### (0) Obligation:

JBC Problem based on JBC Program:
Manifest-Version: 1.0 Created-By: 1.6.0_16 (Sun Microsystems Inc.) Main-Class: PastaB17
`/** * 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 PastaB17 {    public static void main(String[] args) {        Random.args = args;        int x = Random.random();        int y = Random.random();        int z = Random.random();        while (x > z) {            while (y > z) {                y--;            }            x--;        }    }}public class Random {  static String[] args;  static int index = 0;  public static int random() {    String string = args[index];    index++;    return string.length();  }}`

### (1) JBCToGraph (SOUND transformation)

Constructed TerminationGraph.

### (2) Obligation:

Termination Graph based on JBC Program:
PastaB17.main([Ljava/lang/String;)V: Graph of 255 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: PastaB17.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 16 rules for P and 0 rules for R.

P rules:
1299_0_main_Load(EOS(STATIC_1299), i576, i577, i87, i576) → 1300_0_main_LE(EOS(STATIC_1300), i576, i577, i87, i576, i87)
1300_0_main_LE(EOS(STATIC_1300), i576, i577, i87, i576, i87) → 1303_0_main_LE(EOS(STATIC_1303), i576, i577, i87, i576, i87)
1303_0_main_LE(EOS(STATIC_1303), i576, i577, i87, i576, i87) → 1305_0_main_Load(EOS(STATIC_1305), i576, i577, i87) | >(i576, i87)
1308_0_main_Load(EOS(STATIC_1308), i576, i577, i87, i577) → 1310_0_main_LE(EOS(STATIC_1310), i576, i577, i87, i577, i87)
1310_0_main_LE(EOS(STATIC_1310), i576, i577, i87, i577, i87) → 1311_0_main_LE(EOS(STATIC_1311), i576, i577, i87, i577, i87)
1310_0_main_LE(EOS(STATIC_1310), i576, i577, i87, i577, i87) → 1312_0_main_LE(EOS(STATIC_1312), i576, i577, i87, i577, i87)
1311_0_main_LE(EOS(STATIC_1311), i576, i577, i87, i577, i87) → 1313_0_main_Inc(EOS(STATIC_1313), i576, i577, i87) | <=(i577, i87)
1313_0_main_Inc(EOS(STATIC_1313), i576, i577, i87) → 1316_0_main_JMP(EOS(STATIC_1316), +(i576, -1), i577, i87)
1316_0_main_JMP(EOS(STATIC_1316), i581, i577, i87) → 1320_0_main_Load(EOS(STATIC_1320), i581, i577, i87)
1312_0_main_LE(EOS(STATIC_1312), i576, i577, i87, i577, i87) → 1315_0_main_Inc(EOS(STATIC_1315), i576, i577, i87) | >(i577, i87)
1315_0_main_Inc(EOS(STATIC_1315), i576, i577, i87) → 1318_0_main_JMP(EOS(STATIC_1318), i576, +(i577, -1), i87)
1318_0_main_JMP(EOS(STATIC_1318), i576, i582, i87) → 1322_0_main_Load(EOS(STATIC_1322), i576, i582, i87)
R rules:

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

P rules:
1310_0_main_LE(EOS(STATIC_1310), x0, x1, x2, x1, x2) → 1310_0_main_LE(EOS(STATIC_1310), +(x0, -1), x1, x2, x1, x2) | &&(>=(x2, x1), <(x2, +(x0, -1)))
1310_0_main_LE(EOS(STATIC_1310), x0, x1, x2, x1, x2) → 1310_0_main_LE(EOS(STATIC_1310), x0, +(x1, -1), x2, +(x1, -1), x2) | <(x2, x1)
R rules:

Filtered ground terms:

1310_0_main_LE(x1, x2, x3, x4, x5, x6) → 1310_0_main_LE(x2, x3, x4, x5, x6)
EOS(x1) → EOS
Cond_1310_0_main_LE1(x1, x2, x3, x4, x5, x6, x7) → Cond_1310_0_main_LE1(x1, x3, x4, x5, x6, x7)
Cond_1310_0_main_LE(x1, x2, x3, x4, x5, x6, x7) → Cond_1310_0_main_LE(x1, x3, x4, x5, x6, x7)

Filtered duplicate args:

1310_0_main_LE(x1, x2, x3, x4, x5) → 1310_0_main_LE(x1, x4, x5)
Cond_1310_0_main_LE(x1, x2, x3, x4, x5, x6) → Cond_1310_0_main_LE(x1, x2, x5, x6)
Cond_1310_0_main_LE1(x1, x2, x3, x4, x5, x6) → Cond_1310_0_main_LE1(x1, x2, x5, x6)

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

P rules:
1310_0_main_LE(x0, x1, x2) → 1310_0_main_LE(+(x0, -1), x1, x2) | &&(>=(x2, x1), <(x2, +(x0, -1)))
1310_0_main_LE(x0, x1, x2) → 1310_0_main_LE(x0, +(x1, -1), x2) | <(x2, x1)
R rules:

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

P rules:
1310_0_MAIN_LE(x0, x1, x2) → COND_1310_0_MAIN_LE(&&(>=(x2, x1), <(x2, +(x0, -1))), x0, x1, x2)
COND_1310_0_MAIN_LE(TRUE, x0, x1, x2) → 1310_0_MAIN_LE(+(x0, -1), x1, x2)
1310_0_MAIN_LE(x0, x1, x2) → COND_1310_0_MAIN_LE1(<(x2, x1), x0, x1, x2)
COND_1310_0_MAIN_LE1(TRUE, x0, x1, x2) → 1310_0_MAIN_LE(x0, +(x1, -1), x2)
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): 1310_0_MAIN_LE(x0[0], x1[0], x2[0]) → COND_1310_0_MAIN_LE(x2[0] >= x1[0] && x2[0] < x0[0] + -1, x0[0], x1[0], x2[0])
(1): COND_1310_0_MAIN_LE(TRUE, x0[1], x1[1], x2[1]) → 1310_0_MAIN_LE(x0[1] + -1, x1[1], x2[1])
(2): 1310_0_MAIN_LE(x0[2], x1[2], x2[2]) → COND_1310_0_MAIN_LE1(x2[2] < x1[2], x0[2], x1[2], x2[2])
(3): COND_1310_0_MAIN_LE1(TRUE, x0[3], x1[3], x2[3]) → 1310_0_MAIN_LE(x0[3], x1[3] + -1, x2[3])

