### (0) Obligation:

JBC Problem based on JBC Program:
Manifest-Version: 1.0 Created-By: 1.6.0_16 (Sun Microsystems Inc.) Main-Class: PastaC1
`/** * 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 PastaC1 {    public static void main(String[] args) {        Random.args = args;        int x = Random.random();		while (x >= 0) {			int y = 1;			while (x > y) {				y = 2*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:
PastaC1.main([Ljava/lang/String;)V: Graph of 102 nodes with 1 SCC.

### (3) FIGtoITRSProof (SOUND transformation)

Transformed FIGraph SCCs to IDPs. Logs:

Log for SCC 0:

Generated 23 rules for P and 2 rules for R.

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

Filtered ground terms:

397_0_main_LE(x1, x2, x3, x4, x5) → 397_0_main_LE(x2, x3, x4, x5)
Cond_397_0_main_LE1(x1, x2, x3, x4, x5, x6) → Cond_397_0_main_LE1(x1, x3, x4, x5, x6)
Cond_397_0_main_LE(x1, x2, x3, x4, x5, x6) → Cond_397_0_main_LE(x1, x3, x4, x5, x6)

Filtered duplicate args:

397_0_main_LE(x1, x2, x3, x4) → 397_0_main_LE(x3, x4)
Cond_397_0_main_LE1(x1, x2, x3, x4, x5) → Cond_397_0_main_LE1(x1, x4, x5)
Cond_397_0_main_LE(x1, x2, x3, x4, x5) → Cond_397_0_main_LE(x1, x4, x5)

Filtered unneeded arguments:

Cond_397_0_main_LE(x1, x2, x3) → Cond_397_0_main_LE(x1, x2)

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): 397_0_MAIN_LE(x0[0], x1[0]) → COND_397_0_MAIN_LE(x1[0] >= x0[0] && x0[0] >= 0 && 0 <= x0[0] + -1, x0[0], x1[0])
(1): COND_397_0_MAIN_LE(TRUE, x0[1], x1[1]) → 397_0_MAIN_LE(x0[1] + -1, 1)
(2): 397_0_MAIN_LE(x0[2], x1[2]) → COND_397_0_MAIN_LE1(x1[2] >= 1 && x1[2] < x0[2], x0[2], x1[2])
(3): COND_397_0_MAIN_LE1(TRUE, x0[3], x1[3]) → 397_0_MAIN_LE(x0[3], 2 * x1[3])

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

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

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

(2) -> (3), if ((x1[2] >= 1 && x1[2] < x0[2]* TRUE)∧(x0[2]* x0[3])∧(x1[2]* x1[3]))

(3) -> (0), if ((x0[3]* x0[0])∧(2 * x1[3]* x1[0]))

(3) -> (2), if ((x0[3]* x0[2])∧(2 * x1[3]* x1[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 397_0_MAIN_LE(x0, x1) → COND_397_0_MAIN_LE(&&(&&(>=(x1, x0), >=(x0, 0)), <=(0, +(x0, -1))), x0, x1) the following chains were created:
• We consider the chain 397_0_MAIN_LE(x0[0], x1[0]) → COND_397_0_MAIN_LE(&&(&&(>=(x1[0], x0[0]), >=(x0[0], 0)), <=(0, +(x0[0], -1))), x0[0], x1[0]), COND_397_0_MAIN_LE(TRUE, x0[1], x1[1]) → 397_0_MAIN_LE(+(x0[1], -1), 1) which results in the following constraint:

(1)    (&&(&&(>=(x1[0], x0[0]), >=(x0[0], 0)), <=(0, +(x0[0], -1)))=TRUEx0[0]=x0[1]x1[0]=x1[1]397_0_MAIN_LE(x0[0], x1[0])≥NonInfC∧397_0_MAIN_LE(x0[0], x1[0])≥COND_397_0_MAIN_LE(&&(&&(>=(x1[0], x0[0]), >=(x0[0], 0)), <=(0, +(x0[0], -1))), x0[0], x1[0])∧(UIncreasing(COND_397_0_MAIN_LE(&&(&&(>=(x1[0], x0[0]), >=(x0[0], 0)), <=(0, +(x0[0], -1))), x0[0], x1[0])), ≥))

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

(2)    (<=(0, +(x0[0], -1))=TRUE>=(x1[0], x0[0])=TRUE>=(x0[0], 0)=TRUE397_0_MAIN_LE(x0[0], x1[0])≥NonInfC∧397_0_MAIN_LE(x0[0], x1[0])≥COND_397_0_MAIN_LE(&&(&&(>=(x1[0], x0[0]), >=(x0[0], 0)), <=(0, +(x0[0], -1))), x0[0], x1[0])∧(UIncreasing(COND_397_0_MAIN_LE(&&(&&(>=(x1[0], x0[0]), >=(x0[0], 0)), <=(0, +(x0[0], -1))), x0[0], x1[0])), ≥))

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

(3)    (x0[0] + [-1] ≥ 0∧x1[0] + [-1]x0[0] ≥ 0∧x0[0] ≥ 0 ⇒ (UIncreasing(COND_397_0_MAIN_LE(&&(&&(>=(x1[0], x0[0]), >=(x0[0], 0)), <=(0, +(x0[0], -1))), x0[0], x1[0])), ≥)∧[(-1)bni_18 + (-1)Bound*bni_18] + [bni_18]x0[0] ≥ 0∧[(-1)bso_19] ≥ 0)

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

(4)    (x0[0] + [-1] ≥ 0∧x1[0] + [-1]x0[0] ≥ 0∧x0[0] ≥ 0 ⇒ (UIncreasing(COND_397_0_MAIN_LE(&&(&&(>=(x1[0], x0[0]), >=(x0[0], 0)), <=(0, +(x0[0], -1))), x0[0], x1[0])), ≥)∧[(-1)bni_18 + (-1)Bound*bni_18] + [bni_18]x0[0] ≥ 0∧[(-1)bso_19] ≥ 0)

