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 TRUE

(0) Obligation:

JBC Problem based on JBC Program:
Manifest-Version: 1.0 Created-By: 1.6.0_16 (Sun Microsystems Inc.) Main-Class: PastaA1 
/**
 * 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 PastaA1 {
    public static void main(String[] args) {
        Random.args = args;
        int x = Random.random();
        while (x > 0) {
            int y = 0;
            while (y < x) {
                y++;
            }
            x--;
        }
    }
}


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 133 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:
Load323(i20) → Cond_Load323(i20 > 0, i20)
Cond_Load323(TRUE, i20) → Load597(i20, 0)
Load597(i20, i31) → Cond_Load597(i31 >= 0 && i31 < i20 && i31 + 1 > 0, i20, i31)
Cond_Load597(TRUE, i20, i31) → Load597(i20, i31 + 1)
Load597(i20, i31) → Cond_Load5971(i20 > 0 && i31 >= i20, i20, i31)
Cond_Load5971(TRUE, i20, i31) → Load323(i20 + -1)
The set Q consists of the following terms:
Load323(x0)
Cond_Load323(TRUE, x0)
Load597(x0, x1)
Cond_Load597(TRUE, x0, x1)
Cond_Load5971(TRUE, x0, x1)

(5) ITRStoIDPProof (EQUIVALENT transformation)

Added dependency pairs

(6) Obligation:

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


The following domains are used:

Integer, Boolean


The ITRS R consists of the following rules:
Load323(i20) → Cond_Load323(i20 > 0, i20)
Cond_Load323(TRUE, i20) → Load597(i20, 0)
Load597(i20, i31) → Cond_Load597(i31 >= 0 && i31 < i20 && i31 + 1 > 0, i20, i31)
Cond_Load597(TRUE, i20, i31) → Load597(i20, i31 + 1)
Load597(i20, i31) → Cond_Load5971(i20 > 0 && i31 >= i20, i20, i31)
Cond_Load5971(TRUE, i20, i31) → Load323(i20 + -1)

The integer pair graph contains the following rules and edges:
(0): LOAD323(i20[0]) → COND_LOAD323(i20[0] > 0, i20[0])
(1): COND_LOAD323(TRUE, i20[1]) → LOAD597(i20[1], 0)
(2): LOAD597(i20[2], i31[2]) → COND_LOAD597(i31[2] >= 0 && i31[2] < i20[2] && i31[2] + 1 > 0, i20[2], i31[2])
(3): COND_LOAD597(TRUE, i20[3], i31[3]) → LOAD597(i20[3], i31[3] + 1)
(4): LOAD597(i20[4], i31[4]) → COND_LOAD5971(i20[4] > 0 && i31[4] >= i20[4], i20[4], i31[4])
(5): COND_LOAD5971(TRUE, i20[5], i31[5]) → LOAD323(i20[5] + -1)

(0) -> (1), if ((i20[0] > 0 →* TRUE)∧(i20[0] →* i20[1]))


(1) -> (2), if ((i20[1] →* i20[2])∧(0 →* i31[2]))


(1) -> (4), if ((i20[1] →* i20[4])∧(0 →* i31[4]))


(2) -> (3), if ((i20[2] →* i20[3])∧(i31[2] →* i31[3])∧(i31[2] >= 0 && i31[2] < i20[2] && i31[2] + 1 > 0 →* TRUE))


(3) -> (2), if ((i31[3] + 1 →* i31[2])∧(i20[3] →* i20[2]))


(3) -> (4), if ((i20[3] →* i20[4])∧(i31[3] + 1 →* i31[4]))


(4) -> (5), if ((i20[4] > 0 && i31[4] >= i20[4] →* TRUE)∧(i20[4] →* i20[5])∧(i31[4] →* i31[5]))


(5) -> (0), if ((i20[5] + -1 →* i20[0]))



The set Q consists of the following terms:
Load323(x0)
Cond_Load323(TRUE, x0)
Load597(x0, x1)
Cond_Load597(TRUE, x0, x1)
Cond_Load5971(TRUE, x0, x1)

(7) UsableRulesProof (EQUIVALENT transformation)

As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [LPAR04] we can delete all non-usable rules [FROCOS05] from R.

(8) Obligation:

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


The following domains are used:

Integer, Boolean


R is empty.

The integer pair graph contains the following rules and edges:
(0): LOAD323(i20[0]) → COND_LOAD323(i20[0] > 0, i20[0])
(1): COND_LOAD323(TRUE, i20[1]) → LOAD597(i20[1], 0)
(2): LOAD597(i20[2], i31[2]) → COND_LOAD597(i31[2] >= 0 && i31[2] < i20[2] && i31[2] + 1 > 0, i20[2], i31[2])
(3): COND_LOAD597(TRUE, i20[3], i31[3]) → LOAD597(i20[3], i31[3] + 1)
(4): LOAD597(i20[4], i31[4]) → COND_LOAD5971(i20[4] > 0 && i31[4] >= i20[4], i20[4], i31[4])
(5): COND_LOAD5971(TRUE, i20[5], i31[5]) → LOAD323(i20[5] + -1)

(0) -> (1), if ((i20[0] > 0 →* TRUE)∧(i20[0] →* i20[1]))


(1) -> (2), if ((i20[1] →* i20[2])∧(0 →* i31[2]))


(1) -> (4), if ((i20[1] →* i20[4])∧(0 →* i31[4]))


(2) -> (3), if ((i20[2] →* i20[3])∧(i31[2] →* i31[3])∧(i31[2] >= 0 && i31[2] < i20[2] && i31[2] + 1 > 0 →* TRUE))


(3) -> (2), if ((i31[3] + 1 →* i31[2])∧(i20[3] →* i20[2]))


(3) -> (4), if ((i20[3] →* i20[4])∧(i31[3] + 1 →* i31[4]))


(4) -> (5), if ((i20[4] > 0 && i31[4] >= i20[4] →* TRUE)∧(i20[4] →* i20[5])∧(i31[4] →* i31[5]))


(5) -> (0), if ((i20[5] + -1 →* i20[0]))



The set Q consists of the following terms:
Load323(x0)
Cond_Load323(TRUE, x0)
Load597(x0, x1)
Cond_Load597(TRUE, x0, x1)
Cond_Load5971(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 LOAD323(i20) → COND_LOAD323(>(i20, 0), i20) the following chains were created:
  • We consider the chain LOAD323(i20[0]) → COND_LOAD323(>(i20[0], 0), i20[0]), COND_LOAD323(TRUE, i20[1]) → LOAD597(i20[1], 0) which results in the following constraint:

    (1)    (>(i20[0], 0)=TRUE∧i20[0]=i20[1] ⇒ LOAD323(i20[0])≥NonInfC∧LOAD323(i20[0])≥COND_LOAD323(>(i20[0], 0), i20[0])∧(UIncreasing(COND_LOAD323(>(i20[0], 0), i20[0])), ≥))



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

    (2)    (>(i20[0], 0)=TRUE ⇒ LOAD323(i20[0])≥NonInfC∧LOAD323(i20[0])≥COND_LOAD323(>(i20[0], 0), i20[0])∧(UIncreasing(COND_LOAD323(>(i20[0], 0), i20[0])), ≥))



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

    (3)    (i20[0] + [-1] ≥ 0 ⇒ (UIncreasing(COND_LOAD323(>(i20[0], 0), i20[0])), ≥)∧[bni_23 + (-1)Bound*bni_23] + [bni_23]i20[0] ≥ 0∧[1 + (-1)bso_24] ≥ 0)



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

    (4)    (i20[0] + [-1] ≥ 0 ⇒ (UIncreasing(COND_LOAD323(>(i20[0], 0), i20[0])), ≥)∧[bni_23 + (-1)Bound*bni_23] + [bni_23]i20[0] ≥ 0∧[1 + (-1)bso_24] ≥ 0)



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

    (5)    (i20[0] + [-1] ≥ 0 ⇒ (UIncreasing(COND_LOAD323(>(i20[0], 0), i20[0])), ≥)∧[bni_23 + (-1)Bound*bni_23] + [bni_23]i20[0] ≥ 0∧[1 + (-1)bso_24] ≥ 0)



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

    (6)    (i20[0] ≥ 0 ⇒ (UIncreasing(COND_LOAD323(>(i20[0], 0), i20[0])), ≥)∧[(2)bni_23 + (-1)Bound*bni_23] + [bni_23]i20[0] ≥ 0∧[1 + (-1)bso_24] ≥ 0)







For Pair COND_LOAD323(TRUE, i20) → LOAD597(i20, 0) the following chains were created:
  • We consider the chain LOAD323(i20[0]) → COND_LOAD323(>(i20[0], 0), i20[0]), COND_LOAD323(TRUE, i20[1]) → LOAD597(i20[1], 0), LOAD597(i20[2], i31[2]) → COND_LOAD597(&&(&&(>=(i31[2], 0), <(i31[2], i20[2])), >(+(i31[2], 1), 0)), i20[2], i31[2]) which results in the following constraint:

    (7)    (>(i20[0], 0)=TRUE∧i20[0]=i20[1]∧i20[1]=i20[2]∧0=i31[2] ⇒ COND_LOAD323(TRUE, i20[1])≥NonInfC∧COND_LOAD323(TRUE, i20[1])≥LOAD597(i20[1], 0)∧(UIncreasing(LOAD597(i20[1], 0)), ≥))



    We simplified constraint (7) using rules (III), (IV) which results in the following new constraint:

    (8)    (>(i20[0], 0)=TRUE ⇒ COND_LOAD323(TRUE, i20[0])≥NonInfC∧COND_LOAD323(TRUE, i20[0])≥LOAD597(i20[0], 0)∧(UIncreasing(LOAD597(i20[1], 0)), ≥))



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

    (9)    (i20[0] + [-1] ≥ 0 ⇒ (UIncreasing(LOAD597(i20[1], 0)), ≥)∧[(-1)Bound*bni_25] + [bni_25]i20[0] ≥ 0∧[(-1)bso_26] ≥ 0)



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

    (10)    (i20[0] + [-1] ≥ 0 ⇒ (UIncreasing(LOAD597(i20[1], 0)), ≥)∧[(-1)Bound*bni_25] + [bni_25]i20[0] ≥ 0∧[(-1)bso_26] ≥ 0)



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

    (11)    (i20[0] + [-1] ≥ 0 ⇒ (UIncreasing(LOAD597(i20[1], 0)), ≥)∧[(-1)Bound*bni_25] + [bni_25]i20[0] ≥ 0∧[(-1)bso_26] ≥ 0)



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

    (12)    (i20[0] ≥ 0 ⇒ (UIncreasing(LOAD597(i20[1], 0)), ≥)∧[(-1)Bound*bni_25 + bni_25] + [bni_25]i20[0] ≥ 0∧[(-1)bso_26] ≥ 0)



  • We consider the chain LOAD323(i20[0]) → COND_LOAD323(>(i20[0], 0), i20[0]), COND_LOAD323(TRUE, i20[1]) → LOAD597(i20[1], 0), LOAD597(i20[4], i31[4]) → COND_LOAD5971(&&(>(i20[4], 0), >=(i31[4], i20[4])), i20[4], i31[4]) which results in the following constraint:

    (13)    (>(i20[0], 0)=TRUE∧i20[0]=i20[1]∧i20[1]=i20[4]∧0=i31[4] ⇒ COND_LOAD323(TRUE, i20[1])≥NonInfC∧COND_LOAD323(TRUE, i20[1])≥LOAD597(i20[1], 0)∧(UIncreasing(LOAD597(i20[1], 0)), ≥))



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

    (14)    (>(i20[0], 0)=TRUE ⇒ COND_LOAD323(TRUE, i20[0])≥NonInfC∧COND_LOAD323(TRUE, i20[0])≥LOAD597(i20[0], 0)∧(UIncreasing(LOAD597(i20[1], 0)), ≥))



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

    (15)    (i20[0] + [-1] ≥ 0 ⇒ (UIncreasing(LOAD597(i20[1], 0)), ≥)∧[(-1)Bound*bni_25] + [bni_25]i20[0] ≥ 0∧[(-1)bso_26] ≥ 0)



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

    (16)    (i20[0] + [-1] ≥ 0 ⇒ (UIncreasing(LOAD597(i20[1], 0)), ≥)∧[(-1)Bound*bni_25] + [bni_25]i20[0] ≥ 0∧[(-1)bso_26] ≥ 0)



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

    (17)    (i20[0] + [-1] ≥ 0 ⇒ (UIncreasing(LOAD597(i20[1], 0)), ≥)∧[(-1)Bound*bni_25] + [bni_25]i20[0] ≥ 0∧[(-1)bso_26] ≥ 0)



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

    (18)    (i20[0] ≥ 0 ⇒ (UIncreasing(LOAD597(i20[1], 0)), ≥)∧[(-1)Bound*bni_25 + bni_25] + [bni_25]i20[0] ≥ 0∧[(-1)bso_26] ≥ 0)







For Pair LOAD597(i20, i31) → COND_LOAD597(&&(&&(>=(i31, 0), <(i31, i20)), >(+(i31, 1), 0)), i20, i31) the following chains were created:
  • We consider the chain LOAD597(i20[2], i31[2]) → COND_LOAD597(&&(&&(>=(i31[2], 0), <(i31[2], i20[2])), >(+(i31[2], 1), 0)), i20[2], i31[2]), COND_LOAD597(TRUE, i20[3], i31[3]) → LOAD597(i20[3], +(i31[3], 1)) which results in the following constraint:

    (19)    (i20[2]=i20[3]∧i31[2]=i31[3]∧&&(&&(>=(i31[2], 0), <(i31[2], i20[2])), >(+(i31[2], 1), 0))=TRUE ⇒ LOAD597(i20[2], i31[2])≥NonInfC∧LOAD597(i20[2], i31[2])≥COND_LOAD597(&&(&&(>=(i31[2], 0), <(i31[2], i20[2])), >(+(i31[2], 1), 0)), i20[2], i31[2])∧(UIncreasing(COND_LOAD597(&&(&&(>=(i31[2], 0), <(i31[2], i20[2])), >(+(i31[2], 1), 0)), i20[2], i31[2])), ≥))



    We simplified constraint (19) using rules (IV), (IDP_BOOLEAN) which results in the following new constraint:

    (20)    (>(+(i31[2], 1), 0)=TRUE∧>=(i31[2], 0)=TRUE∧<(i31[2], i20[2])=TRUE ⇒ LOAD597(i20[2], i31[2])≥NonInfC∧LOAD597(i20[2], i31[2])≥COND_LOAD597(&&(&&(>=(i31[2], 0), <(i31[2], i20[2])), >(+(i31[2], 1), 0)), i20[2], i31[2])∧(UIncreasing(COND_LOAD597(&&(&&(>=(i31[2], 0), <(i31[2], i20[2])), >(+(i31[2], 1), 0)), i20[2], i31[2])), ≥))



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

    (21)    (i31[2] ≥ 0∧i31[2] ≥ 0∧i20[2] + [-1] + [-1]i31[2] ≥ 0 ⇒ (UIncreasing(COND_LOAD597(&&(&&(>=(i31[2], 0), <(i31[2], i20[2])), >(+(i31[2], 1), 0)), i20[2], i31[2])), ≥)∧[(-1)Bound*bni_27] + [bni_27]i20[2] ≥ 0∧[(-1)bso_28] ≥ 0)



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

    (22)    (i31[2] ≥ 0∧i31[2] ≥ 0∧i20[2] + [-1] + [-1]i31[2] ≥ 0 ⇒ (UIncreasing(COND_LOAD597(&&(&&(>=(i31[2], 0), <(i31[2], i20[2])), >(+(i31[2], 1), 0)), i20[2], i31[2])), ≥)∧[(-1)Bound*bni_27] + [bni_27]i20[2] ≥ 0∧[(-1)bso_28] ≥ 0)



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

    (23)    (i31[2] ≥ 0∧i31[2] ≥ 0∧i20[2] + [-1] + [-1]i31[2] ≥ 0 ⇒ (UIncreasing(COND_LOAD597(&&(&&(>=(i31[2], 0), <(i31[2], i20[2])), >(+(i31[2], 1), 0)), i20[2], i31[2])), ≥)∧[(-1)Bound*bni_27] + [bni_27]i20[2] ≥ 0∧[(-1)bso_28] ≥ 0)



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

    (24)    (i31[2] ≥ 0∧i31[2] ≥ 0∧i20[2] ≥ 0 ⇒ (UIncreasing(COND_LOAD597(&&(&&(>=(i31[2], 0), <(i31[2], i20[2])), >(+(i31[2], 1), 0)), i20[2], i31[2])), ≥)∧[(-1)Bound*bni_27 + bni_27] + [bni_27]i31[2] + [bni_27]i20[2] ≥ 0∧[(-1)bso_28] ≥ 0)







For Pair COND_LOAD597(TRUE, i20, i31) → LOAD597(i20, +(i31, 1)) the following chains were created:
  • We consider the chain LOAD597(i20[2], i31[2]) → COND_LOAD597(&&(&&(>=(i31[2], 0), <(i31[2], i20[2])), >(+(i31[2], 1), 0)), i20[2], i31[2]), COND_LOAD597(TRUE, i20[3], i31[3]) → LOAD597(i20[3], +(i31[3], 1)), LOAD597(i20[2], i31[2]) → COND_LOAD597(&&(&&(>=(i31[2], 0), <(i31[2], i20[2])), >(+(i31[2], 1), 0)), i20[2], i31[2]) which results in the following constraint:

    (25)    (i20[2]=i20[3]∧i31[2]=i31[3]∧&&(&&(>=(i31[2], 0), <(i31[2], i20[2])), >(+(i31[2], 1), 0))=TRUE∧+(i31[3], 1)=i31[2]1∧i20[3]=i20[2]1 ⇒ COND_LOAD597(TRUE, i20[3], i31[3])≥NonInfC∧COND_LOAD597(TRUE, i20[3], i31[3])≥LOAD597(i20[3], +(i31[3], 1))∧(UIncreasing(LOAD597(i20[3], +(i31[3], 1))), ≥))



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

    (26)    (>(+(i31[2], 1), 0)=TRUE∧>=(i31[2], 0)=TRUE∧<(i31[2], i20[2])=TRUE ⇒ COND_LOAD597(TRUE, i20[2], i31[2])≥NonInfC∧COND_LOAD597(TRUE, i20[2], i31[2])≥LOAD597(i20[2], +(i31[2], 1))∧(UIncreasing(LOAD597(i20[3], +(i31[3], 1))), ≥))



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

    (27)    (i31[2] ≥ 0∧i31[2] ≥ 0∧i20[2] + [-1] + [-1]i31[2] ≥ 0 ⇒ (UIncreasing(LOAD597(i20[3], +(i31[3], 1))), ≥)∧[(-1)Bound*bni_29] + [bni_29]i20[2] ≥ 0∧[(-1)bso_30] ≥ 0)



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

    (28)    (i31[2] ≥ 0∧i31[2] ≥ 0∧i20[2] + [-1] + [-1]i31[2] ≥ 0 ⇒ (UIncreasing(LOAD597(i20[3], +(i31[3], 1))), ≥)∧[(-1)Bound*bni_29] + [bni_29]i20[2] ≥ 0∧[(-1)bso_30] ≥ 0)



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

    (29)    (i31[2] ≥ 0∧i31[2] ≥ 0∧i20[2] + [-1] + [-1]i31[2] ≥ 0 ⇒ (UIncreasing(LOAD597(i20[3], +(i31[3], 1))), ≥)∧[(-1)Bound*bni_29] + [bni_29]i20[2] ≥ 0∧[(-1)bso_30] ≥ 0)



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

    (30)    (i31[2] ≥ 0∧i31[2] ≥ 0∧i20[2] ≥ 0 ⇒ (UIncreasing(LOAD597(i20[3], +(i31[3], 1))), ≥)∧[(-1)Bound*bni_29 + bni_29] + [bni_29]i31[2] + [bni_29]i20[2] ≥ 0∧[(-1)bso_30] ≥ 0)



  • We consider the chain LOAD597(i20[2], i31[2]) → COND_LOAD597(&&(&&(>=(i31[2], 0), <(i31[2], i20[2])), >(+(i31[2], 1), 0)), i20[2], i31[2]), COND_LOAD597(TRUE, i20[3], i31[3]) → LOAD597(i20[3], +(i31[3], 1)), LOAD597(i20[4], i31[4]) → COND_LOAD5971(&&(>(i20[4], 0), >=(i31[4], i20[4])), i20[4], i31[4]) which results in the following constraint:

    (31)    (i20[2]=i20[3]∧i31[2]=i31[3]∧&&(&&(>=(i31[2], 0), <(i31[2], i20[2])), >(+(i31[2], 1), 0))=TRUE∧i20[3]=i20[4]∧+(i31[3], 1)=i31[4] ⇒ COND_LOAD597(TRUE, i20[3], i31[3])≥NonInfC∧COND_LOAD597(TRUE, i20[3], i31[3])≥LOAD597(i20[3], +(i31[3], 1))∧(UIncreasing(LOAD597(i20[3], +(i31[3], 1))), ≥))



    We simplified constraint (31) using rules (III), (IV), (IDP_BOOLEAN) which results in the following new constraint:

    (32)    (>(+(i31[2], 1), 0)=TRUE∧>=(i31[2], 0)=TRUE∧<(i31[2], i20[2])=TRUE ⇒ COND_LOAD597(TRUE, i20[2], i31[2])≥NonInfC∧COND_LOAD597(TRUE, i20[2], i31[2])≥LOAD597(i20[2], +(i31[2], 1))∧(UIncreasing(LOAD597(i20[3], +(i31[3], 1))), ≥))



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

    (33)    (i31[2] ≥ 0∧i31[2] ≥ 0∧i20[2] + [-1] + [-1]i31[2] ≥ 0 ⇒ (UIncreasing(LOAD597(i20[3], +(i31[3], 1))), ≥)∧[(-1)Bound*bni_29] + [bni_29]i20[2] ≥ 0∧[(-1)bso_30] ≥ 0)



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

    (34)    (i31[2] ≥ 0∧i31[2] ≥ 0∧i20[2] + [-1] + [-1]i31[2] ≥ 0 ⇒ (UIncreasing(LOAD597(i20[3], +(i31[3], 1))), ≥)∧[(-1)Bound*bni_29] + [bni_29]i20[2] ≥ 0∧[(-1)bso_30] ≥ 0)



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

    (35)    (i31[2] ≥ 0∧i31[2] ≥ 0∧i20[2] + [-1] + [-1]i31[2] ≥ 0 ⇒ (UIncreasing(LOAD597(i20[3], +(i31[3], 1))), ≥)∧[(-1)Bound*bni_29] + [bni_29]i20[2] ≥ 0∧[(-1)bso_30] ≥ 0)



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

    (36)    (i31[2] ≥ 0∧i31[2] ≥ 0∧i20[2] ≥ 0 ⇒ (UIncreasing(LOAD597(i20[3], +(i31[3], 1))), ≥)∧[(-1)Bound*bni_29 + bni_29] + [bni_29]i31[2] + [bni_29]i20[2] ≥ 0∧[(-1)bso_30] ≥ 0)







For Pair LOAD597(i20, i31) → COND_LOAD5971(&&(>(i20, 0), >=(i31, i20)), i20, i31) the following chains were created:
  • We consider the chain LOAD597(i20[4], i31[4]) → COND_LOAD5971(&&(>(i20[4], 0), >=(i31[4], i20[4])), i20[4], i31[4]), COND_LOAD5971(TRUE, i20[5], i31[5]) → LOAD323(+(i20[5], -1)) which results in the following constraint:

    (37)    (&&(>(i20[4], 0), >=(i31[4], i20[4]))=TRUE∧i20[4]=i20[5]∧i31[4]=i31[5] ⇒ LOAD597(i20[4], i31[4])≥NonInfC∧LOAD597(i20[4], i31[4])≥COND_LOAD5971(&&(>(i20[4], 0), >=(i31[4], i20[4])), i20[4], i31[4])∧(UIncreasing(COND_LOAD5971(&&(>(i20[4], 0), >=(i31[4], i20[4])), i20[4], i31[4])), ≥))



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

    (38)    (>(i20[4], 0)=TRUE∧>=(i31[4], i20[4])=TRUE ⇒ LOAD597(i20[4], i31[4])≥NonInfC∧LOAD597(i20[4], i31[4])≥COND_LOAD5971(&&(>(i20[4], 0), >=(i31[4], i20[4])), i20[4], i31[4])∧(UIncreasing(COND_LOAD5971(&&(>(i20[4], 0), >=(i31[4], i20[4])), i20[4], i31[4])), ≥))



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

    (39)    (i20[4] + [-1] ≥ 0∧i31[4] + [-1]i20[4] ≥ 0 ⇒ (UIncreasing(COND_LOAD5971(&&(>(i20[4], 0), >=(i31[4], i20[4])), i20[4], i31[4])), ≥)∧[(-1)Bound*bni_31] + [bni_31]i20[4] ≥ 0∧[(-1)bso_32] ≥ 0)