(0) -> (1), if (x2[0] >= x1[0] && x2[0] < x0[0] + -1x0[0]* x0[1]x1[0]* x1[1]x2[0]* x2[1])

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

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

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

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

(3) -> (2), if (x0[3]* x0[2]x1[3] + -1* x1[2]x2[3]* x2[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@5176f87c 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 1310_0_MAIN_LE(x0, x1, x2) → COND_1310_0_MAIN_LE(&&(>=(x2, x1), <(x2, +(x0, -1))), x0, x1, x2) the following chains were created:
• We consider the chain 1310_0_MAIN_LE(x0[0], x1[0], x2[0]) → COND_1310_0_MAIN_LE(&&(>=(x2[0], x1[0]), <(x2[0], +(x0[0], -1))), x0[0], x1[0], x2[0]), COND_1310_0_MAIN_LE(TRUE, x0[1], x1[1], x2[1]) → 1310_0_MAIN_LE(+(x0[1], -1), x1[1], x2[1]) which results in the following constraint:

(1)    (&&(>=(x2[0], x1[0]), <(x2[0], +(x0[0], -1)))=TRUEx0[0]=x0[1]x1[0]=x1[1]x2[0]=x2[1]1310_0_MAIN_LE(x0[0], x1[0], x2[0])≥NonInfC∧1310_0_MAIN_LE(x0[0], x1[0], x2[0])≥COND_1310_0_MAIN_LE(&&(>=(x2[0], x1[0]), <(x2[0], +(x0[0], -1))), x0[0], x1[0], x2[0])∧(UIncreasing(COND_1310_0_MAIN_LE(&&(>=(x2[0], x1[0]), <(x2[0], +(x0[0], -1))), x0[0], x1[0], x2[0])), ≥))

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

(2)    (>=(x2[0], x1[0])=TRUE<(x2[0], +(x0[0], -1))=TRUE1310_0_MAIN_LE(x0[0], x1[0], x2[0])≥NonInfC∧1310_0_MAIN_LE(x0[0], x1[0], x2[0])≥COND_1310_0_MAIN_LE(&&(>=(x2[0], x1[0]), <(x2[0], +(x0[0], -1))), x0[0], x1[0], x2[0])∧(UIncreasing(COND_1310_0_MAIN_LE(&&(>=(x2[0], x1[0]), <(x2[0], +(x0[0], -1))), x0[0], x1[0], x2[0])), ≥))

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

(3)    (x2[0] + [-1]x1[0] ≥ 0∧x0[0] + [-2] + [-1]x2[0] ≥ 0 ⇒ (UIncreasing(COND_1310_0_MAIN_LE(&&(>=(x2[0], x1[0]), <(x2[0], +(x0[0], -1))), x0[0], x1[0], x2[0])), ≥)∧[(-1)bni_12 + (-1)Bound*bni_12] + [(-1)bni_12]x2[0] + [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)    (x2[0] + [-1]x1[0] ≥ 0∧x0[0] + [-2] + [-1]x2[0] ≥ 0 ⇒ (UIncreasing(COND_1310_0_MAIN_LE(&&(>=(x2[0], x1[0]), <(x2[0], +(x0[0], -1))), x0[0], x1[0], x2[0])), ≥)∧[(-1)bni_12 + (-1)Bound*bni_12] + [(-1)bni_12]x2[0] + [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)    (x2[0] + [-1]x1[0] ≥ 0∧x0[0] + [-2] + [-1]x2[0] ≥ 0 ⇒ (UIncreasing(COND_1310_0_MAIN_LE(&&(>=(x2[0], x1[0]), <(x2[0], +(x0[0], -1))), x0[0], x1[0], x2[0])), ≥)∧[(-1)bni_12 + (-1)Bound*bni_12] + [(-1)bni_12]x2[0] + [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)    (x2[0] ≥ 0∧x0[0] + [-2] + [-1]x1[0] + [-1]x2[0] ≥ 0 ⇒ (UIncreasing(COND_1310_0_MAIN_LE(&&(>=(x2[0], x1[0]), <(x2[0], +(x0[0], -1))), x0[0], x1[0], x2[0])), ≥)∧[(-1)bni_12 + (-1)Bound*bni_12] + [(-1)bni_12]x2[0] ≥ 0∧[(-1)bso_13] ≥ 0)

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

(7)    (x2[0] ≥ 0∧x1[0] ≥ 0 ⇒ (UIncreasing(COND_1310_0_MAIN_LE(&&(>=(x2[0], x1[0]), <(x2[0], +(x0[0], -1))), x0[0], x1[0], x2[0])), ≥)∧[(-1)bni_12 + (-1)Bound*bni_12] + [(-1)bni_12]x2[0] ≥ 0∧[(-1)bso_13] ≥ 0)

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

(8)    (x2[0] ≥ 0∧x1[0] ≥ 0∧x0[0] ≥ 0 ⇒ (UIncreasing(COND_1310_0_MAIN_LE(&&(>=(x2[0], x1[0]), <(x2[0], +(x0[0], -1))), x0[0], x1[0], x2[0])), ≥)∧[(-1)bni_12 + (-1)Bound*bni_12] + [(-1)bni_12]x2[0] ≥ 0∧[(-1)bso_13] ≥ 0)

(9)    (x2[0] ≥ 0∧x1[0] ≥ 0∧x0[0] ≥ 0 ⇒ (UIncreasing(COND_1310_0_MAIN_LE(&&(>=(x2[0], x1[0]), <(x2[0], +(x0[0], -1))), x0[0], x1[0], x2[0])), ≥)∧[(-1)bni_12 + (-1)Bound*bni_12] + [(-1)bni_12]x2[0] ≥ 0∧[(-1)bso_13] ≥ 0)

For Pair COND_1310_0_MAIN_LE(TRUE, x0, x1, x2) → 1310_0_MAIN_LE(+(x0, -1), x1, x2) the following chains were created:
• We consider the chain COND_1310_0_MAIN_LE(TRUE, x0[1], x1[1], x2[1]) → 1310_0_MAIN_LE(+(x0[1], -1), x1[1], x2[1]) which results in the following constraint:

(10)    (COND_1310_0_MAIN_LE(TRUE, x0[1], x1[1], x2[1])≥NonInfC∧COND_1310_0_MAIN_LE(TRUE, x0[1], x1[1], x2[1])≥1310_0_MAIN_LE(+(x0[1], -1), x1[1], x2[1])∧(UIncreasing(1310_0_MAIN_LE(+(x0[1], -1), x1[1], x2[1])), ≥))

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

(11)    ((UIncreasing(1310_0_MAIN_LE(+(x0[1], -1), x1[1], x2[1])), ≥)∧[bni_14] = 0∧[(-1)bso_15] ≥ 0)

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

(12)    ((UIncreasing(1310_0_MAIN_LE(+(x0[1], -1), x1[1], x2[1])), ≥)∧[bni_14] = 0∧[(-1)bso_15] ≥ 0)

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

(13)    ((UIncreasing(1310_0_MAIN_LE(+(x0[1], -1), x1[1], x2[1])), ≥)∧[bni_14] = 0∧[(-1)bso_15] ≥ 0)

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

(14)    ((UIncreasing(1310_0_MAIN_LE(+(x0[1], -1), x1[1], x2[1])), ≥)∧[bni_14] = 0∧0 = 0∧0 = 0∧0 = 0∧[(-1)bso_15] ≥ 0)

For Pair 1310_0_MAIN_LE(x0, x1, x2) → COND_1310_0_MAIN_LE1(<(x2, x1), x0, x1, x2) the following chains were created:
• We consider the chain 1310_0_MAIN_LE(x0[2], x1[2], x2[2]) → COND_1310_0_MAIN_LE1(<(x2[2], x1[2]), x0[2], x1[2], x2[2]), COND_1310_0_MAIN_LE1(TRUE, x0[3], x1[3], x2[3]) → 1310_0_MAIN_LE(x0[3], +(x1[3], -1), x2[3]) which results in the following constraint:

(15)    (<(x2[2], x1[2])=TRUEx0[2]=x0[3]x1[2]=x1[3]x2[2]=x2[3]1310_0_MAIN_LE(x0[2], x1[2], x2[2])≥NonInfC∧1310_0_MAIN_LE(x0[2], x1[2], x2[2])≥COND_1310_0_MAIN_LE1(<(x2[2], x1[2]), x0[2], x1[2], x2[2])∧(UIncreasing(COND_1310_0_MAIN_LE1(<(x2[2], x1[2]), x0[2], x1[2], x2[2])), ≥))

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

(16)    (<(x2[2], x1[2])=TRUE1310_0_MAIN_LE(x0[2], x1[2], x2[2])≥NonInfC∧1310_0_MAIN_LE(x0[2], x1[2], x2[2])≥COND_1310_0_MAIN_LE1(<(x2[2], x1[2]), x0[2], x1[2], x2[2])∧(UIncreasing(COND_1310_0_MAIN_LE1(<(x2[2], x1[2]), x0[2], x1[2], x2[2])), ≥))

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

(17)    (x1[2] + [-1] + [-1]x2[2] ≥ 0 ⇒ (UIncreasing(COND_1310_0_MAIN_LE1(<(x2[2], x1[2]), x0[2], x1[2], x2[2])), ≥)∧[(-1)bni_16 + (-1)Bound*bni_16] + [(-1)bni_16]x2[2] + [bni_16]x1[2] ≥ 0∧[(-1)bso_17] ≥ 0)

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

(18)    (x1[2] + [-1] + [-1]x2[2] ≥ 0 ⇒ (UIncreasing(COND_1310_0_MAIN_LE1(<(x2[2], x1[2]), x0[2], x1[2], x2[2])), ≥)∧[(-1)bni_16 + (-1)Bound*bni_16] + [(-1)bni_16]x2[2] + [bni_16]x1[2] ≥ 0∧[(-1)bso_17] ≥ 0)

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

(19)    (x1[2] + [-1] + [-1]x2[2] ≥ 0 ⇒ (UIncreasing(COND_1310_0_MAIN_LE1(<(x2[2], x1[2]), x0[2], x1[2], x2[2])), ≥)∧[(-1)bni_16 + (-1)Bound*bni_16] + [(-1)bni_16]x2[2] + [bni_16]x1[2] ≥ 0∧[(-1)bso_17] ≥ 0)

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

(20)    (x1[2] + [-1] + [-1]x2[2] ≥ 0 ⇒ (UIncreasing(COND_1310_0_MAIN_LE1(<(x2[2], x1[2]), x0[2], x1[2], x2[2])), ≥)∧0 = 0∧[(-1)bni_16 + (-1)Bound*bni_16] + [(-1)bni_16]x2[2] + [bni_16]x1[2] ≥ 0∧0 = 0∧[(-1)bso_17] ≥ 0)

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

(21)    (x1[2] ≥ 0 ⇒ (UIncreasing(COND_1310_0_MAIN_LE1(<(x2[2], x1[2]), x0[2], x1[2], x2[2])), ≥)∧0 = 0∧[(-1)Bound*bni_16] + [bni_16]x1[2] ≥ 0∧0 = 0∧[(-1)bso_17] ≥ 0)

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

(22)    (x1[2] ≥ 0∧x2[2] ≥ 0 ⇒ (UIncreasing(COND_1310_0_MAIN_LE1(<(x2[2], x1[2]), x0[2], x1[2], x2[2])), ≥)∧0 = 0∧[(-1)Bound*bni_16] + [bni_16]x1[2] ≥ 0∧0 = 0∧[(-1)bso_17] ≥ 0)

(23)    (x1[2] ≥ 0∧x2[2] ≥ 0 ⇒ (UIncreasing(COND_1310_0_MAIN_LE1(<(x2[2], x1[2]), x0[2], x1[2], x2[2])), ≥)∧0 = 0∧[(-1)Bound*bni_16] + [bni_16]x1[2] ≥ 0∧0 = 0∧[(-1)bso_17] ≥ 0)

For Pair COND_1310_0_MAIN_LE1(TRUE, x0, x1, x2) → 1310_0_MAIN_LE(x0, +(x1, -1), x2) the following chains were created:
• We consider the chain COND_1310_0_MAIN_LE1(TRUE, x0[3], x1[3], x2[3]) → 1310_0_MAIN_LE(x0[3], +(x1[3], -1), x2[3]) which results in the following constraint:

