### (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) JBC2FIG (SOUND transformation)

Constructed FIGraph.

### (2) Obligation:

FIGraph based on JBC Program:
PastaB17.main([Ljava/lang/String;)V: Graph of 233 nodes with 1 SCC.

### (3) FIGtoITRSProof (SOUND transformation)

Transformed FIGraph SCCs to IDPs. Logs:

Log for SCC 0:

Generated 16 rules for P and 2 rules for R.

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

Filtered ground terms:

1262_0_main_LE(x1, x2, x3, x4, x5, x6) → 1262_0_main_LE(x2, x3, x4, x5, x6)
Cond_1262_0_main_LE1(x1, x2, x3, x4, x5, x6, x7) → Cond_1262_0_main_LE1(x1, x3, x4, x5, x6, x7)
Cond_1262_0_main_LE(x1, x2, x3, x4, x5, x6, x7) → Cond_1262_0_main_LE(x1, x3, x4, x5, x6, x7)

Filtered duplicate args:

1262_0_main_LE(x1, x2, x3, x4, x5) → 1262_0_main_LE(x1, x4, x5)
Cond_1262_0_main_LE1(x1, x2, x3, x4, x5, x6) → Cond_1262_0_main_LE1(x1, x2, x5, x6)
Cond_1262_0_main_LE(x1, x2, x3, x4, x5, x6) → Cond_1262_0_main_LE(x1, x2, x5, x6)

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

Finished conversion. Obtained 2 rules for P and 0 rules for R. System has predefined symbols.

### (4) Obligation:

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

The following domains are used:

Boolean, Integer

R is empty.

The integer pair graph contains the following rules and edges:
(0): 1262_0_MAIN_LE(x0[0], x1[0], x2[0]) → COND_1262_0_MAIN_LE(x2[0] >= x1[0] && x2[0] < x0[0] + -1, x0[0], x1[0], x2[0])
(1): COND_1262_0_MAIN_LE(TRUE, x0[1], x1[1], x2[1]) → 1262_0_MAIN_LE(x0[1] + -1, x1[1], x2[1])
(2): 1262_0_MAIN_LE(x0[2], x1[2], x2[2]) → COND_1262_0_MAIN_LE1(x2[2] < x1[2], x0[2], x1[2], x2[2])
(3): COND_1262_0_MAIN_LE1(TRUE, x0[3], x1[3], x2[3]) → 1262_0_MAIN_LE(x0[3], x1[3] + -1, x2[3])

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

### (5) IDPNonInfProof (SOUND transformation)

The constraints were generated the following way:
The DP Problem is simplified using the Induction Calculus [NONINF] with the following steps:
Note that final constraints are written in bold face.

For Pair 1262_0_MAIN_LE(x0, x1, x2) → COND_1262_0_MAIN_LE(&&(>=(x2, x1), <(x2, +(x0, -1))), x0, x1, x2) the following chains were created:
• We consider the chain 1262_0_MAIN_LE(x0[0], x1[0], x2[0]) → COND_1262_0_MAIN_LE(&&(>=(x2[0], x1[0]), <(x2[0], +(x0[0], -1))), x0[0], x1[0], x2[0]), COND_1262_0_MAIN_LE(TRUE, x0[1], x1[1], x2[1]) → 1262_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]1262_0_MAIN_LE(x0[0], x1[0], x2[0])≥NonInfC∧1262_0_MAIN_LE(x0[0], x1[0], x2[0])≥COND_1262_0_MAIN_LE(&&(>=(x2[0], x1[0]), <(x2[0], +(x0[0], -1))), x0[0], x1[0], x2[0])∧(UIncreasing(COND_1262_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))=TRUE1262_0_MAIN_LE(x0[0], x1[0], x2[0])≥NonInfC∧1262_0_MAIN_LE(x0[0], x1[0], x2[0])≥COND_1262_0_MAIN_LE(&&(>=(x2[0], x1[0]), <(x2[0], +(x0[0], -1))), x0[0], x1[0], x2[0])∧(UIncreasing(COND_1262_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_1262_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_1262_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_1262_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_1262_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_1262_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_1262_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_1262_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_1262_0_MAIN_LE(TRUE, x0, x1, x2) → 1262_0_MAIN_LE(+(x0, -1), x1, x2) the following chains were created:
• We consider the chain COND_1262_0_MAIN_LE(TRUE, x0[1], x1[1], x2[1]) → 1262_0_MAIN_LE(+(x0[1], -1), x1[1], x2[1]) which results in the following constraint:

(10)    (COND_1262_0_MAIN_LE(TRUE, x0[1], x1[1], x2[1])≥NonInfC∧COND_1262_0_MAIN_LE(TRUE, x0[1], x1[1], x2[1])≥1262_0_MAIN_LE(+(x0[1], -1), x1[1], x2[1])∧(UIncreasing(1262_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(1262_0_MAIN_LE(+(x0[1], -1), x1[1], x2[1])), ≥)∧[(-1)bso_15] ≥ 0)

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

(12)    ((UIncreasing(1262_0_MAIN_LE(+(x0[1], -1), x1[1], x2[1])), ≥)∧[(-1)bso_15] ≥ 0)

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

(13)    ((UIncreasing(1262_0_MAIN_LE(+(x0[1], -1), x1[1], x2[1])), ≥)∧[(-1)bso_15] ≥ 0)

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

(14)    ((UIncreasing(1262_0_MAIN_LE(+(x0[1], -1), x1[1], x2[1])), ≥)∧0 = 0∧0 = 0∧0 = 0∧[(-1)bso_15] ≥ 0)

For Pair 1262_0_MAIN_LE(x0, x1, x2) → COND_1262_0_MAIN_LE1(<(x2, x1), x0, x1, x2) the following chains were created:
• We consider the chain 1262_0_MAIN_LE(x0[2], x1[2], x2[2]) → COND_1262_0_MAIN_LE1(<(x2[2], x1[2]), x0[2], x1[2], x2[2]), COND_1262_0_MAIN_LE1(TRUE, x0[3], x1[3], x2[3]) → 1262_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]1262_0_MAIN_LE(x0[2], x1[2], x2[2])≥NonInfC∧1262_0_MAIN_LE(x0[2], x1[2], x2[2])≥COND_1262_0_MAIN_LE1(<(x2[2], x1[2]), x0[2], x1[2], x2[2])∧(UIncreasing(COND_1262_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])=TRUE1262_0_MAIN_LE(x0[2], x1[2], x2[2])≥NonInfC∧1262_0_MAIN_LE(x0[2], x1[2], x2[2])≥COND_1262_0_MAIN_LE1(<(x2[2], x1[2]), x0[2], x1[2], x2[2])∧(UIncreasing(COND_1262_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_1262_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_1262_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_1262_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_1262_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_1262_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_1262_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_1262_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_1262_0_MAIN_LE1(TRUE, x0, x1, x2) → 1262_0_MAIN_LE(x0, +(x1, -1), x2) the following chains were created:
• We consider the chain COND_1262_0_MAIN_LE1(TRUE, x0[3], x1[3], x2[3]) → 1262_0_MAIN_LE(x0[3], +(x1[3], -1), x2[3]) which results in the following constraint:

(24)    (COND_1262_0_MAIN_LE1(TRUE, x0[3], x1[3], x2[3])≥NonInfC∧COND_1262_0_MAIN_LE1(TRUE, x0[3], x1[3], x2[3])≥1262_0_MAIN_LE(x0[3], +(x1[3], -1), x2[3])∧(UIncreasing(1262_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(1262_0_MAIN_LE(x0[3], +(x1[3], -1), x2[3])), ≥)∧[1 + (-1)bso_19] ≥ 0)

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

(26)    ((UIncreasing(1262_0_MAIN_LE(x0[3], +(x1[3], -1), x2[3])), ≥)∧[1 + (-1)bso_19] ≥ 0)

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

(27)    ((UIncreasing(1262_0_MAIN_LE(x0[3], +(x1[3], -1), x2[3])), ≥)∧[1 + (-1)bso_19] ≥ 0)

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

(28)    ((UIncreasing(1262_0_MAIN_LE(x0[3], +(x1[3], -1), x2[3])), ≥)∧0 = 0∧0 = 0∧0 = 0∧[1 + (-1)bso_19] ≥ 0)

To summarize, we get the following constraints P for the following pairs.
• 1262_0_MAIN_LE(x0, x1, x2) → COND_1262_0_MAIN_LE(&&(>=(x2, x1), <(x2, +(x0, -1))), x0, x1, x2)
• (x2[0] ≥ 0∧x1[0] ≥ 0∧x0[0] ≥ 0 ⇒ (UIncreasing(COND_1262_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_1262_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_1262_0_MAIN_LE(TRUE, x0, x1, x2) → 1262_0_MAIN_LE(+(x0, -1), x1, x2)
• ((UIncreasing(1262_0_MAIN_LE(+(x0[1], -1), x1[1], x2[1])), ≥)∧0 = 0∧0 = 0∧0 = 0∧[(-1)bso_15] ≥ 0)

• 1262_0_MAIN_LE(x0, x1, x2) → COND_1262_0_MAIN_LE1(<(x2, x1), x0, x1, x2)
• (x1[2] ≥ 0∧x2[2] ≥ 0 ⇒ (UIncreasing(COND_1262_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_1262_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_1262_0_MAIN_LE1(TRUE, x0, x1, x2) → 1262_0_MAIN_LE(x0, +(x1, -1), x2)
• ((UIncreasing(1262_0_MAIN_LE(x0[3], +(x1[3], -1), x2[3])), ≥)∧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(1262_0_MAIN_LE(x1, x2, x3)) = [-1] + [-1]x3 + x2
POL(COND_1262_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_1262_0_MAIN_LE1(x1, x2, x3, x4)) = [-1] + [-1]x4 + x3

The following pairs are in P>:

COND_1262_0_MAIN_LE1(TRUE, x0[3], x1[3], x2[3]) → 1262_0_MAIN_LE(x0[3], +(x1[3], -1), x2[3])

The following pairs are in Pbound:

1262_0_MAIN_LE(x0[2], x1[2], x2[2]) → COND_1262_0_MAIN_LE1(<(x2[2], x1[2]), x0[2], x1[2], x2[2])

The following pairs are in P:

1262_0_MAIN_LE(x0[0], x1[0], x2[0]) → COND_1262_0_MAIN_LE(&&(>=(x2[0], x1[0]), <(x2[0], +(x0[0], -1))), x0[0], x1[0], x2[0])
COND_1262_0_MAIN_LE(TRUE, x0[1], x1[1], x2[1]) → 1262_0_MAIN_LE(+(x0[1], -1), x1[1], x2[1])
1262_0_MAIN_LE(x0[2], x1[2], x2[2]) → COND_1262_0_MAIN_LE1(<(x2[2], x1[2]), x0[2], x1[2], x2[2])

There are no usable rules.

### (7) Obligation:

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

The following domains are used:

Boolean, Integer

R is empty.

The integer pair graph contains the following rules and edges:
(0): 1262_0_MAIN_LE(x0[0], x1[0], x2[0]) → COND_1262_0_MAIN_LE(x2[0] >= x1[0] && x2[0] < x0[0] + -1, x0[0], x1[0], x2[0])
(1): COND_1262_0_MAIN_LE(TRUE, x0[1], x1[1], x2[1]) → 1262_0_MAIN_LE(x0[1] + -1, x1[1], x2[1])
(2): 1262_0_MAIN_LE(x0[2], x1[2], x2[2]) → COND_1262_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] + -1* TRUE)∧(x0[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.

### (8) IDependencyGraphProof (EQUIVALENT transformation)

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

### (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:
(1): COND_1262_0_MAIN_LE(TRUE, x0[1], x1[1], x2[1]) → 1262_0_MAIN_LE(x0[1] + -1, x1[1], x2[1])
(0): 1262_0_MAIN_LE(x0[0], x1[0], x2[0]) → COND_1262_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] + -1* TRUE)∧(x0[0]* x0[1])∧(x1[0]* x1[1])∧(x2[0]* x2[1]))

The set Q is empty.

### (10) IDPNonInfProof (SOUND transformation)

The constraints were generated the following way:
The DP Problem is simplified using the Induction Calculus [NONINF] with the following steps:
Note that final constraints are written in bold face.

For Pair COND_1262_0_MAIN_LE(TRUE, x0[1], x1[1], x2[1]) → 1262_0_MAIN_LE(+(x0[1], -1), x1[1], x2[1]) the following chains were created:
• We consider the chain COND_1262_0_MAIN_LE(TRUE, x0[1], x1[1], x2[1]) → 1262_0_MAIN_LE(+(x0[1], -1), x1[1], x2[1]) which results in the following constraint:

(1)    (COND_1262_0_MAIN_LE(TRUE, x0[1], x1[1], x2[1])≥NonInfC∧COND_1262_0_MAIN_LE(TRUE, x0[1], x1[1], x2[1])≥1262_0_MAIN_LE(+(x0[1], -1), x1[1], x2[1])∧(UIncreasing(1262_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(1262_0_MAIN_LE(+(x0[1], -1), x1[1], x2[1])), ≥)∧[(-1)bso_12] ≥ 0)

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

(3)    ((UIncreasing(1262_0_MAIN_LE(+(x0[1], -1), x1[1], x2[1])), ≥)∧[(-1)bso_12] ≥ 0)

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

(4)    ((UIncreasing(1262_0_MAIN_LE(+(x0[1], -1), x1[1], x2[1])), ≥)∧[(-1)bso_12] ≥ 0)

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

(5)    ((UIncreasing(1262_0_MAIN_LE(+(x0[1], -1), x1[1], x2[1])), ≥)∧0 = 0∧0 = 0∧0 = 0∧[(-1)bso_12] ≥ 0)

For Pair 1262_0_MAIN_LE(x0[0], x1[0], x2[0]) → COND_1262_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 1262_0_MAIN_LE(x0[0], x1[0], x2[0]) → COND_1262_0_MAIN_LE(&&(>=(x2[0], x1[0]), <(x2[0], +(x0[0], -1))), x0[0], x1[0], x2[0]), COND_1262_0_MAIN_LE(TRUE, x0[1], x1[1], x2[1]) → 1262_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]1262_0_MAIN_LE(x0[0], x1[0], x2[0])≥NonInfC∧1262_0_MAIN_LE(x0[0], x1[0], x2[0])≥COND_1262_0_MAIN_LE(&&(>=(x2[0], x1[0]), <(x2[0], +(x0[0], -1))), x0[0], x1[0], x2[0])∧(UIncreasing(COND_1262_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))=TRUE1262_0_MAIN_LE(x0[0], x1[0], x2[0])≥NonInfC∧1262_0_MAIN_LE(x0[0], x1[0], x2[0])≥COND_1262_0_MAIN_LE(&&(>=(x2[0], x1[0]), <(x2[0], +(x0[0], -1))), x0[0], x1[0], x2[0])∧(UIncreasing(COND_1262_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_1262_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_1262_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_1262_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_1262_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_1262_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_1262_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_1262_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_1262_0_MAIN_LE(TRUE, x0[1], x1[1], x2[1]) → 1262_0_MAIN_LE(+(x0[1], -1), x1[1], x2[1])
• ((UIncreasing(1262_0_MAIN_LE(+(x0[1], -1), x1[1], x2[1])), ≥)∧0 = 0∧0 = 0∧0 = 0∧[(-1)bso_12] ≥ 0)

• 1262_0_MAIN_LE(x0[0], x1[0], x2[0]) → COND_1262_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_1262_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_1262_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_1262_0_MAIN_LE(x1, x2, x3, x4)) = [-1] + [-1]x4 + [-1]x3 + [2]x2
POL(1262_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>:

1262_0_MAIN_LE(x0[0], x1[0], x2[0]) → COND_1262_0_MAIN_LE(&&(>=(x2[0], x1[0]), <(x2[0], +(x0[0], -1))), x0[0], x1[0], x2[0])

The following pairs are in Pbound:

1262_0_MAIN_LE(x0[0], x1[0], x2[0]) → COND_1262_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_1262_0_MAIN_LE(TRUE, x0[1], x1[1], x2[1]) → 1262_0_MAIN_LE(+(x0[1], -1), x1[1], x2[1])

There are no usable rules.

### (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

R is empty.

The integer pair graph contains the following rules and edges:
(1): COND_1262_0_MAIN_LE(TRUE, x0[1], x1[1], x2[1]) → 1262_0_MAIN_LE(x0[1] + -1, x1[1], x2[1])

The set Q is empty.

### (12) IDependencyGraphProof (EQUIVALENT transformation)

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

### (14) 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): 1262_0_MAIN_LE(x0[0], x1[0], x2[0]) → COND_1262_0_MAIN_LE(x2[0] >= x1[0] && x2[0] < x0[0] + -1, x0[0], x1[0], x2[0])
(1): COND_1262_0_MAIN_LE(TRUE, x0[1], x1[1], x2[1]) → 1262_0_MAIN_LE(x0[1] + -1, x1[1], x2[1])
(3): COND_1262_0_MAIN_LE1(TRUE, x0[3], x1[3], x2[3]) → 1262_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] + -1* TRUE)∧(x0[0]* x0[1])∧(x1[0]* x1[1])∧(x2[0]* x2[1]))

The set Q is empty.

### (15) IDependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC 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:

Integer, Boolean

R is empty.

The integer pair graph contains the following rules and edges:
(1): COND_1262_0_MAIN_LE(TRUE, x0[1], x1[1], x2[1]) → 1262_0_MAIN_LE(x0[1] + -1, x1[1], x2[1])
(0): 1262_0_MAIN_LE(x0[0], x1[0], x2[0]) → COND_1262_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] + -1* TRUE)∧(x0[0]* x0[1])∧(x1[0]* x1[1])∧(x2[0]* x2[1]))

The set Q is empty.

### (17) IDPNonInfProof (SOUND transformation)

The constraints were generated the following way:
The DP Problem is simplified using the Induction Calculus [NONINF] with the following steps:
Note that final constraints are written in bold face.

For Pair COND_1262_0_MAIN_LE(TRUE, x0[1], x1[1], x2[1]) → 1262_0_MAIN_LE(+(x0[1], -1), x1[1], x2[1]) the following chains were created:
• We consider the chain COND_1262_0_MAIN_LE(TRUE, x0[1], x1[1], x2[1]) → 1262_0_MAIN_LE(+(x0[1], -1), x1[1], x2[1]) which results in the following constraint:

(1)    (COND_1262_0_MAIN_LE(TRUE, x0[1], x1[1], x2[1])≥NonInfC∧COND_1262_0_MAIN_LE(TRUE, x0[1], x1[1], x2[1])≥1262_0_MAIN_LE(+(x0[1], -1), x1[1], x2[1])∧(UIncreasing(1262_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(1262_0_MAIN_LE(+(x0[1], -1), x1[1], x2[1])), ≥)∧[1 + (-1)bso_9] ≥ 0)

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

(3)    ((UIncreasing(1262_0_MAIN_LE(+(x0[1], -1), x1[1], x2[1])), ≥)∧[1 + (-1)bso_9] ≥ 0)

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

(4)    ((UIncreasing(1262_0_MAIN_LE(+(x0[1], -1), x1[1], x2[1])), ≥)∧[1 + (-1)bso_9] ≥ 0)

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

(5)    ((UIncreasing(1262_0_MAIN_LE(+(x0[1], -1), x1[1], x2[1])), ≥)∧0 = 0∧0 = 0∧0 = 0∧[1 + (-1)bso_9] ≥ 0)

For Pair 1262_0_MAIN_LE(x0[0], x1[0], x2[0]) → COND_1262_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 1262_0_MAIN_LE(x0[0], x1[0], x2[0]) → COND_1262_0_MAIN_LE(&&(>=(x2[0], x1[0]), <(x2[0], +(x0[0], -1))), x0[0], x1[0], x2[0]), COND_1262_0_MAIN_LE(TRUE, x0[1], x1[1], x2[1]) → 1262_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]1262_0_MAIN_LE(x0[0], x1[0], x2[0])≥NonInfC∧1262_0_MAIN_LE(x0[0], x1[0], x2[0])≥COND_1262_0_MAIN_LE(&&(>=(x2[0], x1[0]), <(x2[0], +(x0[0], -1))), x0[0], x1[0], x2[0])∧(UIncreasing(COND_1262_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))=TRUE1262_0_MAIN_LE(x0[0], x1[0], x2[0])≥NonInfC∧1262_0_MAIN_LE(x0[0], x1[0], x2[0])≥COND_1262_0_MAIN_LE(&&(>=(x2[0], x1[0]), <(x2[0], +(x0[0], -1))), x0[0], x1[0], x2[0])∧(UIncreasing(COND_1262_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_1262_0_MAIN_LE(&&(>=(x2[0], x1[0]), <(x2[0], +(x0[0], -1))), x0[0], x1[0], x2[0])), ≥)∧[(-1)bni_10 + (-1)Bound*bni_10] + [bni_10]x0[0] + [(-1)bni_10]x2[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_1262_0_MAIN_LE(&&(>=(x2[0], x1[0]), <(x2[0], +(x0[0], -1))), x0[0], x1[0], x2[0])), ≥)∧[(-1)bni_10 + (-1)Bound*bni_10] + [bni_10]x0[0] + [(-1)bni_10]x2[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_1262_0_MAIN_LE(&&(>=(x2[0], x1[0]), <(x2[0], +(x0[0], -1))), x0[0], x1[0], x2[0])), ≥)∧[(-1)bni_10 + (-1)Bound*bni_10] + [bni_10]x0[0] + [(-1)bni_10]x2[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_1262_0_MAIN_LE(&&(>=(x2[0], x1[0]), <(x2[0], +(x0[0], -1))), x0[0], x1[0], x2[0])), ≥)∧[(-1)bni_10 + (-1)Bound*bni_10] + [bni_10]x0[0] + [(-1)bni_10]x1[0] + [(-1)bni_10]x2[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_1262_0_MAIN_LE(&&(>=(x2[0], x1[0]), <(x2[0], +(x0[0], -1))), x0[0], x1[0], x2[0])), ≥)∧[bni_10 + (-1)Bound*bni_10] + [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_1262_0_MAIN_LE(&&(>=(x2[0], x1[0]), <(x2[0], +(x0[0], -1))), x0[0], x1[0], x2[0])), ≥)∧[bni_10 + (-1)Bound*bni_10] + [bni_10]x1[0] ≥ 0∧[(-1)bso_11] ≥ 0)

(14)    (x2[0] ≥ 0∧x1[0] ≥ 0∧x0[0] ≥ 0 ⇒ (UIncreasing(COND_1262_0_MAIN_LE(&&(>=(x2[0], x1[0]), <(x2[0], +(x0[0], -1))), x0[0], x1[0], x2[0])), ≥)∧[bni_10 + (-1)Bound*bni_10] + [bni_10]x1[0] ≥ 0∧[(-1)bso_11] ≥ 0)

To summarize, we get the following constraints P for the following pairs.
• COND_1262_0_MAIN_LE(TRUE, x0[1], x1[1], x2[1]) → 1262_0_MAIN_LE(+(x0[1], -1), x1[1], x2[1])
• ((UIncreasing(1262_0_MAIN_LE(+(x0[1], -1), x1[1], x2[1])), ≥)∧0 = 0∧0 = 0∧0 = 0∧[1 + (-1)bso_9] ≥ 0)

• 1262_0_MAIN_LE(x0[0], x1[0], x2[0]) → COND_1262_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_1262_0_MAIN_LE(&&(>=(x2[0], x1[0]), <(x2[0], +(x0[0], -1))), x0[0], x1[0], x2[0])), ≥)∧[bni_10 + (-1)Bound*bni_10] + [bni_10]x1[0] ≥ 0∧[(-1)bso_11] ≥ 0)
• (x2[0] ≥ 0∧x1[0] ≥ 0∧x0[0] ≥ 0 ⇒ (UIncreasing(COND_1262_0_MAIN_LE(&&(>=(x2[0], x1[0]), <(x2[0], +(x0[0], -1))), x0[0], x1[0], x2[0])), ≥)∧[bni_10 + (-1)Bound*bni_10] + [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_1262_0_MAIN_LE(x1, x2, x3, x4)) = [-1] + [-1]x4 + x2
POL(1262_0_MAIN_LE(x1, x2, x3)) = [-1] + x1 + [-1]x3
POL(+(x1, x2)) = x1 + x2
POL(-1) = [-1]
POL(&&(x1, x2)) = 0
POL(>=(x1, x2)) = [1]
POL(<(x1, x2)) = [-1]

The following pairs are in P>:

COND_1262_0_MAIN_LE(TRUE, x0[1], x1[1], x2[1]) → 1262_0_MAIN_LE(+(x0[1], -1), x1[1], x2[1])

The following pairs are in Pbound:

1262_0_MAIN_LE(x0[0], x1[0], x2[0]) → COND_1262_0_MAIN_LE(&&(>=(x2[0], x1[0]), <(x2[0], +(x0[0], -1))), x0[0], x1[0], x2[0])

The following pairs are in P:

1262_0_MAIN_LE(x0[0], x1[0], x2[0]) → COND_1262_0_MAIN_LE(&&(>=(x2[0], x1[0]), <(x2[0], +(x0[0], -1))), x0[0], x1[0], x2[0])

There are no usable rules.

### (19) 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): 1262_0_MAIN_LE(x0[0], x1[0], x2[0]) → COND_1262_0_MAIN_LE(x2[0] >= x1[0] && x2[0] < x0[0] + -1, x0[0], x1[0], x2[0])

The set Q is empty.

### (20) IDependencyGraphProof (EQUIVALENT transformation)

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

### (22) 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_1262_0_MAIN_LE(TRUE, x0[1], x1[1], x2[1]) → 1262_0_MAIN_LE(x0[1] + -1, x1[1], x2[1])

The set Q is empty.

### (23) IDependencyGraphProof (EQUIVALENT transformation)

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