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

    (40)    (i20[4] + [-1] ≥ 0∧i31[4] + [-1]i20[4] ≥ 0 ⇒ (UIncreasing(COND_LOAD5971(&&(>(i20[4], 0), >=(i31[4], i20[4])), i20[4], i31[4])), ≥)∧[(-1)Bound*bni_31] + [bni_31]i20[4] ≥ 0∧[(-1)bso_32] ≥ 0)



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

    (41)    (i20[4] + [-1] ≥ 0∧i31[4] + [-1]i20[4] ≥ 0 ⇒ (UIncreasing(COND_LOAD5971(&&(>(i20[4], 0), >=(i31[4], i20[4])), i20[4], i31[4])), ≥)∧[(-1)Bound*bni_31] + [bni_31]i20[4] ≥ 0∧[(-1)bso_32] ≥ 0)



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

    (42)    (i20[4] ≥ 0∧i31[4] + [-1] + [-1]i20[4] ≥ 0 ⇒ (UIncreasing(COND_LOAD5971(&&(>(i20[4], 0), >=(i31[4], i20[4])), i20[4], i31[4])), ≥)∧[(-1)Bound*bni_31 + bni_31] + [bni_31]i20[4] ≥ 0∧[(-1)bso_32] ≥ 0)



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

    (43)    (i20[4] ≥ 0∧i31[4] ≥ 0 ⇒ (UIncreasing(COND_LOAD5971(&&(>(i20[4], 0), >=(i31[4], i20[4])), i20[4], i31[4])), ≥)∧[(-1)Bound*bni_31 + bni_31] + [bni_31]i20[4] ≥ 0∧[(-1)bso_32] ≥ 0)







For Pair COND_LOAD5971(TRUE, i20, i31) → LOAD323(+(i20, -1)) the following chains were created:
  • We consider the chain LOAD597(i20[4], i31[4]) → COND_LOAD5971(&&(>(i20[4], 0), >=(i31[4], i20[4])), i20[4], i31[4]), COND_LOAD5971(TRUE, i20[5], i31[5]) → LOAD323(+(i20[5], -1)), LOAD323(i20[0]) → COND_LOAD323(>(i20[0], 0), i20[0]) which results in the following constraint:

    (44)    (&&(>(i20[4], 0), >=(i31[4], i20[4]))=TRUE∧i20[4]=i20[5]∧i31[4]=i31[5]∧+(i20[5], -1)=i20[0] ⇒ COND_LOAD5971(TRUE, i20[5], i31[5])≥NonInfC∧COND_LOAD5971(TRUE, i20[5], i31[5])≥LOAD323(+(i20[5], -1))∧(UIncreasing(LOAD323(+(i20[5], -1))), ≥))



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

    (45)    (>(i20[4], 0)=TRUE∧>=(i31[4], i20[4])=TRUE ⇒ COND_LOAD5971(TRUE, i20[4], i31[4])≥NonInfC∧COND_LOAD5971(TRUE, i20[4], i31[4])≥LOAD323(+(i20[4], -1))∧(UIncreasing(LOAD323(+(i20[5], -1))), ≥))



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

    (46)    (i20[4] + [-1] ≥ 0∧i31[4] + [-1]i20[4] ≥ 0 ⇒ (UIncreasing(LOAD323(+(i20[5], -1))), ≥)∧[(-1)Bound*bni_33] + [bni_33]i20[4] ≥ 0∧[(-1)bso_34] ≥ 0)



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

    (47)    (i20[4] + [-1] ≥ 0∧i31[4] + [-1]i20[4] ≥ 0 ⇒ (UIncreasing(LOAD323(+(i20[5], -1))), ≥)∧[(-1)Bound*bni_33] + [bni_33]i20[4] ≥ 0∧[(-1)bso_34] ≥ 0)



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

    (48)    (i20[4] + [-1] ≥ 0∧i31[4] + [-1]i20[4] ≥ 0 ⇒ (UIncreasing(LOAD323(+(i20[5], -1))), ≥)∧[(-1)Bound*bni_33] + [bni_33]i20[4] ≥ 0∧[(-1)bso_34] ≥ 0)



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

    (49)    (i20[4] ≥ 0∧i31[4] + [-1] + [-1]i20[4] ≥ 0 ⇒ (UIncreasing(LOAD323(+(i20[5], -1))), ≥)∧[(-1)Bound*bni_33 + bni_33] + [bni_33]i20[4] ≥ 0∧[(-1)bso_34] ≥ 0)



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

    (50)    (i20[4] ≥ 0∧i31[4] ≥ 0 ⇒ (UIncreasing(LOAD323(+(i20[5], -1))), ≥)∧[(-1)Bound*bni_33 + bni_33] + [bni_33]i20[4] ≥ 0∧[(-1)bso_34] ≥ 0)







To summarize, we get the following constraints P≥ for the following pairs.
  • LOAD323(i20) → COND_LOAD323(>(i20, 0), i20)
    • (i20[0] ≥ 0 ⇒ (UIncreasing(COND_LOAD323(>(i20[0], 0), i20[0])), ≥)∧[(2)bni_23 + (-1)Bound*bni_23] + [bni_23]i20[0] ≥ 0∧[1 + (-1)bso_24] ≥ 0)

  • COND_LOAD323(TRUE, i20) → LOAD597(i20, 0)
    • (i20[0] ≥ 0 ⇒ (UIncreasing(LOAD597(i20[1], 0)), ≥)∧[(-1)Bound*bni_25 + bni_25] + [bni_25]i20[0] ≥ 0∧[(-1)bso_26] ≥ 0)
    • (i20[0] ≥ 0 ⇒ (UIncreasing(LOAD597(i20[1], 0)), ≥)∧[(-1)Bound*bni_25 + bni_25] + [bni_25]i20[0] ≥ 0∧[(-1)bso_26] ≥ 0)

  • LOAD597(i20, i31) → COND_LOAD597(&&(&&(>=(i31, 0), <(i31, i20)), >(+(i31, 1), 0)), i20, i31)
    • (i31[2] ≥ 0∧i31[2] ≥ 0∧i20[2] ≥ 0 ⇒ (UIncreasing(COND_LOAD597(&&(&&(>=(i31[2], 0), <(i31[2], i20[2])), >(+(i31[2], 1), 0)), i20[2], i31[2])), ≥)∧[(-1)Bound*bni_27 + bni_27] + [bni_27]i31[2] + [bni_27]i20[2] ≥ 0∧[(-1)bso_28] ≥ 0)

