### (0) Obligation:

JBC Problem based on JBC Program:
Manifest-Version: 1.0 Created-By: 1.6.0_16 (Sun Microsystems Inc.) Main-Class: PastaB18
`/** * 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 PastaB18 {    public static void main(String[] args) {        Random.args = args;        int x = Random.random();        int y = Random.random();        while (x > 0 && y > 0) {            if (x > y) {                while (x > 0) {                    x--;                }            } else {                while (y > 0) {                    y--;                }            }        }            }}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:
PastaB18.main([Ljava/lang/String;)V: Graph of 177 nodes with 1 SCC.

### (3) FIGtoITRSProof (SOUND transformation)

Transformed FIGraph SCCs to IDPs. Logs:

Log for SCC 0:

Generated 32 rules for P and 5 rules for R.

Combined rules. Obtained 6 rules for P and 1 rules for R.

Filtered ground terms:

816_0_main_LE(x1, x2, x3, x4) → 816_0_main_LE(x2, x3, x4)
Cond_816_0_main_LE(x1, x2, x3, x4, x5) → Cond_816_0_main_LE(x1, x3, x4, x5)
529_0_main_LE(x1, x2, x3, x4) → 529_0_main_LE(x2, x3, x4)
Cond_529_0_main_LE1(x1, x2, x3, x4, x5) → Cond_529_0_main_LE1(x1, x3, x4, x5)
814_0_main_LE(x1, x2, x3, x4) → 814_0_main_LE(x2, x3, x4)
Cond_814_0_main_LE(x1, x2, x3, x4, x5) → Cond_814_0_main_LE(x1, x3, x4, x5)
Cond_529_0_main_LE(x1, x2, x3, x4, x5) → Cond_529_0_main_LE(x1, x3, x4, x5)
593_0_main_Return(x1) → 593_0_main_Return

Filtered duplicate args:

816_0_main_LE(x1, x2, x3) → 816_0_main_LE(x2, x3)
Cond_816_0_main_LE(x1, x2, x3, x4) → Cond_816_0_main_LE(x1, x3, x4)
529_0_main_LE(x1, x2, x3) → 529_0_main_LE(x2, x3)
Cond_529_0_main_LE1(x1, x2, x3, x4) → Cond_529_0_main_LE1(x1, x3, x4)
814_0_main_LE(x1, x2, x3) → 814_0_main_LE(x1, x3)
Cond_814_0_main_LE(x1, x2, x3, x4) → Cond_814_0_main_LE(x1, x2, x4)
Cond_529_0_main_LE(x1, x2, x3, x4) → Cond_529_0_main_LE(x1, x3, x4)

Combined rules. Obtained 6 rules for P and 1 rules for R.

Finished conversion. Obtained 6 rules for P and 1 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

The ITRS R consists of the following rules:
529_0_main_LE(x1, 0) → 593_0_main_Return

The integer pair graph contains the following rules and edges:
(0): 529_0_MAIN_LE(x1[0], x0[0]) → COND_529_0_MAIN_LE(x1[0] >= x0[0] && x1[0] > 0 && x0[0] > 0, x1[0], x0[0])
(1): COND_529_0_MAIN_LE(TRUE, x1[1], x0[1]) → 814_0_MAIN_LE(x0[1], x1[1])
(2): 814_0_MAIN_LE(x0[2], 0) → 529_0_MAIN_LE(0, x0[2])
(3): 814_0_MAIN_LE(x0[3], x1[3]) → COND_814_0_MAIN_LE(x1[3] > 0, x0[3], x1[3])
(4): COND_814_0_MAIN_LE(TRUE, x0[4], x1[4]) → 814_0_MAIN_LE(x0[4], x1[4] + -1)
(5): 529_0_MAIN_LE(x1[5], x0[5]) → COND_529_0_MAIN_LE1(x1[5] > 0 && x1[5] < x0[5] && x0[5] > 0, x1[5], x0[5])
(6): COND_529_0_MAIN_LE1(TRUE, x1[6], x0[6]) → 816_0_MAIN_LE(x1[6], x0[6])
(7): 816_0_MAIN_LE(x1[7], 0) → 529_0_MAIN_LE(x1[7], 0)
(8): 816_0_MAIN_LE(x1[8], x0[8]) → COND_816_0_MAIN_LE(x0[8] > 0, x1[8], x0[8])
(9): COND_816_0_MAIN_LE(TRUE, x1[9], x0[9]) → 816_0_MAIN_LE(x1[9], x0[9] + -1)

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

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

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

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

(2) -> (5), if ((0* x1[5])∧(x0[2]* x0[5]))

(3) -> (4), if ((x1[3] > 0* TRUE)∧(x0[3]* x0[4])∧(x1[3]* x1[4]))

(4) -> (2), if ((x0[4]* x0[2])∧(x1[4] + -1* 0))

(4) -> (3), if ((x0[4]* x0[3])∧(x1[4] + -1* x1[3]))

(5) -> (6), if ((x1[5] > 0 && x1[5] < x0[5] && x0[5] > 0* TRUE)∧(x1[5]* x1[6])∧(x0[5]* x0[6]))

(6) -> (7), if ((x1[6]* x1[7])∧(x0[6]* 0))

(6) -> (8), if ((x1[6]* x1[8])∧(x0[6]* x0[8]))

(7) -> (0), if ((x1[7]* x1[0])∧(0* x0[0]))

(7) -> (5), if ((x1[7]* x1[5])∧(0* x0[5]))

(8) -> (9), if ((x0[8] > 0* TRUE)∧(x1[8]* x1[9])∧(x0[8]* x0[9]))

(9) -> (7), if ((x1[9]* x1[7])∧(x0[9] + -1* 0))

(9) -> (8), if ((x1[9]* x1[8])∧(x0[9] + -1* x0[8]))

The set Q consists of the following terms:
529_0_main_LE(x0, 0)

### (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 529_0_MAIN_LE(x1, x0) → COND_529_0_MAIN_LE(&&(&&(>=(x1, x0), >(x1, 0)), >(x0, 0)), x1, x0) the following chains were created:
• We consider the chain 529_0_MAIN_LE(x1[0], x0[0]) → COND_529_0_MAIN_LE(&&(&&(>=(x1[0], x0[0]), >(x1[0], 0)), >(x0[0], 0)), x1[0], x0[0]), COND_529_0_MAIN_LE(TRUE, x1[1], x0[1]) → 814_0_MAIN_LE(x0[1], x1[1]) which results in the following constraint:

(1)    (&&(&&(>=(x1[0], x0[0]), >(x1[0], 0)), >(x0[0], 0))=TRUEx1[0]=x1[1]x0[0]=x0[1]529_0_MAIN_LE(x1[0], x0[0])≥NonInfC∧529_0_MAIN_LE(x1[0], x0[0])≥COND_529_0_MAIN_LE(&&(&&(>=(x1[0], x0[0]), >(x1[0], 0)), >(x0[0], 0)), x1[0], x0[0])∧(UIncreasing(COND_529_0_MAIN_LE(&&(&&(>=(x1[0], x0[0]), >(x1[0], 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)    (>(x0[0], 0)=TRUE>=(x1[0], x0[0])=TRUE>(x1[0], 0)=TRUE529_0_MAIN_LE(x1[0], x0[0])≥NonInfC∧529_0_MAIN_LE(x1[0], x0[0])≥COND_529_0_MAIN_LE(&&(&&(>=(x1[0], x0[0]), >(x1[0], 0)), >(x0[0], 0)), x1[0], x0[0])∧(UIncreasing(COND_529_0_MAIN_LE(&&(&&(>=(x1[0], x0[0]), >(x1[0], 0)), >(x0[0], 0)), x1[0], x0[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∧x1[0] + [-1] ≥ 0 ⇒ (UIncreasing(COND_529_0_MAIN_LE(&&(&&(>=(x1[0], x0[0]), >(x1[0], 0)), >(x0[0], 0)), x1[0], x0[0])), ≥)∧[(-1)bni_36 + (-1)Bound*bni_36] + [bni_36]x0[0] + [(2)bni_36]x1[0] ≥ 0∧[(-1)bso_37] + x1[0] ≥ 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∧x1[0] + [-1] ≥ 0 ⇒ (UIncreasing(COND_529_0_MAIN_LE(&&(&&(>=(x1[0], x0[0]), >(x1[0], 0)), >(x0[0], 0)), x1[0], x0[0])), ≥)∧[(-1)bni_36 + (-1)Bound*bni_36] + [bni_36]x0[0] + [(2)bni_36]x1[0] ≥ 0∧[(-1)bso_37] + x1[0] ≥ 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∧x1[0] + [-1] ≥ 0 ⇒ (UIncreasing(COND_529_0_MAIN_LE(&&(&&(>=(x1[0], x0[0]), >(x1[0], 0)), >(x0[0], 0)), x1[0], x0[0])), ≥)∧[(-1)bni_36 + (-1)Bound*bni_36] + [bni_36]x0[0] + [(2)bni_36]x1[0] ≥ 0∧[(-1)bso_37] + x1[0] ≥ 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∧x1[0] + [-1] ≥ 0 ⇒ (UIncreasing(COND_529_0_MAIN_LE(&&(&&(>=(x1[0], x0[0]), >(x1[0], 0)), >(x0[0], 0)), x1[0], x0[0])), ≥)∧[(-1)Bound*bni_36] + [bni_36]x0[0] + [(2)bni_36]x1[0] ≥ 0∧[(-1)bso_37] + x1[0] ≥ 0)

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

(7)    (x0[0] ≥ 0∧x1[0] ≥ 0∧x0[0] + x1[0] ≥ 0 ⇒ (UIncreasing(COND_529_0_MAIN_LE(&&(&&(>=(x1[0], x0[0]), >(x1[0], 0)), >(x0[0], 0)), x1[0], x0[0])), ≥)∧[(-1)Bound*bni_36 + (2)bni_36] + [(3)bni_36]x0[0] + [(2)bni_36]x1[0] ≥ 0∧[1 + (-1)bso_37] + x0[0] + x1[0] ≥ 0)

For Pair COND_529_0_MAIN_LE(TRUE, x1, x0) → 814_0_MAIN_LE(x0, x1) the following chains were created:
• We consider the chain COND_529_0_MAIN_LE(TRUE, x1[1], x0[1]) → 814_0_MAIN_LE(x0[1], x1[1]), 814_0_MAIN_LE(x0[2], 0) → 529_0_MAIN_LE(0, x0[2]) which results in the following constraint:

(8)    (x0[1]=x0[2]x1[1]=0COND_529_0_MAIN_LE(TRUE, x1[1], x0[1])≥NonInfC∧COND_529_0_MAIN_LE(TRUE, x1[1], x0[1])≥814_0_MAIN_LE(x0[1], x1[1])∧(UIncreasing(814_0_MAIN_LE(x0[1], x1[1])), ≥))

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

(9)    (COND_529_0_MAIN_LE(TRUE, 0, x0[1])≥NonInfC∧COND_529_0_MAIN_LE(TRUE, 0, x0[1])≥814_0_MAIN_LE(x0[1], 0)∧(UIncreasing(814_0_MAIN_LE(x0[1], x1[1])), ≥))

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

(10)    ((UIncreasing(814_0_MAIN_LE(x0[1], x1[1])), ≥)∧[(-1)bso_39] ≥ 0)

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

(11)    ((UIncreasing(814_0_MAIN_LE(x0[1], x1[1])), ≥)∧[(-1)bso_39] ≥ 0)

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

(12)    ((UIncreasing(814_0_MAIN_LE(x0[1], x1[1])), ≥)∧[(-1)bso_39] ≥ 0)

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

(13)    ((UIncreasing(814_0_MAIN_LE(x0[1], x1[1])), ≥)∧0 = 0∧[(-1)bso_39] ≥ 0)

• We consider the chain COND_529_0_MAIN_LE(TRUE, x1[1], x0[1]) → 814_0_MAIN_LE(x0[1], x1[1]), 814_0_MAIN_LE(x0[3], x1[3]) → COND_814_0_MAIN_LE(>(x1[3], 0), x0[3], x1[3]) which results in the following constraint:

(14)    (x0[1]=x0[3]x1[1]=x1[3]COND_529_0_MAIN_LE(TRUE, x1[1], x0[1])≥NonInfC∧COND_529_0_MAIN_LE(TRUE, x1[1], x0[1])≥814_0_MAIN_LE(x0[1], x1[1])∧(UIncreasing(814_0_MAIN_LE(x0[1], x1[1])), ≥))

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

(15)    (COND_529_0_MAIN_LE(TRUE, x1[1], x0[1])≥NonInfC∧COND_529_0_MAIN_LE(TRUE, x1[1], x0[1])≥814_0_MAIN_LE(x0[1], x1[1])∧(UIncreasing(814_0_MAIN_LE(x0[1], x1[1])), ≥))

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

(16)    ((UIncreasing(814_0_MAIN_LE(x0[1], x1[1])), ≥)∧[(-1)bso_39] ≥ 0)

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

(17)    ((UIncreasing(814_0_MAIN_LE(x0[1], x1[1])), ≥)∧[(-1)bso_39] ≥ 0)

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

(18)    ((UIncreasing(814_0_MAIN_LE(x0[1], x1[1])), ≥)∧[(-1)bso_39] ≥ 0)

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

(19)    ((UIncreasing(814_0_MAIN_LE(x0[1], x1[1])), ≥)∧0 = 0∧0 = 0∧[(-1)bso_39] ≥ 0)

For Pair 814_0_MAIN_LE(x0, 0) → 529_0_MAIN_LE(0, x0) the following chains were created:
• We consider the chain 814_0_MAIN_LE(x0[2], 0) → 529_0_MAIN_LE(0, x0[2]), 529_0_MAIN_LE(x1[0], x0[0]) → COND_529_0_MAIN_LE(&&(&&(>=(x1[0], x0[0]), >(x1[0], 0)), >(x0[0], 0)), x1[0], x0[0]) which results in the following constraint:

(20)    (0=x1[0]x0[2]=x0[0]814_0_MAIN_LE(x0[2], 0)≥NonInfC∧814_0_MAIN_LE(x0[2], 0)≥529_0_MAIN_LE(0, x0[2])∧(UIncreasing(529_0_MAIN_LE(0, x0[2])), ≥))

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

(21)    (814_0_MAIN_LE(x0[2], 0)≥NonInfC∧814_0_MAIN_LE(x0[2], 0)≥529_0_MAIN_LE(0, x0[2])∧(UIncreasing(529_0_MAIN_LE(0, x0[2])), ≥))

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

(22)    ((UIncreasing(529_0_MAIN_LE(0, x0[2])), ≥)∧[(-1)bso_41] ≥ 0)

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

(23)    ((UIncreasing(529_0_MAIN_LE(0, x0[2])), ≥)∧[(-1)bso_41] ≥ 0)

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

(24)    ((UIncreasing(529_0_MAIN_LE(0, x0[2])), ≥)∧[(-1)bso_41] ≥ 0)

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

(25)    ((UIncreasing(529_0_MAIN_LE(0, x0[2])), ≥)∧0 = 0∧[(-1)bso_41] ≥ 0)

• We consider the chain 814_0_MAIN_LE(x0[2], 0) → 529_0_MAIN_LE(0, x0[2]), 529_0_MAIN_LE(x1[5], x0[5]) → COND_529_0_MAIN_LE1(&&(&&(>(x1[5], 0), <(x1[5], x0[5])), >(x0[5], 0)), x1[5], x0[5]) which results in the following constraint:

(26)    (0=x1[5]x0[2]=x0[5]814_0_MAIN_LE(x0[2], 0)≥NonInfC∧814_0_MAIN_LE(x0[2], 0)≥529_0_MAIN_LE(0, x0[2])∧(UIncreasing(529_0_MAIN_LE(0, x0[2])), ≥))

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

(27)    (814_0_MAIN_LE(x0[2], 0)≥NonInfC∧814_0_MAIN_LE(x0[2], 0)≥529_0_MAIN_LE(0, x0[2])∧(UIncreasing(529_0_MAIN_LE(0, x0[2])), ≥))

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

(28)    ((UIncreasing(529_0_MAIN_LE(0, x0[2])), ≥)∧[(-1)bso_41] ≥ 0)

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

(29)    ((UIncreasing(529_0_MAIN_LE(0, x0[2])), ≥)∧[(-1)bso_41] ≥ 0)

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

(30)    ((UIncreasing(529_0_MAIN_LE(0, x0[2])), ≥)∧[(-1)bso_41] ≥ 0)

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

(31)    ((UIncreasing(529_0_MAIN_LE(0, x0[2])), ≥)∧0 = 0∧[(-1)bso_41] ≥ 0)

For Pair 814_0_MAIN_LE(x0, x1) → COND_814_0_MAIN_LE(>(x1, 0), x0, x1) the following chains were created:
• We consider the chain 814_0_MAIN_LE(x0[3], x1[3]) → COND_814_0_MAIN_LE(>(x1[3], 0), x0[3], x1[3]), COND_814_0_MAIN_LE(TRUE, x0[4], x1[4]) → 814_0_MAIN_LE(x0[4], +(x1[4], -1)) which results in the following constraint:

(32)    (>(x1[3], 0)=TRUEx0[3]=x0[4]x1[3]=x1[4]814_0_MAIN_LE(x0[3], x1[3])≥NonInfC∧814_0_MAIN_LE(x0[3], x1[3])≥COND_814_0_MAIN_LE(>(x1[3], 0), x0[3], x1[3])∧(UIncreasing(COND_814_0_MAIN_LE(>(x1[3], 0), x0[3], x1[3])), ≥))

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

(33)    (>(x1[3], 0)=TRUE814_0_MAIN_LE(x0[3], x1[3])≥NonInfC∧814_0_MAIN_LE(x0[3], x1[3])≥COND_814_0_MAIN_LE(>(x1[3], 0), x0[3], x1[3])∧(UIncreasing(COND_814_0_MAIN_LE(>(x1[3], 0), x0[3], x1[3])), ≥))

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

(34)    (x1[3] + [-1] ≥ 0 ⇒ (UIncreasing(COND_814_0_MAIN_LE(>(x1[3], 0), x0[3], x1[3])), ≥)∧[(-1)bni_42 + (-1)Bound*bni_42] + [bni_42]x1[3] + [bni_42]x0[3] ≥ 0∧[(-1)bso_43] ≥ 0)

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

(35)    (x1[3] + [-1] ≥ 0 ⇒ (UIncreasing(COND_814_0_MAIN_LE(>(x1[3], 0), x0[3], x1[3])), ≥)∧[(-1)bni_42 + (-1)Bound*bni_42] + [bni_42]x1[3] + [bni_42]x0[3] ≥ 0∧[(-1)bso_43] ≥ 0)

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

(36)    (x1[3] + [-1] ≥ 0 ⇒ (UIncreasing(COND_814_0_MAIN_LE(>(x1[3], 0), x0[3], x1[3])), ≥)∧[(-1)bni_42 + (-1)Bound*bni_42] + [bni_42]x1[3] + [bni_42]x0[3] ≥ 0∧[(-1)bso_43] ≥ 0)

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

(37)    (x1[3] + [-1] ≥ 0 ⇒ (UIncreasing(COND_814_0_MAIN_LE(>(x1[3], 0), x0[3], x1[3])), ≥)∧[bni_42] = 0∧[(-1)bni_42 + (-1)Bound*bni_42] + [bni_42]x1[3] ≥ 0∧0 = 0∧[(-1)bso_43] ≥ 0)

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

(38)    (x1[3] ≥ 0 ⇒ (UIncreasing(COND_814_0_MAIN_LE(>(x1[3], 0), x0[3], x1[3])), ≥)∧[bni_42] = 0∧[(-1)Bound*bni_42] + [bni_42]x1[3] ≥ 0∧0 = 0∧[(-1)bso_43] ≥ 0)

For Pair COND_814_0_MAIN_LE(TRUE, x0, x1) → 814_0_MAIN_LE(x0, +(x1, -1)) the following chains were created:
• We consider the chain 814_0_MAIN_LE(x0[3], x1[3]) → COND_814_0_MAIN_LE(>(x1[3], 0), x0[3], x1[3]), COND_814_0_MAIN_LE(TRUE, x0[4], x1[4]) → 814_0_MAIN_LE(x0[4], +(x1[4], -1)), 814_0_MAIN_LE(x0[2], 0) → 529_0_MAIN_LE(0, x0[2]) which results in the following constraint:

(39)    (>(x1[3], 0)=TRUEx0[3]=x0[4]x1[3]=x1[4]x0[4]=x0[2]+(x1[4], -1)=0COND_814_0_MAIN_LE(TRUE, x0[4], x1[4])≥NonInfC∧COND_814_0_MAIN_LE(TRUE, x0[4], x1[4])≥814_0_MAIN_LE(x0[4], +(x1[4], -1))∧(UIncreasing(814_0_MAIN_LE(x0[4], +(x1[4], -1))), ≥))

We simplified constraint (39) using rules (III), (IV) which results in the following new constraint:

(40)    (>(x1[3], 0)=TRUE+(x1[3], -1)=0COND_814_0_MAIN_LE(TRUE, x0[3], x1[3])≥NonInfC∧COND_814_0_MAIN_LE(TRUE, x0[3], x1[3])≥814_0_MAIN_LE(x0[3], +(x1[3], -1))∧(UIncreasing(814_0_MAIN_LE(x0[4], +(x1[4], -1))), ≥))

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

(41)    (x1[3] + [-1] ≥ 0∧x1[3] + [-1] ≥ 0 ⇒ (UIncreasing(814_0_MAIN_LE(x0[4], +(x1[4], -1))), ≥)∧[(-1)bni_44 + (-1)Bound*bni_44] + [bni_44]x1[3] + [bni_44]x0[3] ≥ 0∧[1 + (-1)bso_45] ≥ 0)

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

(42)    (x1[3] + [-1] ≥ 0∧x1[3] + [-1] ≥ 0 ⇒ (UIncreasing(814_0_MAIN_LE(x0[4], +(x1[4], -1))), ≥)∧[(-1)bni_44 + (-1)Bound*bni_44] + [bni_44]x1[3] + [bni_44]x0[3] ≥ 0∧[1 + (-1)bso_45] ≥ 0)

We simplified constraint (42) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:

(43)    (x1[3] + [-1] ≥ 0∧x1[3] + [-1] ≥ 0 ⇒ (UIncreasing(814_0_MAIN_LE(x0[4], +(x1[4], -1))), ≥)∧[(-1)bni_44 + (-1)Bound*bni_44] + [bni_44]x1[3] + [bni_44]x0[3] ≥ 0∧[1 + (-1)bso_45] ≥ 0)

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

(44)    (x1[3] + [-1] ≥ 0∧x1[3] + [-1] ≥ 0 ⇒ (UIncreasing(814_0_MAIN_LE(x0[4], +(x1[4], -1))), ≥)∧[bni_44] = 0∧[(-1)bni_44 + (-1)Bound*bni_44] + [bni_44]x1[3] ≥ 0∧0 = 0∧[1 + (-1)bso_45] ≥ 0)

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

(45)    (x1[3] ≥ 0∧x1[3] ≥ 0 ⇒ (UIncreasing(814_0_MAIN_LE(x0[4], +(x1[4], -1))), ≥)∧[bni_44] = 0∧[(-1)Bound*bni_44] + [bni_44]x1[3] ≥ 0∧0 = 0∧[1 + (-1)bso_45] ≥ 0)

• We consider the chain 814_0_MAIN_LE(x0[3], x1[3]) → COND_814_0_MAIN_LE(>(x1[3], 0), x0[3], x1[3]), COND_814_0_MAIN_LE(TRUE, x0[4], x1[4]) → 814_0_MAIN_LE(x0[4], +(x1[4], -1)), 814_0_MAIN_LE(x0[3], x1[3]) → COND_814_0_MAIN_LE(>(x1[3], 0), x0[3], x1[3]) which results in the following constraint:

(46)    (>(x1[3], 0)=TRUEx0[3]=x0[4]x1[3]=x1[4]x0[4]=x0[3]1+(x1[4], -1)=x1[3]1COND_814_0_MAIN_LE(TRUE, x0[4], x1[4])≥NonInfC∧COND_814_0_MAIN_LE(TRUE, x0[4], x1[4])≥814_0_MAIN_LE(x0[4], +(x1[4], -1))∧(UIncreasing(814_0_MAIN_LE(x0[4], +(x1[4], -1))), ≥))

We simplified constraint (46) using rules (III), (IV) which results in the following new constraint:

(47)    (>(x1[3], 0)=TRUECOND_814_0_MAIN_LE(TRUE, x0[3], x1[3])≥NonInfC∧COND_814_0_MAIN_LE(TRUE, x0[3], x1[3])≥814_0_MAIN_LE(x0[3], +(x1[3], -1))∧(UIncreasing(814_0_MAIN_LE(x0[4], +(x1[4], -1))), ≥))

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

(48)    (x1[3] + [-1] ≥ 0 ⇒ (UIncreasing(814_0_MAIN_LE(x0[4], +(x1[4], -1))), ≥)∧[(-1)bni_44 + (-1)Bound*bni_44] + [bni_44]x1[3] + [bni_44]x0[3] ≥ 0∧[1 + (-1)bso_45] ≥ 0)

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

(49)    (x1[3] + [-1] ≥ 0 ⇒ (UIncreasing(814_0_MAIN_LE(x0[4], +(x1[4], -1))), ≥)∧[(-1)bni_44 + (-1)Bound*bni_44] + [bni_44]x1[3] + [bni_44]x0[3] ≥ 0∧[1 + (-1)bso_45] ≥ 0)

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

(50)    (x1[3] + [-1] ≥ 0 ⇒ (UIncreasing(814_0_MAIN_LE(x0[4], +(x1[4], -1))), ≥)∧[(-1)bni_44 + (-1)Bound*bni_44] + [bni_44]x1[3] + [bni_44]x0[3] ≥ 0∧[1 + (-1)bso_45] ≥ 0)

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

(51)    (x1[3] + [-1] ≥ 0 ⇒ (UIncreasing(814_0_MAIN_LE(x0[4], +(x1[4], -1))), ≥)∧[bni_44] = 0∧[(-1)bni_44 + (-1)Bound*bni_44] + [bni_44]x1[3] ≥ 0∧0 = 0∧[1 + (-1)bso_45] ≥ 0)

We simplified constraint (51) using rule (IDP_SMT_SPLIT) which results in the following new constraint:

(52)    (x1[3] ≥ 0 ⇒ (UIncreasing(814_0_MAIN_LE(x0[4], +(x1[4], -1))), ≥)∧[bni_44] = 0∧[(-1)Bound*bni_44] + [bni_44]x1[3] ≥ 0∧0 = 0∧[1 + (-1)bso_45] ≥ 0)

For Pair 529_0_MAIN_LE(x1, x0) → COND_529_0_MAIN_LE1(&&(&&(>(x1, 0), <(x1, x0)), >(x0, 0)), x1, x0) the following chains were created:
• We consider the chain 529_0_MAIN_LE(x1[5], x0[5]) → COND_529_0_MAIN_LE1(&&(&&(>(x1[5], 0), <(x1[5], x0[5])), >(x0[5], 0)), x1[5], x0[5]), COND_529_0_MAIN_LE1(TRUE, x1[6], x0[6]) → 816_0_MAIN_LE(x1[6], x0[6]) which results in the following constraint:

(53)    (&&(&&(>(x1[5], 0), <(x1[5], x0[5])), >(x0[5], 0))=TRUEx1[5]=x1[6]x0[5]=x0[6]529_0_MAIN_LE(x1[5], x0[5])≥NonInfC∧529_0_MAIN_LE(x1[5], x0[5])≥COND_529_0_MAIN_LE1(&&(&&(>(x1[5], 0), <(x1[5], x0[5])), >(x0[5], 0)), x1[5], x0[5])∧(UIncreasing(COND_529_0_MAIN_LE1(&&(&&(>(x1[5], 0), <(x1[5], x0[5])), >(x0[5], 0)), x1[5], x0[5])), ≥))

We simplified constraint (53) using rules (IV), (IDP_BOOLEAN) which results in the following new constraint:

(54)    (>(x0[5], 0)=TRUE>(x1[5], 0)=TRUE<(x1[5], x0[5])=TRUE529_0_MAIN_LE(x1[5], x0[5])≥NonInfC∧529_0_MAIN_LE(x1[5], x0[5])≥COND_529_0_MAIN_LE1(&&(&&(>(x1[5], 0), <(x1[5], x0[5])), >(x0[5], 0)), x1[5], x0[5])∧(UIncreasing(COND_529_0_MAIN_LE1(&&(&&(>(x1[5], 0), <(x1[5], x0[5])), >(x0[5], 0)), x1[5], x0[5])), ≥))

We simplified constraint (54) using rule (POLY_CONSTRAINTS) which results in the following new constraint:

(55)    (x0[5] + [-1] ≥ 0∧x1[5] + [-1] ≥ 0∧x0[5] + [-1] + [-1]x1[5] ≥ 0 ⇒ (UIncreasing(COND_529_0_MAIN_LE1(&&(&&(>(x1[5], 0), <(x1[5], x0[5])), >(x0[5], 0)), x1[5], x0[5])), ≥)∧[(-1)bni_46 + (-1)Bound*bni_46] + [bni_46]x0[5] + [(2)bni_46]x1[5] ≥ 0∧[-2 + (-1)bso_47] + x0[5] ≥ 0)

We simplified constraint (55) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:

(56)    (x0[5] + [-1] ≥ 0∧x1[5] + [-1] ≥ 0∧x0[5] + [-1] + [-1]x1[5] ≥ 0 ⇒ (UIncreasing(COND_529_0_MAIN_LE1(&&(&&(>(x1[5], 0), <(x1[5], x0[5])), >(x0[5], 0)), x1[5], x0[5])), ≥)∧[(-1)bni_46 + (-1)Bound*bni_46] + [bni_46]x0[5] + [(2)bni_46]x1[5] ≥ 0∧[-2 + (-1)bso_47] + x0[5] ≥ 0)

We simplified constraint (56) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:

(57)    (x0[5] + [-1] ≥ 0∧x1[5] + [-1] ≥ 0∧x0[5] + [-1] + [-1]x1[5] ≥ 0 ⇒ (UIncreasing(COND_529_0_MAIN_LE1(&&(&&(>(x1[5], 0), <(x1[5], x0[5])), >(x0[5], 0)), x1[5], x0[5])), ≥)∧[(-1)bni_46 + (-1)Bound*bni_46] + [bni_46]x0[5] + [(2)bni_46]x1[5] ≥ 0∧[-2 + (-1)bso_47] + x0[5] ≥ 0)

We simplified constraint (57) using rule (IDP_SMT_SPLIT) which results in the following new constraint:

(58)    (x0[5] ≥ 0∧x1[5] + [-1] ≥ 0∧x0[5] + [-1]x1[5] ≥ 0 ⇒ (UIncreasing(COND_529_0_MAIN_LE1(&&(&&(>(x1[5], 0), <(x1[5], x0[5])), >(x0[5], 0)), x1[5], x0[5])), ≥)∧[(-1)Bound*bni_46] + [bni_46]x0[5] + [(2)bni_46]x1[5] ≥ 0∧[-1 + (-1)bso_47] + x0[5] ≥ 0)

We simplified constraint (58) using rule (IDP_SMT_SPLIT) which results in the following new constraint:

(59)    (x1[5] + x0[5] ≥ 0∧x1[5] + [-1] ≥ 0∧x0[5] ≥ 0 ⇒ (UIncreasing(COND_529_0_MAIN_LE1(&&(&&(>(x1[5], 0), <(x1[5], x0[5])), >(x0[5], 0)), x1[5], x0[5])), ≥)∧[(-1)Bound*bni_46] + [(3)bni_46]x1[5] + [bni_46]x0[5] ≥ 0∧[-1 + (-1)bso_47] + x1[5] + x0[5] ≥ 0)

We simplified constraint (59) using rule (IDP_SMT_SPLIT) which results in the following new constraint:

(60)    ([1] + x1[5] + x0[5] ≥ 0∧x1[5] ≥ 0∧x0[5] ≥ 0 ⇒ (UIncreasing(COND_529_0_MAIN_LE1(&&(&&(>(x1[5], 0), <(x1[5], x0[5])), >(x0[5], 0)), x1[5], x0[5])), ≥)∧[(3)bni_46 + (-1)Bound*bni_46] + [(3)bni_46]x1[5] + [bni_46]x0[5] ≥ 0∧[(-1)bso_47] + x1[5] + x0[5] ≥ 0)

For Pair COND_529_0_MAIN_LE1(TRUE, x1, x0) → 816_0_MAIN_LE(x1, x0) the following chains were created:
• We consider the chain COND_529_0_MAIN_LE1(TRUE, x1[6], x0[6]) → 816_0_MAIN_LE(x1[6], x0[6]), 816_0_MAIN_LE(x1[7], 0) → 529_0_MAIN_LE(x1[7], 0) which results in the following constraint:

(61)    (x1[6]=x1[7]x0[6]=0COND_529_0_MAIN_LE1(TRUE, x1[6], x0[6])≥NonInfC∧COND_529_0_MAIN_LE1(TRUE, x1[6], x0[6])≥816_0_MAIN_LE(x1[6], x0[6])∧(UIncreasing(816_0_MAIN_LE(x1[6], x0[6])), ≥))

We simplified constraint (61) using rules (III), (IV) which results in the following new constraint:

(62)    (COND_529_0_MAIN_LE1(TRUE, x1[6], 0)≥NonInfC∧COND_529_0_MAIN_LE1(TRUE, x1[6], 0)≥816_0_MAIN_LE(x1[6], 0)∧(UIncreasing(816_0_MAIN_LE(x1[6], x0[6])), ≥))

We simplified constraint (62) using rule (POLY_CONSTRAINTS) which results in the following new constraint:

(63)    ((UIncreasing(816_0_MAIN_LE(x1[6], x0[6])), ≥)∧[2 + (-1)bso_49] ≥ 0)

We simplified constraint (63) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:

(64)    ((UIncreasing(816_0_MAIN_LE(x1[6], x0[6])), ≥)∧[2 + (-1)bso_49] ≥ 0)

We simplified constraint (64) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:

(65)    ((UIncreasing(816_0_MAIN_LE(x1[6], x0[6])), ≥)∧[2 + (-1)bso_49] ≥ 0)

We simplified constraint (65) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint:

(66)    ((UIncreasing(816_0_MAIN_LE(x1[6], x0[6])), ≥)∧0 = 0∧[2 + (-1)bso_49] ≥ 0)

• We consider the chain COND_529_0_MAIN_LE1(TRUE, x1[6], x0[6]) → 816_0_MAIN_LE(x1[6], x0[6]), 816_0_MAIN_LE(x1[8], x0[8]) → COND_816_0_MAIN_LE(>(x0[8], 0), x1[8], x0[8]) which results in the following constraint:

(67)    (x1[6]=x1[8]x0[6]=x0[8]COND_529_0_MAIN_LE1(TRUE, x1[6], x0[6])≥NonInfC∧COND_529_0_MAIN_LE1(TRUE, x1[6], x0[6])≥816_0_MAIN_LE(x1[6], x0[6])∧(UIncreasing(816_0_MAIN_LE(x1[6], x0[6])), ≥))

We simplified constraint (67) using rule (IV) which results in the following new constraint:

(68)    (COND_529_0_MAIN_LE1(TRUE, x1[6], x0[6])≥NonInfC∧COND_529_0_MAIN_LE1(TRUE, x1[6], x0[6])≥816_0_MAIN_LE(x1[6], x0[6])∧(UIncreasing(816_0_MAIN_LE(x1[6], x0[6])), ≥))

We simplified constraint (68) using rule (POLY_CONSTRAINTS) which results in the following new constraint:

(69)    ((UIncreasing(816_0_MAIN_LE(x1[6], x0[6])), ≥)∧[2 + (-1)bso_49] ≥ 0)

We simplified constraint (69) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:

(70)    ((UIncreasing(816_0_MAIN_LE(x1[6], x0[6])), ≥)∧[2 + (-1)bso_49] ≥ 0)

We simplified constraint (70) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:

(71)    ((UIncreasing(816_0_MAIN_LE(x1[6], x0[6])), ≥)∧[2 + (-1)bso_49] ≥ 0)

We simplified constraint (71) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint:

(72)    ((UIncreasing(816_0_MAIN_LE(x1[6], x0[6])), ≥)∧0 = 0∧0 = 0∧[2 + (-1)bso_49] ≥ 0)

For Pair 816_0_MAIN_LE(x1, 0) → 529_0_MAIN_LE(x1, 0) the following chains were created:
• We consider the chain 816_0_MAIN_LE(x1[7], 0) → 529_0_MAIN_LE(x1[7], 0), 529_0_MAIN_LE(x1[0], x0[0]) → COND_529_0_MAIN_LE(&&(&&(>=(x1[0], x0[0]), >(x1[0], 0)), >(x0[0], 0)), x1[0], x0[0]) which results in the following constraint:

(73)    (x1[7]=x1[0]0=x0[0]816_0_MAIN_LE(x1[7], 0)≥NonInfC∧816_0_MAIN_LE(x1[7], 0)≥529_0_MAIN_LE(x1[7], 0)∧(UIncreasing(529_0_MAIN_LE(x1[7], 0)), ≥))

We simplified constraint (73) using rule (IV) which results in the following new constraint:

(74)    (816_0_MAIN_LE(x1[7], 0)≥NonInfC∧816_0_MAIN_LE(x1[7], 0)≥529_0_MAIN_LE(x1[7], 0)∧(UIncreasing(529_0_MAIN_LE(x1[7], 0)), ≥))

We simplified constraint (74) using rule (POLY_CONSTRAINTS) which results in the following new constraint:

(75)    ((UIncreasing(529_0_MAIN_LE(x1[7], 0)), ≥)∧[(-1)bso_51] ≥ 0)

We simplified constraint (75) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:

(76)    ((UIncreasing(529_0_MAIN_LE(x1[7], 0)), ≥)∧[(-1)bso_51] ≥ 0)

We simplified constraint (76) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:

(77)    ((UIncreasing(529_0_MAIN_LE(x1[7], 0)), ≥)∧[(-1)bso_51] ≥ 0)

We simplified constraint (77) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint:

(78)    ((UIncreasing(529_0_MAIN_LE(x1[7], 0)), ≥)∧0 = 0∧[(-1)bso_51] ≥ 0)

• We consider the chain 816_0_MAIN_LE(x1[7], 0) → 529_0_MAIN_LE(x1[7], 0), 529_0_MAIN_LE(x1[5], x0[5]) → COND_529_0_MAIN_LE1(&&(&&(>(x1[5], 0), <(x1[5], x0[5])), >(x0[5], 0)), x1[5], x0[5]) which results in the following constraint:

(79)    (x1[7]=x1[5]0=x0[5]816_0_MAIN_LE(x1[7], 0)≥NonInfC∧816_0_MAIN_LE(x1[7], 0)≥529_0_MAIN_LE(x1[7], 0)∧(UIncreasing(529_0_MAIN_LE(x1[7], 0)), ≥))

We simplified constraint (79) using rule (IV) which results in the following new constraint:

(80)    (816_0_MAIN_LE(x1[7], 0)≥NonInfC∧816_0_MAIN_LE(x1[7], 0)≥529_0_MAIN_LE(x1[7], 0)∧(UIncreasing(529_0_MAIN_LE(x1[7], 0)), ≥))

We simplified constraint (80) using rule (POLY_CONSTRAINTS) which results in the following new constraint:

(81)    ((UIncreasing(529_0_MAIN_LE(x1[7], 0)), ≥)∧[(-1)bso_51] ≥ 0)

We simplified constraint (81) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:

(82)    ((UIncreasing(529_0_MAIN_LE(x1[7], 0)), ≥)∧[(-1)bso_51] ≥ 0)

We simplified constraint (82) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:

(83)    ((UIncreasing(529_0_MAIN_LE(x1[7], 0)), ≥)∧[(-1)bso_51] ≥ 0)

We simplified constraint (83) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint:

(84)    ((UIncreasing(529_0_MAIN_LE(x1[7], 0)), ≥)∧0 = 0∧[(-1)bso_51] ≥ 0)

For Pair 816_0_MAIN_LE(x1, x0) → COND_816_0_MAIN_LE(>(x0, 0), x1, x0) the following chains were created:
• We consider the chain 816_0_MAIN_LE(x1[8], x0[8]) → COND_816_0_MAIN_LE(>(x0[8], 0), x1[8], x0[8]), COND_816_0_MAIN_LE(TRUE, x1[9], x0[9]) → 816_0_MAIN_LE(x1[9], +(x0[9], -1)) which results in the following constraint:

(85)    (>(x0[8], 0)=TRUEx1[8]=x1[9]x0[8]=x0[9]816_0_MAIN_LE(x1[8], x0[8])≥NonInfC∧816_0_MAIN_LE(x1[8], x0[8])≥COND_816_0_MAIN_LE(>(x0[8], 0), x1[8], x0[8])∧(UIncreasing(COND_816_0_MAIN_LE(>(x0[8], 0), x1[8], x0[8])), ≥))

We simplified constraint (85) using rule (IV) which results in the following new constraint:

(86)    (>(x0[8], 0)=TRUE816_0_MAIN_LE(x1[8], x0[8])≥NonInfC∧816_0_MAIN_LE(x1[8], x0[8])≥COND_816_0_MAIN_LE(>(x0[8], 0), x1[8], x0[8])∧(UIncreasing(COND_816_0_MAIN_LE(>(x0[8], 0), x1[8], x0[8])), ≥))

We simplified constraint (86) using rule (POLY_CONSTRAINTS) which results in the following new constraint:

(87)    (x0[8] + [-1] ≥ 0 ⇒ (UIncreasing(COND_816_0_MAIN_LE(>(x0[8], 0), x1[8], x0[8])), ≥)∧[(-1)bni_52 + (-1)Bound*bni_52] + [(2)bni_52]x1[8] ≥ 0∧[(-1)bso_53] ≥ 0)

We simplified constraint (87) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:

(88)    (x0[8] + [-1] ≥ 0 ⇒ (UIncreasing(COND_816_0_MAIN_LE(>(x0[8], 0), x1[8], x0[8])), ≥)∧[(-1)bni_52 + (-1)Bound*bni_52] + [(2)bni_52]x1[8] ≥ 0∧[(-1)bso_53] ≥ 0)

We simplified constraint (88) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:

(89)    (x0[8] + [-1] ≥ 0 ⇒ (UIncreasing(COND_816_0_MAIN_LE(>(x0[8], 0), x1[8], x0[8])), ≥)∧[(-1)bni_52 + (-1)Bound*bni_52] + [(2)bni_52]x1[8] ≥ 0∧[(-1)bso_53] ≥ 0)

We simplified constraint (89) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint:

(90)    (x0[8] + [-1] ≥ 0 ⇒ (UIncreasing(COND_816_0_MAIN_LE(>(x0[8], 0), x1[8], x0[8])), ≥)∧[(2)bni_52] = 0∧[(-1)bni_52 + (-1)Bound*bni_52] ≥ 0∧0 = 0∧[(-1)bso_53] ≥ 0)

We simplified constraint (90) using rule (IDP_SMT_SPLIT) which results in the following new constraint:

(91)    (x0[8] ≥ 0 ⇒ (UIncreasing(COND_816_0_MAIN_LE(>(x0[8], 0), x1[8], x0[8])), ≥)∧[(2)bni_52] = 0∧[(-1)bni_52 + (-1)Bound*bni_52] ≥ 0∧0 = 0∧[(-1)bso_53] ≥ 0)

For Pair COND_816_0_MAIN_LE(TRUE, x1, x0) → 816_0_MAIN_LE(x1, +(x0, -1)) the following chains were created:
• We consider the chain 816_0_MAIN_LE(x1[8], x0[8]) → COND_816_0_MAIN_LE(>(x0[8], 0), x1[8], x0[8]), COND_816_0_MAIN_LE(TRUE, x1[9], x0[9]) → 816_0_MAIN_LE(x1[9], +(x0[9], -1)), 816_0_MAIN_LE(x1[7], 0) → 529_0_MAIN_LE(x1[7], 0) which results in the following constraint:

(92)    (>(x0[8], 0)=TRUEx1[8]=x1[9]x0[8]=x0[9]x1[9]=x1[7]+(x0[9], -1)=0COND_816_0_MAIN_LE(TRUE, x1[9], x0[9])≥NonInfC∧COND_816_0_MAIN_LE(TRUE, x1[9], x0[9])≥816_0_MAIN_LE(x1[9], +(x0[9], -1))∧(UIncreasing(816_0_MAIN_LE(x1[9], +(x0[9], -1))), ≥))

We simplified constraint (92) using rules (III), (IV) which results in the following new constraint:

(93)    (>(x0[8], 0)=TRUE+(x0[8], -1)=0COND_816_0_MAIN_LE(TRUE, x1[8], x0[8])≥NonInfC∧COND_816_0_MAIN_LE(TRUE, x1[8], x0[8])≥816_0_MAIN_LE(x1[8], +(x0[8], -1))∧(UIncreasing(816_0_MAIN_LE(x1[9], +(x0[9], -1))), ≥))

We simplified constraint (93) using rule (POLY_CONSTRAINTS) which results in the following new constraint:

(94)    (x0[8] + [-1] ≥ 0∧x0[8] + [-1] ≥ 0 ⇒ (UIncreasing(816_0_MAIN_LE(x1[9], +(x0[9], -1))), ≥)∧[(-1)bni_54 + (-1)Bound*bni_54] + [(2)bni_54]x1[8] ≥ 0∧[(-1)bso_55] ≥ 0)

We simplified constraint (94) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:

(95)    (x0[8] + [-1] ≥ 0∧x0[8] + [-1] ≥ 0 ⇒ (UIncreasing(816_0_MAIN_LE(x1[9], +(x0[9], -1))), ≥)∧[(-1)bni_54 + (-1)Bound*bni_54] + [(2)bni_54]x1[8] ≥ 0∧[(-1)bso_55] ≥ 0)

We simplified constraint (95) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:

(96)    (x0[8] + [-1] ≥ 0∧x0[8] + [-1] ≥ 0 ⇒ (UIncreasing(816_0_MAIN_LE(x1[9], +(x0[9], -1))), ≥)∧[(-1)bni_54 + (-1)Bound*bni_54] + [(2)bni_54]x1[8] ≥ 0∧[(-1)bso_55] ≥ 0)

We simplified constraint (96) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint:

(97)    (x0[8] + [-1] ≥ 0∧x0[8] + [-1] ≥ 0 ⇒ (UIncreasing(816_0_MAIN_LE(x1[9], +(x0[9], -1))), ≥)∧[(2)bni_54] = 0∧[(-1)bni_54 + (-1)Bound*bni_54] ≥ 0∧0 = 0∧[(-1)bso_55] ≥ 0)

We simplified constraint (97) using rule (IDP_SMT_SPLIT) which results in the following new constraint:

(98)    (x0[8] ≥ 0∧x0[8] ≥ 0 ⇒ (UIncreasing(816_0_MAIN_LE(x1[9], +(x0[9], -1))), ≥)∧[(2)bni_54] = 0∧[(-1)bni_54 + (-1)Bound*bni_54] ≥ 0∧0 = 0∧[(-1)bso_55] ≥ 0)

