### (0) Obligation:

JBC Problem based on JBC Program:
Manifest-Version: 1.0 Created-By: 1.6.0_16 (Sun Microsystems Inc.) Main-Class: PastaB10
`/** * 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 PastaB10 {    public static void main(String[] args) {        Random.args = args;        int x = Random.random();        int y = Random.random();        while (x + y > 0) {            if (x > 0) {                x--;            } else if (y > 0) {                y--;            } else {                continue;            }                    }    }}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:
PastaB10.main([Ljava/lang/String;)V: Graph of 187 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: PastaB10.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 22 rules for P and 0 rules for R.

P rules:
304_0_main_Load(EOS(STATIC_304), i18, i47, i18) → 306_0_main_IntArithmetic(EOS(STATIC_306), i18, i47, i18, i47)
306_0_main_IntArithmetic(EOS(STATIC_306), i18, i47, i18, i47) → 319_0_main_LE(EOS(STATIC_319), i18, i47, +(i18, i47)) | &&(>=(i18, 0), >=(i47, 0))
319_0_main_LE(EOS(STATIC_319), i18, i47, i56) → 328_0_main_LE(EOS(STATIC_328), i18, i47, i56)
328_0_main_LE(EOS(STATIC_328), i18, i47, i56) → 336_0_main_Load(EOS(STATIC_336), i18, i47) | >(i56, 0)
336_0_main_Load(EOS(STATIC_336), i18, i47) → 347_0_main_LE(EOS(STATIC_347), i18, i47, i18)
347_0_main_LE(EOS(STATIC_347), matching1, i47, matching2) → 356_0_main_LE(EOS(STATIC_356), 0, i47, 0) | &&(=(matching1, 0), =(matching2, 0))
347_0_main_LE(EOS(STATIC_347), i60, i47, i60) → 358_0_main_LE(EOS(STATIC_358), i60, i47, i60)
356_0_main_LE(EOS(STATIC_356), matching1, i47, matching2) → 367_0_main_Load(EOS(STATIC_367), 0, i47) | &&(&&(<=(0, 0), =(matching1, 0)), =(matching2, 0))
367_0_main_Load(EOS(STATIC_367), matching1, i47) → 378_0_main_LE(EOS(STATIC_378), 0, i47, i47) | =(matching1, 0)
378_0_main_LE(EOS(STATIC_378), matching1, matching2, matching3) → 393_0_main_LE(EOS(STATIC_393), 0, 0, 0) | &&(&&(=(matching1, 0), =(matching2, 0)), =(matching3, 0))
378_0_main_LE(EOS(STATIC_378), matching1, i67, i67) → 394_0_main_LE(EOS(STATIC_394), 0, i67, i67) | =(matching1, 0)
393_0_main_LE(EOS(STATIC_393), matching1, matching2, matching3) → 428_0_main_Load(EOS(STATIC_428), 0, 0) | &&(&&(&&(<=(0, 0), =(matching1, 0)), =(matching2, 0)), =(matching3, 0))
428_0_main_Load(EOS(STATIC_428), matching1, matching2) → 297_0_main_Load(EOS(STATIC_297), 0, 0) | &&(=(matching1, 0), =(matching2, 0))
394_0_main_LE(EOS(STATIC_394), matching1, i67, i67) → 430_0_main_Inc(EOS(STATIC_430), 0, i67) | &&(>(i67, 0), =(matching1, 0))
430_0_main_Inc(EOS(STATIC_430), matching1, i67) → 993_0_main_JMP(EOS(STATIC_993), 0, +(i67, -1)) | &&(>(i67, 0), =(matching1, 0))
993_0_main_JMP(EOS(STATIC_993), matching1, i218) → 998_0_main_Load(EOS(STATIC_998), 0, i218) | =(matching1, 0)
358_0_main_LE(EOS(STATIC_358), i60, i47, i60) → 369_0_main_Inc(EOS(STATIC_369), i60, i47) | >(i60, 0)
369_0_main_Inc(EOS(STATIC_369), i60, i47) → 380_0_main_JMP(EOS(STATIC_380), +(i60, -1), i47) | >(i60, 0)
380_0_main_JMP(EOS(STATIC_380), i64, i47) → 415_0_main_Load(EOS(STATIC_415), i64, i47)
R rules:

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

P rules:
304_0_main_Load(EOS(STATIC_304), 0, x1, 0) → 304_0_main_Load(EOS(STATIC_304), 0, +(x1, -1), 0) | >(x1, 0)
304_0_main_Load(EOS(STATIC_304), x0, x1, x0) → 304_0_main_Load(EOS(STATIC_304), +(x0, -1), x1, +(x0, -1)) | &&(&&(>(+(x1, 1), 0), >(x0, 0)), <(0, +(x0, x1)))
R rules:

Filtered ground terms:

EOS(x1) → EOS

Filtered duplicate args:

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

P rules:
304_0_main_Load(x1, x0) → 304_0_main_Load(x1, +(x0, -1)) | &&(&&(>(x1, -1), >(x0, 0)), <(0, +(x0, x1)))
R rules:

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

P rules:
304_0_MAIN_LOAD(x1, x0) → COND_304_0_MAIN_LOAD1(&&(&&(>(x1, -1), >(x0, 0)), <(0, +(x0, x1))), x1, x0)
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:
(2): 304_0_MAIN_LOAD(x1[2], x0[2]) → COND_304_0_MAIN_LOAD1(x1[2] > -1 && x0[2] > 0 && 0 < x0[2] + x1[2], x1[2], x0[2])

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

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

(1) -> (2), if (x1[1] + -1* x1[2]0* x0[2])

(2) -> (3), if (x1[2] > -1 && x0[2] > 0 && 0 < x0[2] + x1[2]x1[2]* x1[3]x0[2]* x0[3])

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

(3) -> (2), if (x1[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.IdpCand1ShapeHeuristic@30250179 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 304_0_MAIN_LOAD(x1, 0) → COND_304_0_MAIN_LOAD(>(x1, 0), x1, 0) the following chains were created:
• We consider the chain 304_0_MAIN_LOAD(x1[0], 0) → COND_304_0_MAIN_LOAD(>(x1[0], 0), x1[0], 0), COND_304_0_MAIN_LOAD(TRUE, x1[1], 0) → 304_0_MAIN_LOAD(+(x1[1], -1), 0) which results in the following constraint:

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

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

(3)    (x1[0] + [-1] ≥ 0 ⇒ (UIncreasing(COND_304_0_MAIN_LOAD(>(x1[0], 0), x1[0], 0)), ≥)∧[bni_12 + (-1)Bound*bni_12] + [bni_12]x1[0] ≥ 0∧[(-1)bso_13] ≥ 0)

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

(4)    (x1[0] + [-1] ≥ 0 ⇒ (UIncreasing(COND_304_0_MAIN_LOAD(>(x1[0], 0), x1[0], 0)), ≥)∧[bni_12 + (-1)Bound*bni_12] + [bni_12]x1[0] ≥ 0∧[(-1)bso_13] ≥ 0)

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

(5)    (x1[0] + [-1] ≥ 0 ⇒ (UIncreasing(COND_304_0_MAIN_LOAD(>(x1[0], 0), x1[0], 0)), ≥)∧[bni_12 + (-1)Bound*bni_12] + [bni_12]x1[0] ≥ 0∧[(-1)bso_13] ≥ 0)

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

(6)    (x1[0] ≥ 0 ⇒ (UIncreasing(COND_304_0_MAIN_LOAD(>(x1[0], 0), x1[0], 0)), ≥)∧[(2)bni_12 + (-1)Bound*bni_12] + [bni_12]x1[0] ≥ 0∧[(-1)bso_13] ≥ 0)

For Pair COND_304_0_MAIN_LOAD(TRUE, x1, 0) → 304_0_MAIN_LOAD(+(x1, -1), 0) the following chains were created:
• We consider the chain COND_304_0_MAIN_LOAD(TRUE, x1[1], 0) → 304_0_MAIN_LOAD(+(x1[1], -1), 0) which results in the following constraint:

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

(8)    ((UIncreasing(304_0_MAIN_LOAD(+(x1[1], -1), 0)), ≥)∧[bni_14] = 0∧[1 + (-1)bso_15] ≥ 0)

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

(9)    ((UIncreasing(304_0_MAIN_LOAD(+(x1[1], -1), 0)), ≥)∧[bni_14] = 0∧[1 + (-1)bso_15] ≥ 0)

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

(10)    ((UIncreasing(304_0_MAIN_LOAD(+(x1[1], -1), 0)), ≥)∧[bni_14] = 0∧[1 + (-1)bso_15] ≥ 0)

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

(11)    ((UIncreasing(304_0_MAIN_LOAD(+(x1[1], -1), 0)), ≥)∧[bni_14] = 0∧0 = 0∧[1 + (-1)bso_15] ≥ 0)

