(0) Obligation:

JBC Problem based on JBC Program:
Manifest-Version: 1.0 Created-By: 1.6.0_16 (Sun Microsystems Inc.) Main-Class: PastaA10
`/** * 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 PastaA10 {    public static void main(String[] args) {        Random.args = args;        int x = Random.random();        int y = Random.random();        while (x != y) {            if (x > y) {                y++;            } else {                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:
PastaA10.main([Ljava/lang/String;)V: Graph of 161 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:

629_0_main_Load(x1, x2, x3, x4) → 629_0_main_Load(x2, x3, x4)
Cond_629_0_main_Load1(x1, x2, x3, x4, x5) → Cond_629_0_main_Load1(x1, x3, x4, x5)
Cond_629_0_main_Load(x1, x2, x3, x4, x5) → Cond_629_0_main_Load(x1, x3, x4, x5)

Filtered duplicate args:

Cond_629_0_main_Load1(x1, x2, x3, x4) → Cond_629_0_main_Load1(x1, x3, x4)
Cond_629_0_main_Load(x1, x2, x3, x4) → Cond_629_0_main_Load(x1, x3, x4)

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): 629_0_MAIN_LOAD(x1[0], x0[0]) → COND_629_0_MAIN_LOAD(x1[0] > x0[0] && x0[0] >= 0, x1[0], x0[0])
(1): COND_629_0_MAIN_LOAD(TRUE, x1[1], x0[1]) → 629_0_MAIN_LOAD(x1[1], x0[1] + 1)
(2): 629_0_MAIN_LOAD(x1[2], x0[2]) → COND_629_0_MAIN_LOAD1(x1[2] >= 0 && x1[2] < x0[2], x1[2], x0[2])
(3): COND_629_0_MAIN_LOAD1(TRUE, x1[3], x0[3]) → 629_0_MAIN_LOAD(x1[3] + 1, x0[3])

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

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

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

(2) -> (3), if ((x1[2] >= 0 && x1[2] < x0[2]* TRUE)∧(x1[2]* x1[3])∧(x0[2]* x0[3]))

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

(3) -> (2), if ((x1[3] + 1* x1[2])∧(x0[3]* x0[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 629_0_MAIN_LOAD(x1, x0) → COND_629_0_MAIN_LOAD(&&(>(x1, x0), >=(x0, 0)), x1, x0) the following chains were created:
• We consider the chain 629_0_MAIN_LOAD(x1[0], x0[0]) → COND_629_0_MAIN_LOAD(&&(>(x1[0], x0[0]), >=(x0[0], 0)), x1[0], x0[0]), COND_629_0_MAIN_LOAD(TRUE, x1[1], x0[1]) → 629_0_MAIN_LOAD(x1[1], +(x0[1], 1)) which results in the following constraint:

(1)    (&&(>(x1[0], x0[0]), >=(x0[0], 0))=TRUEx1[0]=x1[1]x0[0]=x0[1]629_0_MAIN_LOAD(x1[0], x0[0])≥NonInfC∧629_0_MAIN_LOAD(x1[0], x0[0])≥COND_629_0_MAIN_LOAD(&&(>(x1[0], x0[0]), >=(x0[0], 0)), x1[0], x0[0])∧(UIncreasing(COND_629_0_MAIN_LOAD(&&(>(x1[0], x0[0]), >=(x0[0], 0)), x1[0], x0[0])), ≥))

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

(2)    (>(x1[0], x0[0])=TRUE>=(x0[0], 0)=TRUE629_0_MAIN_LOAD(x1[0], x0[0])≥NonInfC∧629_0_MAIN_LOAD(x1[0], x0[0])≥COND_629_0_MAIN_LOAD(&&(>(x1[0], x0[0]), >=(x0[0], 0)), x1[0], x0[0])∧(UIncreasing(COND_629_0_MAIN_LOAD(&&(>(x1[0], x0[0]), >=(x0[0], 0)), x1[0], x0[0])), ≥))

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

(3)    (x1[0] + [-1] + [-1]x0[0] ≥ 0∧x0[0] ≥ 0 ⇒ (UIncreasing(COND_629_0_MAIN_LOAD(&&(>(x1[0], x0[0]), >=(x0[0], 0)), x1[0], x0[0])), ≥)∧[(-1)bni_15 + (-1)Bound*bni_15] + [bni_15]max{[-1]x0[0] + x1[0], x0[0] + [-1]x1[0]} ≥ 0∧[(-1)bso_16] + max{[-1]x0[0] + x1[0], x0[0] + [-1]x1[0]} + [-1]max{[-1]x0[0] + x1[0], x0[0] + [-1]x1[0]} ≥ 0)

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

(4)    (x1[0] + [-1] + [-1]x0[0] ≥ 0∧x0[0] ≥ 0 ⇒ (UIncreasing(COND_629_0_MAIN_LOAD(&&(>(x1[0], x0[0]), >=(x0[0], 0)), x1[0], x0[0])), ≥)∧[(-1)bni_15 + (-1)Bound*bni_15] + [bni_15]max{[-1]x0[0] + x1[0], x0[0] + [-1]x1[0]} ≥ 0∧[(-1)bso_16] + max{[-1]x0[0] + x1[0], x0[0] + [-1]x1[0]} + [-1]max{[-1]x0[0] + x1[0], x0[0] + [-1]x1[0]} ≥ 0)