(24)    (COND_1310_0_MAIN_LE1(TRUE, x0[3], x1[3], x2[3])≥NonInfC∧COND_1310_0_MAIN_LE1(TRUE, x0[3], x1[3], x2[3])≥1310_0_MAIN_LE(x0[3], +(x1[3], -1), x2[3])∧(UIncreasing(1310_0_MAIN_LE(x0[3], +(x1[3], -1), x2[3])), ≥))

We simplified constraint (24) using rule (POLY_CONSTRAINTS) which results in the following new constraint:

(25)    ((UIncreasing(1310_0_MAIN_LE(x0[3], +(x1[3], -1), x2[3])), ≥)∧[bni_18] = 0∧[1 + (-1)bso_19] ≥ 0)

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

(26)    ((UIncreasing(1310_0_MAIN_LE(x0[3], +(x1[3], -1), x2[3])), ≥)∧[bni_18] = 0∧[1 + (-1)bso_19] ≥ 0)

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

(27)    ((UIncreasing(1310_0_MAIN_LE(x0[3], +(x1[3], -1), x2[3])), ≥)∧[bni_18] = 0∧[1 + (-1)bso_19] ≥ 0)

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

(28)    ((UIncreasing(1310_0_MAIN_LE(x0[3], +(x1[3], -1), x2[3])), ≥)∧[bni_18] = 0∧0 = 0∧0 = 0∧0 = 0∧[1 + (-1)bso_19] ≥ 0)

To summarize, we get the following constraints P for the following pairs.
• 1310_0_MAIN_LE(x0, x1, x2) → COND_1310_0_MAIN_LE(&&(>=(x2, x1), <(x2, +(x0, -1))), x0, x1, x2)
• (x2[0] ≥ 0∧x1[0] ≥ 0∧x0[0] ≥ 0 ⇒ (UIncreasing(COND_1310_0_MAIN_LE(&&(>=(x2[0], x1[0]), <(x2[0], +(x0[0], -1))), x0[0], x1[0], x2[0])), ≥)∧[(-1)bni_12 + (-1)Bound*bni_12] + [(-1)bni_12]x2[0] ≥ 0∧[(-1)bso_13] ≥ 0)
• (x2[0] ≥ 0∧x1[0] ≥ 0∧x0[0] ≥ 0 ⇒ (UIncreasing(COND_1310_0_MAIN_LE(&&(>=(x2[0], x1[0]), <(x2[0], +(x0[0], -1))), x0[0], x1[0], x2[0])), ≥)∧[(-1)bni_12 + (-1)Bound*bni_12] + [(-1)bni_12]x2[0] ≥ 0∧[(-1)bso_13] ≥ 0)

• COND_1310_0_MAIN_LE(TRUE, x0, x1, x2) → 1310_0_MAIN_LE(+(x0, -1), x1, x2)
• ((UIncreasing(1310_0_MAIN_LE(+(x0[1], -1), x1[1], x2[1])), ≥)∧[bni_14] = 0∧0 = 0∧0 = 0∧0 = 0∧[(-1)bso_15] ≥ 0)

• 1310_0_MAIN_LE(x0, x1, x2) → COND_1310_0_MAIN_LE1(<(x2, x1), x0, x1, x2)
• (x1[2] ≥ 0∧x2[2] ≥ 0 ⇒ (UIncreasing(COND_1310_0_MAIN_LE1(<(x2[2], x1[2]), x0[2], x1[2], x2[2])), ≥)∧0 = 0∧[(-1)Bound*bni_16] + [bni_16]x1[2] ≥ 0∧0 = 0∧[(-1)bso_17] ≥ 0)
• (x1[2] ≥ 0∧x2[2] ≥ 0 ⇒ (UIncreasing(COND_1310_0_MAIN_LE1(<(x2[2], x1[2]), x0[2], x1[2], x2[2])), ≥)∧0 = 0∧[(-1)Bound*bni_16] + [bni_16]x1[2] ≥ 0∧0 = 0∧[(-1)bso_17] ≥ 0)

• COND_1310_0_MAIN_LE1(TRUE, x0, x1, x2) → 1310_0_MAIN_LE(x0, +(x1, -1), x2)
• ((UIncreasing(1310_0_MAIN_LE(x0[3], +(x1[3], -1), x2[3])), ≥)∧[bni_18] = 0∧0 = 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(1310_0_MAIN_LE(x1, x2, x3)) = [-1] + [-1]x3 + x2
POL(COND_1310_0_MAIN_LE(x1, x2, x3, x4)) = [-1] + [-1]x4 + x3
POL(&&(x1, x2)) = [-1]
POL(>=(x1, x2)) = [-1]
POL(<(x1, x2)) = [-1]
POL(+(x1, x2)) = x1 + x2
POL(-1) = [-1]
POL(COND_1310_0_MAIN_LE1(x1, x2, x3, x4)) = [-1] + [-1]x4 + x3

The following pairs are in P>:

COND_1310_0_MAIN_LE1(TRUE, x0[3], x1[3], x2[3]) → 1310_0_MAIN_LE(x0[3], +(x1[3], -1), x2[3])

The following pairs are in Pbound:

1310_0_MAIN_LE(x0[2], x1[2], x2[2]) → COND_1310_0_MAIN_LE1(<(x2[2], x1[2]), x0[2], x1[2], x2[2])

The following pairs are in P:

1310_0_MAIN_LE(x0[0], x1[0], x2[0]) → COND_1310_0_MAIN_LE(&&(>=(x2[0], x1[0]), <(x2[0], +(x0[0], -1))), x0[0], x1[0], x2[0])
COND_1310_0_MAIN_LE(TRUE, x0[1], x1[1], x2[1]) → 1310_0_MAIN_LE(+(x0[1], -1), x1[1], x2[1])
1310_0_MAIN_LE(x0[2], x1[2], x2[2]) → COND_1310_0_MAIN_LE1(<(x2[2], x1[2]), x0[2], x1[2], x2[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:

Boolean, Integer

R is empty.