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

(5)    (x0[0] + [-1] ≥ 0∧x1[0] + [-1]x0[0] ≥ 0∧x0[0] ≥ 0 ⇒ (UIncreasing(COND_397_0_MAIN_LE(&&(&&(>=(x1[0], x0[0]), >=(x0[0], 0)), <=(0, +(x0[0], -1))), x0[0], x1[0])), ≥)∧[(-1)bni_18 + (-1)Bound*bni_18] + [bni_18]x0[0] ≥ 0∧[(-1)bso_19] ≥ 0)

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

(6)    (x0[0] ≥ 0∧x1[0] + [-1] + [-1]x0[0] ≥ 0∧[1] + x0[0] ≥ 0 ⇒ (UIncreasing(COND_397_0_MAIN_LE(&&(&&(>=(x1[0], x0[0]), >=(x0[0], 0)), <=(0, +(x0[0], -1))), x0[0], x1[0])), ≥)∧[(-1)Bound*bni_18] + [bni_18]x0[0] ≥ 0∧[(-1)bso_19] ≥ 0)

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

(7)    (x0[0] ≥ 0∧x1[0] ≥ 0∧[1] + x0[0] ≥ 0 ⇒ (UIncreasing(COND_397_0_MAIN_LE(&&(&&(>=(x1[0], x0[0]), >=(x0[0], 0)), <=(0, +(x0[0], -1))), x0[0], x1[0])), ≥)∧[(-1)Bound*bni_18] + [bni_18]x0[0] ≥ 0∧[(-1)bso_19] ≥ 0)

For Pair COND_397_0_MAIN_LE(TRUE, x0, x1) → 397_0_MAIN_LE(+(x0, -1), 1) the following chains were created:
• We consider the chain 397_0_MAIN_LE(x0[0], x1[0]) → COND_397_0_MAIN_LE(&&(&&(>=(x1[0], x0[0]), >=(x0[0], 0)), <=(0, +(x0[0], -1))), x0[0], x1[0]), COND_397_0_MAIN_LE(TRUE, x0[1], x1[1]) → 397_0_MAIN_LE(+(x0[1], -1), 1), 397_0_MAIN_LE(x0[0], x1[0]) → COND_397_0_MAIN_LE(&&(&&(>=(x1[0], x0[0]), >=(x0[0], 0)), <=(0, +(x0[0], -1))), x0[0], x1[0]) which results in the following constraint:

(8)    (&&(&&(>=(x1[0], x0[0]), >=(x0[0], 0)), <=(0, +(x0[0], -1)))=TRUEx0[0]=x0[1]x1[0]=x1[1]+(x0[1], -1)=x0[0]11=x1[0]1COND_397_0_MAIN_LE(TRUE, x0[1], x1[1])≥NonInfC∧COND_397_0_MAIN_LE(TRUE, x0[1], x1[1])≥397_0_MAIN_LE(+(x0[1], -1), 1)∧(UIncreasing(397_0_MAIN_LE(+(x0[1], -1), 1)), ≥))

We simplified constraint (8) using rules (III), (IV), (IDP_BOOLEAN) which results in the following new constraint:

(9)    (<=(0, +(x0[0], -1))=TRUE>=(x1[0], x0[0])=TRUE>=(x0[0], 0)=TRUECOND_397_0_MAIN_LE(TRUE, x0[0], x1[0])≥NonInfC∧COND_397_0_MAIN_LE(TRUE, x0[0], x1[0])≥397_0_MAIN_LE(+(x0[0], -1), 1)∧(UIncreasing(397_0_MAIN_LE(+(x0[1], -1), 1)), ≥))

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

(10)    (x0[0] + [-1] ≥ 0∧x1[0] + [-1]x0[0] ≥ 0∧x0[0] ≥ 0 ⇒ (UIncreasing(397_0_MAIN_LE(+(x0[1], -1), 1)), ≥)∧[(-1)bni_20 + (-1)Bound*bni_20] + [bni_20]x0[0] ≥ 0∧[1 + (-1)bso_21] ≥ 0)

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

(11)    (x0[0] + [-1] ≥ 0∧x1[0] + [-1]x0[0] ≥ 0∧x0[0] ≥ 0 ⇒ (UIncreasing(397_0_MAIN_LE(+(x0[1], -1), 1)), ≥)∧[(-1)bni_20 + (-1)Bound*bni_20] + [bni_20]x0[0] ≥ 0∧[1 + (-1)bso_21] ≥ 0)

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

(12)    (x0[0] + [-1] ≥ 0∧x1[0] + [-1]x0[0] ≥ 0∧x0[0] ≥ 0 ⇒ (UIncreasing(397_0_MAIN_LE(+(x0[1], -1), 1)), ≥)∧[(-1)bni_20 + (-1)Bound*bni_20] + [bni_20]x0[0] ≥ 0∧[1 + (-1)bso_21] ≥ 0)

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

(13)    (x0[0] ≥ 0∧x1[0] + [-1] + [-1]x0[0] ≥ 0∧[1] + x0[0] ≥ 0 ⇒ (UIncreasing(397_0_MAIN_LE(+(x0[1], -1), 1)), ≥)∧[(-1)Bound*bni_20] + [bni_20]x0[0] ≥ 0∧[1 + (-1)bso_21] ≥ 0)

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

(14)    (x0[0] ≥ 0∧x1[0] ≥ 0∧[1] + x0[0] ≥ 0 ⇒ (UIncreasing(397_0_MAIN_LE(+(x0[1], -1), 1)), ≥)∧[(-1)Bound*bni_20] + [bni_20]x0[0] ≥ 0∧[1 + (-1)bso_21] ≥ 0)

