Termination of the given JBC could be proven:

0 JBC
↳1 JBC2FIG (⇒)
↳2 JBCTerminationGraph
↳3 FIGtoITRSProof (⇒)
↳4 IDP
↳5 IDPNonInfProof (⇒)
↳6 AND
    ↳7 IDP
        ↳8 IDependencyGraphProof (⇔)
        ↳9 TRUE
    ↳10 IDP
        ↳11 IDependencyGraphProof (⇔)
        ↳12 TRUE

(0) Obligation:

JBC Problem based on JBC Program:
Manifest-Version: 1.0 Created-By: 1.6.0_16 (Sun Microsystems Inc.) Main-Class: PastaB11 
/**
 * 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 PastaB11 {
    public static void main(String[] args) {
        Random.args = args;
        int x = Random.random();
        int y = Random.random();

        while (x + y > 0) {
            if (x > y) {
                x--;
            } else if (x == y) {
                x--;
            } else {
                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:
PastaB11.main([Ljava/lang/String;)V: Graph of 176 nodes with 1 SCC.


(3) FIGtoITRSProof (SOUND transformation)

Transformed FIGraph SCCs to IDPs. Logs:

Log for SCC 0:

Generated 26 rules for P and 2 rules for R.


Combined rules. Obtained 3 rules for P and 0 rules for R.


Filtered ground terms:


1063_0_main_Load(x1, x2, x3, x4) → 1063_0_main_Load(x2, x3, x4)
Cond_1063_0_main_Load2(x1, x2, x3, x4, x5) → Cond_1063_0_main_Load2(x1, x3, x4, x5)
Cond_1063_0_main_Load1(x1, x2, x3, x4, x5) → Cond_1063_0_main_Load1(x1, x3, x4, x5)
Cond_1063_0_main_Load(x1, x2, x3, x4, x5) → Cond_1063_0_main_Load(x1, x3, x4, x5)

Filtered duplicate args:


1063_0_main_Load(x1, x2, x3) → 1063_0_main_Load(x2, x3)
Cond_1063_0_main_Load2(x1, x2, x3, x4) → Cond_1063_0_main_Load2(x1, x3, x4)
Cond_1063_0_main_Load1(x1, x2, x3, x4) → Cond_1063_0_main_Load1(x1, x4)
Cond_1063_0_main_Load(x1, x2, x3, x4) → Cond_1063_0_main_Load(x1, x3, x4)

Combined rules. Obtained 3 rules for P and 0 rules for R.


Finished conversion. Obtained 3 rules for P and 0 rules for R. System has predefined symbols.


(4) 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): 1063_0_MAIN_LOAD(x1[0], x0[0]) → COND_1063_0_MAIN_LOAD(x1[0] > x0[0] && 0 < x0[0] + x1[0], x1[0], x0[0])
(1): COND_1063_0_MAIN_LOAD(TRUE, x1[1], x0[1]) → 1063_0_MAIN_LOAD(x1[1] + -1, x0[1])
(2): 1063_0_MAIN_LOAD(x0[2], x0[2]) → COND_1063_0_MAIN_LOAD1(0 < x0[2] + x0[2], x0[2], x0[2])
(3): COND_1063_0_MAIN_LOAD1(TRUE, x0[3], x0[3]) → 1063_0_MAIN_LOAD(x0[3], x0[3] + -1)
(4): 1063_0_MAIN_LOAD(x1[4], x0[4]) → COND_1063_0_MAIN_LOAD2(x1[4] < x0[4] && 0 < x0[4] + x1[4], x1[4], x0[4])
(5): COND_1063_0_MAIN_LOAD2(TRUE, x1[5], x0[5]) → 1063_0_MAIN_LOAD(x1[5], x0[5] + -1)

(0) -> (1), if ((x1[0] > x0[0] && 0 < x0[0] + x1[0] →* TRUE)∧(x1[0] →* x1[1])∧(x0[0] →* x0[1]))


(1) -> (0), if ((x1[1] + -1 →* x1[0])∧(x0[1] →* x0[0]))