The integer pair graph contains the following rules and edges:
(0): 1310_0_MAIN_LE(x0[0], x1[0], x2[0]) → COND_1310_0_MAIN_LE(x2[0] >= x1[0] && x2[0] < x0[0] + -1, x0[0], x1[0], x2[0])
(1): COND_1310_0_MAIN_LE(TRUE, x0[1], x1[1], x2[1]) → 1310_0_MAIN_LE(x0[1] + -1, x1[1], x2[1])
(2): 1310_0_MAIN_LE(x0[2], x1[2], x2[2]) → COND_1310_0_MAIN_LE1(x2[2] < x1[2], x0[2], x1[2], x2[2])

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

(0) -> (1), if (x2[0] >= x1[0] && x2[0] < x0[0] + -1x0[0]* x0[1]x1[0]* x1[1]x2[0]* x2[1])

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

The set Q is empty.

### (10) IDependencyGraphProof (EQUIVALENT transformation)

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

### (11) Obligation:

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

The following domains are used:

Integer, Boolean

R is empty.

The integer pair graph contains the following rules and edges:
(1): COND_1310_0_MAIN_LE(TRUE, x0[1], x1[1], x2[1]) → 1310_0_MAIN_LE(x0[1] + -1, x1[1], x2[1])
(0): 1310_0_MAIN_LE(x0[0], x1[0], x2[0]) → COND_1310_0_MAIN_LE(x2[0] >= x1[0] && x2[0] < x0[0] + -1, x0[0], x1[0], x2[0])

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

(0) -> (1), if (x2[0] >= x1[0] && x2[0] < x0[0] + -1x0[0]* x0[1]x1[0]* x1[1]x2[0]* x2[1])

The set Q is empty.

### (12) 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@5176f87c 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 COND_1310_0_MAIN_LE(TRUE, x0[1], x1[1], x2[1]) → 1310_0_MAIN_LE(+(x0[1], -1), x1[1], x2[1]) the following chains were created:
• We consider the chain COND_1310_0_MAIN_LE(TRUE, x0[1], x1[1], x2[1]) → 1310_0_MAIN_LE(+(x0[1], -1), x1[1], x2[1]) which results in the following constraint:

(1)    (COND_1310_0_MAIN_LE(TRUE, x0[1], x1[1], x2[1])≥NonInfC∧COND_1310_0_MAIN_LE(TRUE, x0[1], x1[1], x2[1])≥1310_0_MAIN_LE(+(x0[1], -1), x1[1], x2[1])∧(UIncreasing(1310_0_MAIN_LE(+(x0[1], -1), x1[1], x2[1])), ≥))

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

(2)    ((UIncreasing(1310_0_MAIN_LE(+(x0[1], -1), x1[1], x2[1])), ≥)∧[bni_11] = 0∧[(-1)bso_12] ≥ 0)

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

(3)    ((UIncreasing(1310_0_MAIN_LE(+(x0[1], -1), x1[1], x2[1])), ≥)∧[bni_11] = 0∧[(-1)bso_12] ≥ 0)

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

(4)    ((UIncreasing(1310_0_MAIN_LE(+(x0[1], -1), x1[1], x2[1])), ≥)∧[bni_11] = 0∧[(-1)bso_12] ≥ 0)

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

(5)    ((UIncreasing(1310_0_MAIN_LE(+(x0[1], -1), x1[1], x2[1])), ≥)∧[bni_11] = 0∧0 = 0∧0 = 0∧0 = 0∧[(-1)bso_12] ≥ 0)

For Pair 1310_0_MAIN_LE(x0[0], x1[0], x2[0]) → COND_1310_0_MAIN_LE(&&(>=(x2[0], x1[0]), <(x2[0], +(x0[0], -1))), x0[0], x1[0], x2[0]) the following chains were created:
• We consider the chain 1310_0_MAIN_LE(x0[0], x1[0], x2[0]) → COND_1310_0_MAIN_LE(&&(>=(x2[0], x1[0]), <(x2[0], +(x0[0], -1))), x0[0], x1[0], x2[0]), COND_1310_0_MAIN_LE(TRUE, x0[1], x1[1], x2[1]) → 1310_0_MAIN_LE(+(x0[1], -1), x1[1], x2[1]) which results in the following constraint:

(6)    (&&(>=(x2[0], x1[0]), <(x2[0], +(x0[0], -1)))=TRUEx0[0]=x0[1]x1[0]=x1[1]x2[0]=x2[1]1310_0_MAIN_LE(x0[0], x1[0], x2[0])≥NonInfC∧1310_0_MAIN_LE(x0[0], x1[0], x2[0])≥COND_1310_0_MAIN_LE(&&(>=(x2[0], x1[0]), <(x2[0], +(x0[0], -1))), x0[0], x1[0], x2[0])∧(UIncreasing(COND_1310_0_MAIN_LE(&&(>=(x2[0], x1[0]), <(x2[0], +(x0[0], -1))), x0[0], x1[0], x2[0])), ≥))

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

(7)    (>=(x2[0], x1[0])=TRUE<(x2[0], +(x0[0], -1))=TRUE1310_0_MAIN_LE(x0[0], x1[0], x2[0])≥NonInfC∧1310_0_MAIN_LE(x0[0], x1[0], x2[0])≥COND_1310_0_MAIN_LE(&&(>=(x2[0], x1[0]), <(x2[0], +(x0[0], -1))), x0[0], x1[0], x2[0])∧(UIncreasing(COND_1310_0_MAIN_LE(&&(>=(x2[0], x1[0]), <(x2[0], +(x0[0], -1))), x0[0], x1[0], x2[0])), ≥))

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

(8)    (x2[0] + [-1]x1[0] ≥ 0∧x0[0] + [-2] + [-1]x2[0] ≥ 0 ⇒ (UIncreasing(COND_1310_0_MAIN_LE(&&(>=(x2[0], x1[0]), <(x2[0], +(x0[0], -1))), x0[0], x1[0], x2[0])), ≥)∧[bni_13 + (-1)Bound*bni_13] + [(2)bni_13]x0[0] + [(-1)bni_13]x2[0] + [(-1)bni_13]x1[0] ≥ 0∧[2 + (-1)bso_14] ≥ 0)

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

(9)    (x2[0] + [-1]x1[0] ≥ 0∧x0[0] + [-2] + [-1]x2[0] ≥ 0 ⇒ (UIncreasing(COND_1310_0_MAIN_LE(&&(>=(x2[0], x1[0]), <(x2[0], +(x0[0], -1))), x0[0], x1[0], x2[0])), ≥)∧[bni_13 + (-1)Bound*bni_13] + [(2)bni_13]x0[0] + [(-1)bni_13]x2[0] + [(-1)bni_13]x1[0] ≥ 0∧[2 + (-1)bso_14] ≥ 0)