  • COND_LOAD597(TRUE, i20, i31) → LOAD597(i20, +(i31, 1))
    • (i31[2] ≥ 0∧i31[2] ≥ 0∧i20[2] ≥ 0 ⇒ (UIncreasing(LOAD597(i20[3], +(i31[3], 1))), ≥)∧[(-1)Bound*bni_29 + bni_29] + [bni_29]i31[2] + [bni_29]i20[2] ≥ 0∧[(-1)bso_30] ≥ 0)
    • (i31[2] ≥ 0∧i31[2] ≥ 0∧i20[2] ≥ 0 ⇒ (UIncreasing(LOAD597(i20[3], +(i31[3], 1))), ≥)∧[(-1)Bound*bni_29 + bni_29] + [bni_29]i31[2] + [bni_29]i20[2] ≥ 0∧[(-1)bso_30] ≥ 0)

  • LOAD597(i20, i31) → COND_LOAD5971(&&(>(i20, 0), >=(i31, i20)), i20, i31)
    • (i20[4] ≥ 0∧i31[4] ≥ 0 ⇒ (UIncreasing(COND_LOAD5971(&&(>(i20[4], 0), >=(i31[4], i20[4])), i20[4], i31[4])), ≥)∧[(-1)Bound*bni_31 + bni_31] + [bni_31]i20[4] ≥ 0∧[(-1)bso_32] ≥ 0)

  • COND_LOAD5971(TRUE, i20, i31) → LOAD323(+(i20, -1))
    • (i20[4] ≥ 0∧i31[4] ≥ 0 ⇒ (UIncreasing(LOAD323(+(i20[5], -1))), ≥)∧[(-1)Bound*bni_33 + bni_33] + [bni_33]i20[4] ≥ 0∧[(-1)bso_34] ≥ 0)




The constraints for P> respective Pbound are constructed from P≥ where we just replace every occurence of "t ≥ s" in P≥ by "t > s" respective "t ≥ c". Here c stands for the fresh constant used for Pbound.
Using the following integer polynomial ordering the resulting constraints can be solved
Polynomial interpretation over integers[POLO]:

POL(TRUE) = 0   
POL(FALSE) = [2]   
POL(LOAD323(x1)) = [1] + x1   
POL(COND_LOAD323(x1, x2)) = x2   
POL(>(x1, x2)) = 0   
POL(0) = 0   
POL(LOAD597(x1, x2)) = x1   
POL(COND_LOAD597(x1, x2, x3)) = x2 + [-1]x1   
POL(&&(x1, x2)) = 0   
POL(>=(x1, x2)) = [-1]   
POL(<(x1, x2)) = [-1]   
POL(+(x1, x2)) = x1 + x2   
POL(1) = [1]   
POL(COND_LOAD5971(x1, x2, x3)) = x2 + [-1]x1   
POL(-1) = [-1]   

The following pairs are in P>:

LOAD323(i20[0]) → COND_LOAD323(>(i20[0], 0), i20[0])

The following pairs are in Pbound:

LOAD323(i20[0]) → COND_LOAD323(>(i20[0], 0), i20[0])
COND_LOAD323(TRUE, i20[1]) → LOAD597(i20[1], 0)
LOAD597(i20[2], i31[2]) → COND_LOAD597(&&(&&(>=(i31[2], 0), <(i31[2], i20[2])), >(+(i31[2], 1), 0)), i20[2], i31[2])
COND_LOAD597(TRUE, i20[3], i31[3]) → LOAD597(i20[3], +(i31[3], 1))
LOAD597(i20[4], i31[4]) → COND_LOAD5971(&&(>(i20[4], 0), >=(i31[4], i20[4])), i20[4], i31[4])
COND_LOAD5971(TRUE, i20[5], i31[5]) → LOAD323(+(i20[5], -1))

The following pairs are in P≥:

COND_LOAD323(TRUE, i20[1]) → LOAD597(i20[1], 0)
LOAD597(i20[2], i31[2]) → COND_LOAD597(&&(&&(>=(i31[2], 0), <(i31[2], i20[2])), >(+(i31[2], 1), 0)), i20[2], i31[2])
COND_LOAD597(TRUE, i20[3], i31[3]) → LOAD597(i20[3], +(i31[3], 1))
LOAD597(i20[4], i31[4]) → COND_LOAD5971(&&(>(i20[4], 0), >=(i31[4], i20[4])), i20[4], i31[4])
COND_LOAD5971(TRUE, i20[5], i31[5]) → LOAD323(+(i20[5], -1))

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

&&(TRUE, TRUE)1 ↔ TRUE1
FALSE1 → &&(TRUE, FALSE)1
FALSE1 → &&(FALSE, TRUE)1
FALSE1 → &&(FALSE, FALSE)1

(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:

Boolean, Integer


R is empty.

The integer pair graph contains the following rules and edges:
(1): COND_LOAD323(TRUE, i20[1]) → LOAD597(i20[1], 0)
(2): LOAD597(i20[2], i31[2]) → COND_LOAD597(i31[2] >= 0 && i31[2] < i20[2] && i31[2] + 1 > 0, i20[2], i31[2])
(3): COND_LOAD597(TRUE, i20[3], i31[3]) → LOAD597(i20[3], i31[3] + 1)
(4): LOAD597(i20[4], i31[4]) → COND_LOAD5971(i20[4] > 0 && i31[4] >= i20[4], i20[4], i31[4])
(5): COND_LOAD5971(TRUE, i20[5], i31[5]) → LOAD323(i20[5] + -1)

(1) -> (2), if ((i20[1] →* i20[2])∧(0 →* i31[2]))


(3) -> (2), if ((i31[3] + 1 →* i31[2])∧(i20[3] →* i20[2]))


(2) -> (3), if ((i20[2] →* i20[3])∧(i31[2] →* i31[3])∧(i31[2] >= 0 && i31[2] < i20[2] && i31[2] + 1 > 0 →* TRUE))


(1) -> (4), if ((i20[1] →* i20[4])∧(0 →* i31[4]))


(3) -> (4), if ((i20[3] →* i20[4])∧(i31[3] + 1 →* i31[4]))


(4) -> (5), if ((i20[4] > 0 && i31[4] >= i20[4] →* TRUE)∧(i20[4] →* i20[5])∧(i31[4] →* i31[5]))



The set Q consists of the following terms:
Load323(x0)
Cond_Load323(TRUE, x0)
Load597(x0, x1)
Cond_Load597(TRUE, x0, x1)
Cond_Load5971(TRUE, x0, x1)

(12) IDependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 3 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, Boolean


R is empty.