(1) -> (2), if ((x1[1] + -1 →* x0[2])∧(x0[1] →* x0[2]))


(1) -> (4), if ((x1[1] + -1 →* x1[4])∧(x0[1] →* x0[4]))


(2) -> (3), if ((0 < x0[2] + x0[2] →* TRUE)∧(x0[2] →* x0[3]))


(3) -> (0), if ((x0[3] →* x1[0])∧(x0[3] + -1 →* x0[0]))


(3) -> (2), if ((x0[3] →* x0[2])∧(x0[3] + -1 →* x0[2]))


(3) -> (4), if ((x0[3] →* x1[4])∧(x0[3] + -1 →* x0[4]))


(4) -> (5), if ((x1[4] < x0[4] && 0 < x0[4] + x1[4] →* TRUE)∧(x1[4] →* x1[5])∧(x0[4] →* x0[5]))


(5) -> (0), if ((x1[5] →* x1[0])∧(x0[5] + -1 →* x0[0]))


(5) -> (2), if ((x1[5] →* x0[2])∧(x0[5] + -1 →* x0[2]))


(5) -> (4), if ((x1[5] →* x1[4])∧(x0[5] + -1 →* x0[4]))



The set Q is empty.

(5) 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 1063_0_MAIN_LOAD(x1, x0) → COND_1063_0_MAIN_LOAD(&&(>(x1, x0), <(0, +(x0, x1))), x1, x0) the following chains were created:
  • We consider the chain 1063_0_MAIN_LOAD(x1[0], x0[0]) → COND_1063_0_MAIN_LOAD(&&(>(x1[0], x0[0]), <(0, +(x0[0], x1[0]))), x1[0], x0[0]), COND_1063_0_MAIN_LOAD(TRUE, x1[1], x0[1]) → 1063_0_MAIN_LOAD(+(x1[1], -1), x0[1]) which results in the following constraint:

    (1)    (&&(>(x1[0], x0[0]), <(0, +(x0[0], x1[0])))=TRUE∧x1[0]=x1[1]∧x0[0]=x0[1] ⇒ 1063_0_MAIN_LOAD(x1[0], x0[0])≥NonInfC∧1063_0_MAIN_LOAD(x1[0], x0[0])≥COND_1063_0_MAIN_LOAD(&&(>(x1[0], x0[0]), <(0, +(x0[0], x1[0]))), x1[0], x0[0])∧(UIncreasing(COND_1063_0_MAIN_LOAD(&&(>(x1[0], x0[0]), <(0, +(x0[0], x1[0]))), x1[0], x0[0])), ≥))



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

    (2)    (>(x1[0], x0[0])=TRUE∧<(0, +(x0[0], x1[0]))=TRUE ⇒ 1063_0_MAIN_LOAD(x1[0], x0[0])≥NonInfC∧1063_0_MAIN_LOAD(x1[0], x0[0])≥COND_1063_0_MAIN_LOAD(&&(>(x1[0], x0[0]), <(0, +(x0[0], x1[0]))), x1[0], x0[0])∧(UIncreasing(COND_1063_0_MAIN_LOAD(&&(>(x1[0], x0[0]), <(0, +(x0[0], x1[0]))), x1[0], x0[0])), ≥))



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

    (3)    (x1[0] + [-1] + [-1]x0[0] ≥ 0∧x0[0] + [-1] + x1[0] ≥ 0 ⇒ (UIncreasing(COND_1063_0_MAIN_LOAD(&&(>(x1[0], x0[0]), <(0, +(x0[0], x1[0]))), x1[0], x0[0])), ≥)∧[(-1)bni_14 + (-1)Bound*bni_14] + [(2)bni_14]x0[0] + [(2)bni_14]x1[0] ≥ 0∧[(-1)bso_15] ≥ 0)



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

    (4)    (x1[0] + [-1] + [-1]x0[0] ≥ 0∧x0[0] + [-1] + x1[0] ≥ 0 ⇒ (UIncreasing(COND_1063_0_MAIN_LOAD(&&(>(x1[0], x0[0]), <(0, +(x0[0], x1[0]))), x1[0], x0[0])), ≥)∧[(-1)bni_14 + (-1)Bound*bni_14] + [(2)bni_14]x0[0] + [(2)bni_14]x1[0] ≥ 0∧[(-1)bso_15] ≥ 0)