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

(10)    (x2[0] + [-1]x1[0] ≥ 0∧x0[0] + [-2] + [-1]x2[0] ≥ 0 ⇒ (UIncreasing(COND_1310_0_MAIN_LE(&&(>=(x2[0], x1[0]), <(x2[0], +(x0[0], -1))), x0[0], x1[0], x2[0])), ≥)∧[bni_13 + (-1)Bound*bni_13] + [(2)bni_13]x0[0] + [(-1)bni_13]x2[0] + [(-1)bni_13]x1[0] ≥ 0∧[2 + (-1)bso_14] ≥ 0)

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

(11)    (x2[0] ≥ 0∧x0[0] + [-2] + [-1]x1[0] + [-1]x2[0] ≥ 0 ⇒ (UIncreasing(COND_1310_0_MAIN_LE(&&(>=(x2[0], x1[0]), <(x2[0], +(x0[0], -1))), x0[0], x1[0], x2[0])), ≥)∧[bni_13 + (-1)Bound*bni_13] + [(2)bni_13]x0[0] + [(-2)bni_13]x1[0] + [(-1)bni_13]x2[0] ≥ 0∧[2 + (-1)bso_14] ≥ 0)

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

(12)    (x2[0] ≥ 0∧x1[0] ≥ 0 ⇒ (UIncreasing(COND_1310_0_MAIN_LE(&&(>=(x2[0], x1[0]), <(x2[0], +(x0[0], -1))), x0[0], x1[0], x2[0])), ≥)∧[(5)bni_13 + (-1)Bound*bni_13] + [bni_13]x2[0] + [(2)bni_13]x1[0] ≥ 0∧[2 + (-1)bso_14] ≥ 0)

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

(13)    (x2[0] ≥ 0∧x1[0] ≥ 0∧x0[0] ≥ 0 ⇒ (UIncreasing(COND_1310_0_MAIN_LE(&&(>=(x2[0], x1[0]), <(x2[0], +(x0[0], -1))), x0[0], x1[0], x2[0])), ≥)∧[(5)bni_13 + (-1)Bound*bni_13] + [bni_13]x2[0] + [(2)bni_13]x1[0] ≥ 0∧[2 + (-1)bso_14] ≥ 0)

(14)    (x2[0] ≥ 0∧x1[0] ≥ 0∧x0[0] ≥ 0 ⇒ (UIncreasing(COND_1310_0_MAIN_LE(&&(>=(x2[0], x1[0]), <(x2[0], +(x0[0], -1))), x0[0], x1[0], x2[0])), ≥)∧[(5)bni_13 + (-1)Bound*bni_13] + [bni_13]x2[0] + [(2)bni_13]x1[0] ≥ 0∧[2 + (-1)bso_14] ≥ 0)

To summarize, we get the following constraints P for the following pairs.
• COND_1310_0_MAIN_LE(TRUE, x0[1], x1[1], x2[1]) → 1310_0_MAIN_LE(+(x0[1], -1), x1[1], x2[1])
• ((UIncreasing(1310_0_MAIN_LE(+(x0[1], -1), x1[1], x2[1])), ≥)∧[bni_11] = 0∧0 = 0∧0 = 0∧0 = 0∧[(-1)bso_12] ≥ 0)

• 1310_0_MAIN_LE(x0[0], x1[0], x2[0]) → COND_1310_0_MAIN_LE(&&(>=(x2[0], x1[0]), <(x2[0], +(x0[0], -1))), x0[0], x1[0], x2[0])
• (x2[0] ≥ 0∧x1[0] ≥ 0∧x0[0] ≥ 0 ⇒ (UIncreasing(COND_1310_0_MAIN_LE(&&(>=(x2[0], x1[0]), <(x2[0], +(x0[0], -1))), x0[0], x1[0], x2[0])), ≥)∧[(5)bni_13 + (-1)Bound*bni_13] + [bni_13]x2[0] + [(2)bni_13]x1[0] ≥ 0∧[2 + (-1)bso_14] ≥ 0)
• (x2[0] ≥ 0∧x1[0] ≥ 0∧x0[0] ≥ 0 ⇒ (UIncreasing(COND_1310_0_MAIN_LE(&&(>=(x2[0], x1[0]), <(x2[0], +(x0[0], -1))), x0[0], x1[0], x2[0])), ≥)∧[(5)bni_13 + (-1)Bound*bni_13] + [bni_13]x2[0] + [(2)bni_13]x1[0] ≥ 0∧[2 + (-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_1310_0_MAIN_LE(x1, x2, x3, x4)) = [-1] + [-1]x4 + [-1]x3 + [2]x2
POL(1310_0_MAIN_LE(x1, x2, x3)) = [1] + [2]x1 + [-1]x3 + [-1]x2
POL(+(x1, x2)) = x1 + x2
POL(-1) = [-1]
POL(&&(x1, x2)) = [-1]
POL(>=(x1, x2)) = [-1]
POL(<(x1, x2)) = [-1]

The following pairs are in P>:

1310_0_MAIN_LE(x0[0], x1[0], x2[0]) → COND_1310_0_MAIN_LE(&&(>=(x2[0], x1[0]), <(x2[0], +(x0[0], -1))), x0[0], x1[0], x2[0])

The following pairs are in Pbound:

1310_0_MAIN_LE(x0[0], x1[0], x2[0]) → COND_1310_0_MAIN_LE(&&(>=(x2[0], x1[0]), <(x2[0], +(x0[0], -1))), x0[0], x1[0], x2[0])

The following pairs are in P:

COND_1310_0_MAIN_LE(TRUE, x0[1], x1[1], x2[1]) → 1310_0_MAIN_LE(+(x0[1], -1), x1[1], x2[1])

There are no usable rules.

### (13) Obligation:

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

The following domains are used:

Integer

R is empty.

The integer pair graph contains the following rules and edges:
(1): COND_1310_0_MAIN_LE(TRUE, x0[1], x1[1], x2[1]) → 1310_0_MAIN_LE(x0[1] + -1, x1[1], x2[1])

The set Q is empty.

### (14) IDependencyGraphProof (EQUIVALENT transformation)

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

### (16) Obligation:

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

The following domains are used:

Boolean, Integer

R is empty.