• We consider the chain 397_0_MAIN_LE(x0[0], x1[0]) → COND_397_0_MAIN_LE(&&(&&(>=(x1[0], x0[0]), >=(x0[0], 0)), <=(0, +(x0[0], -1))), x0[0], x1[0]), COND_397_0_MAIN_LE(TRUE, x0[1], x1[1]) → 397_0_MAIN_LE(+(x0[1], -1), 1), 397_0_MAIN_LE(x0[2], x1[2]) → COND_397_0_MAIN_LE1(&&(>=(x1[2], 1), <(x1[2], x0[2])), x0[2], x1[2]) which results in the following constraint:

(15)    (&&(&&(>=(x1[0], x0[0]), >=(x0[0], 0)), <=(0, +(x0[0], -1)))=TRUEx0[0]=x0[1]x1[0]=x1[1]+(x0[1], -1)=x0[2]1=x1[2]COND_397_0_MAIN_LE(TRUE, x0[1], x1[1])≥NonInfC∧COND_397_0_MAIN_LE(TRUE, x0[1], x1[1])≥397_0_MAIN_LE(+(x0[1], -1), 1)∧(UIncreasing(397_0_MAIN_LE(+(x0[1], -1), 1)), ≥))

We simplified constraint (15) using rules (III), (IV), (IDP_BOOLEAN) which results in the following new constraint:

(16)    (<=(0, +(x0[0], -1))=TRUE>=(x1[0], x0[0])=TRUE>=(x0[0], 0)=TRUECOND_397_0_MAIN_LE(TRUE, x0[0], x1[0])≥NonInfC∧COND_397_0_MAIN_LE(TRUE, x0[0], x1[0])≥397_0_MAIN_LE(+(x0[0], -1), 1)∧(UIncreasing(397_0_MAIN_LE(+(x0[1], -1), 1)), ≥))

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

(17)    (x0[0] + [-1] ≥ 0∧x1[0] + [-1]x0[0] ≥ 0∧x0[0] ≥ 0 ⇒ (UIncreasing(397_0_MAIN_LE(+(x0[1], -1), 1)), ≥)∧[(-1)bni_20 + (-1)Bound*bni_20] + [bni_20]x0[0] ≥ 0∧[1 + (-1)bso_21] ≥ 0)

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

(18)    (x0[0] + [-1] ≥ 0∧x1[0] + [-1]x0[0] ≥ 0∧x0[0] ≥ 0 ⇒ (UIncreasing(397_0_MAIN_LE(+(x0[1], -1), 1)), ≥)∧[(-1)bni_20 + (-1)Bound*bni_20] + [bni_20]x0[0] ≥ 0∧[1 + (-1)bso_21] ≥ 0)

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

(19)    (x0[0] + [-1] ≥ 0∧x1[0] + [-1]x0[0] ≥ 0∧x0[0] ≥ 0 ⇒ (UIncreasing(397_0_MAIN_LE(+(x0[1], -1), 1)), ≥)∧[(-1)bni_20 + (-1)Bound*bni_20] + [bni_20]x0[0] ≥ 0∧[1 + (-1)bso_21] ≥ 0)

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

(20)    (x0[0] ≥ 0∧x1[0] + [-1] + [-1]x0[0] ≥ 0∧[1] + x0[0] ≥ 0 ⇒ (UIncreasing(397_0_MAIN_LE(+(x0[1], -1), 1)), ≥)∧[(-1)Bound*bni_20] + [bni_20]x0[0] ≥ 0∧[1 + (-1)bso_21] ≥ 0)

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

(21)    (x0[0] ≥ 0∧x1[0] ≥ 0∧[1] + x0[0] ≥ 0 ⇒ (UIncreasing(397_0_MAIN_LE(+(x0[1], -1), 1)), ≥)∧[(-1)Bound*bni_20] + [bni_20]x0[0] ≥ 0∧[1 + (-1)bso_21] ≥ 0)

For Pair 397_0_MAIN_LE(x0, x1) → COND_397_0_MAIN_LE1(&&(>=(x1, 1), <(x1, x0)), x0, x1) the following chains were created:
• We consider the chain 397_0_MAIN_LE(x0[2], x1[2]) → COND_397_0_MAIN_LE1(&&(>=(x1[2], 1), <(x1[2], x0[2])), x0[2], x1[2]), COND_397_0_MAIN_LE1(TRUE, x0[3], x1[3]) → 397_0_MAIN_LE(x0[3], *(2, x1[3])) which results in the following constraint:

(22)    (&&(>=(x1[2], 1), <(x1[2], x0[2]))=TRUEx0[2]=x0[3]x1[2]=x1[3]397_0_MAIN_LE(x0[2], x1[2])≥NonInfC∧397_0_MAIN_LE(x0[2], x1[2])≥COND_397_0_MAIN_LE1(&&(>=(x1[2], 1), <(x1[2], x0[2])), x0[2], x1[2])∧(UIncreasing(COND_397_0_MAIN_LE1(&&(>=(x1[2], 1), <(x1[2], x0[2])), x0[2], x1[2])), ≥))

We simplified constraint (22) using rules (IV), (IDP_BOOLEAN) which results in the following new constraint:

(23)    (>=(x1[2], 1)=TRUE<(x1[2], x0[2])=TRUE397_0_MAIN_LE(x0[2], x1[2])≥NonInfC∧397_0_MAIN_LE(x0[2], x1[2])≥COND_397_0_MAIN_LE1(&&(>=(x1[2], 1), <(x1[2], x0[2])), x0[2], x1[2])∧(UIncreasing(COND_397_0_MAIN_LE1(&&(>=(x1[2], 1), <(x1[2], x0[2])), x0[2], x1[2])), ≥))