• We consider the chain 816_0_MAIN_LE(x1[8], x0[8]) → COND_816_0_MAIN_LE(>(x0[8], 0), x1[8], x0[8]), COND_816_0_MAIN_LE(TRUE, x1[9], x0[9]) → 816_0_MAIN_LE(x1[9], +(x0[9], -1)), 816_0_MAIN_LE(x1[8], x0[8]) → COND_816_0_MAIN_LE(>(x0[8], 0), x1[8], x0[8]) which results in the following constraint:

(99)    (>(x0[8], 0)=TRUEx1[8]=x1[9]x0[8]=x0[9]x1[9]=x1[8]1+(x0[9], -1)=x0[8]1COND_816_0_MAIN_LE(TRUE, x1[9], x0[9])≥NonInfC∧COND_816_0_MAIN_LE(TRUE, x1[9], x0[9])≥816_0_MAIN_LE(x1[9], +(x0[9], -1))∧(UIncreasing(816_0_MAIN_LE(x1[9], +(x0[9], -1))), ≥))

We simplified constraint (99) using rules (III), (IV) which results in the following new constraint:

(100)    (>(x0[8], 0)=TRUECOND_816_0_MAIN_LE(TRUE, x1[8], x0[8])≥NonInfC∧COND_816_0_MAIN_LE(TRUE, x1[8], x0[8])≥816_0_MAIN_LE(x1[8], +(x0[8], -1))∧(UIncreasing(816_0_MAIN_LE(x1[9], +(x0[9], -1))), ≥))

We simplified constraint (100) using rule (POLY_CONSTRAINTS) which results in the following new constraint:

(101)    (x0[8] + [-1] ≥ 0 ⇒ (UIncreasing(816_0_MAIN_LE(x1[9], +(x0[9], -1))), ≥)∧[(-1)bni_54 + (-1)Bound*bni_54] + [(2)bni_54]x1[8] ≥ 0∧[(-1)bso_55] ≥ 0)

We simplified constraint (101) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:

(102)    (x0[8] + [-1] ≥ 0 ⇒ (UIncreasing(816_0_MAIN_LE(x1[9], +(x0[9], -1))), ≥)∧[(-1)bni_54 + (-1)Bound*bni_54] + [(2)bni_54]x1[8] ≥ 0∧[(-1)bso_55] ≥ 0)

We simplified constraint (102) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:

(103)    (x0[8] + [-1] ≥ 0 ⇒ (UIncreasing(816_0_MAIN_LE(x1[9], +(x0[9], -1))), ≥)∧[(-1)bni_54 + (-1)Bound*bni_54] + [(2)bni_54]x1[8] ≥ 0∧[(-1)bso_55] ≥ 0)

We simplified constraint (103) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint:

(104)    (x0[8] + [-1] ≥ 0 ⇒ (UIncreasing(816_0_MAIN_LE(x1[9], +(x0[9], -1))), ≥)∧[(2)bni_54] = 0∧[(-1)bni_54 + (-1)Bound*bni_54] ≥ 0∧0 = 0∧[(-1)bso_55] ≥ 0)

We simplified constraint (104) using rule (IDP_SMT_SPLIT) which results in the following new constraint:

(105)    (x0[8] ≥ 0 ⇒ (UIncreasing(816_0_MAIN_LE(x1[9], +(x0[9], -1))), ≥)∧[(2)bni_54] = 0∧[(-1)bni_54 + (-1)Bound*bni_54] ≥ 0∧0 = 0∧[(-1)bso_55] ≥ 0)

To summarize, we get the following constraints P for the following pairs.
• 529_0_MAIN_LE(x1, x0) → COND_529_0_MAIN_LE(&&(&&(>=(x1, x0), >(x1, 0)), >(x0, 0)), x1, x0)
• (x0[0] ≥ 0∧x1[0] ≥ 0∧x0[0] + x1[0] ≥ 0 ⇒ (UIncreasing(COND_529_0_MAIN_LE(&&(&&(>=(x1[0], x0[0]), >(x1[0], 0)), >(x0[0], 0)), x1[0], x0[0])), ≥)∧[(-1)Bound*bni_36 + (2)bni_36] + [(3)bni_36]x0[0] + [(2)bni_36]x1[0] ≥ 0∧[1 + (-1)bso_37] + x0[0] + x1[0] ≥ 0)

• COND_529_0_MAIN_LE(TRUE, x1, x0) → 814_0_MAIN_LE(x0, x1)
• ((UIncreasing(814_0_MAIN_LE(x0[1], x1[1])), ≥)∧0 = 0∧[(-1)bso_39] ≥ 0)
• ((UIncreasing(814_0_MAIN_LE(x0[1], x1[1])), ≥)∧0 = 0∧0 = 0∧[(-1)bso_39] ≥ 0)

• 814_0_MAIN_LE(x0, 0) → 529_0_MAIN_LE(0, x0)
• ((UIncreasing(529_0_MAIN_LE(0, x0[2])), ≥)∧0 = 0∧[(-1)bso_41] ≥ 0)
• ((UIncreasing(529_0_MAIN_LE(0, x0[2])), ≥)∧0 = 0∧[(-1)bso_41] ≥ 0)

• 814_0_MAIN_LE(x0, x1) → COND_814_0_MAIN_LE(>(x1, 0), x0, x1)
• (x1[3] ≥ 0 ⇒ (UIncreasing(COND_814_0_MAIN_LE(>(x1[3], 0), x0[3], x1[3])), ≥)∧[bni_42] = 0∧[(-1)Bound*bni_42] + [bni_42]x1[3] ≥ 0∧0 = 0∧[(-1)bso_43] ≥ 0)

• COND_814_0_MAIN_LE(TRUE, x0, x1) → 814_0_MAIN_LE(x0, +(x1, -1))
• (x1[3] ≥ 0∧x1[3] ≥ 0 ⇒ (UIncreasing(814_0_MAIN_LE(x0[4], +(x1[4], -1))), ≥)∧[bni_44] = 0∧[(-1)Bound*bni_44] + [bni_44]x1[3] ≥ 0∧0 = 0∧[1 + (-1)bso_45] ≥ 0)
• (x1[3] ≥ 0 ⇒ (UIncreasing(814_0_MAIN_LE(x0[4], +(x1[4], -1))), ≥)∧[bni_44] = 0∧[(-1)Bound*bni_44] + [bni_44]x1[3] ≥ 0∧0 = 0∧[1 + (-1)bso_45] ≥ 0)

• 529_0_MAIN_LE(x1, x0) → COND_529_0_MAIN_LE1(&&(&&(>(x1, 0), <(x1, x0)), >(x0, 0)), x1, x0)
• ([1] + x1[5] + x0[5] ≥ 0∧x1[5] ≥ 0∧x0[5] ≥ 0 ⇒ (UIncreasing(COND_529_0_MAIN_LE1(&&(&&(>(x1[5], 0), <(x1[5], x0[5])), >(x0[5], 0)), x1[5], x0[5])), ≥)∧[(3)bni_46 + (-1)Bound*bni_46] + [(3)bni_46]x1[5] + [bni_46]x0[5] ≥ 0∧[(-1)bso_47] + x1[5] + x0[5] ≥ 0)

• COND_529_0_MAIN_LE1(TRUE, x1, x0) → 816_0_MAIN_LE(x1, x0)
• ((UIncreasing(816_0_MAIN_LE(x1[6], x0[6])), ≥)∧0 = 0∧[2 + (-1)bso_49] ≥ 0)
• ((UIncreasing(816_0_MAIN_LE(x1[6], x0[6])), ≥)∧0 = 0∧0 = 0∧[2 + (-1)bso_49] ≥ 0)

• 816_0_MAIN_LE(x1, 0) → 529_0_MAIN_LE(x1, 0)
• ((UIncreasing(529_0_MAIN_LE(x1[7], 0)), ≥)∧0 = 0∧[(-1)bso_51] ≥ 0)
• ((UIncreasing(529_0_MAIN_LE(x1[7], 0)), ≥)∧0 = 0∧[(-1)bso_51] ≥ 0)

• 816_0_MAIN_LE(x1, x0) → COND_816_0_MAIN_LE(>(x0, 0), x1, x0)
• (x0[8] ≥ 0 ⇒ (UIncreasing(COND_816_0_MAIN_LE(>(x0[8], 0), x1[8], x0[8])), ≥)∧[(2)bni_52] = 0∧[(-1)bni_52 + (-1)Bound*bni_52] ≥ 0∧0 = 0∧[(-1)bso_53] ≥ 0)

• COND_816_0_MAIN_LE(TRUE, x1, x0) → 816_0_MAIN_LE(x1, +(x0, -1))
• (x0[8] ≥ 0∧x0[8] ≥ 0 ⇒ (UIncreasing(816_0_MAIN_LE(x1[9], +(x0[9], -1))), ≥)∧[(2)bni_54] = 0∧[(-1)bni_54 + (-1)Bound*bni_54] ≥ 0∧0 = 0∧[(-1)bso_55] ≥ 0)
• (x0[8] ≥ 0 ⇒ (UIncreasing(816_0_MAIN_LE(x1[9], +(x0[9], -1))), ≥)∧[(2)bni_54] = 0∧[(-1)bni_54 + (-1)Bound*bni_54] ≥ 0∧0 = 0∧[(-1)bso_55] ≥ 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(529_0_main_LE(x1, x2)) = [-1] + [-1]x2 + [-1]x1
POL(0) = 0
POL(593_0_main_Return) = [-1]
POL(529_0_MAIN_LE(x1, x2)) = [-1] + x2 + [2]x1
POL(COND_529_0_MAIN_LE(x1, x2, x3)) = [-1] + x3 + x2
POL(&&(x1, x2)) = [-1]
POL(>=(x1, x2)) = [-1]
POL(>(x1, x2)) = [-1]
POL(814_0_MAIN_LE(x1, x2)) = [-1] + x2 + x1
POL(COND_814_0_MAIN_LE(x1, x2, x3)) = [-1] + x3 + x2
POL(+(x1, x2)) = x1 + x2
POL(-1) = [-1]
POL(COND_529_0_MAIN_LE1(x1, x2, x3)) = [1] + [2]x2
POL(<(x1, x2)) = [-1]
POL(816_0_MAIN_LE(x1, x2)) = [-1] + [2]x1
POL(COND_816_0_MAIN_LE(x1, x2, x3)) = [-1] + [2]x2

The following pairs are in P>:

529_0_MAIN_LE(x1[0], x0[0]) → COND_529_0_MAIN_LE(&&(&&(>=(x1[0], x0[0]), >(x1[0], 0)), >(x0[0], 0)), x1[0], x0[0])
COND_814_0_MAIN_LE(TRUE, x0[4], x1[4]) → 814_0_MAIN_LE(x0[4], +(x1[4], -1))
COND_529_0_MAIN_LE1(TRUE, x1[6], x0[6]) → 816_0_MAIN_LE(x1[6], x0[6])

The following pairs are in Pbound:

529_0_MAIN_LE(x1[0], x0[0]) → COND_529_0_MAIN_LE(&&(&&(>=(x1[0], x0[0]), >(x1[0], 0)), >(x0[0], 0)), x1[0], x0[0])
529_0_MAIN_LE(x1[5], x0[5]) → COND_529_0_MAIN_LE1(&&(&&(>(x1[5], 0), <(x1[5], x0[5])), >(x0[5], 0)), x1[5], x0[5])

The following pairs are in P:

COND_529_0_MAIN_LE(TRUE, x1[1], x0[1]) → 814_0_MAIN_LE(x0[1], x1[1])
814_0_MAIN_LE(x0[2], 0) → 529_0_MAIN_LE(0, x0[2])
814_0_MAIN_LE(x0[3], x1[3]) → COND_814_0_MAIN_LE(>(x1[3], 0), x0[3], x1[3])
529_0_MAIN_LE(x1[5], x0[5]) → COND_529_0_MAIN_LE1(&&(&&(>(x1[5], 0), <(x1[5], x0[5])), >(x0[5], 0)), x1[5], x0[5])
816_0_MAIN_LE(x1[7], 0) → 529_0_MAIN_LE(x1[7], 0)
816_0_MAIN_LE(x1[8], x0[8]) → COND_816_0_MAIN_LE(>(x0[8], 0), x1[8], x0[8])
COND_816_0_MAIN_LE(TRUE, x1[9], x0[9]) → 816_0_MAIN_LE(x1[9], +(x0[9], -1))

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

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

### (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:

Integer, Boolean

The ITRS R consists of the following rules:
529_0_main_LE(x1, 0) → 593_0_main_Return

The integer pair graph contains the following rules and edges:
(1): COND_529_0_MAIN_LE(TRUE, x1[1], x0[1]) → 814_0_MAIN_LE(x0[1], x1[1])
(2): 814_0_MAIN_LE(x0[2], 0) → 529_0_MAIN_LE(0, x0[2])
(3): 814_0_MAIN_LE(x0[3], x1[3]) → COND_814_0_MAIN_LE(x1[3] > 0, x0[3], x1[3])
(5): 529_0_MAIN_LE(x1[5], x0[5]) → COND_529_0_MAIN_LE1(x1[5] > 0 && x1[5] < x0[5] && x0[5] > 0, x1[5], x0[5])
(7): 816_0_MAIN_LE(x1[7], 0) → 529_0_MAIN_LE(x1[7], 0)
(8): 816_0_MAIN_LE(x1[8], x0[8]) → COND_816_0_MAIN_LE(x0[8] > 0, x1[8], x0[8])
(9): COND_816_0_MAIN_LE(TRUE, x1[9], x0[9]) → 816_0_MAIN_LE(x1[9], x0[9] + -1)

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

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

(2) -> (5), if ((0* x1[5])∧(x0[2]* x0[5]))

(7) -> (5), if ((x1[7]* x1[5])∧(0* x0[5]))

(9) -> (7), if ((x1[9]* x1[7])∧(x0[9] + -1* 0))

(9) -> (8), if ((x1[9]* x1[8])∧(x0[9] + -1* x0[8]))

(8) -> (9), if ((x0[8] > 0* TRUE)∧(x1[8]* x1[9])∧(x0[8]* x0[9]))

The set Q consists of the following terms:
529_0_main_LE(x0, 0)

### (8) IDependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 5 less nodes.

### (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

The ITRS R consists of the following rules:
529_0_main_LE(x1, 0) → 593_0_main_Return

The integer pair graph contains the following rules and edges:
(9): COND_816_0_MAIN_LE(TRUE, x1[9], x0[9]) → 816_0_MAIN_LE(x1[9], x0[9] + -1)
(8): 816_0_MAIN_LE(x1[8], x0[8]) → COND_816_0_MAIN_LE(x0[8] > 0, x1[8], x0[8])

(9) -> (8), if ((x1[9]* x1[8])∧(x0[9] + -1* x0[8]))

(8) -> (9), if ((x0[8] > 0* TRUE)∧(x1[8]* x1[9])∧(x0[8]* x0[9]))

The set Q consists of the following terms:
529_0_main_LE(x0, 0)

### (10) UsableRulesProof (EQUIVALENT transformation)

As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [LPAR04] we can delete all non-usable rules [FROCOS05] from R.

### (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:
(9): COND_816_0_MAIN_LE(TRUE, x1[9], x0[9]) → 816_0_MAIN_LE(x1[9], x0[9] + -1)
(8): 816_0_MAIN_LE(x1[8], x0[8]) → COND_816_0_MAIN_LE(x0[8] > 0, x1[8], x0[8])

(9) -> (8), if ((x1[9]* x1[8])∧(x0[9] + -1* x0[8]))

(8) -> (9), if ((x0[8] > 0* TRUE)∧(x1[8]* x1[9])∧(x0[8]* x0[9]))

The set Q consists of the following terms:
529_0_main_LE(x0, 0)

### (12) 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_816_0_MAIN_LE(TRUE, x1[9], x0[9]) → 816_0_MAIN_LE(x1[9], +(x0[9], -1)) the following chains were created:
• We consider the chain 816_0_MAIN_LE(x1[8], x0[8]) → COND_816_0_MAIN_LE(>(x0[8], 0), x1[8], x0[8]), COND_816_0_MAIN_LE(TRUE, x1[9], x0[9]) → 816_0_MAIN_LE(x1[9], +(x0[9], -1)), 816_0_MAIN_LE(x1[8], x0[8]) → COND_816_0_MAIN_LE(>(x0[8], 0), x1[8], x0[8]) which results in the following constraint:

(1)    (>(x0[8], 0)=TRUEx1[8]=x1[9]x0[8]=x0[9]x1[9]=x1[8]1+(x0[9], -1)=x0[8]1COND_816_0_MAIN_LE(TRUE, x1[9], x0[9])≥NonInfC∧COND_816_0_MAIN_LE(TRUE, x1[9], x0[9])≥816_0_MAIN_LE(x1[9], +(x0[9], -1))∧(UIncreasing(816_0_MAIN_LE(x1[9], +(x0[9], -1))), ≥))

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

(2)    (>(x0[8], 0)=TRUECOND_816_0_MAIN_LE(TRUE, x1[8], x0[8])≥NonInfC∧COND_816_0_MAIN_LE(TRUE, x1[8], x0[8])≥816_0_MAIN_LE(x1[8], +(x0[8], -1))∧(UIncreasing(816_0_MAIN_LE(x1[9], +(x0[9], -1))), ≥))

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

(3)    (x0[8] + [-1] ≥ 0 ⇒ (UIncreasing(816_0_MAIN_LE(x1[9], +(x0[9], -1))), ≥)∧[(-1)bni_11 + (-1)Bound*bni_11] + [bni_11]x0[8] ≥ 0∧[1 + (-1)bso_12] ≥ 0)

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

(4)    (x0[8] + [-1] ≥ 0 ⇒ (UIncreasing(816_0_MAIN_LE(x1[9], +(x0[9], -1))), ≥)∧[(-1)bni_11 + (-1)Bound*bni_11] + [bni_11]x0[8] ≥ 0∧[1 + (-1)bso_12] ≥ 0)

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

(5)    (x0[8] + [-1] ≥ 0 ⇒ (UIncreasing(816_0_MAIN_LE(x1[9], +(x0[9], -1))), ≥)∧[(-1)bni_11 + (-1)Bound*bni_11] + [bni_11]x0[8] ≥ 0∧[1 + (-1)bso_12] ≥ 0)

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

(6)    (x0[8] + [-1] ≥ 0 ⇒ (UIncreasing(816_0_MAIN_LE(x1[9], +(x0[9], -1))), ≥)∧0 = 0∧[(-1)bni_11 + (-1)Bound*bni_11] + [bni_11]x0[8] ≥ 0∧0 = 0∧[1 + (-1)bso_12] ≥ 0)

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

(7)    (x0[8] ≥ 0 ⇒ (UIncreasing(816_0_MAIN_LE(x1[9], +(x0[9], -1))), ≥)∧0 = 0∧[(-1)Bound*bni_11] + [bni_11]x0[8] ≥ 0∧0 = 0∧[1 + (-1)bso_12] ≥ 0)

For Pair 816_0_MAIN_LE(x1[8], x0[8]) → COND_816_0_MAIN_LE(>(x0[8], 0), x1[8], x0[8]) the following chains were created:
• We consider the chain 816_0_MAIN_LE(x1[8], x0[8]) → COND_816_0_MAIN_LE(>(x0[8], 0), x1[8], x0[8]), COND_816_0_MAIN_LE(TRUE, x1[9], x0[9]) → 816_0_MAIN_LE(x1[9], +(x0[9], -1)) which results in the following constraint:

(8)    (>(x0[8], 0)=TRUEx1[8]=x1[9]x0[8]=x0[9]816_0_MAIN_LE(x1[8], x0[8])≥NonInfC∧816_0_MAIN_LE(x1[8], x0[8])≥COND_816_0_MAIN_LE(>(x0[8], 0), x1[8], x0[8])∧(UIncreasing(COND_816_0_MAIN_LE(>(x0[8], 0), x1[8], x0[8])), ≥))

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

(9)    (>(x0[8], 0)=TRUE816_0_MAIN_LE(x1[8], x0[8])≥NonInfC∧816_0_MAIN_LE(x1[8], x0[8])≥COND_816_0_MAIN_LE(>(x0[8], 0), x1[8], x0[8])∧(UIncreasing(COND_816_0_MAIN_LE(>(x0[8], 0), x1[8], x0[8])), ≥))

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

(10)    (x0[8] + [-1] ≥ 0 ⇒ (UIncreasing(COND_816_0_MAIN_LE(>(x0[8], 0), x1[8], x0[8])), ≥)∧[(-1)bni_13 + (-1)Bound*bni_13] + [bni_13]x0[8] ≥ 0∧[(-1)bso_14] ≥ 0)

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

(11)    (x0[8] + [-1] ≥ 0 ⇒ (UIncreasing(COND_816_0_MAIN_LE(>(x0[8], 0), x1[8], x0[8])), ≥)∧[(-1)bni_13 + (-1)Bound*bni_13] + [bni_13]x0[8] ≥ 0∧[(-1)bso_14] ≥ 0)

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

(12)    (x0[8] + [-1] ≥ 0 ⇒ (UIncreasing(COND_816_0_MAIN_LE(>(x0[8], 0), x1[8], x0[8])), ≥)∧[(-1)bni_13 + (-1)Bound*bni_13] + [bni_13]x0[8] ≥ 0∧[(-1)bso_14] ≥ 0)

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

(13)    (x0[8] + [-1] ≥ 0 ⇒ (UIncreasing(COND_816_0_MAIN_LE(>(x0[8], 0), x1[8], x0[8])), ≥)∧0 = 0∧[(-1)bni_13 + (-1)Bound*bni_13] + [bni_13]x0[8] ≥ 0∧0 = 0∧[(-1)bso_14] ≥ 0)

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

(14)    (x0[8] ≥ 0 ⇒ (UIncreasing(COND_816_0_MAIN_LE(>(x0[8], 0), x1[8], x0[8])), ≥)∧0 = 0∧[(-1)Bound*bni_13] + [bni_13]x0[8] ≥ 0∧0 = 0∧[(-1)bso_14] ≥ 0)

To summarize, we get the following constraints P for the following pairs.
• COND_816_0_MAIN_LE(TRUE, x1[9], x0[9]) → 816_0_MAIN_LE(x1[9], +(x0[9], -1))
• (x0[8] ≥ 0 ⇒ (UIncreasing(816_0_MAIN_LE(x1[9], +(x0[9], -1))), ≥)∧0 = 0∧[(-1)Bound*bni_11] + [bni_11]x0[8] ≥ 0∧0 = 0∧[1 + (-1)bso_12] ≥ 0)

• 816_0_MAIN_LE(x1[8], x0[8]) → COND_816_0_MAIN_LE(>(x0[8], 0), x1[8], x0[8])
• (x0[8] ≥ 0 ⇒ (UIncreasing(COND_816_0_MAIN_LE(>(x0[8], 0), x1[8], x0[8])), ≥)∧0 = 0∧[(-1)Bound*bni_13] + [bni_13]x0[8] ≥ 0∧0 = 0∧[(-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_816_0_MAIN_LE(x1, x2, x3)) = [-1] + x3
POL(816_0_MAIN_LE(x1, x2)) = [-1] + x2
POL(+(x1, x2)) = x1 + x2
POL(-1) = [-1]
POL(>(x1, x2)) = [-1]
POL(0) = 0

The following pairs are in P>:

COND_816_0_MAIN_LE(TRUE, x1[9], x0[9]) → 816_0_MAIN_LE(x1[9], +(x0[9], -1))

The following pairs are in Pbound:

COND_816_0_MAIN_LE(TRUE, x1[9], x0[9]) → 816_0_MAIN_LE(x1[9], +(x0[9], -1))
816_0_MAIN_LE(x1[8], x0[8]) → COND_816_0_MAIN_LE(>(x0[8], 0), x1[8], x0[8])

The following pairs are in P:

816_0_MAIN_LE(x1[8], x0[8]) → COND_816_0_MAIN_LE(>(x0[8], 0), x1[8], x0[8])

There are no usable rules.

