(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) JBCToGraph (SOUND transformation)

Constructed TerminationGraph.

(2) Obligation:

Termination Graph based on JBC Program:
PastaB11.main([Ljava/lang/String;)V: Graph of 199 nodes with 1 SCC.


(3) TerminationGraphToSCCProof (SOUND transformation)

Splitted TerminationGraph to 1 SCCs.

(4) Obligation:

SCC of termination graph based on JBC Program.
SCC contains nodes from the following methods: PastaB11.main([Ljava/lang/String;)V
SCC calls the following helper methods:
Performed SCC analyses: UsedFieldsAnalysis

(5) SCCToIDPv1Proof (SOUND transformation)

Transformed FIGraph SCCs to IDPs. Log:

Generated 26 rules for P and 0 rules for R.


P rules:
828_0_main_Load(EOS(STATIC_828), i117, i191, i117) → 830_0_main_IntArithmetic(EOS(STATIC_830), i117, i191, i117, i191)
830_0_main_IntArithmetic(EOS(STATIC_830), i117, i191, i117, i191) → 832_0_main_LE(EOS(STATIC_832), i117, i191, +(i117, i191))
832_0_main_LE(EOS(STATIC_832), i117, i191, i204) → 836_0_main_LE(EOS(STATIC_836), i117, i191, i204)
836_0_main_LE(EOS(STATIC_836), i117, i191, i204) → 840_0_main_Load(EOS(STATIC_840), i117, i191) | >(i204, 0)
840_0_main_Load(EOS(STATIC_840), i117, i191) → 843_0_main_Load(EOS(STATIC_843), i117, i191, i117)
843_0_main_Load(EOS(STATIC_843), i117, i191, i117) → 846_0_main_LE(EOS(STATIC_846), i117, i191, i117, i191)
846_0_main_LE(EOS(STATIC_846), i117, i191, i117, i191) → 848_0_main_LE(EOS(STATIC_848), i117, i191, i117, i191)
846_0_main_LE(EOS(STATIC_846), i117, i191, i117, i191) → 849_0_main_LE(EOS(STATIC_849), i117, i191, i117, i191)
848_0_main_LE(EOS(STATIC_848), i117, i191, i117, i191) → 851_0_main_Load(EOS(STATIC_851), i117, i191) | <=(i117, i191)
851_0_main_Load(EOS(STATIC_851), i117, i191) → 855_0_main_Load(EOS(STATIC_855), i117, i191, i117)
855_0_main_Load(EOS(STATIC_855), i117, i191, i117) → 859_0_main_NE(EOS(STATIC_859), i117, i191, i117, i191)
859_0_main_NE(EOS(STATIC_859), i117, i191, i117, i191) → 865_0_main_NE(EOS(STATIC_865), i117, i191, i117, i191)
859_0_main_NE(EOS(STATIC_859), i191, i191, i191, i191) → 867_0_main_NE(EOS(STATIC_867), i191, i191, i191, i191)
865_0_main_NE(EOS(STATIC_865), i117, i191, i117, i191) → 869_0_main_Inc(EOS(STATIC_869), i117, i191) | !(=(i117, i191))
869_0_main_Inc(EOS(STATIC_869), i117, i191) → 873_0_main_JMP(EOS(STATIC_873), i117, +(i191, -1))
873_0_main_JMP(EOS(STATIC_873), i117, i213) → 879_0_main_Load(EOS(STATIC_879), i117, i213)
879_0_main_Load(EOS(STATIC_879), i117, i213) → 824_0_main_Load(EOS(STATIC_824), i117, i213)
824_0_main_Load(EOS(STATIC_824), i117, i191) → 828_0_main_Load(EOS(STATIC_828), i117, i191, i117)
867_0_main_NE(EOS(STATIC_867), i191, i191, i191, i191) → 871_0_main_Inc(EOS(STATIC_871), i191, i191)
871_0_main_Inc(EOS(STATIC_871), i191, i191) → 876_0_main_JMP(EOS(STATIC_876), +(i191, -1), i191)
876_0_main_JMP(EOS(STATIC_876), i216, i191) → 884_0_main_Load(EOS(STATIC_884), i216, i191)
884_0_main_Load(EOS(STATIC_884), i216, i191) → 824_0_main_Load(EOS(STATIC_824), i216, i191)
849_0_main_LE(EOS(STATIC_849), i117, i191, i117, i191) → 853_0_main_Inc(EOS(STATIC_853), i117, i191) | >(i117, i191)
853_0_main_Inc(EOS(STATIC_853), i117, i191) → 857_0_main_JMP(EOS(STATIC_857), +(i117, -1), i191)
857_0_main_JMP(EOS(STATIC_857), i207, i191) → 864_0_main_Load(EOS(STATIC_864), i207, i191)
864_0_main_Load(EOS(STATIC_864), i207, i191) → 824_0_main_Load(EOS(STATIC_824), i207, i191)
R rules:

