+(i139[3], -1), +(i141[3], -1))), ≥)∧0 = 0∧[(-1)bni_11 + (-1)Bound*bni_11] + [bni_11]i141[2] ≥ 0∧0 = 0∧[1 + (-1)bso_12] ≥ 0)



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

    (7)    (i141[2] ≥ 0 ⇒ (UIncreasing(LOAD1161(+(i139[3], -1), +(i141[3], -1))), ≥)∧0 = 0∧[(-1)Bound*bni_11] + [bni_11]i141[2] ≥ 0∧0 = 0∧[1 + (-1)bso_12] ≥ 0)







For Pair LOAD1161(i139[2], i141[2]) → COND_LOAD1161(>(i141[2], 0), i139[2], i141[2]) the following chains were created:
  • We consider the chain LOAD1161(i139[2], i141[2]) → COND_LOAD1161(>(i141[2], 0), i139[2], i141[2]), COND_LOAD1161(TRUE, i139[3], i141[3]) → LOAD1161(+(i139[3], -1), +(i141[3], -1)) which results in the following constraint:

    (8)    (i139[2]=i139[3]∧>(i141[2], 0)=TRUE∧i141[2]=i141[3] ⇒ LOAD1161(i139[2], i141[2])≥NonInfC∧LOAD1161(i139[2], i141[2])≥COND_LOAD1161(>(i141[2], 0), i139[2], i141[2])∧(UIncreasing(COND_LOAD1161(>(i141[2], 0), i139[2], i141[2])), ≥))



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

    (9)    (>(i141[2], 0)=TRUE ⇒ LOAD1161(i139[2], i141[2])≥NonInfC∧LOAD1161(i139[2], i141[2])≥COND_LOAD1161(>(i141[2], 0), i139[2], i141[2])∧(UIncreasing(COND_LOAD1161(>(i141[2], 0), i139[2], i141[2])), ≥))



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

    (10)    (i141[2] + [-1] ≥ 0 ⇒ (UIncreasing(COND_LOAD1161(>(i141[2], 0), i139[2], i141[2])), ≥)∧[(-1)bni_13 + (-1)Bound*bni_13] + [bni_13]i141[2] ≥ 0∧[(-1)bso_14] ≥ 0)



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

    (11)    (i141[2] + [-1] ≥ 0 ⇒ (UIncreasing(COND_LOAD1161(>(i141[2], 0), i139[2], i141[2])), ≥)∧[(-1)bni_13 + (-1)Bound*bni_13] + [bni_13]i141[2] ≥ 0∧[(-1)bso_14] ≥ 0)



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

    (12)    (i141[2] + [-1] ≥ 0 ⇒ (UIncreasing(COND_LOAD1161(>(i141[2], 0), i139[2], i141[2])), ≥)∧[(-1)bni_13 + (-1)Bound*bni_13] + [bni_13]i141[2] ≥ 0∧[(-1)bso_14] ≥ 0)



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

    (13)    (i141[2] + [-1] ≥ 0 ⇒ (UIncreasing(COND_LOAD1161(>(i141[2], 0), i139[2], i141[2])), ≥)∧0 = 0∧[(-1)bni_13 + (-1)Bound*bni_13] + [bni_13]i141[2] ≥ 0∧0 = 0∧[(-1)bso_14] ≥ 0)



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

    (14)    (i141[2] ≥ 0 ⇒ (UIncreasing(COND_LOAD1161(>(i141[2], 0), i139[2], i141[2])), ≥)∧0 = 0∧[(-1)Bound*bni_13] + [bni_13]i141[2] ≥ 0∧0 = 0∧[(-1)bso_14] ≥ 0)







To summarize, we get the following constraints P≥ for the following pairs.
  • COND_LOAD1161(TRUE, i139[3], i141[3]) → LOAD1161(+(i139[3], -1), +(i141[3], -1))
    • (i141[2] ≥ 0 ⇒ (UIncreasing(LOAD1161(+(i139[3], -1), +(i141[3], -1))), ≥)∧0 = 0∧[(-1)Bound*bni_11] + [bni_11]i141[2] ≥ 0∧0 = 0∧[1 + (-1)bso_12] ≥ 0)

  • LOAD1161(i139[2], i141[2]) → COND_LOAD1161(>(i141[2], 0), i139[2], i141[2])
    • (i141[2] ≥ 0 ⇒ (UIncreasing(COND_LOAD1161(>(i141[2], 0), i139[2], i141[2])), ≥)∧0 = 0∧[(-1)Bound*bni_13] + [bni_13]i141[2] ≥ 0∧0 = 0∧[(-1)bso_14] ≥ 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_LOAD1161(x1, x2, x3)) = [-1] + x3   
POL(LOAD1161(x1, x2)) = [-1] + x2   
POL(+(x1, x2)) = x1 + x2   
POL(-1) = [-1]   
POL(>(x1, x2)) = [2]   
POL(0) = 0   

The following pairs are in P>:

COND_LOAD1161(TRUE, i139[3], i141[3]) → LOAD1161(+(i139[3], -1), +(i141[3], -1))

The following pairs are in Pbound:

COND_LOAD1161(TRUE, i139[3], i141[3]) → LOAD1161(+(i139[3], -1), +(i141[3], -1))
LOAD1161(i139[2], i141[2]) → COND_LOAD1161(>(i141[2], 0), i139[2], i141[2])

The following pairs are in P≥:

LOAD1161(i139[2], i141[2]) → COND_LOAD1161(>(i141[2], 0), i139[2], i141[2])

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:
(2): LOAD1161(i139[2], i141[2]) → COND_LOAD1161(i141[2] > 0, i139[2], i141[2])


The set Q consists of the following terms:
Load804(x0)
Cond_Load804(TRUE, x0)
Load1161(x0, x1)
Cond_Load1161(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:
none


R is empty.

The integer pair graph is empty.

The set Q consists of the following terms:
Load804(x0)
Cond_Load804(TRUE, x0)
Load1161(x0, x1)
Cond_Load1161(TRUE, x0, x1)

(31) IDependencyGraphProof (EQUIVALENT transformation)

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

(32) TRUE