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

(24)    (x1[2] + [-1] ≥ 0∧x0[2] + [-1] + [-1]x1[2] ≥ 0 ⇒ (UIncreasing(COND_397_0_MAIN_LE1(&&(>=(x1[2], 1), <(x1[2], x0[2])), x0[2], x1[2])), ≥)∧[(-1)bni_22 + (-1)Bound*bni_22] + [bni_22]x0[2] ≥ 0∧[(-1)bso_23] ≥ 0)

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

(25)    (x1[2] + [-1] ≥ 0∧x0[2] + [-1] + [-1]x1[2] ≥ 0 ⇒ (UIncreasing(COND_397_0_MAIN_LE1(&&(>=(x1[2], 1), <(x1[2], x0[2])), x0[2], x1[2])), ≥)∧[(-1)bni_22 + (-1)Bound*bni_22] + [bni_22]x0[2] ≥ 0∧[(-1)bso_23] ≥ 0)

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

(26)    (x1[2] + [-1] ≥ 0∧x0[2] + [-1] + [-1]x1[2] ≥ 0 ⇒ (UIncreasing(COND_397_0_MAIN_LE1(&&(>=(x1[2], 1), <(x1[2], x0[2])), x0[2], x1[2])), ≥)∧[(-1)bni_22 + (-1)Bound*bni_22] + [bni_22]x0[2] ≥ 0∧[(-1)bso_23] ≥ 0)

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

(27)    (x1[2] ≥ 0∧x0[2] + [-2] + [-1]x1[2] ≥ 0 ⇒ (UIncreasing(COND_397_0_MAIN_LE1(&&(>=(x1[2], 1), <(x1[2], x0[2])), x0[2], x1[2])), ≥)∧[(-1)bni_22 + (-1)Bound*bni_22] + [bni_22]x0[2] ≥ 0∧[(-1)bso_23] ≥ 0)

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

(28)    (x1[2] ≥ 0∧x0[2] ≥ 0 ⇒ (UIncreasing(COND_397_0_MAIN_LE1(&&(>=(x1[2], 1), <(x1[2], x0[2])), x0[2], x1[2])), ≥)∧[bni_22 + (-1)Bound*bni_22] + [bni_22]x1[2] + [bni_22]x0[2] ≥ 0∧[(-1)bso_23] ≥ 0)

For Pair COND_397_0_MAIN_LE1(TRUE, x0, x1) → 397_0_MAIN_LE(x0, *(2, x1)) the following chains were created:
• We consider the chain 397_0_MAIN_LE(x0[2], x1[2]) → COND_397_0_MAIN_LE1(&&(>=(x1[2], 1), <(x1[2], x0[2])), x0[2], x1[2]), COND_397_0_MAIN_LE1(TRUE, x0[3], x1[3]) → 397_0_MAIN_LE(x0[3], *(2, x1[3])), 397_0_MAIN_LE(x0[0], x1[0]) → COND_397_0_MAIN_LE(&&(&&(>=(x1[0], x0[0]), >=(x0[0], 0)), <=(0, +(x0[0], -1))), x0[0], x1[0]) which results in the following constraint:

(29)    (&&(>=(x1[2], 1), <(x1[2], x0[2]))=TRUEx0[2]=x0[3]x1[2]=x1[3]x0[3]=x0[0]*(2, x1[3])=x1[0]COND_397_0_MAIN_LE1(TRUE, x0[3], x1[3])≥NonInfC∧COND_397_0_MAIN_LE1(TRUE, x0[3], x1[3])≥397_0_MAIN_LE(x0[3], *(2, x1[3]))∧(UIncreasing(397_0_MAIN_LE(x0[3], *(2, x1[3]))), ≥))

We simplified constraint (29) using rules (III), (IV), (IDP_BOOLEAN) which results in the following new constraint:

(30)    (>=(x1[2], 1)=TRUE<(x1[2], x0[2])=TRUECOND_397_0_MAIN_LE1(TRUE, x0[2], x1[2])≥NonInfC∧COND_397_0_MAIN_LE1(TRUE, x0[2], x1[2])≥397_0_MAIN_LE(x0[2], *(2, x1[2]))∧(UIncreasing(397_0_MAIN_LE(x0[3], *(2, x1[3]))), ≥))

We simplified constraint (30) using rule (POLY_CONSTRAINTS) which results in the following new constraint:

(31)    (x1[2] + [-1] ≥ 0∧x0[2] + [-1] + [-1]x1[2] ≥ 0 ⇒ (UIncreasing(397_0_MAIN_LE(x0[3], *(2, x1[3]))), ≥)∧[(-1)bni_24 + (-1)Bound*bni_24] + [bni_24]x0[2] ≥ 0∧[(-1)bso_25] ≥ 0)

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

(32)    (x1[2] + [-1] ≥ 0∧x0[2] + [-1] + [-1]x1[2] ≥ 0 ⇒ (UIncreasing(397_0_MAIN_LE(x0[3], *(2, x1[3]))), ≥)∧[(-1)bni_24 + (-1)Bound*bni_24] + [bni_24]x0[2] ≥ 0∧[(-1)bso_25] ≥ 0)

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

(33)    (x1[2] + [-1] ≥ 0∧x0[2] + [-1] + [-1]x1[2] ≥ 0 ⇒ (UIncreasing(397_0_MAIN_LE(x0[3], *(2, x1[3]))), ≥)∧[(-1)bni_24 + (-1)Bound*bni_24] + [bni_24]x0[2] ≥ 0∧[(-1)bso_25] ≥ 0)

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

(34)    (x1[2] ≥ 0∧x0[2] + [-2] + [-1]x1[2] ≥ 0 ⇒ (UIncreasing(397_0_MAIN_LE(x0[3], *(2, x1[3]))), ≥)∧[(-1)bni_24 + (-1)Bound*bni_24] + [bni_24]x0[2] ≥ 0∧[(-1)bso_25] ≥ 0)