For Pair 304_0_MAIN_LOAD(x1, x0) → COND_304_0_MAIN_LOAD1(&&(&&(>(x1, -1), >(x0, 0)), <(0, +(x0, x1))), x1, x0) the following chains were created:
• We consider the chain 304_0_MAIN_LOAD(x1[2], x0[2]) → COND_304_0_MAIN_LOAD1(&&(&&(>(x1[2], -1), >(x0[2], 0)), <(0, +(x0[2], x1[2]))), x1[2], x0[2]), COND_304_0_MAIN_LOAD1(TRUE, x1[3], x0[3]) → 304_0_MAIN_LOAD(x1[3], +(x0[3], -1)) which results in the following constraint:

(12)    (&&(&&(>(x1[2], -1), >(x0[2], 0)), <(0, +(x0[2], x1[2])))=TRUEx1[2]=x1[3]x0[2]=x0[3]304_0_MAIN_LOAD(x1[2], x0[2])≥NonInfC∧304_0_MAIN_LOAD(x1[2], x0[2])≥COND_304_0_MAIN_LOAD1(&&(&&(>(x1[2], -1), >(x0[2], 0)), <(0, +(x0[2], x1[2]))), x1[2], x0[2])∧(UIncreasing(COND_304_0_MAIN_LOAD1(&&(&&(>(x1[2], -1), >(x0[2], 0)), <(0, +(x0[2], x1[2]))), x1[2], x0[2])), ≥))

We simplified constraint (12) using rules (IV), (IDP_BOOLEAN) which results in the following new constraint:

(13)    (<(0, +(x0[2], x1[2]))=TRUE>(x1[2], -1)=TRUE>(x0[2], 0)=TRUE304_0_MAIN_LOAD(x1[2], x0[2])≥NonInfC∧304_0_MAIN_LOAD(x1[2], x0[2])≥COND_304_0_MAIN_LOAD1(&&(&&(>(x1[2], -1), >(x0[2], 0)), <(0, +(x0[2], x1[2]))), x1[2], x0[2])∧(UIncreasing(COND_304_0_MAIN_LOAD1(&&(&&(>(x1[2], -1), >(x0[2], 0)), <(0, +(x0[2], x1[2]))), x1[2], x0[2])), ≥))

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

(14)    (x0[2] + [-1] + x1[2] ≥ 0∧x1[2] ≥ 0∧x0[2] + [-1] ≥ 0 ⇒ (UIncreasing(COND_304_0_MAIN_LOAD1(&&(&&(>(x1[2], -1), >(x0[2], 0)), <(0, +(x0[2], x1[2]))), x1[2], x0[2])), ≥)∧[bni_16 + (-1)Bound*bni_16] + [bni_16]x0[2] + [bni_16]x1[2] ≥ 0∧[(-1)bso_17] ≥ 0)

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

(15)    (x0[2] + [-1] + x1[2] ≥ 0∧x1[2] ≥ 0∧x0[2] + [-1] ≥ 0 ⇒ (UIncreasing(COND_304_0_MAIN_LOAD1(&&(&&(>(x1[2], -1), >(x0[2], 0)), <(0, +(x0[2], x1[2]))), x1[2], x0[2])), ≥)∧[bni_16 + (-1)Bound*bni_16] + [bni_16]x0[2] + [bni_16]x1[2] ≥ 0∧[(-1)bso_17] ≥ 0)

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

(16)    (x0[2] + [-1] + x1[2] ≥ 0∧x1[2] ≥ 0∧x0[2] + [-1] ≥ 0 ⇒ (UIncreasing(COND_304_0_MAIN_LOAD1(&&(&&(>(x1[2], -1), >(x0[2], 0)), <(0, +(x0[2], x1[2]))), x1[2], x0[2])), ≥)∧[bni_16 + (-1)Bound*bni_16] + [bni_16]x0[2] + [bni_16]x1[2] ≥ 0∧[(-1)bso_17] ≥ 0)

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

(17)    (x0[2] + x1[2] ≥ 0∧x1[2] ≥ 0∧x0[2] ≥ 0 ⇒ (UIncreasing(COND_304_0_MAIN_LOAD1(&&(&&(>(x1[2], -1), >(x0[2], 0)), <(0, +(x0[2], x1[2]))), x1[2], x0[2])), ≥)∧[(2)bni_16 + (-1)Bound*bni_16] + [bni_16]x0[2] + [bni_16]x1[2] ≥ 0∧[(-1)bso_17] ≥ 0)