The integer pair graph contains the following rules and edges:
(3): COND_LOAD597(TRUE, i20[3], i31[3]) → LOAD597(i20[3], i31[3] + 1)
(2): LOAD597(i20[2], i31[2]) → COND_LOAD597(i31[2] >= 0 && i31[2] < i20[2] && i31[2] + 1 > 0, i20[2], i31[2])

(3) -> (2), if ((i31[3] + 1 →* i31[2])∧(i20[3] →* i20[2]))


(2) -> (3), if ((i20[2] →* i20[3])∧(i31[2] →* i31[3])∧(i31[2] >= 0 && i31[2] < i20[2] && i31[2] + 1 > 0 →* TRUE))



The set Q consists of the following terms:
Load323(x0)
Cond_Load323(TRUE, x0)
Load597(x0, x1)
Cond_Load597(TRUE, x0, x1)
Cond_Load5971(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_LOAD597(TRUE, i20[3], i31[3]) → LOAD597(i20[3], +(i31[3], 1)) the following chains were created:
  • We consider the chain LOAD597(i20[2], i31[2]) → COND_LOAD597(&&(&&(>=(i31[2], 0), <(i31[2], i20[2])), >(+(i31[2], 1), 0)), i20[2], i31[2]), COND_LOAD597(TRUE, i20[3], i31[3]) → LOAD597(i20[3], +(i31[3], 1)), LOAD597(i20[2], i31[2]) → COND_LOAD597(&&(&&(>=(i31[2], 0), <(i31[2], i20[2])), >(+(i31[2], 1), 0)), i20[2], i31[2]) which results in the following constraint:

    (1)    (i20[2]=i20[3]∧i31[2]=i31[3]∧&&(&&(>=(i31[2], 0), <(i31[2], i20[2])), >(+(i31[2], 1), 0))=TRUE∧+(i31[3], 1)=i31[2]1∧i20[3]=i20[2]1 ⇒ COND_LOAD597(TRUE, i20[3], i31[3])≥NonInfC∧COND_LOAD597(TRUE, i20[3], i31[3])≥LOAD597(i20[3], +(i31[3], 1))∧(UIncreasing(LOAD597(i20[3], +(i31[3], 1))), ≥))



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

    (2)    (>(+(i31[2], 1), 0)=TRUE∧>=(i31[2], 0)=TRUE∧<(i31[2], i20[2])=TRUE ⇒ COND_LOAD597(TRUE, i20[2], i31[2])≥NonInfC∧COND_LOAD597(TRUE, i20[2], i31[2])≥LOAD597(i20[2], +(i31[2], 1))∧(UIncreasing(LOAD597(i20[3], +(i31[3], 1))), ≥))



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

    (3)    (i31[2] ≥ 0∧i31[2] ≥ 0∧i20[2] + [-1] + [-1]i31[2] ≥ 0 ⇒ (UIncreasing(LOAD597(i20[3], +(i31[3], 1))), ≥)∧[(-1)bni_14 + (-1)Bound*bni_14] + [(-1)bni_14]i31[2] + [bni_14]i20[2] ≥ 0∧[1 + (-1)bso_15] ≥ 0)



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

    (4)    (i31[2] ≥ 0∧i31[2] ≥ 0∧i20[2] + [-1] + [-1]i31[2] ≥ 0 ⇒ (UIncreasing(LOAD597(i20[3], +(i31[3], 1))), ≥)∧[(-1)bni_14 + (-1)Bound*bni_14] + [(-1)bni_14]i31[2] + [bni_14]i20[2] ≥ 0∧[1 + (-1)bso_15] ≥ 0)



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

    (5)    (i31[2] ≥ 0∧i31[2] ≥ 0∧i20[2] + [-1] + [-1]i31[2] ≥ 0 ⇒ (UIncreasing(LOAD597(i20[3], +(i31[3], 1))), ≥)∧[(-1)bni_14 + (-1)Bound*bni_14] + [(-1)bni_14]i31[2] + [bni_14]i20[2] ≥ 0∧[1 + (-1)bso_15] ≥ 0)



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

    (6)    (i31[2] ≥ 0∧i31[2] ≥ 0∧i20[2] ≥ 0 ⇒ (UIncreasing(LOAD597(i20[3], +(i31[3], 1))), ≥)∧[(-1)Bound*bni_14] + [bni_14]i20[2] ≥ 0∧[1 + (-1)bso_15] ≥ 0)







For Pair LOAD597(i20[2], i31[2]) → COND_LOAD597(&&(&&(>=(i31[2], 0), <(i31[2], i20[2])), >(+(i31[2], 1), 0)), i20[2], i31[2]) the following chains were created:
  • We consider the chain LOAD597(i20[2], i31[2]) → COND_LOAD597(&&(&&(>=(i31[2], 0), <(i31[2], i20[2])), >(+(i31[2], 1), 0)), i20[2], i31[2]), COND_LOAD597(TRUE, i20[3], i31[3]) → LOAD597(i20[3], +(i31[3], 1)) which results in the following constraint:

    (7)    (i20[2]=i20[3]∧i31[2]=i31[3]∧&&(&&(>=(i31[2], 0), <(i31[2], i20[2])), >(+(i31[2], 1), 0))=TRUE ⇒ LOAD597(i20[2], i31[2])≥NonInfC∧LOAD597(i20[2], i31[2])≥COND_LOAD597(&&(&&(>=(i31[2], 0), <(i31[2], i20[2])), >(+(i31[2], 1), 0)), i20[2], i31[2])∧(UIncreasing(COND_LOAD597(&&(&&(>=(i31[2], 0), <(i31[2], i20[2])), >(+(i31[2], 1), 0)), i20[2], i31[2])), ≥))



    We simplified constraint (7) using rules (IV), (IDP_BOOLEAN) which results in the following new constraint:

    (8)    (>(+(i31[2], 1), 0)=TRUE∧>=(i31[2], 0)=TRUE∧<(i31[2], i20[2])=TRUE ⇒ LOAD597(i20[2], i31[2])≥NonInfC∧LOAD597(i20[2], i31[2])≥COND_LOAD597(&&(&&(>=(i31[2], 0), <(i31[2], i20[2])), >(+(i31[2], 1), 0)), i20[2], i31[2])∧(UIncreasing(COND_LOAD597(&&(&&(>=(i31[2], 0), <(i31[2], i20[2])), >(+(i31[2], 1), 0)), i20[2], i31[2])), ≥))



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

    (9)    (i31[2] ≥ 0∧i31[2] ≥ 0∧i20[2] + [-1] + [-1]i31[2] ≥ 0 ⇒ (UIncreasing(COND_LOAD597(&&(&&(>=(i31[2], 0), <(i31[2], i20[2])), >(+(i31[2], 1), 0)), i20[2], i31[2])), ≥)∧[(-1)bni_16 + (-1)Bound*bni_16] + [(-1)bni_16]i31[2] + [bni_16]i20[2] ≥ 0∧[(-1)bso_17] ≥ 0)



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

    (10)    (i31[2] ≥ 0∧i31[2] ≥ 0∧i20[2] + [-1] + [-1]i31[2] ≥ 0 ⇒ (UIncreasing(COND_LOAD597(&&(&&(>=(i31[2], 0), <(i31[2], i20[2])), >(+(i31[2], 1), 0)), i20[2], i31[2])), ≥)∧[(-1)bni_16 + (-1)Bound*bni_16] + [(-1)bni_16]i31[2] + [bni_16]i20[2] ≥ 0∧[(-1)bso_17] ≥ 0)



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

    (11)    (i31[2] ≥ 0∧i31[2] ≥ 0∧i20[2] + [-1] + [-1]i31[2] ≥ 0 ⇒ (UIncreasing(COND_LOAD597(&&(&&(>=(i31[2], 0), <(i31[2], i20[2])), >(+(i31[2], 1), 0)), i20[2], i31[2])), ≥)∧[(-1)bni_16 + (-1)Bound*bni_16] + [(-1)bni_16]i31[2] + [bni_16]i20[2] ≥ 0∧[(-1)bso_17] ≥ 0)



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

    (12)    (i31[2] ≥ 0∧i31[2] ≥ 0∧i20[2] ≥ 0 ⇒ (UIncreasing(COND_LOAD597(&&(&&(>=(i31[2], 0), <(i31[2], i20[2])), >(+(i31[2], 1), 0)), i20[2], i31[2])), ≥)∧[(-1)Bound*bni_16] + [bni_16]i20[2] ≥ 0∧[(-1)bso_17] ≥ 0)







To summarize, we get the following constraints P≥ for the following pairs.
  • COND_LOAD597(TRUE, i20[3], i31[3]) → LOAD597(i20[3], +(i31[3], 1))
    • (i31[2] ≥ 0∧i31[2] ≥ 0∧i20[2] ≥ 0 ⇒ (UIncreasing(LOAD597(i20[3], +(i31[3], 1))), ≥)∧[(-1)Bound*bni_14] + [bni_14]i20[2] ≥ 0∧[1 + (-1)bso_15] ≥ 0)

  • LOAD597(i20[2], i31[2]) → COND_LOAD597(&&(&&(>=(i31[2], 0), <(i31[2], i20[2])), >(+(i31[2], 1), 0)), i20[2], i31[2])
    • (i31[2] ≥ 0∧i31[2] ≥ 0∧i20[2] ≥ 0 ⇒ (UIncreasing(COND_LOAD597(&&(&&(>=(i31[2], 0), <(i31[2], i20[2])), >(+(i31[2], 1), 0)), i20[2], i31[2])), ≥)∧[(-1)Bound*bni_16] + [bni_16]i20[2] ≥ 0∧[(-1)bso_17] ≥ 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_LOAD597(x1, x2, x3)) = [-1] + [-1]x3 + x2   
POL(LOAD597(x1, x2)) = [-1] + [-1]x2 + x1   
POL(+(x1, x2)) = x1 + x2   
POL(1) = [1]   
POL(&&(x1, x2)) = [-1]   
POL(>=(x1, x2)) = [-1]   
POL(0) = 0   
POL(<(x1, x2)) = [-1]   
POL(>(x1, x2)) = [-1]   

The following pairs are in P>:

COND_LOAD597(TRUE, i20[3], i31[3]) → LOAD597(i20[3], +(i31[3], 1))

The following pairs are in Pbound:

COND_LOAD597(TRUE, i20[3], i31[3]) → LOAD597(i20[3], +(i31[3], 1))
LOAD597(i20[2], i31[2]) → COND_LOAD597(&&(&&(>=(i31[2], 0), <(i31[2], i20[2])), >(+(i31[2], 1), 0)), i20[2], i31[2])

The following pairs are in P≥:

LOAD597(i20[2], i31[2]) → COND_LOAD597(&&(&&(>=(i31[2], 0), <(i31[2], i20[2])), >(+(i31[2], 1), 0)), i20[2], i31[2])

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

TRUE1 → &&(TRUE, TRUE)1
FALSE1 → &&(TRUE, FALSE)1
FALSE1 → &&(FALSE, TRUE)1
FALSE1 → &&(FALSE, FALSE)1

(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:

Boolean, Integer


R is empty.

The integer pair graph contains the following rules and edges:
(2): LOAD597(i20[2], i31[2]) → COND_LOAD597(i31[2] >= 0 && i31[2] < i20[2] && i31[2] + 1 > 0, i20[2], i31[2])


The set Q consists of the following terms:
Load323(x0)
Cond_Load323(TRUE, x0)
Load597(x0, x1)
Cond_Load597(TRUE, x0, x1)
Cond_Load5971(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:
Load323(x0)
Cond_Load323(TRUE, x0)
Load597(x0, x1)
Cond_Load597(TRUE, x0, x1)
Cond_Load5971(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:
none


R is empty.

The integer pair graph is empty.

The set Q consists of the following terms:
Load323(x0)
Cond_Load323(TRUE, x0)
Load597(x0, x1)
Cond_Load597(TRUE, x0, x1)
Cond_Load5971(TRUE, x0, x1)

(23) IDependencyGraphProof (EQUIVALENT transformation)

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

(24) TRUE