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


P rules:
828_0_main_Load(EOS(STATIC_828), x0, x1, x0) → 828_0_main_Load(EOS(STATIC_828), x0, +(x1, -1), x0) | &&(>(x1, x0), <(0, +(x0, x1)))
828_0_main_Load(EOS(STATIC_828), x0, x0, x0) → 828_0_main_Load(EOS(STATIC_828), +(x0, -1), x0, +(x0, -1)) | <(0, +(x0, x0))
828_0_main_Load(EOS(STATIC_828), x0, x1, x0) → 828_0_main_Load(EOS(STATIC_828), +(x0, -1), x1, +(x0, -1)) | &&(<(x1, x0), <(0, +(x0, x1)))
R rules:

Filtered ground terms:



828_0_main_Load(x1, x2, x3, x4) → 828_0_main_Load(x2, x3, x4)
EOS(x1) → EOS
Cond_828_0_main_Load2(x1, x2, x3, x4, x5) → Cond_828_0_main_Load2(x1, x3, x4, x5)
Cond_828_0_main_Load1(x1, x2, x3, x4, x5) → Cond_828_0_main_Load1(x1, x3, x4, x5)
Cond_828_0_main_Load(x1, x2, x3, x4, x5) → Cond_828_0_main_Load(x1, x3, x4, x5)

Filtered duplicate args:



828_0_main_Load(x1, x2, x3) → 828_0_main_Load(x2, x3)
Cond_828_0_main_Load(x1, x2, x3, x4) → Cond_828_0_main_Load(x1, x3, x4)
Cond_828_0_main_Load1(x1, x2, x3, x4) → Cond_828_0_main_Load1(x1, x4)
Cond_828_0_main_Load2(x1, x2, x3, x4) → Cond_828_0_main_Load2(x1, x3, x4)

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


P rules:
828_0_main_Load(x1, x0) → 828_0_main_Load(+(x1, -1), x0) | &&(>(x1, x0), <(0, +(x0, x1)))
828_0_main_Load(x0, x0) → 828_0_main_Load(x0, +(x0, -1)) | <(0, +(x0, x0))
828_0_main_Load(x1, x0) → 828_0_main_Load(x1, +(x0, -1)) | &&(<(x1, x0), <(0, +(x0, x1)))
R rules:

Finished conversion. Obtained 6 rules for P and 0 rules for R. System has predefined symbols.


P rules:
828_0_MAIN_LOAD(x1, x0) → COND_828_0_MAIN_LOAD(&&(>(x1, x0), <(0, +(x0, x1))), x1, x0)
COND_828_0_MAIN_LOAD(TRUE, x1, x0) → 828_0_MAIN_LOAD(+(x1, -1), x0)
828_0_MAIN_LOAD(x0, x0) → COND_828_0_MAIN_LOAD1(<(0, +(x0, x0)), x0, x0)
COND_828_0_MAIN_LOAD1(TRUE, x0, x0) → 828_0_MAIN_LOAD(x0, +(x0, -1))
828_0_MAIN_LOAD(x1, x0) → COND_828_0_MAIN_LOAD2(&&(<(x1, x0), <(0, +(x0, x1))), x1, x0)
COND_828_0_MAIN_LOAD2(TRUE, x1, x0) → 828_0_MAIN_LOAD(x1, +(x0, -1))
R rules:

(6) Obligation:

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


The following domains are used:

Boolean, Integer


R is empty.