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

    (5)    (x1[0] + [-1] + [-1]x0[0] ≥ 0∧x0[0] + [-1] + x1[0] ≥ 0 ⇒ (UIncreasing(COND_1063_0_MAIN_LOAD(&&(>(x1[0], x0[0]), <(0, +(x0[0], x1[0]))), x1[0], x0[0])), ≥)∧[(-1)bni_14 + (-1)Bound*bni_14] + [(2)bni_14]x0[0] + [(2)bni_14]x1[0] ≥ 0∧[(-1)bso_15] ≥ 0)



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

    (6)    (x1[0] ≥ 0∧[2]x0[0] + x1[0] ≥ 0 ⇒ (UIncreasing(COND_1063_0_MAIN_LOAD(&&(>(x1[0], x0[0]), <(0, +(x0[0], x1[0]))), x1[0], x0[0])), ≥)∧[bni_14 + (-1)Bound*bni_14] + [(4)bni_14]x0[0] + [(2)bni_14]x1[0] ≥ 0∧[(-1)bso_15] ≥ 0)



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

    (7)    (x1[0] ≥ 0∧[2]x0[0] + x1[0] ≥ 0∧x0[0] ≥ 0 ⇒ (UIncreasing(COND_1063_0_MAIN_LOAD(&&(>(x1[0], x0[0]), <(0, +(x0[0], x1[0]))), x1[0], x0[0])), ≥)∧[bni_14 + (-1)Bound*bni_14] + [(4)bni_14]x0[0] + [(2)bni_14]x1[0] ≥ 0∧[(-1)bso_15] ≥ 0)


    (8)    (x1[0] ≥ 0∧[-2]x0[0] + x1[0] ≥ 0∧x0[0] ≥ 0 ⇒ (UIncreasing(COND_1063_0_MAIN_LOAD(&&(>(x1[0], x0[0]), <(0, +(x0[0], x1[0]))), x1[0], x0[0])), ≥)∧[bni_14 + (-1)Bound*bni_14] + [(-4)bni_14]x0[0] + [(2)bni_14]x1[0] ≥ 0∧[(-1)bso_15] ≥ 0)



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

    (9)    ([2]x0[0] + x1[0] ≥ 0∧x1[0] ≥ 0∧x0[0] ≥ 0 ⇒ (UIncreasing(COND_1063_0_MAIN_LOAD(&&(>(x1[0], x0[0]), <(0, +(x0[0], x1[0]))), x1[0], x0[0])), ≥)∧[bni_14 + (-1)Bound*bni_14] + [(2)bni_14]x1[0] ≥ 0∧[(-1)bso_15] ≥ 0)







For Pair COND_1063_0_MAIN_LOAD(TRUE, x1, x0) → 1063_0_MAIN_LOAD(+(x1, -1), x0) the following chains were created:
  • We consider the chain COND_1063_0_MAIN_LOAD(TRUE, x1[1], x0[1]) → 1063_0_MAIN_LOAD(+(x1[1], -1), x0[1]) which results in the following constraint:

    (10)    (COND_1063_0_MAIN_LOAD(TRUE, x1[1], x0[1])≥NonInfC∧COND_1063_0_MAIN_LOAD(TRUE, x1[1], x0[1])≥1063_0_MAIN_LOAD(+(x1[1], -1), x0[1])∧(UIncreasing(1063_0_MAIN_LOAD(+(x1[1], -1), x0[1])), ≥))



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

    (11)    ((UIncreasing(1063_0_MAIN_LOAD(+(x1[1], -1), x0[1])), ≥)∧[2 + (-1)bso_17] ≥ 0)



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

    (12)    ((UIncreasing(1063_0_MAIN_LOAD(+(x1[1], -1), x0[1])), ≥)∧[2 + (-1)bso_17] ≥ 0)