### (13) Obligation:

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

The following domains are used:

Integer

R is empty.

The integer pair graph contains the following rules and edges:
(8): 816_0_MAIN_LE(x1[8], x0[8]) → COND_816_0_MAIN_LE(x0[8] > 0, x1[8], x0[8])

The set Q consists of the following terms:
529_0_main_LE(x0, 0)

### (14) IDependencyGraphProof (EQUIVALENT transformation)

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

### (16) Obligation:

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

The following domains are used:

Integer

The ITRS R consists of the following rules:
529_0_main_LE(x1, 0) → 593_0_main_Return

The integer pair graph contains the following rules and edges:
(1): COND_529_0_MAIN_LE(TRUE, x1[1], x0[1]) → 814_0_MAIN_LE(x0[1], x1[1])
(2): 814_0_MAIN_LE(x0[2], 0) → 529_0_MAIN_LE(0, x0[2])
(3): 814_0_MAIN_LE(x0[3], x1[3]) → COND_814_0_MAIN_LE(x1[3] > 0, x0[3], x1[3])
(4): COND_814_0_MAIN_LE(TRUE, x0[4], x1[4]) → 814_0_MAIN_LE(x0[4], x1[4] + -1)
(6): COND_529_0_MAIN_LE1(TRUE, x1[6], x0[6]) → 816_0_MAIN_LE(x1[6], x0[6])
(7): 816_0_MAIN_LE(x1[7], 0) → 529_0_MAIN_LE(x1[7], 0)
(8): 816_0_MAIN_LE(x1[8], x0[8]) → COND_816_0_MAIN_LE(x0[8] > 0, x1[8], x0[8])
(9): COND_816_0_MAIN_LE(TRUE, x1[9], x0[9]) → 816_0_MAIN_LE(x1[9], x0[9] + -1)

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

(4) -> (2), if ((x0[4]* x0[2])∧(x1[4] + -1* 0))

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

(4) -> (3), if ((x0[4]* x0[3])∧(x1[4] + -1* x1[3]))

(3) -> (4), if ((x1[3] > 0* TRUE)∧(x0[3]* x0[4])∧(x1[3]* x1[4]))

(6) -> (7), if ((x1[6]* x1[7])∧(x0[6]* 0))

(9) -> (7), if ((x1[9]* x1[7])∧(x0[9] + -1* 0))

(6) -> (8), if ((x1[6]* x1[8])∧(x0[6]* x0[8]))

(9) -> (8), if ((x1[9]* x1[8])∧(x0[9] + -1* x0[8]))

(8) -> (9), if ((x0[8] > 0* TRUE)∧(x1[8]* x1[9])∧(x0[8]* x0[9]))

The set Q consists of the following terms:
529_0_main_LE(x0, 0)

### (17) IDependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 2 SCCs with 4 less nodes.

### (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:

Integer

The ITRS R consists of the following rules:
529_0_main_LE(x1, 0) → 593_0_main_Return

The integer pair graph contains the following rules and edges:
(4): COND_814_0_MAIN_LE(TRUE, x0[4], x1[4]) → 814_0_MAIN_LE(x0[4], x1[4] + -1)
(3): 814_0_MAIN_LE(x0[3], x1[3]) → COND_814_0_MAIN_LE(x1[3] > 0, x0[3], x1[3])

(4) -> (3), if ((x0[4]* x0[3])∧(x1[4] + -1* x1[3]))

(3) -> (4), if ((x1[3] > 0* TRUE)∧(x0[3]* x0[4])∧(x1[3]* x1[4]))

The set Q consists of the following terms:
529_0_main_LE(x0, 0)

### (20) UsableRulesProof (EQUIVALENT transformation)

As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [LPAR04] we can delete all non-usable rules [FROCOS05] from R.

### (21) Obligation:

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

The following domains are used:

Integer

R is empty.

The integer pair graph contains the following rules and edges:
(4): COND_814_0_MAIN_LE(TRUE, x0[4], x1[4]) → 814_0_MAIN_LE(x0[4], x1[4] + -1)
(3): 814_0_MAIN_LE(x0[3], x1[3]) → COND_814_0_MAIN_LE(x1[3] > 0, x0[3], x1[3])

(4) -> (3), if ((x0[4]* x0[3])∧(x1[4] + -1* x1[3]))

(3) -> (4), if ((x1[3] > 0* TRUE)∧(x0[3]* x0[4])∧(x1[3]* x1[4]))

The set Q consists of the following terms:
529_0_main_LE(x0, 0)

### (22) 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_814_0_MAIN_LE(TRUE, x0[4], x1[4]) → 814_0_MAIN_LE(x0[4], +(x1[4], -1)) the following chains were created:
• We consider the chain 814_0_MAIN_LE(x0[3], x1[3]) → COND_814_0_MAIN_LE(>(x1[3], 0), x0[3], x1[3]), COND_814_0_MAIN_LE(TRUE, x0[4], x1[4]) → 814_0_MAIN_LE(x0[4], +(x1[4], -1)), 814_0_MAIN_LE(x0[3], x1[3]) → COND_814_0_MAIN_LE(>(x1[3], 0), x0[3], x1[3]) which results in the following constraint:

(1)    (>(x1[3], 0)=TRUEx0[3]=x0[4]x1[3]=x1[4]x0[4]=x0[3]1+(x1[4], -1)=x1[3]1COND_814_0_MAIN_LE(TRUE, x0[4], x1[4])≥NonInfC∧COND_814_0_MAIN_LE(TRUE, x0[4], x1[4])≥814_0_MAIN_LE(x0[4], +(x1[4], -1))∧(UIncreasing(814_0_MAIN_LE(x0[4], +(x1[4], -1))), ≥))

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

(2)    (>(x1[3], 0)=TRUECOND_814_0_MAIN_LE(TRUE, x0[3], x1[3])≥NonInfC∧COND_814_0_MAIN_LE(TRUE, x0[3], x1[3])≥814_0_MAIN_LE(x0[3], +(x1[3], -1))∧(UIncreasing(814_0_MAIN_LE(x0[4], +(x1[4], -1))), ≥))

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

(3)    (x1[3] + [-1] ≥ 0 ⇒ (UIncreasing(814_0_MAIN_LE(x0[4], +(x1[4], -1))), ≥)∧[(-1)bni_11 + (-1)Bound*bni_11] + [bni_11]x1[3] ≥ 0∧[1 + (-1)bso_12] ≥ 0)

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

(4)    (x1[3] + [-1] ≥ 0 ⇒ (UIncreasing(814_0_MAIN_LE(x0[4], +(x1[4], -1))), ≥)∧[(-1)bni_11 + (-1)Bound*bni_11] + [bni_11]x1[3] ≥ 0∧[1 + (-1)bso_12] ≥ 0)

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

(5)    (x1[3] + [-1] ≥ 0 ⇒ (UIncreasing(814_0_MAIN_LE(x0[4], +(x1[4], -1))), ≥)∧[(-1)bni_11 + (-1)Bound*bni_11] + [bni_11]x1[3] ≥ 0∧[1 + (-1)bso_12] ≥ 0)

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

(6)    (x1[3] + [-1] ≥ 0 ⇒ (UIncreasing(814_0_MAIN_LE(x0[4], +(x1[4], -1))), ≥)∧0 = 0∧[(-1)bni_11 + (-1)Bound*bni_11] + [bni_11]x1[3] ≥ 0∧0 = 0∧[1 + (-1)bso_12] ≥ 0)

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

(7)    (x1[3] ≥ 0 ⇒ (UIncreasing(814_0_MAIN_LE(x0[4], +(x1[4], -1))), ≥)∧0 = 0∧[(-1)Bound*bni_11] + [bni_11]x1[3] ≥ 0∧0 = 0∧[1 + (-1)bso_12] ≥ 0)

For Pair 814_0_MAIN_LE(x0[3], x1[3]) → COND_814_0_MAIN_LE(>(x1[3], 0), x0[3], x1[3]) the following chains were created:
• We consider the chain 814_0_MAIN_LE(x0[3], x1[3]) → COND_814_0_MAIN_LE(>(x1[3], 0), x0[3], x1[3]), COND_814_0_MAIN_LE(TRUE, x0[4], x1[4]) → 814_0_MAIN_LE(x0[4], +(x1[4], -1)) which results in the following constraint:

(8)    (>(x1[3], 0)=TRUEx0[3]=x0[4]x1[3]=x1[4]814_0_MAIN_LE(x0[3], x1[3])≥NonInfC∧814_0_MAIN_LE(x0[3], x1[3])≥COND_814_0_MAIN_LE(>(x1[3], 0), x0[3], x1[3])∧(UIncreasing(COND_814_0_MAIN_LE(>(x1[3], 0), x0[3], x1[3])), ≥))

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

(9)    (>(x1[3], 0)=TRUE814_0_MAIN_LE(x0[3], x1[3])≥NonInfC∧814_0_MAIN_LE(x0[3], x1[3])≥COND_814_0_MAIN_LE(>(x1[3], 0), x0[3], x1[3])∧(UIncreasing(COND_814_0_MAIN_LE(>(x1[3], 0), x0[3], x1[3])), ≥))

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

(10)    (x1[3] + [-1] ≥ 0 ⇒ (UIncreasing(COND_814_0_MAIN_LE(>(x1[3], 0), x0[3], x1[3])), ≥)∧[(-1)bni_13 + (-1)Bound*bni_13] + [bni_13]x1[3] ≥ 0∧[(-1)bso_14] ≥ 0)

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

(11)    (x1[3] + [-1] ≥ 0 ⇒ (UIncreasing(COND_814_0_MAIN_LE(>(x1[3], 0), x0[3], x1[3])), ≥)∧[(-1)bni_13 + (-1)Bound*bni_13] + [bni_13]x1[3] ≥ 0∧[(-1)bso_14] ≥ 0)

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

(12)    (x1[3] + [-1] ≥ 0 ⇒ (UIncreasing(COND_814_0_MAIN_LE(>(x1[3], 0), x0[3], x1[3])), ≥)∧[(-1)bni_13 + (-1)Bound*bni_13] + [bni_13]x1[3] ≥ 0∧[(-1)bso_14] ≥ 0)

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

(13)    (x1[3] + [-1] ≥ 0 ⇒ (UIncreasing(COND_814_0_MAIN_LE(>(x1[3], 0), x0[3], x1[3])), ≥)∧0 = 0∧[(-1)bni_13 + (-1)Bound*bni_13] + [bni_13]x1[3] ≥ 0∧0 = 0∧[(-1)bso_14] ≥ 0)

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

(14)    (x1[3] ≥ 0 ⇒ (UIncreasing(COND_814_0_MAIN_LE(>(x1[3], 0), x0[3], x1[3])), ≥)∧0 = 0∧[(-1)Bound*bni_13] + [bni_13]x1[3] ≥ 0∧0 = 0∧[(-1)bso_14] ≥ 0)

To summarize, we get the following constraints P for the following pairs.
• COND_814_0_MAIN_LE(TRUE, x0[4], x1[4]) → 814_0_MAIN_LE(x0[4], +(x1[4], -1))
• (x1[3] ≥ 0 ⇒ (UIncreasing(814_0_MAIN_LE(x0[4], +(x1[4], -1))), ≥)∧0 = 0∧[(-1)Bound*bni_11] + [bni_11]x1[3] ≥ 0∧0 = 0∧[1 + (-1)bso_12] ≥ 0)

• 814_0_MAIN_LE(x0[3], x1[3]) → COND_814_0_MAIN_LE(>(x1[3], 0), x0[3], x1[3])
• (x1[3] ≥ 0 ⇒ (UIncreasing(COND_814_0_MAIN_LE(>(x1[3], 0), x0[3], x1[3])), ≥)∧0 = 0∧[(-1)Bound*bni_13] + [bni_13]x1[3] ≥ 0∧0 = 0∧[(-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_814_0_MAIN_LE(x1, x2, x3)) = [-1] + x3
POL(814_0_MAIN_LE(x1, x2)) = [-1] + x2
POL(+(x1, x2)) = x1 + x2
POL(-1) = [-1]
POL(>(x1, x2)) = [-1]
POL(0) = 0

The following pairs are in P>:

COND_814_0_MAIN_LE(TRUE, x0[4], x1[4]) → 814_0_MAIN_LE(x0[4], +(x1[4], -1))

The following pairs are in Pbound:

COND_814_0_MAIN_LE(TRUE, x0[4], x1[4]) → 814_0_MAIN_LE(x0[4], +(x1[4], -1))
814_0_MAIN_LE(x0[3], x1[3]) → COND_814_0_MAIN_LE(>(x1[3], 0), x0[3], x1[3])

The following pairs are in P:

814_0_MAIN_LE(x0[3], x1[3]) → COND_814_0_MAIN_LE(>(x1[3], 0), x0[3], x1[3])

There are no usable rules.