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

(5)    (x1[0] + [-1] + [-1]x0[0] ≥ 0∧x0[0] ≥ 0∧[-2]x0[0] + [2]x1[0] ≥ 0 ⇒ (UIncreasing(COND_629_0_MAIN_LOAD(&&(>(x1[0], x0[0]), >=(x0[0], 0)), x1[0], x0[0])), ≥)∧[(-1)bni_15 + (-1)Bound*bni_15] + [(-1)bni_15]x0[0] + [bni_15]x1[0] ≥ 0∧[(-1)bso_16] ≥ 0)

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

(6)    (x1[0] ≥ 0∧x0[0] ≥ 0∧[2] + [2]x1[0] ≥ 0 ⇒ (UIncreasing(COND_629_0_MAIN_LOAD(&&(>(x1[0], x0[0]), >=(x0[0], 0)), x1[0], x0[0])), ≥)∧[(-1)Bound*bni_15] + [bni_15]x1[0] ≥ 0∧[(-1)bso_16] ≥ 0)

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

(7)    (x1[0] ≥ 0∧x0[0] ≥ 0∧[1] + x1[0] ≥ 0 ⇒ (UIncreasing(COND_629_0_MAIN_LOAD(&&(>(x1[0], x0[0]), >=(x0[0], 0)), x1[0], x0[0])), ≥)∧[(-1)Bound*bni_15] + [bni_15]x1[0] ≥ 0∧[(-1)bso_16] ≥ 0)

For Pair COND_629_0_MAIN_LOAD(TRUE, x1, x0) → 629_0_MAIN_LOAD(x1, +(x0, 1)) the following chains were created:
• We consider the chain COND_629_0_MAIN_LOAD(TRUE, x1[1], x0[1]) → 629_0_MAIN_LOAD(x1[1], +(x0[1], 1)), 629_0_MAIN_LOAD(x1[0], x0[0]) → COND_629_0_MAIN_LOAD(&&(>(x1[0], x0[0]), >=(x0[0], 0)), x1[0], x0[0]), COND_629_0_MAIN_LOAD(TRUE, x1[1], x0[1]) → 629_0_MAIN_LOAD(x1[1], +(x0[1], 1)) which results in the following constraint:

(8)    (x1[1]=x1[0]+(x0[1], 1)=x0[0]&&(>(x1[0], x0[0]), >=(x0[0], 0))=TRUEx1[0]=x1[1]1x0[0]=x0[1]1COND_629_0_MAIN_LOAD(TRUE, x1[1]1, x0[1]1)≥NonInfC∧COND_629_0_MAIN_LOAD(TRUE, x1[1]1, x0[1]1)≥629_0_MAIN_LOAD(x1[1]1, +(x0[1]1, 1))∧(UIncreasing(629_0_MAIN_LOAD(x1[1]1, +(x0[1]1, 1))), ≥))

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

(9)    (>(x1[0], +(x0[1], 1))=TRUE>=(+(x0[1], 1), 0)=TRUECOND_629_0_MAIN_LOAD(TRUE, x1[0], +(x0[1], 1))≥NonInfC∧COND_629_0_MAIN_LOAD(TRUE, x1[0], +(x0[1], 1))≥629_0_MAIN_LOAD(x1[0], +(+(x0[1], 1), 1))∧(UIncreasing(629_0_MAIN_LOAD(x1[1]1, +(x0[1]1, 1))), ≥))

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

(10)    (x1[0] + [-2] + [-1]x0[1] ≥ 0∧x0[1] + [1] ≥ 0 ⇒ (UIncreasing(629_0_MAIN_LOAD(x1[1]1, +(x0[1]1, 1))), ≥)∧[(-1)bni_17 + (-1)Bound*bni_17] + [bni_17]max{[-1] + [-1]x0[1] + x1[0], [1] + x0[1] + [-1]x1[0]} ≥ 0∧[(-1)bso_18] + max{[-1] + [-1]x0[1] + x1[0], [1] + x0[1] + [-1]x1[0]} + [-1]max{[-2] + [-1]x0[1] + x1[0], [2] + x0[1] + [-1]x1[0]} ≥ 0)

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

(11)    (x1[0] + [-2] + [-1]x0[1] ≥ 0∧x0[1] + [1] ≥ 0 ⇒ (UIncreasing(629_0_MAIN_LOAD(x1[1]1, +(x0[1]1, 1))), ≥)∧[(-1)bni_17 + (-1)Bound*bni_17] + [bni_17]max{[-1] + [-1]x0[1] + x1[0], [1] + x0[1] + [-1]x1[0]} ≥ 0∧[(-1)bso_18] + max{[-1] + [-1]x0[1] + x1[0], [1] + x0[1] + [-1]x1[0]} + [-1]max{[-2] + [-1]x0[1] + x1[0], [2] + x0[1] + [-1]x1[0]} ≥ 0)

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

(12)    (x1[0] + [-2] + [-1]x0[1] ≥ 0∧x0[1] + [1] ≥ 0∧[-2] + [-2]x0[1] + [2]x1[0] ≥ 0∧[-4] + [-2]x0[1] + [2]x1[0] ≥ 0 ⇒ (UIncreasing(629_0_MAIN_LOAD(x1[1]1, +(x0[1]1, 1))), ≥)∧[(-2)bni_17 + (-1)Bound*bni_17] + [(-1)bni_17]x0[1] + [bni_17]x1[0] ≥ 0∧[1 + (-1)bso_18] ≥ 0)