The integer pair graph contains the following rules and edges:
(0): 828_0_MAIN_LOAD(x1[0], x0[0]) → COND_828_0_MAIN_LOAD(x1[0] > x0[0] && 0 < x0[0] + x1[0], x1[0], x0[0])
(1): COND_828_0_MAIN_LOAD(TRUE, x1[1], x0[1]) → 828_0_MAIN_LOAD(x1[1] + -1, x0[1])
(2): 828_0_MAIN_LOAD(x0[2], x0[2]) → COND_828_0_MAIN_LOAD1(0 < x0[2] + x0[2], x0[2], x0[2])
(3): COND_828_0_MAIN_LOAD1(TRUE, x0[3], x0[3]) → 828_0_MAIN_LOAD(x0[3], x0[3] + -1)
(4): 828_0_MAIN_LOAD(x1[4], x0[4]) → COND_828_0_MAIN_LOAD2(x1[4] < x0[4] && 0 < x0[4] + x1[4], x1[4], x0[4])
(5): COND_828_0_MAIN_LOAD2(TRUE, x1[5], x0[5]) → 828_0_MAIN_LOAD(x1[5], x0[5] + -1)

(0) -> (1), if (x1[0] > x0[0] && 0 < x0[0] + x1[0]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]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]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.

(7) IDPNonInfProof (SOUND transformation)

Used the following options for this NonInfProof:
IDPGPoloSolver: Range: [(-1,2)] IsNat: false Interpretation Shape Heuristic: aprove.DPFramework.IDPProblem.Processors.nonInf.poly.IdpCand1ShapeHeuristic@2ef2a026 Constraint Generator: NonInfConstraintGenerator: PathGenerator: MetricPathGenerator: Max Left Steps: 0 Max Right Steps: 0

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 828_0_MAIN_LOAD(x1, x0) → COND_828_0_MAIN_LOAD(&&(>(x1, x0), <(0, +(x0, x1))), x1, x0) the following chains were created:
  • We consider the chain 828_0_MAIN_LOAD(x1[0], x0[0]) → COND_828_0_MAIN_LOAD(&&(>(x1[0], x0[0]), <(0, +(x0[0], x1[0]))), x1[0], x0[0]), COND_828_0_MAIN_LOAD(TRUE, x1[1], x0[1]) → 828_0_MAIN_LOAD(+(x1[1], -1), x0[1]) which results in the following constraint:

    (1)    (&&(>(x1[0], x0[0]), <(0, +(x0[0], x1[0])))=TRUEx1[0]=x1[1]x0[0]=x0[1]828_0_MAIN_LOAD(x1[0], x0[0])≥NonInfC∧828_0_MAIN_LOAD(x1[0], x0[0])≥COND_828_0_MAIN_LOAD(&&(>(x1[0], x0[0]), <(0, +(x0[0], x1[0]))), x1[0], x0[0])∧(UIncreasing(COND_828_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]))=TRUE828_0_MAIN_LOAD(x1[0], x0[0])≥NonInfC∧828_0_MAIN_LOAD(x1[0], x0[0])≥COND_828_0_MAIN_LOAD(&&(>(x1[0], x0[0]), <(0, +(x0[0], x1[0]))), x1[0], x0[0])∧(UIncreasing(COND_828_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_828_0_MAIN_LOAD(&&(>(x1[0], x0[0]), <(0, +(x0[0], x1[0]))), x1[0], x0[0])), ≥)∧[(-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_828_0_MAIN_LOAD(&&(>(x1[0], x0[0]), <(0, +(x0[0], x1[0]))), x1[0], x0[0])), ≥)∧[(-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_828_0_MAIN_LOAD(&&(>(x1[0], x0[0]), <(0, +(x0[0], x1[0]))), x1[0], x0[0])), ≥)∧[(-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_828_0_MAIN_LOAD(&&(>(x1[0], x0[0]), <(0, +(x0[0], x1[0]))), x1[0], x0[0])), ≥)∧[(-1)Bound*bni_14 + (2)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_828_0_MAIN_LOAD(&&(>(x1[0], x0[0]), <(0, +(x0[0], x1[0]))), x1[0], x0[0])), ≥)∧[(-1)Bound*bni_14 + (2)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_828_0_MAIN_LOAD(&&(>(x1[0], x0[0]), <(0, +(x0[0], x1[0]))), x1[0], x0[0])), ≥)∧[(-1)Bound*bni_14 + (2)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_828_0_MAIN_LOAD(&&(>(x1[0], x0[0]), <(0, +(x0[0], x1[0]))), x1[0], x0[0])), ≥)∧[(-1)Bound*bni_14 + (2)bni_14] + [(2)bni_14]x1[0] ≥ 0∧[(-1)bso_15] ≥ 0)