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

    (13)    ((UIncreasing(1063_0_MAIN_LOAD(+(x1[1], -1), x0[1])), ≥)∧[2 + (-1)bso_17] ≥ 0)



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

    (14)    ((UIncreasing(1063_0_MAIN_LOAD(+(x1[1], -1), x0[1])), ≥)∧0 = 0∧0 = 0∧[2 + (-1)bso_17] ≥ 0)







For Pair 1063_0_MAIN_LOAD(x0, x0) → COND_1063_0_MAIN_LOAD1(<(0, +(x0, x0)), x0, x0) the following chains were created:
  • We consider the chain 1063_0_MAIN_LOAD(x0[2], x0[2]) → COND_1063_0_MAIN_LOAD1(<(0, +(x0[2], x0[2])), x0[2], x0[2]), COND_1063_0_MAIN_LOAD1(TRUE, x0[3], x0[3]) → 1063_0_MAIN_LOAD(x0[3], +(x0[3], -1)) which results in the following constraint:

    (15)    (<(0, +(x0[2], x0[2]))=TRUE∧x0[2]=x0[3] ⇒ 1063_0_MAIN_LOAD(x0[2], x0[2])≥NonInfC∧1063_0_MAIN_LOAD(x0[2], x0[2])≥COND_1063_0_MAIN_LOAD1(<(0, +(x0[2], x0[2])), x0[2], x0[2])∧(UIncreasing(COND_1063_0_MAIN_LOAD1(<(0, +(x0[2], x0[2])), x0[2], x0[2])), ≥))



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

    (16)    (<(0, +(x0[2], x0[2]))=TRUE ⇒ 1063_0_MAIN_LOAD(x0[2], x0[2])≥NonInfC∧1063_0_MAIN_LOAD(x0[2], x0[2])≥COND_1063_0_MAIN_LOAD1(<(0, +(x0[2], x0[2])), x0[2], x0[2])∧(UIncreasing(COND_1063_0_MAIN_LOAD1(<(0, +(x0[2], x0[2])), x0[2], x0[2])), ≥))



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

    (17)    ([2]x0[2] + [-1] ≥ 0 ⇒ (UIncreasing(COND_1063_0_MAIN_LOAD1(<(0, +(x0[2], x0[2])), x0[2], x0[2])), ≥)∧[(-1)bni_18 + (-1)Bound*bni_18] + [(4)bni_18]x0[2] ≥ 0∧[(-1)bso_19] ≥ 0)



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

    (18)    ([2]x0[2] + [-1] ≥ 0 ⇒ (UIncreasing(COND_1063_0_MAIN_LOAD1(<(0, +(x0[2], x0[2])), x0[2], x0[2])), ≥)∧[(-1)bni_18 + (-1)Bound*bni_18] + [(4)bni_18]x0[2] ≥ 0∧[(-1)bso_19] ≥ 0)



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

    (19)    ([2]x0[2] + [-1] ≥ 0 ⇒ (UIncreasing(COND_1063_0_MAIN_LOAD1(<(0, +(x0[2], x0[2])), x0[2], x0[2])), ≥)∧[(-1)bni_18 + (-1)Bound*bni_18] + [(4)bni_18]x0[2] ≥ 0∧[(-1)bso_19] ≥ 0)







For Pair COND_1063_0_MAIN_LOAD1(TRUE, x0, x0) → 1063_0_MAIN_LOAD(x0, +(x0, -1)) the following chains were created:
  • We consider the chain COND_1063_0_MAIN_LOAD1(TRUE, x0[3], x0[3]) → 1063_0_MAIN_LOAD(x0[3], +(x0[3], -1)) which results in the following constraint:

    (20)    (COND_1063_0_MAIN_LOAD1(TRUE, x0[3], x0[3])≥NonInfC∧COND_1063_0_MAIN_LOAD1(TRUE, x0[3], x0[3])≥1063_0_MAIN_LOAD(x0[3], +(x0[3], -1))∧(UIncreasing(1063_0_MAIN_LOAD(x0[3], +(x0[3], -1))), ≥))