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

(35)    (x1[2] ≥ 0∧x0[2] ≥ 0 ⇒ (UIncreasing(397_0_MAIN_LE(x0[3], *(2, x1[3]))), ≥)∧[bni_24 + (-1)Bound*bni_24] + [bni_24]x1[2] + [bni_24]x0[2] ≥ 0∧[(-1)bso_25] ≥ 0)

• We consider the chain 397_0_MAIN_LE(x0[2], x1[2]) → COND_397_0_MAIN_LE1(&&(>=(x1[2], 1), <(x1[2], x0[2])), x0[2], x1[2]), COND_397_0_MAIN_LE1(TRUE, x0[3], x1[3]) → 397_0_MAIN_LE(x0[3], *(2, x1[3])), 397_0_MAIN_LE(x0[2], x1[2]) → COND_397_0_MAIN_LE1(&&(>=(x1[2], 1), <(x1[2], x0[2])), x0[2], x1[2]) which results in the following constraint:

(36)    (&&(>=(x1[2], 1), <(x1[2], x0[2]))=TRUEx0[2]=x0[3]x1[2]=x1[3]x0[3]=x0[2]1*(2, x1[3])=x1[2]1COND_397_0_MAIN_LE1(TRUE, x0[3], x1[3])≥NonInfC∧COND_397_0_MAIN_LE1(TRUE, x0[3], x1[3])≥397_0_MAIN_LE(x0[3], *(2, x1[3]))∧(UIncreasing(397_0_MAIN_LE(x0[3], *(2, x1[3]))), ≥))

We simplified constraint (36) using rules (III), (IV), (IDP_BOOLEAN) which results in the following new constraint:

(37)    (>=(x1[2], 1)=TRUE<(x1[2], x0[2])=TRUECOND_397_0_MAIN_LE1(TRUE, x0[2], x1[2])≥NonInfC∧COND_397_0_MAIN_LE1(TRUE, x0[2], x1[2])≥397_0_MAIN_LE(x0[2], *(2, x1[2]))∧(UIncreasing(397_0_MAIN_LE(x0[3], *(2, x1[3]))), ≥))

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

(38)    (x1[2] + [-1] ≥ 0∧x0[2] + [-1] + [-1]x1[2] ≥ 0 ⇒ (UIncreasing(397_0_MAIN_LE(x0[3], *(2, x1[3]))), ≥)∧[(-1)bni_24 + (-1)Bound*bni_24] + [bni_24]x0[2] ≥ 0∧[(-1)bso_25] ≥ 0)

We simplified constraint (38) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:

(39)    (x1[2] + [-1] ≥ 0∧x0[2] + [-1] + [-1]x1[2] ≥ 0 ⇒ (UIncreasing(397_0_MAIN_LE(x0[3], *(2, x1[3]))), ≥)∧[(-1)bni_24 + (-1)Bound*bni_24] + [bni_24]x0[2] ≥ 0∧[(-1)bso_25] ≥ 0)

We simplified constraint (39) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:

(40)    (x1[2] + [-1] ≥ 0∧x0[2] + [-1] + [-1]x1[2] ≥ 0 ⇒ (UIncreasing(397_0_MAIN_LE(x0[3], *(2, x1[3]))), ≥)∧[(-1)bni_24 + (-1)Bound*bni_24] + [bni_24]x0[2] ≥ 0∧[(-1)bso_25] ≥ 0)

We simplified constraint (40) using rule (IDP_SMT_SPLIT) which results in the following new constraint:

(41)    (x1[2] ≥ 0∧x0[2] + [-2] + [-1]x1[2] ≥ 0 ⇒ (UIncreasing(397_0_MAIN_LE(x0[3], *(2, x1[3]))), ≥)∧[(-1)bni_24 + (-1)Bound*bni_24] + [bni_24]x0[2] ≥ 0∧[(-1)bso_25] ≥ 0)

We simplified constraint (41) using rule (IDP_SMT_SPLIT) which results in the following new constraint:

(42)    (x1[2] ≥ 0∧x0[2] ≥ 0 ⇒ (UIncreasing(397_0_MAIN_LE(x0[3], *(2, x1[3]))), ≥)∧[bni_24 + (-1)Bound*bni_24] + [bni_24]x1[2] + [bni_24]x0[2] ≥ 0∧[(-1)bso_25] ≥ 0)

To summarize, we get the following constraints P for the following pairs.
• 397_0_MAIN_LE(x0, x1) → COND_397_0_MAIN_LE(&&(&&(>=(x1, x0), >=(x0, 0)), <=(0, +(x0, -1))), x0, x1)
• (x0[0] ≥ 0∧x1[0] ≥ 0∧[1] + x0[0] ≥ 0 ⇒ (UIncreasing(COND_397_0_MAIN_LE(&&(&&(>=(x1[0], x0[0]), >=(x0[0], 0)), <=(0, +(x0[0], -1))), x0[0], x1[0])), ≥)∧[(-1)Bound*bni_18] + [bni_18]x0[0] ≥ 0∧[(-1)bso_19] ≥ 0)

• COND_397_0_MAIN_LE(TRUE, x0, x1) → 397_0_MAIN_LE(+(x0, -1), 1)
• (x0[0] ≥ 0∧x1[0] ≥ 0∧[1] + x0[0] ≥ 0 ⇒ (UIncreasing(397_0_MAIN_LE(+(x0[1], -1), 1)), ≥)∧[(-1)Bound*bni_20] + [bni_20]x0[0] ≥ 0∧[1 + (-1)bso_21] ≥ 0)
• (x0[0] ≥ 0∧x1[0] ≥ 0∧[1] + x0[0] ≥ 0 ⇒ (UIncreasing(397_0_MAIN_LE(+(x0[1], -1), 1)), ≥)∧[(-1)Bound*bni_20] + [bni_20]x0[0] ≥ 0∧[1 + (-1)bso_21] ≥ 0)