For Pair COND_828_0_MAIN_LOAD(TRUE, x1, x0) → 828_0_MAIN_LOAD(+(x1, -1), x0) the following chains were created:
  • We consider the chain COND_828_0_MAIN_LOAD(TRUE, x1[1], x0[1]) → 828_0_MAIN_LOAD(+(x1[1], -1), x0[1]) which results in the following constraint:

    (10)    (COND_828_0_MAIN_LOAD(TRUE, x1[1], x0[1])≥NonInfC∧COND_828_0_MAIN_LOAD(TRUE, x1[1], x0[1])≥828_0_MAIN_LOAD(+(x1[1], -1), x0[1])∧(UIncreasing(828_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(828_0_MAIN_LOAD(+(x1[1], -1), x0[1])), ≥)∧[bni_16] = 0∧[2 + (-1)bso_17] ≥ 0)



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

    (12)    ((UIncreasing(828_0_MAIN_LOAD(+(x1[1], -1), x0[1])), ≥)∧[bni_16] = 0∧[2 + (-1)bso_17] ≥ 0)



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

    (13)    ((UIncreasing(828_0_MAIN_LOAD(+(x1[1], -1), x0[1])), ≥)∧[bni_16] = 0∧[2 + (-1)bso_17] ≥ 0)



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

    (14)    ((UIncreasing(828_0_MAIN_LOAD(+(x1[1], -1), x0[1])), ≥)∧[bni_16] = 0∧0 = 0∧0 = 0∧[2 + (-1)bso_17] ≥ 0)







For Pair 828_0_MAIN_LOAD(x0, x0) → COND_828_0_MAIN_LOAD1(<(0, +(x0, x0)), x0, x0) the following chains were created:
  • We consider the chain 828_0_MAIN_LOAD(x0[2], x0[2]) → COND_828_0_MAIN_LOAD1(<(0, +(x0[2], x0[2])), x0[2], x0[2]), COND_828_0_MAIN_LOAD1(TRUE, x0[3], x0[3]) → 828_0_MAIN_LOAD(x0[3], +(x0[3], -1)) which results in the following constraint:

    (15)    (<(0, +(x0[2], x0[2]))=TRUEx0[2]=x0[3]828_0_MAIN_LOAD(x0[2], x0[2])≥NonInfC∧828_0_MAIN_LOAD(x0[2], x0[2])≥COND_828_0_MAIN_LOAD1(<(0, +(x0[2], x0[2])), x0[2], x0[2])∧(UIncreasing(COND_828_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]))=TRUE828_0_MAIN_LOAD(x0[2], x0[2])≥NonInfC∧828_0_MAIN_LOAD(x0[2], x0[2])≥COND_828_0_MAIN_LOAD1(<(0, +(x0[2], x0[2])), x0[2], x0[2])∧(UIncreasing(COND_828_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_828_0_MAIN_LOAD1(<(0, +(x0[2], x0[2])), x0[2], x0[2])), ≥)∧[(-1)Bound*bni_18] + [(4)bni_18]x0[2] ≥ 0∧[1 + (-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_828_0_MAIN_LOAD1(<(0, +(x0[2], x0[2])), x0[2], x0[2])), ≥)∧[(-1)Bound*bni_18] + [(4)bni_18]x0[2] ≥ 0∧[1 + (-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_828_0_MAIN_LOAD1(<(0, +(x0[2], x0[2])), x0[2], x0[2])), ≥)∧[(-1)Bound*bni_18] + [(4)bni_18]x0[2] ≥ 0∧[1 + (-1)bso_19] ≥ 0)







For Pair COND_828_0_MAIN_LOAD1(TRUE, x0, x0) → 828_0_MAIN_LOAD(x0, +(x0, -1)) the following chains were created:
  • We consider the chain COND_828_0_MAIN_LOAD1(TRUE, x0[3], x0[3]) → 828_0_MAIN_LOAD(x0[3], +(x0[3], -1)) which results in the following constraint:

    (20)    (COND_828_0_MAIN_LOAD1(TRUE, x0[3], x0[3])≥NonInfC∧COND_828_0_MAIN_LOAD1(TRUE, x0[3], x0[3])≥828_0_MAIN_LOAD(x0[3], +(x0[3], -1))∧(UIncreasing(828_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(828_0_MAIN_LOAD(x0[3], +(x0[3], -1))), ≥)∧[bni_20] = 0∧[1 + (-1)bso_21] ≥ 0)