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

    (21)    ((UIncreasing(1063_0_MAIN_LOAD(x0[3], +(x0[3], -1))), ≥)∧[2 + (-1)bso_21] ≥ 0)



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

    (22)    ((UIncreasing(1063_0_MAIN_LOAD(x0[3], +(x0[3], -1))), ≥)∧[2 + (-1)bso_21] ≥ 0)



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

    (23)    ((UIncreasing(1063_0_MAIN_LOAD(x0[3], +(x0[3], -1))), ≥)∧[2 + (-1)bso_21] ≥ 0)



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

    (24)    ((UIncreasing(1063_0_MAIN_LOAD(x0[3], +(x0[3], -1))), ≥)∧0 = 0∧[2 + (-1)bso_21] ≥ 0)







For Pair 1063_0_MAIN_LOAD(x1, x0) → COND_1063_0_MAIN_LOAD2(&&(<(x1, x0), <(0, +(x0, x1))), x1, x0) the following chains were created:
  • We consider the chain 1063_0_MAIN_LOAD(x1[4], x0[4]) → COND_1063_0_MAIN_LOAD2(&&(<(x1[4], x0[4]), <(0, +(x0[4], x1[4]))), x1[4], x0[4]), COND_1063_0_MAIN_LOAD2(TRUE, x1[5], x0[5]) → 1063_0_MAIN_LOAD(x1[5], +(x0[5], -1)) which results in the following constraint:

    (25)    (&&(<(x1[4], x0[4]), <(0, +(x0[4], x1[4])))=TRUE∧x1[4]=x1[5]∧x0[4]=x0[5] ⇒ 1063_0_MAIN_LOAD(x1[4], x0[4])≥NonInfC∧1063_0_MAIN_LOAD(x1[4], x0[4])≥COND_1063_0_MAIN_LOAD2(&&(<(x1[4], x0[4]), <(0, +(x0[4], x1[4]))), x1[4], x0[4])∧(UIncreasing(COND_1063_0_MAIN_LOAD2(&&(<(x1[4], x0[4]), <(0, +(x0[4], x1[4]))), x1[4], x0[4])), ≥))



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

    (26)    (<(x1[4], x0[4])=TRUE∧<(0, +(x0[4], x1[4]))=TRUE ⇒ 1063_0_MAIN_LOAD(x1[4], x0[4])≥NonInfC∧1063_0_MAIN_LOAD(x1[4], x0[4])≥COND_1063_0_MAIN_LOAD2(&&(<(x1[4], x0[4]), <(0, +(x0[4], x1[4]))), x1[4], x0[4])∧(UIncreasing(COND_1063_0_MAIN_LOAD2(&&(<(x1[4], x0[4]), <(0, +(x0[4], x1[4]))), x1[4], x0[4])), ≥))



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

    (27)    (x0[4] + [-1] + [-1]x1[4] ≥ 0∧x0[4] + [-1] + x1[4] ≥ 0 ⇒ (UIncreasing(COND_1063_0_MAIN_LOAD2(&&(<(x1[4], x0[4]), <(0, +(x0[4], x1[4]))), x1[4], x0[4])), ≥)∧[(-1)bni_22 + (-1)Bound*bni_22] + [(2)bni_22]x0[4] + [(2)bni_22]x1[4] ≥ 0∧[(-1)bso_23] ≥ 0)



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

    (28)    (x0[4] + [-1] + [-1]x1[4] ≥ 0∧x0[4] + [-1] + x1[4] ≥ 0 ⇒ (UIncreasing(COND_1063_0_MAIN_LOAD2(&&(<(x1[4], x0[4]), <(0, +(x0[4], x1[4]))), x1[4], x0[4])), ≥)∧[(-1)bni_22 + (-1)Bound*bni_22] + [(2)bni_22]x0[4] + [(2)bni_22]x1[4] ≥ 0∧[(-1)bso_23] ≥ 0)



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

    (29)    (x0[4] + [-1] + [-1]x1[4] ≥ 0∧x0[4] + [-1] + x1[4] ≥ 0 ⇒ (UIncreasing(COND_1063_0_MAIN_LOAD2(&&(<(x1[4], x0[4]), <(0, +(x0[4], x1[4]))), x1[4], x0[4])), ≥)∧[(-1)bni_22 + (-1)Bound*bni_22] + [(2)bni_22]x0[4] + [(2)bni_22]x1[4] ≥ 0∧[(-1)bso_23] ≥ 0)



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

    (30)    (x0[4] ≥ 0∧[2]x1[4] + x0[4] ≥ 0 ⇒ (UIncreasing(COND_1063_0_MAIN_LOAD2(&&(<(x1[4], x0[4]), <(0, +(x0[4], x1[4]))), x1[4], x0[4])), ≥)∧[bni_22 + (-1)Bound*bni_22] + [(4)bni_22]x1[4] + [(2)bni_22]x0[4] ≥ 0∧[(-1)bso_23] ≥ 0)



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

    (31)    (x0[4] ≥ 0∧[2]x1[4] + x0[4] ≥ 0∧x1[4] ≥ 0 ⇒ (UIncreasing(COND_1063_0_MAIN_LOAD2(&&(<(x1[4], x0[4]), <(0, +(x0[4], x1[4]))), x1[4], x0[4])), ≥)∧[bni_22 + (-1)Bound*bni_22] + [(4)bni_22]x1[4] + [(2)bni_22]x0[4] ≥ 0∧[(-1)bso_23] ≥ 0)


    (32)    (x0[4] ≥ 0∧[-2]x1[4] + x0[4] ≥ 0∧x1[4] ≥ 0 ⇒ (UIncreasing(COND_1063_0_MAIN_LOAD2(&&(<(x1[4], x0[4]), <(0, +(x0[4], x1[4]))), x1[4], x0[4])), ≥)∧[bni_22 + (-1)Bound*bni_22] + [(-4)bni_22]x1[4] + [(2)bni_22]x0[4] ≥ 0∧[(-1)bso_23] ≥ 0)



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

    (33)    ([2]x1[4] + x0[4] ≥ 0∧x0[4] ≥ 0∧x1[4] ≥ 0 ⇒ (UIncreasing(COND_1063_0_MAIN_LOAD2(&&(<(x1[4], x0[4]), <(0, +(x0[4], x1[4]))), x1[4], x0[4])), ≥)∧[bni_22 + (-1)Bound*bni_22] + [(2)bni_22]x0[4] ≥ 0∧[(-1)bso_23] ≥ 0)







For Pair COND_1063_0_MAIN_LOAD2(TRUE, x1, x0) → 1063_0_MAIN_LOAD(x1, +(x0, -1)) the following chains were created:
  • We consider the chain COND_1063_0_MAIN_LOAD2(TRUE, x1[5], x0[5]) → 1063_0_MAIN_LOAD(x1[5], +(x0[5], -1)) which results in the following constraint:

    (34)    (COND_1063_0_MAIN_LOAD2(TRUE, x1[5], x0[5])≥NonInfC∧COND_1063_0_MAIN_LOAD2(TRUE, x1[5], x0[5])≥1063_0_MAIN_LOAD(x1[5], +(x0[5], -1))∧(UIncreasing(1063_0_MAIN_LOAD(x1[5], +(x0[5], -1))), ≥))



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

    (35)    ((UIncreasing(1063_0_MAIN_LOAD(x1[5], +(x0[5], -1))), ≥)∧[2 + (-1)bso_25] ≥ 0)



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

    (36)    ((UIncreasing(1063_0_MAIN_LOAD(x1[5], +(x0[5], -1))), ≥)∧[2 + (-1)bso_25] ≥ 0)



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

    (37)    ((UIncreasing(1063_0_MAIN_LOAD(x1[5], +(x0[5], -1))), ≥)∧[2 + (-1)bso_25] ≥ 0)



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

    (38)    ((UIncreasing(1063_0_MAIN_LOAD(x1[5], +(x0[5], -1))), ≥)∧0 = 0∧0 = 0∧[2 + (-1)bso_25] ≥ 0)