The integer pair graph contains the following rules and edges:
(0): 1310_0_MAIN_LE(x0[0], x1[0], x2[0]) → COND_1310_0_MAIN_LE(x2[0] >= x1[0] && x2[0] < x0[0] + -1, x0[0], x1[0], x2[0])
(1): COND_1310_0_MAIN_LE(TRUE, x0[1], x1[1], x2[1]) → 1310_0_MAIN_LE(x0[1] + -1, x1[1], x2[1])
(3): COND_1310_0_MAIN_LE1(TRUE, x0[3], x1[3], x2[3]) → 1310_0_MAIN_LE(x0[3], x1[3] + -1, x2[3])

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

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

(0) -> (1), if (x2[0] >= x1[0] && x2[0] < x0[0] + -1x0[0]* x0[1]x1[0]* x1[1]x2[0]* x2[1])

The set Q is empty.

### (17) IDependencyGraphProof (EQUIVALENT transformation)

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

### (18) 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:
(1): COND_1310_0_MAIN_LE(TRUE, x0[1], x1[1], x2[1]) → 1310_0_MAIN_LE(x0[1] + -1, x1[1], x2[1])
(0): 1310_0_MAIN_LE(x0[0], x1[0], x2[0]) → COND_1310_0_MAIN_LE(x2[0] >= x1[0] && x2[0] < x0[0] + -1, x0[0], x1[0], x2[0])

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

(0) -> (1), if (x2[0] >= x1[0] && x2[0] < x0[0] + -1x0[0]* x0[1]x1[0]* x1[1]x2[0]* x2[1])

The set Q is empty.

### (19) 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@5176f87c 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 COND_1310_0_MAIN_LE(TRUE, x0[1], x1[1], x2[1]) → 1310_0_MAIN_LE(+(x0[1], -1), x1[1], x2[1]) the following chains were created:
• We consider the chain COND_1310_0_MAIN_LE(TRUE, x0[1], x1[1], x2[1]) → 1310_0_MAIN_LE(+(x0[1], -1), x1[1], x2[1]) which results in the following constraint:

(1)    (COND_1310_0_MAIN_LE(TRUE, x0[1], x1[1], x2[1])≥NonInfC∧COND_1310_0_MAIN_LE(TRUE, x0[1], x1[1], x2[1])≥1310_0_MAIN_LE(+(x0[1], -1), x1[1], x2[1])∧(UIncreasing(1310_0_MAIN_LE(+(x0[1], -1), x1[1], x2[1])), ≥))

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

(2)    ((UIncreasing(1310_0_MAIN_LE(+(x0[1], -1), x1[1], x2[1])), ≥)∧[bni_8] = 0∧[1 + (-1)bso_9] ≥ 0)

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

(3)    ((UIncreasing(1310_0_MAIN_LE(+(x0[1], -1), x1[1], x2[1])), ≥)∧[bni_8] = 0∧[1 + (-1)bso_9] ≥ 0)

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

(4)    ((UIncreasing(1310_0_MAIN_LE(+(x0[1], -1), x1[1], x2[1])), ≥)∧[bni_8] = 0∧[1 + (-1)bso_9] ≥ 0)

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

(5)    ((UIncreasing(1310_0_MAIN_LE(+(x0[1], -1), x1[1], x2[1])), ≥)∧[bni_8] = 0∧0 = 0∧0 = 0∧0 = 0∧[1 + (-1)bso_9] ≥ 0)

For Pair 1310_0_MAIN_LE(x0[0], x1[0], x2[0]) → COND_1310_0_MAIN_LE(&&(>=(x2[0], x1[0]), <(x2[0], +(x0[0], -1))), x0[0], x1[0], x2[0]) the following chains were created:
• We consider the chain 1310_0_MAIN_LE(x0[0], x1[0], x2[0]) → COND_1310_0_MAIN_LE(&&(>=(x2[0], x1[0]), <(x2[0], +(x0[0], -1))), x0[0], x1[0], x2[0]), COND_1310_0_MAIN_LE(TRUE, x0[1], x1[1], x2[1]) → 1310_0_MAIN_LE(+(x0[1], -1), x1[1], x2[1]) which results in the following constraint:

(6)    (&&(>=(x2[0], x1[0]), <(x2[0], +(x0[0], -1)))=TRUEx0[0]=x0[1]x1[0]=x1[1]x2[0]=x2[1]1310_0_MAIN_LE(x0[0], x1[0], x2[0])≥NonInfC∧1310_0_MAIN_LE(x0[0], x1[0], x2[0])≥COND_1310_0_MAIN_LE(&&(>=(x2[0], x1[0]), <(x2[0], +(x0[0], -1))), x0[0], x1[0], x2[0])∧(UIncreasing(COND_1310_0_MAIN_LE(&&(>=(x2[0], x1[0]), <(x2[0], +(x0[0], -1))), x0[0], x1[0], x2[0])), ≥))

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

(7)    (>=(x2[0], x1[0])=TRUE<(x2[0], +(x0[0], -1))=TRUE1310_0_MAIN_LE(x0[0], x1[0], x2[0])≥NonInfC∧1310_0_MAIN_LE(x0[0], x1[0], x2[0])≥COND_1310_0_MAIN_LE(&&(>=(x2[0], x1[0]), <(x2[0], +(x0[0], -1))), x0[0], x1[0], x2[0])∧(UIncreasing(COND_1310_0_MAIN_LE(&&(>=(x2[0], x1[0]), <(x2[0], +(x0[0], -1))), x0[0], x1[0], x2[0])), ≥))

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

(8)    (x2[0] + [-1]x1[0] ≥ 0∧x0[0] + [-2] + [-1]x2[0] ≥ 0 ⇒ (UIncreasing(COND_1310_0_MAIN_LE(&&(>=(x2[0], x1[0]), <(x2[0], +(x0[0], -1))), x0[0], x1[0], x2[0])), ≥)∧[(-1)Bound*bni_10] + [bni_10]x0[0] + [(-1)bni_10]x1[0] ≥ 0∧[(-1)bso_11] ≥ 0)

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

(9)    (x2[0] + [-1]x1[0] ≥ 0∧x0[0] + [-2] + [-1]x2[0] ≥ 0 ⇒ (UIncreasing(COND_1310_0_MAIN_LE(&&(>=(x2[0], x1[0]), <(x2[0], +(x0[0], -1))), x0[0], x1[0], x2[0])), ≥)∧[(-1)Bound*bni_10] + [bni_10]x0[0] + [(-1)bni_10]x1[0] ≥ 0∧[(-1)bso_11] ≥ 0)