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

    (22)    ((UIncreasing(828_0_MAIN_LOAD(x0[3], +(x0[3], -1))), ≥)∧[bni_20] = 0∧[1 + (-1)bso_21] ≥ 0)



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

    (23)    ((UIncreasing(828_0_MAIN_LOAD(x0[3], +(x0[3], -1))), ≥)∧[bni_20] = 0∧[1 + (-1)bso_21] ≥ 0)



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

    (24)    ((UIncreasing(828_0_MAIN_LOAD(x0[3], +(x0[3], -1))), ≥)∧[bni_20] = 0∧0 = 0∧[1 + (-1)bso_21] ≥ 0)







For Pair 828_0_MAIN_LOAD(x1, x0) → COND_828_0_MAIN_LOAD2(&&(<(x1, x0), <(0, +(x0, x1))), x1, x0) the following chains were created:
  • We consider the chain 828_0_MAIN_LOAD(x1[4], x0[4]) → COND_828_0_MAIN_LOAD2(&&(<(x1[4], x0[4]), <(0, +(x0[4], x1[4]))), x1[4], x0[4]), COND_828_0_MAIN_LOAD2(TRUE, x1[5], x0[5]) → 828_0_MAIN_LOAD(x1[5], +(x0[5], -1)) which results in the following constraint:

    (25)    (&&(<(x1[4], x0[4]), <(0, +(x0[4], x1[4])))=TRUEx1[4]=x1[5]x0[4]=x0[5]828_0_MAIN_LOAD(x1[4], x0[4])≥NonInfC∧828_0_MAIN_LOAD(x1[4], x0[4])≥COND_828_0_MAIN_LOAD2(&&(<(x1[4], x0[4]), <(0, +(x0[4], x1[4]))), x1[4], x0[4])∧(UIncreasing(COND_828_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]))=TRUE828_0_MAIN_LOAD(x1[4], x0[4])≥NonInfC∧828_0_MAIN_LOAD(x1[4], x0[4])≥COND_828_0_MAIN_LOAD2(&&(<(x1[4], x0[4]), <(0, +(x0[4], x1[4]))), x1[4], x0[4])∧(UIncreasing(COND_828_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_828_0_MAIN_LOAD2(&&(<(x1[4], x0[4]), <(0, +(x0[4], x1[4]))), x1[4], x0[4])), ≥)∧[(-1)Bound*bni_22] + [(2)bni_22]x0[4] + [(2)bni_22]x1[4] ≥ 0∧[1 + (-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_828_0_MAIN_LOAD2(&&(<(x1[4], x0[4]), <(0, +(x0[4], x1[4]))), x1[4], x0[4])), ≥)∧[(-1)Bound*bni_22] + [(2)bni_22]x0[4] + [(2)bni_22]x1[4] ≥ 0∧[1 + (-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_828_0_MAIN_LOAD2(&&(<(x1[4], x0[4]), <(0, +(x0[4], x1[4]))), x1[4], x0[4])), ≥)∧[(-1)Bound*bni_22] + [(2)bni_22]x0[4] + [(2)bni_22]x1[4] ≥ 0∧[1 + (-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_828_0_MAIN_LOAD2(&&(<(x1[4], x0[4]), <(0, +(x0[4], x1[4]))), x1[4], x0[4])), ≥)∧[(-1)Bound*bni_22 + (2)bni_22] + [(4)bni_22]x1[4] + [(2)bni_22]x0[4] ≥ 0∧[1 + (-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_828_0_MAIN_LOAD2(&&(<(x1[4], x0[4]), <(0, +(x0[4], x1[4]))), x1[4], x0[4])), ≥)∧[(-1)Bound*bni_22 + (2)bni_22] + [(4)bni_22]x1[4] + [(2)bni_22]x0[4] ≥ 0∧[1 + (-1)bso_23] ≥ 0)


    (32)    (x0[4] ≥ 0∧[-2]x1[4] + x0[4] ≥ 0∧x1[4] ≥ 0 ⇒ (UIncreasing(COND_828_0_MAIN_LOAD2(&&(<(x1[4], x0[4]), <(0, +(x0[4], x1[4]))), x1[4], x0[4])), ≥)∧[(-1)Bound*bni_22 + (2)bni_22] + [(-4)bni_22]x1[4] + [(2)bni_22]x0[4] ≥ 0∧[1 + (-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_828_0_MAIN_LOAD2(&&(<(x1[4], x0[4]), <(0, +(x0[4], x1[4]))), x1[4], x0[4])), ≥)∧[(-1)Bound*bni_22 + (2)bni_22] + [(2)bni_22]x0[4] ≥ 0∧[1 + (-1)bso_23] ≥ 0)