To summarize, we get the following constraints P≥ for the following pairs.
  • 1063_0_MAIN_LOAD(x1, x0) → COND_1063_0_MAIN_LOAD(&&(>(x1, x0), <(0, +(x0, x1))), x1, x0)
    • (x1[0] ≥ 0∧[2]x0[0] + x1[0] ≥ 0∧x0[0] ≥ 0 ⇒ (UIncreasing(COND_1063_0_MAIN_LOAD(&&(>(x1[0], x0[0]), <(0, +(x0[0], x1[0]))), x1[0], x0[0])), ≥)∧[bni_14 + (-1)Bound*bni_14] + [(4)bni_14]x0[0] + [(2)bni_14]x1[0] ≥ 0∧[(-1)bso_15] ≥ 0)
    • ([2]x0[0] + x1[0] ≥ 0∧x1[0] ≥ 0∧x0[0] ≥ 0 ⇒ (UIncreasing(COND_1063_0_MAIN_LOAD(&&(>(x1[0], x0[0]), <(0, +(x0[0], x1[0]))), x1[0], x0[0])), ≥)∧[bni_14 + (-1)Bound*bni_14] + [(2)bni_14]x1[0] ≥ 0∧[(-1)bso_15] ≥ 0)

  • COND_1063_0_MAIN_LOAD(TRUE, x1, x0) → 1063_0_MAIN_LOAD(+(x1, -1), x0)
    • ((UIncreasing(1063_0_MAIN_LOAD(+(x1[1], -1), x0[1])), ≥)∧0 = 0∧0 = 0∧[2 + (-1)bso_17] ≥ 0)

  • 1063_0_MAIN_LOAD(x0, x0) → COND_1063_0_MAIN_LOAD1(<(0, +(x0, x0)), x0, x0)
    • ([2]x0[2] + [-1] ≥ 0 ⇒ (UIncreasing(COND_1063_0_MAIN_LOAD1(<(0, +(x0[2], x0[2])), x0[2], x0[2])), ≥)∧[(-1)bni_18 + (-1)Bound*bni_18] + [(4)bni_18]x0[2] ≥ 0∧[(-1)bso_19] ≥ 0)

  • COND_1063_0_MAIN_LOAD1(TRUE, x0, x0) → 1063_0_MAIN_LOAD(x0, +(x0, -1))
    • ((UIncreasing(1063_0_MAIN_LOAD(x0[3], +(x0[3], -1))), ≥)∧0 = 0∧[2 + (-1)bso_21] ≥ 0)

  • 1063_0_MAIN_LOAD(x1, x0) → COND_1063_0_MAIN_LOAD2(&&(<(x1, x0), <(0, +(x0, x1))), x1, x0)
    • (x0[4] ≥ 0∧[2]x1[4] + x0[4] ≥ 0∧x1[4] ≥ 0 ⇒ (UIncreasing(COND_1063_0_MAIN_LOAD2(&&(<(x1[4], x0[4]), <(0, +(x0[4], x1[4]))), x1[4], x0[4])), ≥)∧[bni_22 + (-1)Bound*bni_22] + [(4)bni_22]x1[4] + [(2)bni_22]x0[4] ≥ 0∧[(-1)bso_23] ≥ 0)
    • ([2]x1[4] + x0[4] ≥ 0∧x0[4] ≥ 0∧x1[4] ≥ 0 ⇒ (UIncreasing(COND_1063_0_MAIN_LOAD2(&&(<(x1[4], x0[4]), <(0, +(x0[4], x1[4]))), x1[4], x0[4])), ≥)∧[bni_22 + (-1)Bound*bni_22] + [(2)bni_22]x0[4] ≥ 0∧[(-1)bso_23] ≥ 0)