### (23) 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:
(3): 814_0_MAIN_LE(x0[3], x1[3]) → COND_814_0_MAIN_LE(x1[3] > 0, x0[3], x1[3])

The set Q consists of the following terms:
529_0_main_LE(x0, 0)

### (24) IDependencyGraphProof (EQUIVALENT transformation)

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

### (26) 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

The ITRS R consists of the following rules:
529_0_main_LE(x1, 0) → 593_0_main_Return

The integer pair graph contains the following rules and edges:
(9): COND_816_0_MAIN_LE(TRUE, x1[9], x0[9]) → 816_0_MAIN_LE(x1[9], x0[9] + -1)
(8): 816_0_MAIN_LE(x1[8], x0[8]) → COND_816_0_MAIN_LE(x0[8] > 0, x1[8], x0[8])

(9) -> (8), if ((x1[9]* x1[8])∧(x0[9] + -1* x0[8]))

(8) -> (9), if ((x0[8] > 0* TRUE)∧(x1[8]* x1[9])∧(x0[8]* x0[9]))

The set Q consists of the following terms:
529_0_main_LE(x0, 0)

### (27) UsableRulesProof (EQUIVALENT transformation)

As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [LPAR04] we can delete all non-usable rules [FROCOS05] from R.

### (28) 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:
(9): COND_816_0_MAIN_LE(TRUE, x1[9], x0[9]) → 816_0_MAIN_LE(x1[9], x0[9] + -1)
(8): 816_0_MAIN_LE(x1[8], x0[8]) → COND_816_0_MAIN_LE(x0[8] > 0, x1[8], x0[8])

(9) -> (8), if ((x1[9]* x1[8])∧(x0[9] + -1* x0[8]))

(8) -> (9), if ((x0[8] > 0* TRUE)∧(x1[8]* x1[9])∧(x0[8]* x0[9]))

The set Q consists of the following terms:
529_0_main_LE(x0, 0)

### (29) 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_816_0_MAIN_LE(TRUE, x1[9], x0[9]) → 816_0_MAIN_LE(x1[9], +(x0[9], -1)) the following chains were created:
• We consider the chain 816_0_MAIN_LE(x1[8], x0[8]) → COND_816_0_MAIN_LE(>(x0[8], 0), x1[8], x0[8]), COND_816_0_MAIN_LE(TRUE, x1[9], x0[9]) → 816_0_MAIN_LE(x1[9], +(x0[9], -1)), 816_0_MAIN_LE(x1[8], x0[8]) → COND_816_0_MAIN_LE(>(x0[8], 0), x1[8], x0[8]) which results in the following constraint:

(1)    (>(x0[8], 0)=TRUEx1[8]=x1[9]x0[8]=x0[9]x1[9]=x1[8]1+(x0[9], -1)=x0[8]1COND_816_0_MAIN_LE(TRUE, x1[9], x0[9])≥NonInfC∧COND_816_0_MAIN_LE(TRUE, x1[9], x0[9])≥816_0_MAIN_LE(x1[9], +(x0[9], -1))∧(UIncreasing(816_0_MAIN_LE(x1[9], +(x0[9], -1))), ≥))

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

(2)    (>(x0[8], 0)=TRUECOND_816_0_MAIN_LE(TRUE, x1[8], x0[8])≥NonInfC∧COND_816_0_MAIN_LE(TRUE, x1[8], x0[8])≥816_0_MAIN_LE(x1[8], +(x0[8], -1))∧(UIncreasing(816_0_MAIN_LE(x1[9], +(x0[9], -1))), ≥))

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

(3)    (x0[8] + [-1] ≥ 0 ⇒ (UIncreasing(816_0_MAIN_LE(x1[9], +(x0[9], -1))), ≥)∧[(-1)bni_11 + (-1)Bound*bni_11] + [bni_11]x0[8] ≥ 0∧[1 + (-1)bso_12] ≥ 0)

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

(4)    (x0[8] + [-1] ≥ 0 ⇒ (UIncreasing(816_0_MAIN_LE(x1[9], +(x0[9], -1))), ≥)∧[(-1)bni_11 + (-1)Bound*bni_11] + [bni_11]x0[8] ≥ 0∧[1 + (-1)bso_12] ≥ 0)

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

(5)    (x0[8] + [-1] ≥ 0 ⇒ (UIncreasing(816_0_MAIN_LE(x1[9], +(x0[9], -1))), ≥)∧[(-1)bni_11 + (-1)Bound*bni_11] + [bni_11]x0[8] ≥ 0∧[1 + (-1)bso_12] ≥ 0)

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

(6)    (x0[8] + [-1] ≥ 0 ⇒ (UIncreasing(816_0_MAIN_LE(x1[9], +(x0[9], -1))), ≥)∧0 = 0∧[(-1)bni_11 + (-1)Bound*bni_11] + [bni_11]x0[8] ≥ 0∧0 = 0∧[1 + (-1)bso_12] ≥ 0)

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

(7)    (x0[8] ≥ 0 ⇒ (UIncreasing(816_0_MAIN_LE(x1[9], +(x0[9], -1))), ≥)∧0 = 0∧[(-1)Bound*bni_11] + [bni_11]x0[8] ≥ 0∧0 = 0∧[1 + (-1)bso_12] ≥ 0)

For Pair 816_0_MAIN_LE(x1[8], x0[8]) → COND_816_0_MAIN_LE(>(x0[8], 0), x1[8], x0[8]) the following chains were created:
• We consider the chain 816_0_MAIN_LE(x1[8], x0[8]) → COND_816_0_MAIN_LE(>(x0[8], 0), x1[8], x0[8]), COND_816_0_MAIN_LE(TRUE, x1[9], x0[9]) → 816_0_MAIN_LE(x1[9], +(x0[9], -1)) which results in the following constraint:

(8)    (>(x0[8], 0)=TRUEx1[8]=x1[9]x0[8]=x0[9]816_0_MAIN_LE(x1[8], x0[8])≥NonInfC∧816_0_MAIN_LE(x1[8], x0[8])≥COND_816_0_MAIN_LE(>(x0[8], 0), x1[8], x0[8])∧(UIncreasing(COND_816_0_MAIN_LE(>(x0[8], 0), x1[8], x0[8])), ≥))

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

(9)    (>(x0[8], 0)=TRUE816_0_MAIN_LE(x1[8], x0[8])≥NonInfC∧816_0_MAIN_LE(x1[8], x0[8])≥COND_816_0_MAIN_LE(>(x0[8], 0), x1[8], x0[8])∧(UIncreasing(COND_816_0_MAIN_LE(>(x0[8], 0), x1[8], x0[8])), ≥))

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

(10)    (x0[8] + [-1] ≥ 0 ⇒ (UIncreasing(COND_816_0_MAIN_LE(>(x0[8], 0), x1[8], x0[8])), ≥)∧[(-1)bni_13 + (-1)Bound*bni_13] + [bni_13]x0[8] ≥ 0∧[(-1)bso_14] ≥ 0)

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

(11)    (x0[8] + [-1] ≥ 0 ⇒ (UIncreasing(COND_816_0_MAIN_LE(>(x0[8], 0), x1[8], x0[8])), ≥)∧[(-1)bni_13 + (-1)Bound*bni_13] + [bni_13]x0[8] ≥ 0∧[(-1)bso_14] ≥ 0)

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

(12)    (x0[8] + [-1] ≥ 0 ⇒ (UIncreasing(COND_816_0_MAIN_LE(>(x0[8], 0), x1[8], x0[8])), ≥)∧[(-1)bni_13 + (-1)Bound*bni_13] + [bni_13]x0[8] ≥ 0∧[(-1)bso_14] ≥ 0)

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

(13)    (x0[8] + [-1] ≥ 0 ⇒ (UIncreasing(COND_816_0_MAIN_LE(>(x0[8], 0), x1[8], x0[8])), ≥)∧0 = 0∧[(-1)bni_13 + (-1)Bound*bni_13] + [bni_13]x0[8] ≥ 0∧0 = 0∧[(-1)bso_14] ≥ 0)

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

(14)    (x0[8] ≥ 0 ⇒ (UIncreasing(COND_816_0_MAIN_LE(>(x0[8], 0), x1[8], x0[8])), ≥)∧0 = 0∧[(-1)Bound*bni_13] + [bni_13]x0[8] ≥ 0∧0 = 0∧[(-1)bso_14] ≥ 0)

To summarize, we get the following constraints P for the following pairs.
• COND_816_0_MAIN_LE(TRUE, x1[9], x0[9]) → 816_0_MAIN_LE(x1[9], +(x0[9], -1))
• (x0[8] ≥ 0 ⇒ (UIncreasing(816_0_MAIN_LE(x1[9], +(x0[9], -1))), ≥)∧0 = 0∧[(-1)Bound*bni_11] + [bni_11]x0[8] ≥ 0∧0 = 0∧[1 + (-1)bso_12] ≥ 0)

• 816_0_MAIN_LE(x1[8], x0[8]) → COND_816_0_MAIN_LE(>(x0[8], 0), x1[8], x0[8])
• (x0[8] ≥ 0 ⇒ (UIncreasing(COND_816_0_MAIN_LE(>(x0[8], 0), x1[8], x0[8])), ≥)∧0 = 0∧[(-1)Bound*bni_13] + [bni_13]x0[8] ≥ 0∧0 = 0∧[(-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_816_0_MAIN_LE(x1, x2, x3)) = [-1] + x3
POL(816_0_MAIN_LE(x1, x2)) = [-1] + x2
POL(+(x1, x2)) = x1 + x2
POL(-1) = [-1]
POL(>(x1, x2)) = [-1]
POL(0) = 0

The following pairs are in P>:

COND_816_0_MAIN_LE(TRUE, x1[9], x0[9]) → 816_0_MAIN_LE(x1[9], +(x0[9], -1))

The following pairs are in Pbound:

COND_816_0_MAIN_LE(TRUE, x1[9], x0[9]) → 816_0_MAIN_LE(x1[9], +(x0[9], -1))
816_0_MAIN_LE(x1[8], x0[8]) → COND_816_0_MAIN_LE(>(x0[8], 0), x1[8], x0[8])

The following pairs are in P:

816_0_MAIN_LE(x1[8], x0[8]) → COND_816_0_MAIN_LE(>(x0[8], 0), x1[8], x0[8])

There are no usable rules.

### (30) 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:
(8): 816_0_MAIN_LE(x1[8], x0[8]) → COND_816_0_MAIN_LE(x0[8] > 0, x1[8], x0[8])

The set Q consists of the following terms:
529_0_main_LE(x0, 0)

### (31) IDependencyGraphProof (EQUIVALENT transformation)

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