For Pair COND_828_0_MAIN_LOAD2(TRUE, x1, x0) → 828_0_MAIN_LOAD(x1, +(x0, -1)) the following chains were created:
  • We consider the chain COND_828_0_MAIN_LOAD2(TRUE, x1[5], x0[5]) → 828_0_MAIN_LOAD(x1[5], +(x0[5], -1)) which results in the following constraint:

    (34)    (COND_828_0_MAIN_LOAD2(TRUE, x1[5], x0[5])≥NonInfC∧COND_828_0_MAIN_LOAD2(TRUE, x1[5], x0[5])≥828_0_MAIN_LOAD(x1[5], +(x0[5], -1))∧(UIncreasing(828_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(828_0_MAIN_LOAD(x1[5], +(x0[5], -1))), ≥)∧[bni_24] = 0∧[1 + (-1)bso_25] ≥ 0)



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

    (36)    ((UIncreasing(828_0_MAIN_LOAD(x1[5], +(x0[5], -1))), ≥)∧[bni_24] = 0∧[1 + (-1)bso_25] ≥ 0)



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

    (37)    ((UIncreasing(828_0_MAIN_LOAD(x1[5], +(x0[5], -1))), ≥)∧[bni_24] = 0∧[1 + (-1)bso_25] ≥ 0)



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

    (38)    ((UIncreasing(828_0_MAIN_LOAD(x1[5], +(x0[5], -1))), ≥)∧[bni_24] = 0∧0 = 0∧0 = 0∧[1 + (-1)bso_25] ≥ 0)







To summarize, we get the following constraints P for the following pairs.
  • 828_0_MAIN_LOAD(x1, x0) → COND_828_0_MAIN_LOAD(&&(>(x1, x0), <(0, +(x0, x1))), x1, x0)
    • (x1[0] ≥ 0∧[2]x0[0] + x1[0] ≥ 0∧x0[0] ≥ 0 ⇒ (UIncreasing(COND_828_0_MAIN_LOAD(&&(>(x1[0], x0[0]), <(0, +(x0[0], x1[0]))), x1[0], x0[0])), ≥)∧[(-1)Bound*bni_14 + (2)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_828_0_MAIN_LOAD(&&(>(x1[0], x0[0]), <(0, +(x0[0], x1[0]))), x1[0], x0[0])), ≥)∧[(-1)Bound*bni_14 + (2)bni_14] + [(2)bni_14]x1[0] ≥ 0∧[(-1)bso_15] ≥ 0)

  • COND_828_0_MAIN_LOAD(TRUE, x1, x0) → 828_0_MAIN_LOAD(+(x1, -1), x0)
    • ((UIncreasing(828_0_MAIN_LOAD(+(x1[1], -1), x0[1])), ≥)∧[bni_16] = 0∧0 = 0∧0 = 0∧[2 + (-1)bso_17] ≥ 0)

  • 828_0_MAIN_LOAD(x0, x0) → COND_828_0_MAIN_LOAD1(<(0, +(x0, x0)), x0, x0)
    • ([2]x0[2] + [-1] ≥ 0 ⇒ (UIncreasing(COND_828_0_MAIN_LOAD1(<(0, +(x0[2], x0[2])), x0[2], x0[2])), ≥)∧[(-1)Bound*bni_18] + [(4)bni_18]x0[2] ≥ 0∧[1 + (-1)bso_19] ≥ 0)

  • COND_828_0_MAIN_LOAD1(TRUE, x0, x0) → 828_0_MAIN_LOAD(x0, +(x0, -1))
    • ((UIncreasing(828_0_MAIN_LOAD(x0[3], +(x0[3], -1))), ≥)∧[bni_20] = 0∧0 = 0∧[1 + (-1)bso_21] ≥ 0)