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

(10)    (x2[0] + [-1]x1[0] ≥ 0∧x0[0] + [-2] + [-1]x2[0] ≥ 0 ⇒ (UIncreasing(COND_1310_0_MAIN_LE(&&(>=(x2[0], x1[0]), <(x2[0], +(x0[0], -1))), x0[0], x1[0], x2[0])), ≥)∧[(-1)Bound*bni_10] + [bni_10]x0[0] + [(-1)bni_10]x1[0] ≥ 0∧[(-1)bso_11] ≥ 0)

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

(11)    (x2[0] ≥ 0∧x0[0] + [-2] + [-1]x1[0] + [-1]x2[0] ≥ 0 ⇒ (UIncreasing(COND_1310_0_MAIN_LE(&&(>=(x2[0], x1[0]), <(x2[0], +(x0[0], -1))), x0[0], x1[0], x2[0])), ≥)∧[(-1)Bound*bni_10] + [bni_10]x0[0] + [(-1)bni_10]x1[0] ≥ 0∧[(-1)bso_11] ≥ 0)

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

(12)    (x2[0] ≥ 0∧x1[0] ≥ 0 ⇒ (UIncreasing(COND_1310_0_MAIN_LE(&&(>=(x2[0], x1[0]), <(x2[0], +(x0[0], -1))), x0[0], x1[0], x2[0])), ≥)∧[(-1)Bound*bni_10 + (2)bni_10] + [bni_10]x2[0] + [bni_10]x1[0] ≥ 0∧[(-1)bso_11] ≥ 0)

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

(13)    (x2[0] ≥ 0∧x1[0] ≥ 0∧x0[0] ≥ 0 ⇒ (UIncreasing(COND_1310_0_MAIN_LE(&&(>=(x2[0], x1[0]), <(x2[0], +(x0[0], -1))), x0[0], x1[0], x2[0])), ≥)∧[(-1)Bound*bni_10 + (2)bni_10] + [bni_10]x2[0] + [bni_10]x1[0] ≥ 0∧[(-1)bso_11] ≥ 0)

(14)    (x2[0] ≥ 0∧x1[0] ≥ 0∧x0[0] ≥ 0 ⇒ (UIncreasing(COND_1310_0_MAIN_LE(&&(>=(x2[0], x1[0]), <(x2[0], +(x0[0], -1))), x0[0], x1[0], x2[0])), ≥)∧[(-1)Bound*bni_10 + (2)bni_10] + [bni_10]x2[0] + [bni_10]x1[0] ≥ 0∧[(-1)bso_11] ≥ 0)

To summarize, we get the following constraints P for the following pairs.
• COND_1310_0_MAIN_LE(TRUE, x0[1], x1[1], x2[1]) → 1310_0_MAIN_LE(+(x0[1], -1), x1[1], x2[1])
• ((UIncreasing(1310_0_MAIN_LE(+(x0[1], -1), x1[1], x2[1])), ≥)∧[bni_8] = 0∧0 = 0∧0 = 0∧0 = 0∧[1 + (-1)bso_9] ≥ 0)

• 1310_0_MAIN_LE(x0[0], x1[0], x2[0]) → COND_1310_0_MAIN_LE(&&(>=(x2[0], x1[0]), <(x2[0], +(x0[0], -1))), x0[0], x1[0], x2[0])
• (x2[0] ≥ 0∧x1[0] ≥ 0∧x0[0] ≥ 0 ⇒ (UIncreasing(COND_1310_0_MAIN_LE(&&(>=(x2[0], x1[0]), <(x2[0], +(x0[0], -1))), x0[0], x1[0], x2[0])), ≥)∧[(-1)Bound*bni_10 + (2)bni_10] + [bni_10]x2[0] + [bni_10]x1[0] ≥ 0∧[(-1)bso_11] ≥ 0)
• (x2[0] ≥ 0∧x1[0] ≥ 0∧x0[0] ≥ 0 ⇒ (UIncreasing(COND_1310_0_MAIN_LE(&&(>=(x2[0], x1[0]), <(x2[0], +(x0[0], -1))), x0[0], x1[0], x2[0])), ≥)∧[(-1)Bound*bni_10 + (2)bni_10] + [bni_10]x2[0] + [bni_10]x1[0] ≥ 0∧[(-1)bso_11] ≥ 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_1310_0_MAIN_LE(x1, x2, x3, x4)) = [-1]x3 + x2
POL(1310_0_MAIN_LE(x1, x2, x3)) = x1 + [-1]x2
POL(+(x1, x2)) = x1 + x2
POL(-1) = [-1]
POL(&&(x1, x2)) = [-1]
POL(>=(x1, x2)) = [1]
POL(<(x1, x2)) = [-1]

The following pairs are in P>:

COND_1310_0_MAIN_LE(TRUE, x0[1], x1[1], x2[1]) → 1310_0_MAIN_LE(+(x0[1], -1), x1[1], x2[1])

The following pairs are in Pbound:

1310_0_MAIN_LE(x0[0], x1[0], x2[0]) → COND_1310_0_MAIN_LE(&&(>=(x2[0], x1[0]), <(x2[0], +(x0[0], -1))), x0[0], x1[0], x2[0])

The following pairs are in P:

1310_0_MAIN_LE(x0[0], x1[0], x2[0]) → COND_1310_0_MAIN_LE(&&(>=(x2[0], x1[0]), <(x2[0], +(x0[0], -1))), x0[0], x1[0], x2[0])

There are no usable rules.

### (21) 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): 1310_0_MAIN_LE(x0[0], x1[0], x2[0]) → COND_1310_0_MAIN_LE(x2[0] >= x1[0] && x2[0] < x0[0] + -1, x0[0], x1[0], x2[0])

The set Q is empty.

### (22) IDependencyGraphProof (EQUIVALENT transformation)

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

### (24) 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_1310_0_MAIN_LE(TRUE, x0[1], x1[1], x2[1]) → 1310_0_MAIN_LE(x0[1] + -1, x1[1], x2[1])

The set Q is empty.

### (25) IDependencyGraphProof (EQUIVALENT transformation)

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