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

(13)    (x1[0] ≥ 0∧x0[1] + [1] ≥ 0∧[2] + [2]x1[0] ≥ 0∧[2]x1[0] ≥ 0 ⇒ (UIncreasing(629_0_MAIN_LOAD(x1[1]1, +(x0[1]1, 1))), ≥)∧[(-1)Bound*bni_17] + [bni_17]x1[0] ≥ 0∧[1 + (-1)bso_18] ≥ 0)

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

(14)    (x1[0] ≥ 0∧x0[1] + [1] ≥ 0∧[2] + [2]x1[0] ≥ 0∧[2]x1[0] ≥ 0∧x0[1] ≥ 0 ⇒ (UIncreasing(629_0_MAIN_LOAD(x1[1]1, +(x0[1]1, 1))), ≥)∧[(-1)Bound*bni_17] + [bni_17]x1[0] ≥ 0∧[1 + (-1)bso_18] ≥ 0)

(15)    (x1[0] ≥ 0∧[-1]x0[1] + [1] ≥ 0∧[2] + [2]x1[0] ≥ 0∧[2]x1[0] ≥ 0∧x0[1] ≥ 0 ⇒ (UIncreasing(629_0_MAIN_LOAD(x1[1]1, +(x0[1]1, 1))), ≥)∧[(-1)Bound*bni_17] + [bni_17]x1[0] ≥ 0∧[1 + (-1)bso_18] ≥ 0)

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

(16)    (x1[0] ≥ 0∧x0[1] + [1] ≥ 0∧x0[1] ≥ 0∧[1] + x1[0] ≥ 0∧x1[0] ≥ 0 ⇒ (UIncreasing(629_0_MAIN_LOAD(x1[1]1, +(x0[1]1, 1))), ≥)∧[(-1)Bound*bni_17] + [bni_17]x1[0] ≥ 0∧[1 + (-1)bso_18] ≥ 0)

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

(17)    (x1[0] ≥ 0∧[-1]x0[1] + [1] ≥ 0∧x0[1] ≥ 0∧[1] + x1[0] ≥ 0∧x1[0] ≥ 0 ⇒ (UIncreasing(629_0_MAIN_LOAD(x1[1]1, +(x0[1]1, 1))), ≥)∧[(-1)Bound*bni_17] + [bni_17]x1[0] ≥ 0∧[1 + (-1)bso_18] ≥ 0)

• We consider the chain COND_629_0_MAIN_LOAD1(TRUE, x1[3], x0[3]) → 629_0_MAIN_LOAD(+(x1[3], 1), x0[3]), 629_0_MAIN_LOAD(x1[0], x0[0]) → COND_629_0_MAIN_LOAD(&&(>(x1[0], x0[0]), >=(x0[0], 0)), x1[0], x0[0]), COND_629_0_MAIN_LOAD(TRUE, x1[1], x0[1]) → 629_0_MAIN_LOAD(x1[1], +(x0[1], 1)) which results in the following constraint:

(18)    (+(x1[3], 1)=x1[0]x0[3]=x0[0]&&(>(x1[0], x0[0]), >=(x0[0], 0))=TRUEx1[0]=x1[1]x0[0]=x0[1]COND_629_0_MAIN_LOAD(TRUE, x1[1], x0[1])≥NonInfC∧COND_629_0_MAIN_LOAD(TRUE, x1[1], x0[1])≥629_0_MAIN_LOAD(x1[1], +(x0[1], 1))∧(UIncreasing(629_0_MAIN_LOAD(x1[1], +(x0[1], 1))), ≥))

We simplified constraint (18) using rules (III), (IDP_BOOLEAN) which results in the following new constraint:

(19)    (>(+(x1[3], 1), x0[0])=TRUE>=(x0[0], 0)=TRUECOND_629_0_MAIN_LOAD(TRUE, +(x1[3], 1), x0[0])≥NonInfC∧COND_629_0_MAIN_LOAD(TRUE, +(x1[3], 1), x0[0])≥629_0_MAIN_LOAD(+(x1[3], 1), +(x0[0], 1))∧(UIncreasing(629_0_MAIN_LOAD(x1[1], +(x0[1], 1))), ≥))

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

(20)    (x1[3] + [-1]x0[0] ≥ 0∧x0[0] ≥ 0 ⇒ (UIncreasing(629_0_MAIN_LOAD(x1[1], +(x0[1], 1))), ≥)∧[(-1)bni_17 + (-1)Bound*bni_17] + [bni_17]max{[1] + [-1]x0[0] + x1[3], [-1] + x0[0] + [-1]x1[3]} ≥ 0∧[(-1)bso_18] + max{[1] + [-1]x0[0] + x1[3], [-1] + x0[0] + [-1]x1[3]} + [-1]max{[-1]x0[0] + x1[3], x0[0] + [-1]x1[3]} ≥ 0)

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

(21)    (x1[3] + [-1]x0[0] ≥ 0∧x0[0] ≥ 0 ⇒ (UIncreasing(629_0_MAIN_LOAD(x1[1], +(x0[1], 1))), ≥)∧[(-1)bni_17 + (-1)Bound*bni_17] + [bni_17]max{[1] + [-1]x0[0] + x1[3], [-1] + x0[0] + [-1]x1[3]} ≥ 0∧[(-1)bso_18] + max{[1] + [-1]x0[0] + x1[3], [-1] + x0[0] + [-1]x1[3]} + [-1]max{[-1]x0[0] + x1[3], x0[0] + [-1]x1[3]} ≥ 0)