• 397_0_MAIN_LE(x0, x1) → COND_397_0_MAIN_LE1(&&(>=(x1, 1), <(x1, x0)), x0, x1)
• (x1[2] ≥ 0∧x0[2] ≥ 0 ⇒ (UIncreasing(COND_397_0_MAIN_LE1(&&(>=(x1[2], 1), <(x1[2], x0[2])), x0[2], x1[2])), ≥)∧[bni_22 + (-1)Bound*bni_22] + [bni_22]x1[2] + [bni_22]x0[2] ≥ 0∧[(-1)bso_23] ≥ 0)

• COND_397_0_MAIN_LE1(TRUE, x0, x1) → 397_0_MAIN_LE(x0, *(2, x1))
• (x1[2] ≥ 0∧x0[2] ≥ 0 ⇒ (UIncreasing(397_0_MAIN_LE(x0[3], *(2, x1[3]))), ≥)∧[bni_24 + (-1)Bound*bni_24] + [bni_24]x1[2] + [bni_24]x0[2] ≥ 0∧[(-1)bso_25] ≥ 0)
• (x1[2] ≥ 0∧x0[2] ≥ 0 ⇒ (UIncreasing(397_0_MAIN_LE(x0[3], *(2, x1[3]))), ≥)∧[bni_24 + (-1)Bound*bni_24] + [bni_24]x1[2] + [bni_24]x0[2] ≥ 0∧[(-1)bso_25] ≥ 0)

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

POL(TRUE) = [2]
POL(FALSE) = [1]
POL(397_0_MAIN_LE(x1, x2)) = [-1] + x1
POL(COND_397_0_MAIN_LE(x1, x2, x3)) = [-1] + x2
POL(&&(x1, x2)) = [-1]
POL(>=(x1, x2)) = [-1]
POL(0) = 0
POL(<=(x1, x2)) = [-1]
POL(+(x1, x2)) = x1 + x2
POL(-1) = [-1]
POL(1) = [1]
POL(COND_397_0_MAIN_LE1(x1, x2, x3)) = [-1] + x2
POL(<(x1, x2)) = [-1]
POL(*(x1, x2)) = x1·x2
POL(2) = [2]

The following pairs are in P>:

COND_397_0_MAIN_LE(TRUE, x0[1], x1[1]) → 397_0_MAIN_LE(+(x0[1], -1), 1)

The following pairs are in Pbound:

397_0_MAIN_LE(x0[0], x1[0]) → COND_397_0_MAIN_LE(&&(&&(>=(x1[0], x0[0]), >=(x0[0], 0)), <=(0, +(x0[0], -1))), x0[0], x1[0])
COND_397_0_MAIN_LE(TRUE, x0[1], x1[1]) → 397_0_MAIN_LE(+(x0[1], -1), 1)
397_0_MAIN_LE(x0[2], x1[2]) → COND_397_0_MAIN_LE1(&&(>=(x1[2], 1), <(x1[2], x0[2])), x0[2], x1[2])
COND_397_0_MAIN_LE1(TRUE, x0[3], x1[3]) → 397_0_MAIN_LE(x0[3], *(2, x1[3]))

The following pairs are in P:

397_0_MAIN_LE(x0[0], x1[0]) → COND_397_0_MAIN_LE(&&(&&(>=(x1[0], x0[0]), >=(x0[0], 0)), <=(0, +(x0[0], -1))), x0[0], x1[0])
397_0_MAIN_LE(x0[2], x1[2]) → COND_397_0_MAIN_LE1(&&(>=(x1[2], 1), <(x1[2], x0[2])), x0[2], x1[2])
COND_397_0_MAIN_LE1(TRUE, x0[3], x1[3]) → 397_0_MAIN_LE(x0[3], *(2, x1[3]))

At least the following rules have been oriented under context sensitive arithmetic replacement:

TRUE1&&(TRUE, TRUE)1
FALSE1&&(TRUE, FALSE)1
FALSE1&&(FALSE, TRUE)1
FALSE1&&(FALSE, FALSE)1

### (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): 397_0_MAIN_LE(x0[0], x1[0]) → COND_397_0_MAIN_LE(x1[0] >= x0[0] && x0[0] >= 0 && 0 <= x0[0] + -1, x0[0], x1[0])
(2): 397_0_MAIN_LE(x0[2], x1[2]) → COND_397_0_MAIN_LE1(x1[2] >= 1 && x1[2] < x0[2], x0[2], x1[2])
(3): COND_397_0_MAIN_LE1(TRUE, x0[3], x1[3]) → 397_0_MAIN_LE(x0[3], 2 * x1[3])

(3) -> (0), if ((x0[3]* x0[0])∧(2 * x1[3]* x1[0]))

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

(2) -> (3), if ((x1[2] >= 1 && x1[2] < x0[2]* TRUE)∧(x0[2]* x0[3])∧(x1[2]* x1[3]))

The set Q is empty.

### (7) IDependencyGraphProof (EQUIVALENT transformation)

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

### (8) Obligation:

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

The following domains are used:

Integer, Boolean

R is empty.

The integer pair graph contains the following rules and edges:
(3): COND_397_0_MAIN_LE1(TRUE, x0[3], x1[3]) → 397_0_MAIN_LE(x0[3], 2 * x1[3])
(2): 397_0_MAIN_LE(x0[2], x1[2]) → COND_397_0_MAIN_LE1(x1[2] >= 1 && x1[2] < x0[2], x0[2], x1[2])

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

(2) -> (3), if ((x1[2] >= 1 && x1[2] < x0[2]* TRUE)∧(x0[2]* x0[3])∧(x1[2]* x1[3]))

The set Q is empty.