  • 828_0_MAIN_LOAD(x1, x0) → COND_828_0_MAIN_LOAD2(&&(<(x1, x0), <(0, +(x0, x1))), x1, x0)
    • (x0[4] ≥ 0∧[2]x1[4] + x0[4] ≥ 0∧x1[4] ≥ 0 ⇒ (UIncreasing(COND_828_0_MAIN_LOAD2(&&(<(x1[4], x0[4]), <(0, +(x0[4], x1[4]))), x1[4], x0[4])), ≥)∧[(-1)Bound*bni_22 + (2)bni_22] + [(4)bni_22]x1[4] + [(2)bni_22]x0[4] ≥ 0∧[1 + (-1)bso_23] ≥ 0)
    • ([2]x1[4] + x0[4] ≥ 0∧x0[4] ≥ 0∧x1[4] ≥ 0 ⇒ (UIncreasing(COND_828_0_MAIN_LOAD2(&&(<(x1[4], x0[4]), <(0, +(x0[4], x1[4]))), x1[4], x0[4])), ≥)∧[(-1)Bound*bni_22 + (2)bni_22] + [(2)bni_22]x0[4] ≥ 0∧[1 + (-1)bso_23] ≥ 0)

  • COND_828_0_MAIN_LOAD2(TRUE, x1, x0) → 828_0_MAIN_LOAD(x1, +(x0, -1))
    • ((UIncreasing(828_0_MAIN_LOAD(x1[5], +(x0[5], -1))), ≥)∧[bni_24] = 0∧0 = 0∧0 = 0∧[1 + (-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(828_0_MAIN_LOAD(x1, x2)) = [2]x2 + [2]x1   
POL(COND_828_0_MAIN_LOAD(x1, x2, x3)) = [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_828_0_MAIN_LOAD1(x1, x2, x3)) = [-1] + [2]x3 + [2]x2   
POL(COND_828_0_MAIN_LOAD2(x1, x2, x3)) = [-1] + [2]x3 + [2]x2   

The following pairs are in P>:

COND_828_0_MAIN_LOAD(TRUE, x1[1], x0[1]) → 828_0_MAIN_LOAD(+(x1[1], -1), x0[1])
828_0_MAIN_LOAD(x0[2], x0[2]) → COND_828_0_MAIN_LOAD1(<(0, +(x0[2], x0[2])), x0[2], x0[2])
COND_828_0_MAIN_LOAD1(TRUE, x0[3], x0[3]) → 828_0_MAIN_LOAD(x0[3], +(x0[3], -1))
828_0_MAIN_LOAD(x1[4], x0[4]) → COND_828_0_MAIN_LOAD2(&&(<(x1[4], x0[4]), <(0, +(x0[4], x1[4]))), x1[4], x0[4])
COND_828_0_MAIN_LOAD2(TRUE, x1[5], x0[5]) → 828_0_MAIN_LOAD(x1[5], +(x0[5], -1))

The following pairs are in Pbound:

828_0_MAIN_LOAD(x1[0], x0[0]) → COND_828_0_MAIN_LOAD(&&(>(x1[0], x0[0]), <(0, +(x0[0], x1[0]))), x1[0], x0[0])
828_0_MAIN_LOAD(x0[2], x0[2]) → COND_828_0_MAIN_LOAD1(<(0, +(x0[2], x0[2])), x0[2], x0[2])
828_0_MAIN_LOAD(x1[4], x0[4]) → COND_828_0_MAIN_LOAD2(&&(<(x1[4], x0[4]), <(0, +(x0[4], x1[4]))), x1[4], x0[4])

The following pairs are in P:

828_0_MAIN_LOAD(x1[0], x0[0]) → COND_828_0_MAIN_LOAD(&&(>(x1[0], x0[0]), <(0, +(x0[0], x1[0]))), x1[0], x0[0])

There are no usable rules.

(8) Complex Obligation (AND)

(9) 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): 828_0_MAIN_LOAD(x1[0], x0[0]) → COND_828_0_MAIN_LOAD(x1[0] > x0[0] && 0 < x0[0] + x1[0], x1[0], x0[0])


The set Q is empty.

(10) IDependencyGraphProof (EQUIVALENT transformation)

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

(11) TRUE

(12) 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_828_0_MAIN_LOAD(TRUE, x1[1], x0[1]) → 828_0_MAIN_LOAD(x1[1] + -1, x0[1])
(3): COND_828_0_MAIN_LOAD1(TRUE, x0[3], x0[3]) → 828_0_MAIN_LOAD(x0[3], x0[3] + -1)
(5): COND_828_0_MAIN_LOAD2(TRUE, x1[5], x0[5]) → 828_0_MAIN_LOAD(x1[5], x0[5] + -1)


The set Q is empty.

(13) IDependencyGraphProof (EQUIVALENT transformation)

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

(14) TRUE