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

(22)    (x1[3] + [-1]x0[0] ≥ 0∧x0[0] ≥ 0∧[2] + [-2]x0[0] + [2]x1[3] ≥ 0∧[-2]x0[0] + [2]x1[3] ≥ 0 ⇒ (UIncreasing(629_0_MAIN_LOAD(x1[1], +(x0[1], 1))), ≥)∧[(-1)Bound*bni_17] + [(-1)bni_17]x0[0] + [bni_17]x1[3] ≥ 0∧[1 + (-1)bso_18] ≥ 0)

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

(23)    (x1[3] ≥ 0∧x0[0] ≥ 0∧[2] + [2]x1[3] ≥ 0∧[2]x1[3] ≥ 0 ⇒ (UIncreasing(629_0_MAIN_LOAD(x1[1], +(x0[1], 1))), ≥)∧[(-1)Bound*bni_17] + [bni_17]x1[3] ≥ 0∧[1 + (-1)bso_18] ≥ 0)

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

(24)    (x1[3] ≥ 0∧x0[0] ≥ 0∧[1] + x1[3] ≥ 0∧x1[3] ≥ 0 ⇒ (UIncreasing(629_0_MAIN_LOAD(x1[1], +(x0[1], 1))), ≥)∧[(-1)Bound*bni_17] + [bni_17]x1[3] ≥ 0∧[1 + (-1)bso_18] ≥ 0)

For Pair 629_0_MAIN_LOAD(x1, x0) → COND_629_0_MAIN_LOAD1(&&(>=(x1, 0), <(x1, x0)), x1, x0) the following chains were created:
• We consider the chain 629_0_MAIN_LOAD(x1[2], x0[2]) → COND_629_0_MAIN_LOAD1(&&(>=(x1[2], 0), <(x1[2], x0[2])), x1[2], x0[2]), COND_629_0_MAIN_LOAD1(TRUE, x1[3], x0[3]) → 629_0_MAIN_LOAD(+(x1[3], 1), x0[3]) which results in the following constraint:

(25)    (&&(>=(x1[2], 0), <(x1[2], x0[2]))=TRUEx1[2]=x1[3]x0[2]=x0[3]629_0_MAIN_LOAD(x1[2], x0[2])≥NonInfC∧629_0_MAIN_LOAD(x1[2], x0[2])≥COND_629_0_MAIN_LOAD1(&&(>=(x1[2], 0), <(x1[2], x0[2])), x1[2], x0[2])∧(UIncreasing(COND_629_0_MAIN_LOAD1(&&(>=(x1[2], 0), <(x1[2], x0[2])), x1[2], x0[2])), ≥))

We simplified constraint (25) using rules (IV), (IDP_BOOLEAN) which results in the following new constraint:

(26)    (>=(x1[2], 0)=TRUE<(x1[2], x0[2])=TRUE629_0_MAIN_LOAD(x1[2], x0[2])≥NonInfC∧629_0_MAIN_LOAD(x1[2], x0[2])≥COND_629_0_MAIN_LOAD1(&&(>=(x1[2], 0), <(x1[2], x0[2])), x1[2], x0[2])∧(UIncreasing(COND_629_0_MAIN_LOAD1(&&(>=(x1[2], 0), <(x1[2], x0[2])), x1[2], x0[2])), ≥))

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

(27)    (x1[2] ≥ 0∧x0[2] + [-1] + [-1]x1[2] ≥ 0 ⇒ (UIncreasing(COND_629_0_MAIN_LOAD1(&&(>=(x1[2], 0), <(x1[2], x0[2])), x1[2], x0[2])), ≥)∧[(-1)bni_19 + (-1)Bound*bni_19] + [bni_19]max{[-1]x0[2] + x1[2], x0[2] + [-1]x1[2]} ≥ 0∧[(-1)bso_20] + max{[-1]x0[2] + x1[2], x0[2] + [-1]x1[2]} + [-1]max{[-1]x0[2] + x1[2], x0[2] + [-1]x1[2]} ≥ 0)

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

(28)    (x1[2] ≥ 0∧x0[2] + [-1] + [-1]x1[2] ≥ 0 ⇒ (UIncreasing(COND_629_0_MAIN_LOAD1(&&(>=(x1[2], 0), <(x1[2], x0[2])), x1[2], x0[2])), ≥)∧[(-1)bni_19 + (-1)Bound*bni_19] + [bni_19]max{[-1]x0[2] + x1[2], x0[2] + [-1]x1[2]} ≥ 0∧[(-1)bso_20] + max{[-1]x0[2] + x1[2], x0[2] + [-1]x1[2]} + [-1]max{[-1]x0[2] + x1[2], x0[2] + [-1]x1[2]} ≥ 0)

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

(29)    (x1[2] ≥ 0∧x0[2] + [-1] + [-1]x1[2] ≥ 0∧[-1] + [2]x0[2] + [-2]x1[2] ≥ 0 ⇒ (UIncreasing(COND_629_0_MAIN_LOAD1(&&(>=(x1[2], 0), <(x1[2], x0[2])), x1[2], x0[2])), ≥)∧[(-1)bni_19 + (-1)Bound*bni_19] + [bni_19]x0[2] + [(-1)bni_19]x1[2] ≥ 0∧[(-1)bso_20] ≥ 0)

We simplified constraint (29) using rule (IDP_SMT_SPLIT) which results in the following new constraint:

(30)    (x1[2] ≥ 0∧x0[2] ≥ 0∧[1] + [2]x0[2] ≥ 0 ⇒ (UIncreasing(COND_629_0_MAIN_LOAD1(&&(>=(x1[2], 0), <(x1[2], x0[2])), x1[2], x0[2])), ≥)∧[(-1)Bound*bni_19] + [bni_19]x0[2] ≥ 0∧[(-1)bso_20] ≥ 0)

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

(31)    (x1[2] ≥ 0∧x0[2] ≥ 0∧x0[2] ≥ 0 ⇒ (UIncreasing(COND_629_0_MAIN_LOAD1(&&(>=(x1[2], 0), <(x1[2], x0[2])), x1[2], x0[2])), ≥)∧[(-1)Bound*bni_19] + [bni_19]x0[2] ≥ 0∧[(-1)bso_20] ≥ 0)

For Pair COND_629_0_MAIN_LOAD1(TRUE, x1, x0) → 629_0_MAIN_LOAD(+(x1, 1), x0) the following chains were created:
• We consider the chain COND_629_0_MAIN_LOAD(TRUE, x1[1], x0[1]) → 629_0_MAIN_LOAD(x1[1], +(x0[1], 1)), 629_0_MAIN_LOAD(x1[2], x0[2]) → COND_629_0_MAIN_LOAD1(&&(>=(x1[2], 0), <(x1[2], x0[2])), x1[2], x0[2]), COND_629_0_MAIN_LOAD1(TRUE, x1[3], x0[3]) → 629_0_MAIN_LOAD(+(x1[3], 1), x0[3]) which results in the following constraint:

(32)    (x1[1]=x1[2]+(x0[1], 1)=x0[2]&&(>=(x1[2], 0), <(x1[2], x0[2]))=TRUEx1[2]=x1[3]x0[2]=x0[3]COND_629_0_MAIN_LOAD1(TRUE, x1[3], x0[3])≥NonInfC∧COND_629_0_MAIN_LOAD1(TRUE, x1[3], x0[3])≥629_0_MAIN_LOAD(+(x1[3], 1), x0[3])∧(UIncreasing(629_0_MAIN_LOAD(+(x1[3], 1), x0[3])), ≥))

We simplified constraint (32) using rules (III), (IDP_BOOLEAN) which results in the following new constraint:

(33)    (>=(x1[2], 0)=TRUE<(x1[2], +(x0[1], 1))=TRUECOND_629_0_MAIN_LOAD1(TRUE, x1[2], +(x0[1], 1))≥NonInfC∧COND_629_0_MAIN_LOAD1(TRUE, x1[2], +(x0[1], 1))≥629_0_MAIN_LOAD(+(x1[2], 1), +(x0[1], 1))∧(UIncreasing(629_0_MAIN_LOAD(+(x1[3], 1), x0[3])), ≥))

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

(34)    (x1[2] ≥ 0∧x0[1] + [-1]x1[2] ≥ 0 ⇒ (UIncreasing(629_0_MAIN_LOAD(+(x1[3], 1), x0[3])), ≥)∧[(-1)bni_21 + (-1)Bound*bni_21] + [bni_21]max{[-1] + [-1]x0[1] + x1[2], [1] + x0[1] + [-1]x1[2]} ≥ 0∧[(-1)bso_22] + max{[-1] + [-1]x0[1] + x1[2], [1] + x0[1] + [-1]x1[2]} + [-1]max{[-1]x0[1] + x1[2], x0[1] + [-1]x1[2]} ≥ 0)

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

(35)    (x1[2] ≥ 0∧x0[1] + [-1]x1[2] ≥ 0 ⇒ (UIncreasing(629_0_MAIN_LOAD(+(x1[3], 1), x0[3])), ≥)∧[(-1)bni_21 + (-1)Bound*bni_21] + [bni_21]max{[-1] + [-1]x0[1] + x1[2], [1] + x0[1] + [-1]x1[2]} ≥ 0∧[(-1)bso_22] + max{[-1] + [-1]x0[1] + x1[2], [1] + x0[1] + [-1]x1[2]} + [-1]max{[-1]x0[1] + x1[2], x0[1] + [-1]x1[2]} ≥ 0)

We simplified constraint (35) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraints:

(36)    (x1[2] ≥ 0∧x0[1] + [-1]x1[2] ≥ 0∧[1] + [2]x0[1] + [-2]x1[2] ≥ 0∧[-2]x0[1] + [2]x1[2] ≥ 0 ⇒ (UIncreasing(629_0_MAIN_LOAD(+(x1[3], 1), x0[3])), ≥)∧[(-1)Bound*bni_21] + [bni_21]x0[1] + [(-1)bni_21]x1[2] ≥ 0∧[1 + (-1)bso_22] + [2]x0[1] + [-2]x1[2] ≥ 0)

(37)    (x1[2] ≥ 0∧x0[1] + [-1]x1[2] ≥ 0∧[1] + [2]x0[1] + [-2]x1[2] ≥ 0∧[-1] + [2]x0[1] + [-2]x1[2] ≥ 0 ⇒ (UIncreasing(629_0_MAIN_LOAD(+(x1[3], 1), x0[3])), ≥)∧[(-1)Bound*bni_21] + [bni_21]x0[1] + [(-1)bni_21]x1[2] ≥ 0∧[1 + (-1)bso_22] ≥ 0)

We simplified constraint (36) using rule (IDP_SMT_SPLIT) which results in the following new constraint:

(38)    (x0[1] + x1[2] ≥ 0∧[-1]x1[2] ≥ 0∧[1] + [-2]x1[2] ≥ 0∧[2]x1[2] ≥ 0 ⇒ (UIncreasing(629_0_MAIN_LOAD(+(x1[3], 1), x0[3])), ≥)∧[(-1)Bound*bni_21] + [(-1)bni_21]x1[2] ≥ 0∧[1 + (-1)bso_22] + [-2]x1[2] ≥ 0)

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

(39)    (x1[2] ≥ 0∧x0[1] ≥ 0∧[1] + [2]x0[1] ≥ 0∧[-1] + [2]x0[1] ≥ 0 ⇒ (UIncreasing(629_0_MAIN_LOAD(+(x1[3], 1), x0[3])), ≥)∧[(-1)Bound*bni_21] + [bni_21]x0[1] ≥ 0∧[1 + (-1)bso_22] ≥ 0)

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

(40)    (x0[1] ≥ 0∧0 ≥ 0∧[1] ≥ 0∧0 ≥ 0 ⇒ (UIncreasing(629_0_MAIN_LOAD(+(x1[3], 1), x0[3])), ≥)∧[(-1)Bound*bni_21] ≥ 0∧[1 + (-1)bso_22] ≥ 0)

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

(41)    (x1[2] ≥ 0∧x0[1] ≥ 0∧x0[1] ≥ 0∧x0[1] ≥ 0 ⇒ (UIncreasing(629_0_MAIN_LOAD(+(x1[3], 1), x0[3])), ≥)∧[(-1)Bound*bni_21] + [bni_21]x0[1] ≥ 0∧[1 + (-1)bso_22] ≥ 0)

• We consider the chain COND_629_0_MAIN_LOAD1(TRUE, x1[3], x0[3]) → 629_0_MAIN_LOAD(+(x1[3], 1), x0[3]), 629_0_MAIN_LOAD(x1[2], x0[2]) → COND_629_0_MAIN_LOAD1(&&(>=(x1[2], 0), <(x1[2], x0[2])), x1[2], x0[2]), COND_629_0_MAIN_LOAD1(TRUE, x1[3], x0[3]) → 629_0_MAIN_LOAD(+(x1[3], 1), x0[3]) which results in the following constraint:

(42)    (+(x1[3], 1)=x1[2]x0[3]=x0[2]&&(>=(x1[2], 0), <(x1[2], x0[2]))=TRUEx1[2]=x1[3]1x0[2]=x0[3]1COND_629_0_MAIN_LOAD1(TRUE, x1[3]1, x0[3]1)≥NonInfC∧COND_629_0_MAIN_LOAD1(TRUE, x1[3]1, x0[3]1)≥629_0_MAIN_LOAD(+(x1[3]1, 1), x0[3]1)∧(UIncreasing(629_0_MAIN_LOAD(+(x1[3]1, 1), x0[3]1)), ≥))

We simplified constraint (42) using rules (III), (IDP_BOOLEAN) which results in the following new constraint:

(43)    (>=(+(x1[3], 1), 0)=TRUE<(+(x1[3], 1), x0[2])=TRUECOND_629_0_MAIN_LOAD1(TRUE, +(x1[3], 1), x0[2])≥NonInfC∧COND_629_0_MAIN_LOAD1(TRUE, +(x1[3], 1), x0[2])≥629_0_MAIN_LOAD(+(+(x1[3], 1), 1), x0[2])∧(UIncreasing(629_0_MAIN_LOAD(+(x1[3]1, 1), x0[3]1)), ≥))

We simplified constraint (43) using rule (POLY_CONSTRAINTS) which results in the following new constraint:

(44)    (x1[3] + [1] ≥ 0∧x0[2] + [-2] + [-1]x1[3] ≥ 0 ⇒ (UIncreasing(629_0_MAIN_LOAD(+(x1[3]1, 1), x0[3]1)), ≥)∧[(-1)bni_21 + (-1)Bound*bni_21] + [bni_21]max{[1] + [-1]x0[2] + x1[3], [-1] + x0[2] + [-1]x1[3]} ≥ 0∧[(-1)bso_22] + max{[1] + [-1]x0[2] + x1[3], [-1] + x0[2] + [-1]x1[3]} + [-1]max{[2] + [-1]x0[2] + x1[3], [-2] + x0[2] + [-1]x1[3]} ≥ 0)

We simplified constraint (44) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:

(45)    (x1[3] + [1] ≥ 0∧x0[2] + [-2] + [-1]x1[3] ≥ 0 ⇒ (UIncreasing(629_0_MAIN_LOAD(+(x1[3]1, 1), x0[3]1)), ≥)∧[(-1)bni_21 + (-1)Bound*bni_21] + [bni_21]max{[1] + [-1]x0[2] + x1[3], [-1] + x0[2] + [-1]x1[3]} ≥ 0∧[(-1)bso_22] + max{[1] + [-1]x0[2] + x1[3], [-1] + x0[2] + [-1]x1[3]} + [-1]max{[2] + [-1]x0[2] + x1[3], [-2] + x0[2] + [-1]x1[3]} ≥ 0)

We simplified constraint (45) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraints:

(46)    (x1[3] + [1] ≥ 0∧x0[2] + [-2] + [-1]x1[3] ≥ 0∧[-3] + [2]x0[2] + [-2]x1[3] ≥ 0∧[4] + [-2]x0[2] + [2]x1[3] ≥ 0 ⇒ (UIncreasing(629_0_MAIN_LOAD(+(x1[3]1, 1), x0[3]1)), ≥)∧[(-2)bni_21 + (-1)Bound*bni_21] + [bni_21]x0[2] + [(-1)bni_21]x1[3] ≥ 0∧[-3 + (-1)bso_22] + [2]x0[2] + [-2]x1[3] ≥ 0)

(47)    (x1[3] + [1] ≥ 0∧x0[2] + [-2] + [-1]x1[3] ≥ 0∧[-3] + [2]x0[2] + [-2]x1[3] ≥ 0∧[-5] + [2]x0[2] + [-2]x1[3] ≥ 0 ⇒ (UIncreasing(629_0_MAIN_LOAD(+(x1[3]1, 1), x0[3]1)), ≥)∧[(-2)bni_21 + (-1)Bound*bni_21] + [bni_21]x0[2] + [(-1)bni_21]x1[3] ≥ 0∧[1 + (-1)bso_22] ≥ 0)

We simplified constraint (46) using rule (IDP_SMT_SPLIT) which results in the following new constraint:

(48)    (x0[2] + [-1] + [-1]x1[3] ≥ 0∧x1[3] ≥ 0∧[1] + [2]x1[3] ≥ 0∧[-2]x1[3] ≥ 0 ⇒ (UIncreasing(629_0_MAIN_LOAD(+(x1[3]1, 1), x0[3]1)), ≥)∧[(-1)Bound*bni_21] + [bni_21]x1[3] ≥ 0∧[1 + (-1)bso_22] + [2]x1[3] ≥ 0)

We simplified constraint (47) using rule (IDP_SMT_SPLIT) which results in the following new constraint:

(49)    (x0[2] + [-1] + [-1]x1[3] ≥ 0∧x1[3] ≥ 0∧[1] + [2]x1[3] ≥ 0∧[-1] + [2]x1[3] ≥ 0 ⇒ (UIncreasing(629_0_MAIN_LOAD(+(x1[3]1, 1), x0[3]1)), ≥)∧[(-1)Bound*bni_21] + [bni_21]x1[3] ≥ 0∧[1 + (-1)bso_22] ≥ 0)

We simplified constraint (48) using rule (IDP_SMT_SPLIT) which results in the following new constraint:

(50)    (x0[2] + [-1]x1[3] ≥ 0∧x1[3] ≥ 0∧[1] + [2]x1[3] ≥ 0∧[-2]x1[3] ≥ 0 ⇒ (UIncreasing(629_0_MAIN_LOAD(+(x1[3]1, 1), x0[3]1)), ≥)∧[(-1)Bound*bni_21] + [bni_21]x1[3] ≥ 0∧[1 + (-1)bso_22] + [2]x1[3] ≥ 0)

We simplified constraint (50) using rule (IDP_SMT_SPLIT) which results in the following new constraint:

(51)    (x0[2] ≥ 0∧0 ≥ 0∧[1] ≥ 0∧0 ≥ 0 ⇒ (UIncreasing(629_0_MAIN_LOAD(+(x1[3]1, 1), x0[3]1)), ≥)∧[(-1)Bound*bni_21] ≥ 0∧[1 + (-1)bso_22] ≥ 0)

We simplified constraint (49) using rule (IDP_SMT_SPLIT) which results in the following new constraint:

(52)    (x0[2] ≥ 0∧x1[3] ≥ 0∧[1] + [2]x1[3] ≥ 0∧[-1] + [2]x1[3] ≥ 0 ⇒ (UIncreasing(629_0_MAIN_LOAD(+(x1[3]1, 1), x0[3]1)), ≥)∧[(-1)Bound*bni_21] + [bni_21]x1[3] ≥ 0∧[1 + (-1)bso_22] ≥ 0)

We simplified constraint (52) using rule (IDP_POLY_GCD) which results in the following new constraint:

(53)    (x0[2] ≥ 0∧x1[3] ≥ 0∧x1[3] ≥ 0∧x1[3] ≥ 0 ⇒ (UIncreasing(629_0_MAIN_LOAD(+(x1[3]1, 1), x0[3]1)), ≥)∧[(-1)Bound*bni_21] + [bni_21]x1[3] ≥ 0∧[1 + (-1)bso_22] ≥ 0)

To summarize, we get the following constraints P for the following pairs.
• 629_0_MAIN_LOAD(x1, x0) → COND_629_0_MAIN_LOAD(&&(>(x1, x0), >=(x0, 0)), x1, x0)
• (x1[0] ≥ 0∧x0[0] ≥ 0∧[1] + x1[0] ≥ 0 ⇒ (UIncreasing(COND_629_0_MAIN_LOAD(&&(>(x1[0], x0[0]), >=(x0[0], 0)), x1[0], x0[0])), ≥)∧[(-1)Bound*bni_15] + [bni_15]x1[0] ≥ 0∧[(-1)bso_16] ≥ 0)

• COND_629_0_MAIN_LOAD(TRUE, x1, x0) → 629_0_MAIN_LOAD(x1, +(x0, 1))
• (x1[0] ≥ 0∧x0[1] + [1] ≥ 0∧x0[1] ≥ 0∧[1] + x1[0] ≥ 0∧x1[0] ≥ 0 ⇒ (UIncreasing(629_0_MAIN_LOAD(x1[1]1, +(x0[1]1, 1))), ≥)∧[(-1)Bound*bni_17] + [bni_17]x1[0] ≥ 0∧[1 + (-1)bso_18] ≥ 0)
• (x1[0] ≥ 0∧[-1]x0[1] + [1] ≥ 0∧x0[1] ≥ 0∧[1] + x1[0] ≥ 0∧x1[0] ≥ 0 ⇒ (UIncreasing(629_0_MAIN_LOAD(x1[1]1, +(x0[1]1, 1))), ≥)∧[(-1)Bound*bni_17] + [bni_17]x1[0] ≥ 0∧[1 + (-1)bso_18] ≥ 0)
• (x1[3] ≥ 0∧x0[0] ≥ 0∧[1] + x1[3] ≥ 0∧x1[3] ≥ 0 ⇒ (UIncreasing(629_0_MAIN_LOAD(x1[1], +(x0[1], 1))), ≥)∧[(-1)Bound*bni_17] + [bni_17]x1[3] ≥ 0∧[1 + (-1)bso_18] ≥ 0)

• 629_0_MAIN_LOAD(x1, x0) → COND_629_0_MAIN_LOAD1(&&(>=(x1, 0), <(x1, x0)), x1, x0)
• (x1[2] ≥ 0∧x0[2] ≥ 0∧x0[2] ≥ 0 ⇒ (UIncreasing(COND_629_0_MAIN_LOAD1(&&(>=(x1[2], 0), <(x1[2], x0[2])), x1[2], x0[2])), ≥)∧[(-1)Bound*bni_19] + [bni_19]x0[2] ≥ 0∧[(-1)bso_20] ≥ 0)

• COND_629_0_MAIN_LOAD1(TRUE, x1, x0) → 629_0_MAIN_LOAD(+(x1, 1), x0)
• (x0[1] ≥ 0∧0 ≥ 0∧[1] ≥ 0∧0 ≥ 0 ⇒ (UIncreasing(629_0_MAIN_LOAD(+(x1[3], 1), x0[3])), ≥)∧[(-1)Bound*bni_21] ≥ 0∧[1 + (-1)bso_22] ≥ 0)
• (x1[2] ≥ 0∧x0[1] ≥ 0∧x0[1] ≥ 0∧x0[1] ≥ 0 ⇒ (UIncreasing(629_0_MAIN_LOAD(+(x1[3], 1), x0[3])), ≥)∧[(-1)Bound*bni_21] + [bni_21]x0[1] ≥ 0∧[1 + (-1)bso_22] ≥ 0)
• (x0[2] ≥ 0∧0 ≥ 0∧[1] ≥ 0∧0 ≥ 0 ⇒ (UIncreasing(629_0_MAIN_LOAD(+(x1[3]1, 1), x0[3]1)), ≥)∧[(-1)Bound*bni_21] ≥ 0∧[1 + (-1)bso_22] ≥ 0)
• (x0[2] ≥ 0∧x1[3] ≥ 0∧x1[3] ≥ 0∧x1[3] ≥ 0 ⇒ (UIncreasing(629_0_MAIN_LOAD(+(x1[3]1, 1), x0[3]1)), ≥)∧[(-1)Bound*bni_21] + [bni_21]x1[3] ≥ 0∧[1 + (-1)bso_22] ≥ 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(629_0_MAIN_LOAD(x1, x2)) = [-1] + max{[-1]x2 + x1, x2 + [-1]x1}
POL(COND_629_0_MAIN_LOAD(x1, x2, x3)) = [-1] + max{[-1]x3 + x2, x3 + [-1]x2}
POL(&&(x1, x2)) = [-1]
POL(>(x1, x2)) = [-1]
POL(>=(x1, x2)) = [-1]
POL(0) = 0
POL(+(x1, x2)) = x1 + x2
POL(1) = [1]
POL(COND_629_0_MAIN_LOAD1(x1, x2, x3)) = [-1] + max{[-1]x3 + x2, x3 + [-1]x2}
POL(<(x1, x2)) = [-1]

The following pairs are in P>:

COND_629_0_MAIN_LOAD(TRUE, x1[1], x0[1]) → 629_0_MAIN_LOAD(x1[1], +(x0[1], 1))
COND_629_0_MAIN_LOAD1(TRUE, x1[3], x0[3]) → 629_0_MAIN_LOAD(+(x1[3], 1), x0[3])

The following pairs are in Pbound:

629_0_MAIN_LOAD(x1[0], x0[0]) → COND_629_0_MAIN_LOAD(&&(>(x1[0], x0[0]), >=(x0[0], 0)), x1[0], x0[0])
COND_629_0_MAIN_LOAD(TRUE, x1[1], x0[1]) → 629_0_MAIN_LOAD(x1[1], +(x0[1], 1))
629_0_MAIN_LOAD(x1[2], x0[2]) → COND_629_0_MAIN_LOAD1(&&(>=(x1[2], 0), <(x1[2], x0[2])), x1[2], x0[2])
COND_629_0_MAIN_LOAD1(TRUE, x1[3], x0[3]) → 629_0_MAIN_LOAD(+(x1[3], 1), x0[3])

The following pairs are in P:

629_0_MAIN_LOAD(x1[0], x0[0]) → COND_629_0_MAIN_LOAD(&&(>(x1[0], x0[0]), >=(x0[0], 0)), x1[0], x0[0])
629_0_MAIN_LOAD(x1[2], x0[2]) → COND_629_0_MAIN_LOAD1(&&(>=(x1[2], 0), <(x1[2], x0[2])), x1[2], x0[2])

There are no usable 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): 629_0_MAIN_LOAD(x1[0], x0[0]) → COND_629_0_MAIN_LOAD(x1[0] > x0[0] && x0[0] >= 0, x1[0], x0[0])
(2): 629_0_MAIN_LOAD(x1[2], x0[2]) → COND_629_0_MAIN_LOAD1(x1[2] >= 0 && x1[2] < x0[2], x1[2], x0[2])

The set Q is empty.

(7) IDependencyGraphProof (EQUIVALENT transformation)

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