### (0) Obligation:

JBC Problem based on JBC Program:
Manifest-Version: 1.0 Created-By: 1.6.0_16 (Sun Microsystems Inc.) Main-Class: PastaC11
`/** * 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 PastaC11 {    public static void main(String[] args) {        Random.args = args;        int x = Random.random();        int y = Random.random();        while (true) {			if (x >= 0) {				x--;				y = Random.random();			} else if (y >= 0) {				y--;			} else {				break;			}        }    } }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:
PastaC11.main([Ljava/lang/String;)V: Graph of 252 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: PastaC11.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 33 rules for P and 0 rules for R.

P rules:
933_0_main_LT(EOS(STATIC_933), matching1, i347, matching2) → 935_0_main_LT(EOS(STATIC_935), -1, i347, -1) | &&(=(matching1, -1), =(matching2, -1))
933_0_main_LT(EOS(STATIC_933), i353, i347, i353) → 936_0_main_LT(EOS(STATIC_936), i353, i347, i353)
935_0_main_LT(EOS(STATIC_935), matching1, i347, matching2) → 938_0_main_Load(EOS(STATIC_938), -1, i347) | &&(&&(<(-1, 0), =(matching1, -1)), =(matching2, -1))
938_0_main_Load(EOS(STATIC_938), matching1, i347) → 943_0_main_LT(EOS(STATIC_943), -1, i347, i347) | =(matching1, -1)
943_0_main_LT(EOS(STATIC_943), matching1, i360, i360) → 948_0_main_LT(EOS(STATIC_948), -1, i360, i360) | =(matching1, -1)
948_0_main_LT(EOS(STATIC_948), matching1, i360, i360) → 954_0_main_Inc(EOS(STATIC_954), -1, i360) | &&(>=(i360, 0), =(matching1, -1))
954_0_main_Inc(EOS(STATIC_954), matching1, i360) → 957_0_main_JMP(EOS(STATIC_957), -1, +(i360, -1)) | &&(>=(i360, 0), =(matching1, -1))
957_0_main_JMP(EOS(STATIC_957), matching1, i361) → 963_0_main_Load(EOS(STATIC_963), -1, i361) | =(matching1, -1)
929_0_main_Load(EOS(STATIC_929), i346, i347) → 933_0_main_LT(EOS(STATIC_933), i346, i347, i346)
936_0_main_LT(EOS(STATIC_936), i353, i347, i353) → 940_0_main_Inc(EOS(STATIC_940), i353) | >=(i353, 0)
940_0_main_Inc(EOS(STATIC_940), i353) → 945_0_main_InvokeMethod(EOS(STATIC_945), +(i353, -1)) | >=(i353, 0)
945_0_main_InvokeMethod(EOS(STATIC_945), i357) → 950_0_random_FieldAccess(EOS(STATIC_950), i357)
950_0_random_FieldAccess(EOS(STATIC_950), i357) → 960_0_random_FieldAccess(EOS(STATIC_960), i357)
960_0_random_FieldAccess(EOS(STATIC_960), i357) → 970_0_random_ArrayAccess(EOS(STATIC_970), i357)
970_0_random_ArrayAccess(EOS(STATIC_970), i357) → 972_0_random_ArrayAccess(EOS(STATIC_972), i357)
972_0_random_ArrayAccess(EOS(STATIC_972), i357) → 976_0_random_Store(EOS(STATIC_976), i357, o242)
976_0_random_Store(EOS(STATIC_976), i357, o242) → 981_0_random_FieldAccess(EOS(STATIC_981), i357, o242)
981_0_random_FieldAccess(EOS(STATIC_981), i357, o242) → 984_0_random_ConstantStackPush(EOS(STATIC_984), i357, o242)
984_0_random_ConstantStackPush(EOS(STATIC_984), i357, o242) → 989_0_random_IntArithmetic(EOS(STATIC_989), i357, o242)
989_0_random_IntArithmetic(EOS(STATIC_989), i357, o242) → 994_0_random_FieldAccess(EOS(STATIC_994), i357, o242)
994_0_random_FieldAccess(EOS(STATIC_994), i357, o242) → 997_0_random_Load(EOS(STATIC_997), i357, o242)
997_0_random_Load(EOS(STATIC_997), i357, o242) → 1009_0_random_InvokeMethod(EOS(STATIC_1009), i357, o242)
1009_0_random_InvokeMethod(EOS(STATIC_1009), i357, java.lang.Object(o262sub)) → 1012_0_random_InvokeMethod(EOS(STATIC_1012), i357, java.lang.Object(o262sub))
1012_0_random_InvokeMethod(EOS(STATIC_1012), i357, java.lang.Object(o262sub)) → 1016_0_length_Load(EOS(STATIC_1016), i357, java.lang.Object(o262sub), java.lang.Object(o262sub))
1016_0_length_Load(EOS(STATIC_1016), i357, java.lang.Object(o262sub), java.lang.Object(o262sub)) → 1030_0_length_FieldAccess(EOS(STATIC_1030), i357, java.lang.Object(o262sub), java.lang.Object(o262sub))
1030_0_length_FieldAccess(EOS(STATIC_1030), i357, java.lang.Object(java.lang.String(o271sub, i408)), java.lang.Object(java.lang.String(o271sub, i408))) → 1033_0_length_FieldAccess(EOS(STATIC_1033), i357, java.lang.Object(java.lang.String(o271sub, i408)), java.lang.Object(java.lang.String(o271sub, i408))) | &&(>=(i408, 0), >=(i409, 0))
1033_0_length_FieldAccess(EOS(STATIC_1033), i357, java.lang.Object(java.lang.String(o271sub, i408)), java.lang.Object(java.lang.String(o271sub, i408))) → 1041_0_length_Return(EOS(STATIC_1041), i357, java.lang.Object(java.lang.String(o271sub, i408)), i408)
1041_0_length_Return(EOS(STATIC_1041), i357, java.lang.Object(java.lang.String(o271sub, i408)), i408) → 1047_0_random_Return(EOS(STATIC_1047), i357, i408)
1047_0_random_Return(EOS(STATIC_1047), i357, i408) → 1050_0_main_Store(EOS(STATIC_1050), i357, i408)
1050_0_main_Store(EOS(STATIC_1050), i357, i408) → 1059_0_main_JMP(EOS(STATIC_1059), i357, i408)
1059_0_main_JMP(EOS(STATIC_1059), i357, i408) → 1067_0_main_Load(EOS(STATIC_1067), i357, i408)
R rules:

Combined rules. Obtained 2 conditional rules for P and 0 conditional rules for R.

P rules:
933_0_main_LT(EOS(STATIC_933), -1, x1, -1) → 933_0_main_LT(EOS(STATIC_933), -1, +(x1, -1), -1) | >(+(x1, 1), 0)
933_0_main_LT(EOS(STATIC_933), x0, x1, x0) → 933_0_main_LT(EOS(STATIC_933), +(x0, -1), x2, +(x0, -1)) | &&(>(+(x2, 1), 0), >(+(x0, 1), 0))
R rules:

Filtered ground terms:

933_0_main_LT(x1, x2, x3, x4) → 933_0_main_LT(x2, x3, x4)
EOS(x1) → EOS
Cond_933_0_main_LT1(x1, x2, x3, x4, x5, x6) → Cond_933_0_main_LT1(x1, x3, x4, x5, x6)
Cond_933_0_main_LT(x1, x2, x3, x4, x5) → Cond_933_0_main_LT(x1, x4)

Filtered duplicate args:

933_0_main_LT(x1, x2, x3) → 933_0_main_LT(x2, x3)
Cond_933_0_main_LT1(x1, x2, x3, x4, x5) → Cond_933_0_main_LT1(x1, x3, x4, x5)

Filtered unneeded arguments:

Cond_933_0_main_LT1(x1, x2, x3, x4) → Cond_933_0_main_LT1(x1, x3, x4)

Combined rules. Obtained 2 conditional rules for P and 0 conditional rules for R.

P rules:
933_0_main_LT(x1, -1) → 933_0_main_LT(+(x1, -1), -1) | >(x1, -1)
933_0_main_LT(x1, x0) → 933_0_main_LT(x2, +(x0, -1)) | &&(>(x2, -1), >(x0, -1))
R rules:

Finished conversion. Obtained 4 rules for P and 0 rules for R. System has predefined symbols.

P rules:
933_0_MAIN_LT(x1, -1) → COND_933_0_MAIN_LT(>(x1, -1), x1, -1)
COND_933_0_MAIN_LT(TRUE, x1, -1) → 933_0_MAIN_LT(+(x1, -1), -1)
933_0_MAIN_LT(x1, x0) → COND_933_0_MAIN_LT1(&&(>(x2, -1), >(x0, -1)), x1, x0, x2)
COND_933_0_MAIN_LT1(TRUE, x1, x0, x2) → 933_0_MAIN_LT(x2, +(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:

Integer, Boolean

R is empty.

The integer pair graph contains the following rules and edges:
(0): 933_0_MAIN_LT(x1[0], -1) → COND_933_0_MAIN_LT(x1[0] > -1, x1[0], -1)
(1): COND_933_0_MAIN_LT(TRUE, x1[1], -1) → 933_0_MAIN_LT(x1[1] + -1, -1)
(2): 933_0_MAIN_LT(x1[2], x0[2]) → COND_933_0_MAIN_LT1(x2[2] > -1 && x0[2] > -1, x1[2], x0[2], x2[2])
(3): COND_933_0_MAIN_LT1(TRUE, x1[3], x0[3], x2[3]) → 933_0_MAIN_LT(x2[3], x0[3] + -1)

(0) -> (1), if (x1[0] > -1x1[0]* x1[1])

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

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

(2) -> (3), if (x2[2] > -1 && x0[2] > -1x1[2]* x1[3]x0[2]* x0[3]x2[2]* x2[3])

(3) -> (0), if (x2[3]* x1[0]x0[3] + -1* -1)

(3) -> (2), if (x2[3]* x1[2]x0[3] + -1* x0[2])

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.IdpDefaultShapeHeuristic@e1f8dfe Constraint Generator: NonInfConstraintGenerator: PathGenerator: MetricPathGenerator: Max Left Steps: 1 Max Right Steps: 1

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 933_0_MAIN_LT(x1, -1) → COND_933_0_MAIN_LT(>(x1, -1), x1, -1) the following chains were created:
• We consider the chain 933_0_MAIN_LT(x1[0], -1) → COND_933_0_MAIN_LT(>(x1[0], -1), x1[0], -1), COND_933_0_MAIN_LT(TRUE, x1[1], -1) → 933_0_MAIN_LT(+(x1[1], -1), -1) which results in the following constraint:

(1)    (>(x1[0], -1)=TRUEx1[0]=x1[1]933_0_MAIN_LT(x1[0], -1)≥NonInfC∧933_0_MAIN_LT(x1[0], -1)≥COND_933_0_MAIN_LT(>(x1[0], -1), x1[0], -1)∧(UIncreasing(COND_933_0_MAIN_LT(>(x1[0], -1), x1[0], -1)), ≥))

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

(2)    (>(x1[0], -1)=TRUE933_0_MAIN_LT(x1[0], -1)≥NonInfC∧933_0_MAIN_LT(x1[0], -1)≥COND_933_0_MAIN_LT(>(x1[0], -1), x1[0], -1)∧(UIncreasing(COND_933_0_MAIN_LT(>(x1[0], -1), x1[0], -1)), ≥))

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

(3)    (x1[0] ≥ 0 ⇒ (UIncreasing(COND_933_0_MAIN_LT(>(x1[0], -1), x1[0], -1)), ≥)∧[(-2)bni_17 + (-1)Bound*bni_17] ≥ 0∧[(-1)bso_18] ≥ 0)

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

(4)    (x1[0] ≥ 0 ⇒ (UIncreasing(COND_933_0_MAIN_LT(>(x1[0], -1), x1[0], -1)), ≥)∧[(-2)bni_17 + (-1)Bound*bni_17] ≥ 0∧[(-1)bso_18] ≥ 0)

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

(5)    (x1[0] ≥ 0 ⇒ (UIncreasing(COND_933_0_MAIN_LT(>(x1[0], -1), x1[0], -1)), ≥)∧[(-2)bni_17 + (-1)Bound*bni_17] ≥ 0∧[(-1)bso_18] ≥ 0)

For Pair COND_933_0_MAIN_LT(TRUE, x1, -1) → 933_0_MAIN_LT(+(x1, -1), -1) the following chains were created:
• We consider the chain 933_0_MAIN_LT(x1[0], -1) → COND_933_0_MAIN_LT(>(x1[0], -1), x1[0], -1), COND_933_0_MAIN_LT(TRUE, x1[1], -1) → 933_0_MAIN_LT(+(x1[1], -1), -1), 933_0_MAIN_LT(x1[0], -1) → COND_933_0_MAIN_LT(>(x1[0], -1), x1[0], -1) which results in the following constraint:

(6)    (>(x1[0], -1)=TRUEx1[0]=x1[1]+(x1[1], -1)=x1[0]1COND_933_0_MAIN_LT(TRUE, x1[1], -1)≥NonInfC∧COND_933_0_MAIN_LT(TRUE, x1[1], -1)≥933_0_MAIN_LT(+(x1[1], -1), -1)∧(UIncreasing(933_0_MAIN_LT(+(x1[1], -1), -1)), ≥))

We simplified constraint (6) using rules (III), (IV) which results in the following new constraint:

(7)    (>(x1[0], -1)=TRUECOND_933_0_MAIN_LT(TRUE, x1[0], -1)≥NonInfC∧COND_933_0_MAIN_LT(TRUE, x1[0], -1)≥933_0_MAIN_LT(+(x1[0], -1), -1)∧(UIncreasing(933_0_MAIN_LT(+(x1[1], -1), -1)), ≥))

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

(8)    (x1[0] ≥ 0 ⇒ (UIncreasing(933_0_MAIN_LT(+(x1[1], -1), -1)), ≥)∧[(-2)bni_19 + (-1)Bound*bni_19] ≥ 0∧[(-1)bso_20] ≥ 0)

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

(9)    (x1[0] ≥ 0 ⇒ (UIncreasing(933_0_MAIN_LT(+(x1[1], -1), -1)), ≥)∧[(-2)bni_19 + (-1)Bound*bni_19] ≥ 0∧[(-1)bso_20] ≥ 0)

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

(10)    (x1[0] ≥ 0 ⇒ (UIncreasing(933_0_MAIN_LT(+(x1[1], -1), -1)), ≥)∧[(-2)bni_19 + (-1)Bound*bni_19] ≥ 0∧[(-1)bso_20] ≥ 0)

• We consider the chain 933_0_MAIN_LT(x1[0], -1) → COND_933_0_MAIN_LT(>(x1[0], -1), x1[0], -1), COND_933_0_MAIN_LT(TRUE, x1[1], -1) → 933_0_MAIN_LT(+(x1[1], -1), -1), 933_0_MAIN_LT(x1[2], x0[2]) → COND_933_0_MAIN_LT1(&&(>(x2[2], -1), >(x0[2], -1)), x1[2], x0[2], x2[2]) which results in the following constraint:

(11)    (>(x1[0], -1)=TRUEx1[0]=x1[1]+(x1[1], -1)=x1[2]-1=x0[2]COND_933_0_MAIN_LT(TRUE, x1[1], -1)≥NonInfC∧COND_933_0_MAIN_LT(TRUE, x1[1], -1)≥933_0_MAIN_LT(+(x1[1], -1), -1)∧(UIncreasing(933_0_MAIN_LT(+(x1[1], -1), -1)), ≥))

We simplified constraint (11) using rules (III), (IV) which results in the following new constraint:

(12)    (>(x1[0], -1)=TRUECOND_933_0_MAIN_LT(TRUE, x1[0], -1)≥NonInfC∧COND_933_0_MAIN_LT(TRUE, x1[0], -1)≥933_0_MAIN_LT(+(x1[0], -1), -1)∧(UIncreasing(933_0_MAIN_LT(+(x1[1], -1), -1)), ≥))

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

(13)    (x1[0] ≥ 0 ⇒ (UIncreasing(933_0_MAIN_LT(+(x1[1], -1), -1)), ≥)∧[(-2)bni_19 + (-1)Bound*bni_19] ≥ 0∧[(-1)bso_20] ≥ 0)

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

(14)    (x1[0] ≥ 0 ⇒ (UIncreasing(933_0_MAIN_LT(+(x1[1], -1), -1)), ≥)∧[(-2)bni_19 + (-1)Bound*bni_19] ≥ 0∧[(-1)bso_20] ≥ 0)

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

(15)    (x1[0] ≥ 0 ⇒ (UIncreasing(933_0_MAIN_LT(+(x1[1], -1), -1)), ≥)∧[(-2)bni_19 + (-1)Bound*bni_19] ≥ 0∧[(-1)bso_20] ≥ 0)

For Pair 933_0_MAIN_LT(x1, x0) → COND_933_0_MAIN_LT1(&&(>(x2, -1), >(x0, -1)), x1, x0, x2) the following chains were created:
• We consider the chain COND_933_0_MAIN_LT(TRUE, x1[1], -1) → 933_0_MAIN_LT(+(x1[1], -1), -1), 933_0_MAIN_LT(x1[2], x0[2]) → COND_933_0_MAIN_LT1(&&(>(x2[2], -1), >(x0[2], -1)), x1[2], x0[2], x2[2]), COND_933_0_MAIN_LT1(TRUE, x1[3], x0[3], x2[3]) → 933_0_MAIN_LT(x2[3], +(x0[3], -1)) which results in the following constraint:

(16)    (+(x1[1], -1)=x1[2]-1=x0[2]&&(>(x2[2], -1), >(x0[2], -1))=TRUEx1[2]=x1[3]x0[2]=x0[3]x2[2]=x2[3]933_0_MAIN_LT(x1[2], x0[2])≥NonInfC∧933_0_MAIN_LT(x1[2], x0[2])≥COND_933_0_MAIN_LT1(&&(>(x2[2], -1), >(x0[2], -1)), x1[2], x0[2], x2[2])∧(UIncreasing(COND_933_0_MAIN_LT1(&&(>(x2[2], -1), >(x0[2], -1)), x1[2], x0[2], x2[2])), ≥))

We solved constraint (16) using rules (I), (II), (III), (IV), (IDP_CONSTANT_FOLD), (IDP_BOOLEAN).
• We consider the chain COND_933_0_MAIN_LT1(TRUE, x1[3], x0[3], x2[3]) → 933_0_MAIN_LT(x2[3], +(x0[3], -1)), 933_0_MAIN_LT(x1[2], x0[2]) → COND_933_0_MAIN_LT1(&&(>(x2[2], -1), >(x0[2], -1)), x1[2], x0[2], x2[2]), COND_933_0_MAIN_LT1(TRUE, x1[3], x0[3], x2[3]) → 933_0_MAIN_LT(x2[3], +(x0[3], -1)) which results in the following constraint:

(17)    (x2[3]=x1[2]+(x0[3], -1)=x0[2]&&(>(x2[2], -1), >(x0[2], -1))=TRUEx1[2]=x1[3]1x0[2]=x0[3]1x2[2]=x2[3]1933_0_MAIN_LT(x1[2], x0[2])≥NonInfC∧933_0_MAIN_LT(x1[2], x0[2])≥COND_933_0_MAIN_LT1(&&(>(x2[2], -1), >(x0[2], -1)), x1[2], x0[2], x2[2])∧(UIncreasing(COND_933_0_MAIN_LT1(&&(>(x2[2], -1), >(x0[2], -1)), x1[2], x0[2], x2[2])), ≥))

We simplified constraint (17) using rules (III), (IV), (IDP_BOOLEAN) which results in the following new constraint:

(18)    (>(x2[2], -1)=TRUE>(+(x0[3], -1), -1)=TRUE933_0_MAIN_LT(x2[3], +(x0[3], -1))≥NonInfC∧933_0_MAIN_LT(x2[3], +(x0[3], -1))≥COND_933_0_MAIN_LT1(&&(>(x2[2], -1), >(+(x0[3], -1), -1)), x2[3], +(x0[3], -1), x2[2])∧(UIncreasing(COND_933_0_MAIN_LT1(&&(>(x2[2], -1), >(x0[2], -1)), x1[2], x0[2], x2[2])), ≥))

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

(19)    (x2[2] ≥ 0∧x0[3] + [-1] ≥ 0 ⇒ (UIncreasing(COND_933_0_MAIN_LT1(&&(>(x2[2], -1), >(x0[2], -1)), x1[2], x0[2], x2[2])), ≥)∧[(-2)bni_21 + (-1)Bound*bni_21] + [(2)bni_21]x0[3] ≥ 0∧[(-1)bso_22] ≥ 0)

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

(20)    (x2[2] ≥ 0∧x0[3] + [-1] ≥ 0 ⇒ (UIncreasing(COND_933_0_MAIN_LT1(&&(>(x2[2], -1), >(x0[2], -1)), x1[2], x0[2], x2[2])), ≥)∧[(-2)bni_21 + (-1)Bound*bni_21] + [(2)bni_21]x0[3] ≥ 0∧[(-1)bso_22] ≥ 0)

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

(21)    (x2[2] ≥ 0∧x0[3] + [-1] ≥ 0 ⇒ (UIncreasing(COND_933_0_MAIN_LT1(&&(>(x2[2], -1), >(x0[2], -1)), x1[2], x0[2], x2[2])), ≥)∧[(-2)bni_21 + (-1)Bound*bni_21] + [(2)bni_21]x0[3] ≥ 0∧[(-1)bso_22] ≥ 0)

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

(22)    (x2[2] ≥ 0∧x0[3] + [-1] ≥ 0 ⇒ (UIncreasing(COND_933_0_MAIN_LT1(&&(>(x2[2], -1), >(x0[2], -1)), x1[2], x0[2], x2[2])), ≥)∧0 = 0∧[(-2)bni_21 + (-1)Bound*bni_21] + [(2)bni_21]x0[3] ≥ 0∧0 = 0∧[(-1)bso_22] ≥ 0)

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

(23)    (x2[2] ≥ 0∧x0[3] ≥ 0 ⇒ (UIncreasing(COND_933_0_MAIN_LT1(&&(>(x2[2], -1), >(x0[2], -1)), x1[2], x0[2], x2[2])), ≥)∧0 = 0∧[(-1)Bound*bni_21] + [(2)bni_21]x0[3] ≥ 0∧0 = 0∧[(-1)bso_22] ≥ 0)

For Pair COND_933_0_MAIN_LT1(TRUE, x1, x0, x2) → 933_0_MAIN_LT(x2, +(x0, -1)) the following chains were created:
• We consider the chain 933_0_MAIN_LT(x1[2], x0[2]) → COND_933_0_MAIN_LT1(&&(>(x2[2], -1), >(x0[2], -1)), x1[2], x0[2], x2[2]), COND_933_0_MAIN_LT1(TRUE, x1[3], x0[3], x2[3]) → 933_0_MAIN_LT(x2[3], +(x0[3], -1)), 933_0_MAIN_LT(x1[0], -1) → COND_933_0_MAIN_LT(>(x1[0], -1), x1[0], -1) which results in the following constraint:

(24)    (&&(>(x2[2], -1), >(x0[2], -1))=TRUEx1[2]=x1[3]x0[2]=x0[3]x2[2]=x2[3]x2[3]=x1[0]+(x0[3], -1)=-1COND_933_0_MAIN_LT1(TRUE, x1[3], x0[3], x2[3])≥NonInfC∧COND_933_0_MAIN_LT1(TRUE, x1[3], x0[3], x2[3])≥933_0_MAIN_LT(x2[3], +(x0[3], -1))∧(UIncreasing(933_0_MAIN_LT(x2[3], +(x0[3], -1))), ≥))

We simplified constraint (24) using rules (III), (IV), (IDP_BOOLEAN) which results in the following new constraint:

(25)    (+(x0[2], -1)=-1>(x2[2], -1)=TRUE>(x0[2], -1)=TRUECOND_933_0_MAIN_LT1(TRUE, x1[2], x0[2], x2[2])≥NonInfC∧COND_933_0_MAIN_LT1(TRUE, x1[2], x0[2], x2[2])≥933_0_MAIN_LT(x2[2], +(x0[2], -1))∧(UIncreasing(933_0_MAIN_LT(x2[3], +(x0[3], -1))), ≥))

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

(26)    (x0[2] ≥ 0∧x2[2] ≥ 0∧x0[2] ≥ 0 ⇒ (UIncreasing(933_0_MAIN_LT(x2[3], +(x0[3], -1))), ≥)∧[(-1)bni_23 + (-1)Bound*bni_23] + [(2)bni_23]x0[2] ≥ 0∧[1 + (-1)bso_24] ≥ 0)

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

(27)    (x0[2] ≥ 0∧x2[2] ≥ 0∧x0[2] ≥ 0 ⇒ (UIncreasing(933_0_MAIN_LT(x2[3], +(x0[3], -1))), ≥)∧[(-1)bni_23 + (-1)Bound*bni_23] + [(2)bni_23]x0[2] ≥ 0∧[1 + (-1)bso_24] ≥ 0)

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

(28)    (x0[2] ≥ 0∧x2[2] ≥ 0∧x0[2] ≥ 0 ⇒ (UIncreasing(933_0_MAIN_LT(x2[3], +(x0[3], -1))), ≥)∧[(-1)bni_23 + (-1)Bound*bni_23] + [(2)bni_23]x0[2] ≥ 0∧[1 + (-1)bso_24] ≥ 0)

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

(29)    (x0[2] ≥ 0∧x2[2] ≥ 0∧x0[2] ≥ 0 ⇒ (UIncreasing(933_0_MAIN_LT(x2[3], +(x0[3], -1))), ≥)∧0 = 0∧[(-1)bni_23 + (-1)Bound*bni_23] + [(2)bni_23]x0[2] ≥ 0∧0 = 0∧[1 + (-1)bso_24] ≥ 0)

• We consider the chain 933_0_MAIN_LT(x1[2], x0[2]) → COND_933_0_MAIN_LT1(&&(>(x2[2], -1), >(x0[2], -1)), x1[2], x0[2], x2[2]), COND_933_0_MAIN_LT1(TRUE, x1[3], x0[3], x2[3]) → 933_0_MAIN_LT(x2[3], +(x0[3], -1)), 933_0_MAIN_LT(x1[2], x0[2]) → COND_933_0_MAIN_LT1(&&(>(x2[2], -1), >(x0[2], -1)), x1[2], x0[2], x2[2]) which results in the following constraint:

(30)    (&&(>(x2[2], -1), >(x0[2], -1))=TRUEx1[2]=x1[3]x0[2]=x0[3]x2[2]=x2[3]x2[3]=x1[2]1+(x0[3], -1)=x0[2]1COND_933_0_MAIN_LT1(TRUE, x1[3], x0[3], x2[3])≥NonInfC∧COND_933_0_MAIN_LT1(TRUE, x1[3], x0[3], x2[3])≥933_0_MAIN_LT(x2[3], +(x0[3], -1))∧(UIncreasing(933_0_MAIN_LT(x2[3], +(x0[3], -1))), ≥))

We simplified constraint (30) using rules (III), (IV), (IDP_BOOLEAN) which results in the following new constraint:

(31)    (>(x2[2], -1)=TRUE>(x0[2], -1)=TRUECOND_933_0_MAIN_LT1(TRUE, x1[2], x0[2], x2[2])≥NonInfC∧COND_933_0_MAIN_LT1(TRUE, x1[2], x0[2], x2[2])≥933_0_MAIN_LT(x2[2], +(x0[2], -1))∧(UIncreasing(933_0_MAIN_LT(x2[3], +(x0[3], -1))), ≥))

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

(32)    (x2[2] ≥ 0∧x0[2] ≥ 0 ⇒ (UIncreasing(933_0_MAIN_LT(x2[3], +(x0[3], -1))), ≥)∧[(-1)bni_23 + (-1)Bound*bni_23] + [(2)bni_23]x0[2] ≥ 0∧[1 + (-1)bso_24] ≥ 0)

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

(33)    (x2[2] ≥ 0∧x0[2] ≥ 0 ⇒ (UIncreasing(933_0_MAIN_LT(x2[3], +(x0[3], -1))), ≥)∧[(-1)bni_23 + (-1)Bound*bni_23] + [(2)bni_23]x0[2] ≥ 0∧[1 + (-1)bso_24] ≥ 0)

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

(34)    (x2[2] ≥ 0∧x0[2] ≥ 0 ⇒ (UIncreasing(933_0_MAIN_LT(x2[3], +(x0[3], -1))), ≥)∧[(-1)bni_23 + (-1)Bound*bni_23] + [(2)bni_23]x0[2] ≥ 0∧[1 + (-1)bso_24] ≥ 0)

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

(35)    (x2[2] ≥ 0∧x0[2] ≥ 0 ⇒ (UIncreasing(933_0_MAIN_LT(x2[3], +(x0[3], -1))), ≥)∧0 = 0∧[(-1)bni_23 + (-1)Bound*bni_23] + [(2)bni_23]x0[2] ≥ 0∧0 = 0∧[1 + (-1)bso_24] ≥ 0)

To summarize, we get the following constraints P for the following pairs.
• 933_0_MAIN_LT(x1, -1) → COND_933_0_MAIN_LT(>(x1, -1), x1, -1)
• (x1[0] ≥ 0 ⇒ (UIncreasing(COND_933_0_MAIN_LT(>(x1[0], -1), x1[0], -1)), ≥)∧[(-2)bni_17 + (-1)Bound*bni_17] ≥ 0∧[(-1)bso_18] ≥ 0)

• COND_933_0_MAIN_LT(TRUE, x1, -1) → 933_0_MAIN_LT(+(x1, -1), -1)
• (x1[0] ≥ 0 ⇒ (UIncreasing(933_0_MAIN_LT(+(x1[1], -1), -1)), ≥)∧[(-2)bni_19 + (-1)Bound*bni_19] ≥ 0∧[(-1)bso_20] ≥ 0)
• (x1[0] ≥ 0 ⇒ (UIncreasing(933_0_MAIN_LT(+(x1[1], -1), -1)), ≥)∧[(-2)bni_19 + (-1)Bound*bni_19] ≥ 0∧[(-1)bso_20] ≥ 0)

• 933_0_MAIN_LT(x1, x0) → COND_933_0_MAIN_LT1(&&(>(x2, -1), >(x0, -1)), x1, x0, x2)
• (x2[2] ≥ 0∧x0[3] ≥ 0 ⇒ (UIncreasing(COND_933_0_MAIN_LT1(&&(>(x2[2], -1), >(x0[2], -1)), x1[2], x0[2], x2[2])), ≥)∧0 = 0∧[(-1)Bound*bni_21] + [(2)bni_21]x0[3] ≥ 0∧0 = 0∧[(-1)bso_22] ≥ 0)

• COND_933_0_MAIN_LT1(TRUE, x1, x0, x2) → 933_0_MAIN_LT(x2, +(x0, -1))
• (x0[2] ≥ 0∧x2[2] ≥ 0∧x0[2] ≥ 0 ⇒ (UIncreasing(933_0_MAIN_LT(x2[3], +(x0[3], -1))), ≥)∧0 = 0∧[(-1)bni_23 + (-1)Bound*bni_23] + [(2)bni_23]x0[2] ≥ 0∧0 = 0∧[1 + (-1)bso_24] ≥ 0)
• (x2[2] ≥ 0∧x0[2] ≥ 0 ⇒ (UIncreasing(933_0_MAIN_LT(x2[3], +(x0[3], -1))), ≥)∧0 = 0∧[(-1)bni_23 + (-1)Bound*bni_23] + [(2)bni_23]x0[2] ≥ 0∧0 = 0∧[1 + (-1)bso_24] ≥ 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(933_0_MAIN_LT(x1, x2)) = [2]x2
POL(-1) = [-1]
POL(COND_933_0_MAIN_LT(x1, x2, x3)) = [2]x3
POL(>(x1, x2)) = [-1]
POL(+(x1, x2)) = x1 + x2
POL(COND_933_0_MAIN_LT1(x1, x2, x3, x4)) = [-1] + [2]x3 + [-1]x1
POL(&&(x1, x2)) = [-1]

The following pairs are in P>:

COND_933_0_MAIN_LT1(TRUE, x1[3], x0[3], x2[3]) → 933_0_MAIN_LT(x2[3], +(x0[3], -1))

The following pairs are in Pbound:

933_0_MAIN_LT(x1[0], -1) → COND_933_0_MAIN_LT(>(x1[0], -1), x1[0], -1)
COND_933_0_MAIN_LT(TRUE, x1[1], -1) → 933_0_MAIN_LT(+(x1[1], -1), -1)
933_0_MAIN_LT(x1[2], x0[2]) → COND_933_0_MAIN_LT1(&&(>(x2[2], -1), >(x0[2], -1)), x1[2], x0[2], x2[2])
COND_933_0_MAIN_LT1(TRUE, x1[3], x0[3], x2[3]) → 933_0_MAIN_LT(x2[3], +(x0[3], -1))

The following pairs are in P:

933_0_MAIN_LT(x1[0], -1) → COND_933_0_MAIN_LT(>(x1[0], -1), x1[0], -1)
COND_933_0_MAIN_LT(TRUE, x1[1], -1) → 933_0_MAIN_LT(+(x1[1], -1), -1)
933_0_MAIN_LT(x1[2], x0[2]) → COND_933_0_MAIN_LT1(&&(>(x2[2], -1), >(x0[2], -1)), x1[2], x0[2], x2[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

### (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): 933_0_MAIN_LT(x1[0], -1) → COND_933_0_MAIN_LT(x1[0] > -1, x1[0], -1)
(1): COND_933_0_MAIN_LT(TRUE, x1[1], -1) → 933_0_MAIN_LT(x1[1] + -1, -1)
(2): 933_0_MAIN_LT(x1[2], x0[2]) → COND_933_0_MAIN_LT1(x2[2] > -1 && x0[2] > -1, x1[2], x0[2], x2[2])

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

(0) -> (1), if (x1[0] > -1x1[0]* x1[1])

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

The set Q is empty.

### (9) IDependencyGraphProof (EQUIVALENT transformation)

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

### (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_933_0_MAIN_LT(TRUE, x1[1], -1) → 933_0_MAIN_LT(x1[1] + -1, -1)
(0): 933_0_MAIN_LT(x1[0], -1) → COND_933_0_MAIN_LT(x1[0] > -1, x1[0], -1)

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

(0) -> (1), if (x1[0] > -1x1[0]* x1[1])

The set Q is empty.

### (11) 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.IdpDefaultShapeHeuristic@e1f8dfe Constraint Generator: NonInfConstraintGenerator: PathGenerator: MetricPathGenerator: Max Left Steps: 1 Max Right Steps: 1

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_933_0_MAIN_LT(TRUE, x1[1], -1) → 933_0_MAIN_LT(+(x1[1], -1), -1) the following chains were created:
• We consider the chain 933_0_MAIN_LT(x1[0], -1) → COND_933_0_MAIN_LT(>(x1[0], -1), x1[0], -1), COND_933_0_MAIN_LT(TRUE, x1[1], -1) → 933_0_MAIN_LT(+(x1[1], -1), -1), 933_0_MAIN_LT(x1[0], -1) → COND_933_0_MAIN_LT(>(x1[0], -1), x1[0], -1) which results in the following constraint:

(1)    (>(x1[0], -1)=TRUEx1[0]=x1[1]+(x1[1], -1)=x1[0]1COND_933_0_MAIN_LT(TRUE, x1[1], -1)≥NonInfC∧COND_933_0_MAIN_LT(TRUE, x1[1], -1)≥933_0_MAIN_LT(+(x1[1], -1), -1)∧(UIncreasing(933_0_MAIN_LT(+(x1[1], -1), -1)), ≥))

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

(2)    (>(x1[0], -1)=TRUECOND_933_0_MAIN_LT(TRUE, x1[0], -1)≥NonInfC∧COND_933_0_MAIN_LT(TRUE, x1[0], -1)≥933_0_MAIN_LT(+(x1[0], -1), -1)∧(UIncreasing(933_0_MAIN_LT(+(x1[1], -1), -1)), ≥))

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

(3)    (x1[0] ≥ 0 ⇒ (UIncreasing(933_0_MAIN_LT(+(x1[1], -1), -1)), ≥)∧[(-1)Bound*bni_11] + [bni_11]x1[0] ≥ 0∧[1 + (-1)bso_12] ≥ 0)

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

(4)    (x1[0] ≥ 0 ⇒ (UIncreasing(933_0_MAIN_LT(+(x1[1], -1), -1)), ≥)∧[(-1)Bound*bni_11] + [bni_11]x1[0] ≥ 0∧[1 + (-1)bso_12] ≥ 0)

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

(5)    (x1[0] ≥ 0 ⇒ (UIncreasing(933_0_MAIN_LT(+(x1[1], -1), -1)), ≥)∧[(-1)Bound*bni_11] + [bni_11]x1[0] ≥ 0∧[1 + (-1)bso_12] ≥ 0)

For Pair 933_0_MAIN_LT(x1[0], -1) → COND_933_0_MAIN_LT(>(x1[0], -1), x1[0], -1) the following chains were created:
• We consider the chain 933_0_MAIN_LT(x1[0], -1) → COND_933_0_MAIN_LT(>(x1[0], -1), x1[0], -1), COND_933_0_MAIN_LT(TRUE, x1[1], -1) → 933_0_MAIN_LT(+(x1[1], -1), -1) which results in the following constraint:

(6)    (>(x1[0], -1)=TRUEx1[0]=x1[1]933_0_MAIN_LT(x1[0], -1)≥NonInfC∧933_0_MAIN_LT(x1[0], -1)≥COND_933_0_MAIN_LT(>(x1[0], -1), x1[0], -1)∧(UIncreasing(COND_933_0_MAIN_LT(>(x1[0], -1), x1[0], -1)), ≥))

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

(7)    (>(x1[0], -1)=TRUE933_0_MAIN_LT(x1[0], -1)≥NonInfC∧933_0_MAIN_LT(x1[0], -1)≥COND_933_0_MAIN_LT(>(x1[0], -1), x1[0], -1)∧(UIncreasing(COND_933_0_MAIN_LT(>(x1[0], -1), x1[0], -1)), ≥))

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

(8)    (x1[0] ≥ 0 ⇒ (UIncreasing(COND_933_0_MAIN_LT(>(x1[0], -1), x1[0], -1)), ≥)∧[(-1)Bound*bni_13] + [bni_13]x1[0] ≥ 0∧[(-1)bso_14] ≥ 0)

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

(9)    (x1[0] ≥ 0 ⇒ (UIncreasing(COND_933_0_MAIN_LT(>(x1[0], -1), x1[0], -1)), ≥)∧[(-1)Bound*bni_13] + [bni_13]x1[0] ≥ 0∧[(-1)bso_14] ≥ 0)

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

(10)    (x1[0] ≥ 0 ⇒ (UIncreasing(COND_933_0_MAIN_LT(>(x1[0], -1), x1[0], -1)), ≥)∧[(-1)Bound*bni_13] + [bni_13]x1[0] ≥ 0∧[(-1)bso_14] ≥ 0)

To summarize, we get the following constraints P for the following pairs.
• COND_933_0_MAIN_LT(TRUE, x1[1], -1) → 933_0_MAIN_LT(+(x1[1], -1), -1)
• (x1[0] ≥ 0 ⇒ (UIncreasing(933_0_MAIN_LT(+(x1[1], -1), -1)), ≥)∧[(-1)Bound*bni_11] + [bni_11]x1[0] ≥ 0∧[1 + (-1)bso_12] ≥ 0)

• 933_0_MAIN_LT(x1[0], -1) → COND_933_0_MAIN_LT(>(x1[0], -1), x1[0], -1)
• (x1[0] ≥ 0 ⇒ (UIncreasing(COND_933_0_MAIN_LT(>(x1[0], -1), x1[0], -1)), ≥)∧[(-1)Bound*bni_13] + [bni_13]x1[0] ≥ 0∧[(-1)bso_14] ≥ 0)

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

POL(TRUE) = 0
POL(FALSE) = 0
POL(COND_933_0_MAIN_LT(x1, x2, x3)) = [-1] + [-1]x3 + x2
POL(-1) = [-1]
POL(933_0_MAIN_LT(x1, x2)) = [-1] + [-1]x2 + x1
POL(+(x1, x2)) = x1 + x2
POL(>(x1, x2)) = [-1]

The following pairs are in P>:

COND_933_0_MAIN_LT(TRUE, x1[1], -1) → 933_0_MAIN_LT(+(x1[1], -1), -1)

The following pairs are in Pbound:

COND_933_0_MAIN_LT(TRUE, x1[1], -1) → 933_0_MAIN_LT(+(x1[1], -1), -1)
933_0_MAIN_LT(x1[0], -1) → COND_933_0_MAIN_LT(>(x1[0], -1), x1[0], -1)

The following pairs are in P:

933_0_MAIN_LT(x1[0], -1) → COND_933_0_MAIN_LT(>(x1[0], -1), x1[0], -1)

There are no usable rules.

### (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:
(0): 933_0_MAIN_LT(x1[0], -1) → COND_933_0_MAIN_LT(x1[0] > -1, x1[0], -1)

The set Q is empty.

### (13) IDependencyGraphProof (EQUIVALENT transformation)

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