  • COND_1063_0_MAIN_LOAD2(TRUE, x1, x0) → 1063_0_MAIN_LOAD(x1, +(x0, -1))
    • ((UIncreasing(1063_0_MAIN_LOAD(x1[5], +(x0[5], -1))), ≥)∧0 = 0∧0 = 0∧[2 + (-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(1063_0_MAIN_LOAD(x1, x2)) = [-1] + [2]x2 + [2]x1   
POL(COND_1063_0_MAIN_LOAD(x1, x2, x3)) = [-1] + [2]x3 + [2]x2   
POL(&&(x1, x2)) = [-1]   
POL(>(x1, x2)) = [-1]   
POL(<(x1, x2)) = [-1]   
POL(0) = 0   
POL(+(x1, x2)) = x1 + x2   
POL(-1) = [-1]   
POL(COND_1063_0_MAIN_LOAD1(x1, x2, x3)) = [-1] + [2]x3 + [2]x2   
POL(COND_1063_0_MAIN_LOAD2(x1, x2, x3)) = [-1] + [2]x3 + [2]x2   

The following pairs are in P>:

COND_1063_0_MAIN_LOAD(TRUE, x1[1], x0[1]) → 1063_0_MAIN_LOAD(+(x1[1], -1), x0[1])
COND_1063_0_MAIN_LOAD1(TRUE, x0[3], x0[3]) → 1063_0_MAIN_LOAD(x0[3], +(x0[3], -1))
COND_1063_0_MAIN_LOAD2(TRUE, x1[5], x0[5]) → 1063_0_MAIN_LOAD(x1[5], +(x0[5], -1))

The following pairs are in Pbound:

1063_0_MAIN_LOAD(x1[0], x0[0]) → COND_1063_0_MAIN_LOAD(&&(>(x1[0], x0[0]), <(0, +(x0[0], x1[0]))), x1[0], x0[0])
1063_0_MAIN_LOAD(x0[2], x0[2]) → COND_1063_0_MAIN_LOAD1(<(0, +(x0[2], x0[2])), x0[2], x0[2])
1063_0_MAIN_LOAD(x1[4], x0[4]) → COND_1063_0_MAIN_LOAD2(&&(<(x1[4], x0[4]), <(0, +(x0[4], x1[4]))), x1[4], x0[4])

The following pairs are in P≥:

1063_0_MAIN_LOAD(x1[0], x0[0]) → COND_1063_0_MAIN_LOAD(&&(>(x1[0], x0[0]), <(0, +(x0[0], x1[0]))), x1[0], x0[0])
1063_0_MAIN_LOAD(x0[2], x0[2]) → COND_1063_0_MAIN_LOAD1(<(0, +(x0[2], x0[2])), x0[2], x0[2])
1063_0_MAIN_LOAD(x1[4], x0[4]) → COND_1063_0_MAIN_LOAD2(&&(<(x1[4], x0[4]), <(0, +(x0[4], x1[4]))), x1[4], x0[4])

There are no usable rules.

(6) Complex Obligation (AND)

(7) 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): 1063_0_MAIN_LOAD(x1[0], x0[0]) → COND_1063_0_MAIN_LOAD(x1[0] > x0[0] && 0 < x0[0] + x1[0], x1[0], x0[0])
(2): 1063_0_MAIN_LOAD(x0[2], x0[2]) → COND_1063_0_MAIN_LOAD1(0 < x0[2] + x0[2], x0[2], x0[2])
(4): 1063_0_MAIN_LOAD(x1[4], x0[4]) → COND_1063_0_MAIN_LOAD2(x1[4] < x0[4] && 0 < x0[4] + x1[4], x1[4], x0[4])


The set Q is empty.

(8) IDependencyGraphProof (EQUIVALENT transformation)

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

(9) TRUE

(10) Obligation:

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


The following domains are used:

Integer


R is empty.

The integer pair graph contains the following rules and edges:
(1): COND_1063_0_MAIN_LOAD(TRUE, x1[1], x0[1]) → 1063_0_MAIN_LOAD(x1[1] + -1, x0[1])
(3): COND_1063_0_MAIN_LOAD1(TRUE, x0[3], x0[3]) → 1063_0_MAIN_LOAD(x0[3], x0[3] + -1)
(5): COND_1063_0_MAIN_LOAD2(TRUE, x1[5], x0[5]) → 1063_0_MAIN_LOAD(x1[5], x0[5] + -1)


The set Q is empty.

(11) IDependencyGraphProof (EQUIVALENT transformation)

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

(12) TRUE