### (9) 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_397_0_MAIN_LE1(TRUE, x0[3], x1[3]) → 397_0_MAIN_LE(x0[3], *(2, x1[3])) the following chains were created:
• We consider the chain 397_0_MAIN_LE(x0[2], x1[2]) → COND_397_0_MAIN_LE1(&&(>=(x1[2], 1), <(x1[2], x0[2])), x0[2], x1[2]), COND_397_0_MAIN_LE1(TRUE, x0[3], x1[3]) → 397_0_MAIN_LE(x0[3], *(2, x1[3])), 397_0_MAIN_LE(x0[2], x1[2]) → COND_397_0_MAIN_LE1(&&(>=(x1[2], 1), <(x1[2], x0[2])), x0[2], x1[2]) which results in the following constraint:

(1)    (&&(>=(x1[2], 1), <(x1[2], x0[2]))=TRUEx0[2]=x0[3]x1[2]=x1[3]x0[3]=x0[2]1*(2, x1[3])=x1[2]1COND_397_0_MAIN_LE1(TRUE, x0[3], x1[3])≥NonInfC∧COND_397_0_MAIN_LE1(TRUE, x0[3], x1[3])≥397_0_MAIN_LE(x0[3], *(2, x1[3]))∧(UIncreasing(397_0_MAIN_LE(x0[3], *(2, x1[3]))), ≥))

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

(2)    (>=(x1[2], 1)=TRUE<(x1[2], x0[2])=TRUECOND_397_0_MAIN_LE1(TRUE, x0[2], x1[2])≥NonInfC∧COND_397_0_MAIN_LE1(TRUE, x0[2], x1[2])≥397_0_MAIN_LE(x0[2], *(2, x1[2]))∧(UIncreasing(397_0_MAIN_LE(x0[3], *(2, x1[3]))), ≥))

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

(3)    (x1[2] + [-1] ≥ 0∧x0[2] + [-1] + [-1]x1[2] ≥ 0 ⇒ (UIncreasing(397_0_MAIN_LE(x0[3], *(2, x1[3]))), ≥)∧[(-1)bni_13 + (-1)Bound*bni_13] + [(-1)bni_13]x1[2] + [bni_13]x0[2] ≥ 0∧[(-1)bso_14] + x1[2] ≥ 0)

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

(4)    (x1[2] + [-1] ≥ 0∧x0[2] + [-1] + [-1]x1[2] ≥ 0 ⇒ (UIncreasing(397_0_MAIN_LE(x0[3], *(2, x1[3]))), ≥)∧[(-1)bni_13 + (-1)Bound*bni_13] + [(-1)bni_13]x1[2] + [bni_13]x0[2] ≥ 0∧[(-1)bso_14] + x1[2] ≥ 0)

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

(5)    (x1[2] + [-1] ≥ 0∧x0[2] + [-1] + [-1]x1[2] ≥ 0 ⇒ (UIncreasing(397_0_MAIN_LE(x0[3], *(2, x1[3]))), ≥)∧[(-1)bni_13 + (-1)Bound*bni_13] + [(-1)bni_13]x1[2] + [bni_13]x0[2] ≥ 0∧[(-1)bso_14] + x1[2] ≥ 0)

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

(6)    (x1[2] ≥ 0∧x0[2] + [-2] + [-1]x1[2] ≥ 0 ⇒ (UIncreasing(397_0_MAIN_LE(x0[3], *(2, x1[3]))), ≥)∧[(-2)bni_13 + (-1)Bound*bni_13] + [(-1)bni_13]x1[2] + [bni_13]x0[2] ≥ 0∧[1 + (-1)bso_14] + x1[2] ≥ 0)

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

(7)    (x1[2] ≥ 0∧x0[2] ≥ 0 ⇒ (UIncreasing(397_0_MAIN_LE(x0[3], *(2, x1[3]))), ≥)∧[(-1)Bound*bni_13] + [bni_13]x0[2] ≥ 0∧[1 + (-1)bso_14] + x1[2] ≥ 0)

For Pair 397_0_MAIN_LE(x0[2], x1[2]) → COND_397_0_MAIN_LE1(&&(>=(x1[2], 1), <(x1[2], x0[2])), x0[2], x1[2]) the following chains were created:
• We consider the chain 397_0_MAIN_LE(x0[2], x1[2]) → COND_397_0_MAIN_LE1(&&(>=(x1[2], 1), <(x1[2], x0[2])), x0[2], x1[2]), COND_397_0_MAIN_LE1(TRUE, x0[3], x1[3]) → 397_0_MAIN_LE(x0[3], *(2, x1[3])) which results in the following constraint:

(8)    (&&(>=(x1[2], 1), <(x1[2], x0[2]))=TRUEx0[2]=x0[3]x1[2]=x1[3]397_0_MAIN_LE(x0[2], x1[2])≥NonInfC∧397_0_MAIN_LE(x0[2], x1[2])≥COND_397_0_MAIN_LE1(&&(>=(x1[2], 1), <(x1[2], x0[2])), x0[2], x1[2])∧(UIncreasing(COND_397_0_MAIN_LE1(&&(>=(x1[2], 1), <(x1[2], x0[2])), x0[2], x1[2])), ≥))

We simplified constraint (8) using rules (IV), (IDP_BOOLEAN) which results in the following new constraint:

(9)    (>=(x1[2], 1)=TRUE<(x1[2], x0[2])=TRUE397_0_MAIN_LE(x0[2], x1[2])≥NonInfC∧397_0_MAIN_LE(x0[2], x1[2])≥COND_397_0_MAIN_LE1(&&(>=(x1[2], 1), <(x1[2], x0[2])), x0[2], x1[2])∧(UIncreasing(COND_397_0_MAIN_LE1(&&(>=(x1[2], 1), <(x1[2], x0[2])), x0[2], x1[2])), ≥))

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

(10)    (x1[2] + [-1] ≥ 0∧x0[2] + [-1] + [-1]x1[2] ≥ 0 ⇒ (UIncreasing(COND_397_0_MAIN_LE1(&&(>=(x1[2], 1), <(x1[2], x0[2])), x0[2], x1[2])), ≥)∧[(-1)bni_15 + (-1)Bound*bni_15] + [(-1)bni_15]x1[2] + [bni_15]x0[2] ≥ 0∧[(-1)bso_16] ≥ 0)

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

(11)    (x1[2] + [-1] ≥ 0∧x0[2] + [-1] + [-1]x1[2] ≥ 0 ⇒ (UIncreasing(COND_397_0_MAIN_LE1(&&(>=(x1[2], 1), <(x1[2], x0[2])), x0[2], x1[2])), ≥)∧[(-1)bni_15 + (-1)Bound*bni_15] + [(-1)bni_15]x1[2] + [bni_15]x0[2] ≥ 0∧[(-1)bso_16] ≥ 0)

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

(12)    (x1[2] + [-1] ≥ 0∧x0[2] + [-1] + [-1]x1[2] ≥ 0 ⇒ (UIncreasing(COND_397_0_MAIN_LE1(&&(>=(x1[2], 1), <(x1[2], x0[2])), x0[2], x1[2])), ≥)∧[(-1)bni_15 + (-1)Bound*bni_15] + [(-1)bni_15]x1[2] + [bni_15]x0[2] ≥ 0∧[(-1)bso_16] ≥ 0)

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

(13)    (x1[2] ≥ 0∧x0[2] + [-2] + [-1]x1[2] ≥ 0 ⇒ (UIncreasing(COND_397_0_MAIN_LE1(&&(>=(x1[2], 1), <(x1[2], x0[2])), x0[2], x1[2])), ≥)∧[(-2)bni_15 + (-1)Bound*bni_15] + [(-1)bni_15]x1[2] + [bni_15]x0[2] ≥ 0∧[(-1)bso_16] ≥ 0)

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

(14)    (x1[2] ≥ 0∧x0[2] ≥ 0 ⇒ (UIncreasing(COND_397_0_MAIN_LE1(&&(>=(x1[2], 1), <(x1[2], x0[2])), x0[2], x1[2])), ≥)∧[(-1)Bound*bni_15] + [bni_15]x0[2] ≥ 0∧[(-1)bso_16] ≥ 0)

To summarize, we get the following constraints P for the following pairs.
• COND_397_0_MAIN_LE1(TRUE, x0[3], x1[3]) → 397_0_MAIN_LE(x0[3], *(2, x1[3]))
• (x1[2] ≥ 0∧x0[2] ≥ 0 ⇒ (UIncreasing(397_0_MAIN_LE(x0[3], *(2, x1[3]))), ≥)∧[(-1)Bound*bni_13] + [bni_13]x0[2] ≥ 0∧[1 + (-1)bso_14] + x1[2] ≥ 0)

• 397_0_MAIN_LE(x0[2], x1[2]) → COND_397_0_MAIN_LE1(&&(>=(x1[2], 1), <(x1[2], x0[2])), x0[2], x1[2])
• (x1[2] ≥ 0∧x0[2] ≥ 0 ⇒ (UIncreasing(COND_397_0_MAIN_LE1(&&(>=(x1[2], 1), <(x1[2], x0[2])), x0[2], x1[2])), ≥)∧[(-1)Bound*bni_15] + [bni_15]x0[2] ≥ 0∧[(-1)bso_16] ≥ 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) = [2]
POL(COND_397_0_MAIN_LE1(x1, x2, x3)) = [-1] + [-1]x3 + x2
POL(397_0_MAIN_LE(x1, x2)) = [-1] + [-1]x2 + x1
POL(*(x1, x2)) = x1·x2
POL(2) = [2]
POL(&&(x1, x2)) = [-1]
POL(>=(x1, x2)) = [-1]
POL(1) = [1]
POL(<(x1, x2)) = [-1]

The following pairs are in P>:

COND_397_0_MAIN_LE1(TRUE, x0[3], x1[3]) → 397_0_MAIN_LE(x0[3], *(2, x1[3]))

The following pairs are in Pbound:

COND_397_0_MAIN_LE1(TRUE, x0[3], x1[3]) → 397_0_MAIN_LE(x0[3], *(2, x1[3]))
397_0_MAIN_LE(x0[2], x1[2]) → COND_397_0_MAIN_LE1(&&(>=(x1[2], 1), <(x1[2], x0[2])), x0[2], x1[2])

The following pairs are in P:

397_0_MAIN_LE(x0[2], x1[2]) → COND_397_0_MAIN_LE1(&&(>=(x1[2], 1), <(x1[2], x0[2])), x0[2], x1[2])

At least the following rules have been oriented under context sensitive arithmetic replacement:

TRUE1&&(TRUE, TRUE)1
FALSE1&&(TRUE, FALSE)1
FALSE1&&(FALSE, TRUE)1
FALSE1&&(FALSE, FALSE)1

### (10) Obligation:

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

The following domains are used:

Boolean, Integer

R is empty.

The integer pair graph contains the following rules and edges:
(2): 397_0_MAIN_LE(x0[2], x1[2]) → COND_397_0_MAIN_LE1(x1[2] >= 1 && x1[2] < x0[2], x0[2], x1[2])

The set Q is empty.

### (11) IDependencyGraphProof (EQUIVALENT transformation)

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