For Pair COND_304_0_MAIN_LOAD1(TRUE, x1, x0) → 304_0_MAIN_LOAD(x1, +(x0, -1)) the following chains were created:
• We consider the chain COND_304_0_MAIN_LOAD1(TRUE, x1[3], x0[3]) → 304_0_MAIN_LOAD(x1[3], +(x0[3], -1)) which results in the following constraint:

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

(19)    ((UIncreasing(304_0_MAIN_LOAD(x1[3], +(x0[3], -1))), ≥)∧[bni_18] = 0∧[1 + (-1)bso_19] ≥ 0)

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

(20)    ((UIncreasing(304_0_MAIN_LOAD(x1[3], +(x0[3], -1))), ≥)∧[bni_18] = 0∧[1 + (-1)bso_19] ≥ 0)

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

(21)    ((UIncreasing(304_0_MAIN_LOAD(x1[3], +(x0[3], -1))), ≥)∧[bni_18] = 0∧[1 + (-1)bso_19] ≥ 0)

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

(22)    ((UIncreasing(304_0_MAIN_LOAD(x1[3], +(x0[3], -1))), ≥)∧[bni_18] = 0∧0 = 0∧0 = 0∧[1 + (-1)bso_19] ≥ 0)

To summarize, we get the following constraints P for the following pairs.
• (x1[0] ≥ 0 ⇒ (UIncreasing(COND_304_0_MAIN_LOAD(>(x1[0], 0), x1[0], 0)), ≥)∧[(2)bni_12 + (-1)Bound*bni_12] + [bni_12]x1[0] ≥ 0∧[(-1)bso_13] ≥ 0)

• ((UIncreasing(304_0_MAIN_LOAD(+(x1[1], -1), 0)), ≥)∧[bni_14] = 0∧0 = 0∧[1 + (-1)bso_15] ≥ 0)

• 304_0_MAIN_LOAD(x1, x0) → COND_304_0_MAIN_LOAD1(&&(&&(>(x1, -1), >(x0, 0)), <(0, +(x0, x1))), x1, x0)
• (x0[2] + x1[2] ≥ 0∧x1[2] ≥ 0∧x0[2] ≥ 0 ⇒ (UIncreasing(COND_304_0_MAIN_LOAD1(&&(&&(>(x1[2], -1), >(x0[2], 0)), <(0, +(x0[2], x1[2]))), x1[2], x0[2])), ≥)∧[(2)bni_16 + (-1)Bound*bni_16] + [bni_16]x0[2] + [bni_16]x1[2] ≥ 0∧[(-1)bso_17] ≥ 0)

• ((UIncreasing(304_0_MAIN_LOAD(x1[3], +(x0[3], -1))), ≥)∧[bni_18] = 0∧0 = 0∧0 = 0∧[1 + (-1)bso_19] ≥ 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(304_0_MAIN_LOAD(x1, x2)) = [1] + x2 + x1
POL(0) = 0
POL(COND_304_0_MAIN_LOAD(x1, x2, x3)) = [1] + x2
POL(>(x1, x2)) = [-1]
POL(+(x1, x2)) = x1 + x2
POL(-1) = [-1]
POL(COND_304_0_MAIN_LOAD1(x1, x2, x3)) = [1] + x3 + x2
POL(&&(x1, x2)) = [-1]
POL(<(x1, x2)) = [-1]

The following pairs are in P>:

The following pairs are in Pbound:

304_0_MAIN_LOAD(x1[2], x0[2]) → COND_304_0_MAIN_LOAD1(&&(&&(>(x1[2], -1), >(x0[2], 0)), <(0, +(x0[2], x1[2]))), x1[2], x0[2])

The following pairs are in P:

304_0_MAIN_LOAD(x1[2], x0[2]) → COND_304_0_MAIN_LOAD1(&&(&&(>(x1[2], -1), >(x0[2], 0)), <(0, +(x0[2], x1[2]))), x1[2], x0[2])

There are no usable rules.

### (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:

Integer, Boolean

R is empty.

The integer pair graph contains the following rules and edges:
(2): 304_0_MAIN_LOAD(x1[2], x0[2]) → COND_304_0_MAIN_LOAD1(x1[2] > -1 && x0[2] > 0 && 0 < x0[2] + x1[2], x1[2], x0[2])

The set Q is empty.

### (10) IDependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 2 less nodes.

### (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: