### (0) Obligation:

JBC Problem based on JBC Program:
Manifest-Version: 1.0 Created-By: 1.6.0_16 (Sun Microsystems Inc.) Main-Class: Ackermann
`/** * The classical Ackermann function. * * All calls terminate. * * Julia + BinTerm prove that all calls terminate * * Note that we have to express the basic cases as m <= 0 and n <= 0 * in order to prove termination. * * @author <A HREF="mailto:fausto.spoto@univr.it">Fausto Spoto</A> */public class Ackermann {    public static int ack(int m, int n) {	if (m <= 0) return n + 1;	else if (n <= 0) return ack(m - 1,1);	else return ack(m - 1,ack(m,n - 1));    }    public static void main(String[] args) {	ack(10,12);    }}`

### (1) JBC2FIG (SOUND transformation)

Constructed FIGraph.

### (2) Obligation:

FIGraph based on JBC Program:
Ackermann.main([Ljava/lang/String;)V: Graph of 13 nodes with 0 SCCs.

Ackermann.ack(II)I: Graph of 50 nodes with 0 SCCs.

### (3) FIGtoITRSProof (SOUND transformation)

Transformed FIGraph SCCs to IDPs. Logs:

Log for SCC 0:

Generated 33 rules for P and 20 rules for R.

Combined rules. Obtained 4 rules for P and 6 rules for R.

Filtered ground terms:

215_0_ack_GT(x1, x2, x3, x4) → 215_0_ack_GT(x2, x3, x4)
295_0_ack_Return(x1, x2) → 295_0_ack_Return(x2)
266_0_ack_Return(x1, x2, x3, x4) → 266_0_ack_Return(x2, x3, x4)
Cond_215_0_ack_GT1(x1, x2, x3, x4, x5) → Cond_215_0_ack_GT1(x1, x3, x4, x5)
258_1_ack_InvokeMethod(x1, x2, x3, x4, x5) → 258_1_ack_InvokeMethod(x1, x2, x3, x4)
Cond_215_0_ack_GT(x1, x2, x3, x4, x5) → Cond_215_0_ack_GT(x1, x3, x4, x5)
231_0_ack_Return(x1, x2, x3, x4) → 231_0_ack_Return(x3, x4)

Filtered duplicate args:

215_0_ack_GT(x1, x2, x3) → 215_0_ack_GT(x2, x3)
Cond_215_0_ack_GT1(x1, x2, x3, x4) → Cond_215_0_ack_GT1(x1, x3, x4)
Cond_215_0_ack_GT(x1, x2, x3, x4) → Cond_215_0_ack_GT(x1, x3, x4)

Filtered unneeded arguments:

Cond_215_0_ack_GT(x1, x2, x3) → Cond_215_0_ack_GT(x1, x3)
258_1_ack_InvokeMethod(x1, x2, x3, x4) → 258_1_ack_InvokeMethod(x1, x4)

Combined rules. Obtained 4 rules for P and 6 rules for R.

Finished conversion. Obtained 4 rules for P and 6 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:

Integer, Boolean

The ITRS R consists of the following rules:
215_0_ack_GT(x1, 0) → 231_0_ack_Return(x1, x1 + 1)
258_1_ack_InvokeMethod(231_0_ack_Return(1, x2), 0) → 266_0_ack_Return(x3, x4, x2)
258_1_ack_InvokeMethod(295_0_ack_Return(x0), x3) → 266_0_ack_Return(x1, x2, x0)
287_1_ack_InvokeMethod(231_0_ack_Return(x1, x2), 0, x1) → 295_0_ack_Return(x2)
287_1_ack_InvokeMethod(266_0_ack_Return(x0, x1, x2), x0, x1) → 295_0_ack_Return(x2)
287_1_ack_InvokeMethod(295_0_ack_Return(x0), x1, x2) → 295_0_ack_Return(x0)

The integer pair graph contains the following rules and edges:
(0): 215_0_ACK_GT(x1[0], x0[0]) → COND_215_0_ACK_GT(x1[0] <= 0 && x0[0] > 0, x1[0], x0[0])
(1): COND_215_0_ACK_GT(TRUE, x1[1], x0[1]) → 215_0_ACK_GT(1, x0[1] - 1)
(2): 215_0_ACK_GT(x1[2], x0[2]) → COND_215_0_ACK_GT1(x1[2] > 0 && x0[2] > 0, x1[2], x0[2])
(3): COND_215_0_ACK_GT1(TRUE, x1[3], x0[3]) → 277_1_ACK_INVOKEMETHOD(215_0_ack_GT(x1[3] - 1, x0[3]), x0[3] - 1, x0[3], x1[3] - 1)
(4): COND_215_0_ACK_GT1(TRUE, x1[4], x0[4]) → 215_0_ACK_GT(x1[4] - 1, x0[4])
(5): 277_1_ACK_INVOKEMETHOD(266_0_ack_Return(x0[5], 0, x2[5]), x3[5], x0[5], 0) → 215_0_ACK_GT(x2[5], x3[5])
(6): 277_1_ACK_INVOKEMETHOD(295_0_ack_Return(x0[6]), x1[6], x2[6], x3[6]) → 215_0_ACK_GT(x0[6], x1[6])

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

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

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

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

(2) -> (4), if ((x1[2] > 0 && x0[2] > 0* TRUE)∧(x1[2]* x1[4])∧(x0[2]* x0[4]))

(3) -> (5), if ((215_0_ack_GT(x1[3] - 1, x0[3]) →* 266_0_ack_Return(x0[5], 0, x2[5]))∧(x0[3] - 1* x3[5])∧(x0[3]* x0[5])∧(x1[3] - 1* 0))

(3) -> (6), if ((215_0_ack_GT(x1[3] - 1, x0[3]) →* 295_0_ack_Return(x0[6]))∧(x0[3] - 1* x1[6])∧(x0[3]* x2[6])∧(x1[3] - 1* x3[6]))

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

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

(5) -> (0), if ((x2[5]* x1[0])∧(x3[5]* x0[0]))

(5) -> (2), if ((x2[5]* x1[2])∧(x3[5]* x0[2]))

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

(6) -> (2), if ((x0[6]* x1[2])∧(x1[6]* x0[2]))

The set Q consists of the following terms:
215_0_ack_GT(x0, 0)
258_1_ack_InvokeMethod(231_0_ack_Return(1, x0), 0)
258_1_ack_InvokeMethod(295_0_ack_Return(x0), x1)
287_1_ack_InvokeMethod(231_0_ack_Return(x0, x1), 0, x0)
287_1_ack_InvokeMethod(266_0_ack_Return(x0, x1, x2), x0, x1)
287_1_ack_InvokeMethod(295_0_ack_Return(x0), x1, x2)

### (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 215_0_ACK_GT(x1, x0) → COND_215_0_ACK_GT(&&(<=(x1, 0), >(x0, 0)), x1, x0) the following chains were created:
• We consider the chain 215_0_ACK_GT(x1[0], x0[0]) → COND_215_0_ACK_GT(&&(<=(x1[0], 0), >(x0[0], 0)), x1[0], x0[0]), COND_215_0_ACK_GT(TRUE, x1[1], x0[1]) → 215_0_ACK_GT(1, -(x0[1], 1)) which results in the following constraint:

(1)    (&&(<=(x1[0], 0), >(x0[0], 0))=TRUEx1[0]=x1[1]x0[0]=x0[1]215_0_ACK_GT(x1[0], x0[0])≥NonInfC∧215_0_ACK_GT(x1[0], x0[0])≥COND_215_0_ACK_GT(&&(<=(x1[0], 0), >(x0[0], 0)), x1[0], x0[0])∧(UIncreasing(COND_215_0_ACK_GT(&&(<=(x1[0], 0), >(x0[0], 0)), x1[0], x0[0])), ≥))

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

(2)    (&&(<=(x1[0], 0), >(x0[0], 0))=TRUE215_0_ACK_GT(x1[0], x0[0])≥NonInfC∧215_0_ACK_GT(x1[0], x0[0])≥COND_215_0_ACK_GT(&&(<=(x1[0], 0), >(x0[0], 0)), x1[0], x0[0])∧(UIncreasing(COND_215_0_ACK_GT(&&(<=(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)    (0 ≥ 0 ⇒ (UIncreasing(COND_215_0_ACK_GT(&&(<=(x1[0], 0), >(x0[0], 0)), x1[0], x0[0])), ≥)∧[(-1)bni_36 + (-1)Bound*bni_36] ≥ 0∧[(-1)bso_37] ≥ 0)

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

(4)    (0 ≥ 0 ⇒ (UIncreasing(COND_215_0_ACK_GT(&&(<=(x1[0], 0), >(x0[0], 0)), x1[0], x0[0])), ≥)∧[(-1)bni_36 + (-1)Bound*bni_36] ≥ 0∧[(-1)bso_37] ≥ 0)

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

(5)    (0 ≥ 0 ⇒ (UIncreasing(COND_215_0_ACK_GT(&&(<=(x1[0], 0), >(x0[0], 0)), x1[0], x0[0])), ≥)∧[(-1)bni_36 + (-1)Bound*bni_36] ≥ 0∧[(-1)bso_37] ≥ 0)

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

(6)    (0 ≥ 0 ⇒ (UIncreasing(COND_215_0_ACK_GT(&&(<=(x1[0], 0), >(x0[0], 0)), x1[0], x0[0])), ≥)∧0 ≥ 0∧0 ≥ 0∧[(-1)bni_36 + (-1)Bound*bni_36] ≥ 0∧0 ≥ 0∧0 ≥ 0∧[(-1)bso_37] ≥ 0)

For Pair COND_215_0_ACK_GT(TRUE, x1, x0) → 215_0_ACK_GT(1, -(x0, 1)) the following chains were created:
• We consider the chain 215_0_ACK_GT(x1[0], x0[0]) → COND_215_0_ACK_GT(&&(<=(x1[0], 0), >(x0[0], 0)), x1[0], x0[0]), COND_215_0_ACK_GT(TRUE, x1[1], x0[1]) → 215_0_ACK_GT(1, -(x0[1], 1)), 215_0_ACK_GT(x1[0], x0[0]) → COND_215_0_ACK_GT(&&(<=(x1[0], 0), >(x0[0], 0)), x1[0], x0[0]) which results in the following constraint:

(7)    (&&(<=(x1[0], 0), >(x0[0], 0))=TRUEx1[0]=x1[1]x0[0]=x0[1]1=x1[0]1-(x0[1], 1)=x0[0]1COND_215_0_ACK_GT(TRUE, x1[1], x0[1])≥NonInfC∧COND_215_0_ACK_GT(TRUE, x1[1], x0[1])≥215_0_ACK_GT(1, -(x0[1], 1))∧(UIncreasing(215_0_ACK_GT(1, -(x0[1], 1))), ≥))

We simplified constraint (7) using rules (III), (IV) which results in the following new constraint:

(8)    (&&(<=(x1[0], 0), >(x0[0], 0))=TRUECOND_215_0_ACK_GT(TRUE, x1[0], x0[0])≥NonInfC∧COND_215_0_ACK_GT(TRUE, x1[0], x0[0])≥215_0_ACK_GT(1, -(x0[0], 1))∧(UIncreasing(215_0_ACK_GT(1, -(x0[1], 1))), ≥))

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

(9)    (0 ≥ 0 ⇒ (UIncreasing(215_0_ACK_GT(1, -(x0[1], 1))), ≥)∧[(-1)bni_38 + (-1)Bound*bni_38] ≥ 0∧[(-1)bso_39] ≥ 0)

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

(10)    (0 ≥ 0 ⇒ (UIncreasing(215_0_ACK_GT(1, -(x0[1], 1))), ≥)∧[(-1)bni_38 + (-1)Bound*bni_38] ≥ 0∧[(-1)bso_39] ≥ 0)

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

(11)    (0 ≥ 0 ⇒ (UIncreasing(215_0_ACK_GT(1, -(x0[1], 1))), ≥)∧[(-1)bni_38 + (-1)Bound*bni_38] ≥ 0∧[(-1)bso_39] ≥ 0)

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

(12)    (0 ≥ 0 ⇒ (UIncreasing(215_0_ACK_GT(1, -(x0[1], 1))), ≥)∧0 ≥ 0∧0 ≥ 0∧[(-1)bni_38 + (-1)Bound*bni_38] ≥ 0∧0 ≥ 0∧0 ≥ 0∧[(-1)bso_39] ≥ 0)

• We consider the chain 215_0_ACK_GT(x1[0], x0[0]) → COND_215_0_ACK_GT(&&(<=(x1[0], 0), >(x0[0], 0)), x1[0], x0[0]), COND_215_0_ACK_GT(TRUE, x1[1], x0[1]) → 215_0_ACK_GT(1, -(x0[1], 1)), 215_0_ACK_GT(x1[2], x0[2]) → COND_215_0_ACK_GT1(&&(>(x1[2], 0), >(x0[2], 0)), x1[2], x0[2]) which results in the following constraint:

(13)    (&&(<=(x1[0], 0), >(x0[0], 0))=TRUEx1[0]=x1[1]x0[0]=x0[1]1=x1[2]-(x0[1], 1)=x0[2]COND_215_0_ACK_GT(TRUE, x1[1], x0[1])≥NonInfC∧COND_215_0_ACK_GT(TRUE, x1[1], x0[1])≥215_0_ACK_GT(1, -(x0[1], 1))∧(UIncreasing(215_0_ACK_GT(1, -(x0[1], 1))), ≥))

We simplified constraint (13) using rules (III), (IV) which results in the following new constraint:

(14)    (&&(<=(x1[0], 0), >(x0[0], 0))=TRUECOND_215_0_ACK_GT(TRUE, x1[0], x0[0])≥NonInfC∧COND_215_0_ACK_GT(TRUE, x1[0], x0[0])≥215_0_ACK_GT(1, -(x0[0], 1))∧(UIncreasing(215_0_ACK_GT(1, -(x0[1], 1))), ≥))

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

(15)    (0 ≥ 0 ⇒ (UIncreasing(215_0_ACK_GT(1, -(x0[1], 1))), ≥)∧[(-1)bni_38 + (-1)Bound*bni_38] ≥ 0∧[(-1)bso_39] ≥ 0)

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

(16)    (0 ≥ 0 ⇒ (UIncreasing(215_0_ACK_GT(1, -(x0[1], 1))), ≥)∧[(-1)bni_38 + (-1)Bound*bni_38] ≥ 0∧[(-1)bso_39] ≥ 0)

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

(17)    (0 ≥ 0 ⇒ (UIncreasing(215_0_ACK_GT(1, -(x0[1], 1))), ≥)∧[(-1)bni_38 + (-1)Bound*bni_38] ≥ 0∧[(-1)bso_39] ≥ 0)

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

(18)    (0 ≥ 0 ⇒ (UIncreasing(215_0_ACK_GT(1, -(x0[1], 1))), ≥)∧0 ≥ 0∧0 ≥ 0∧[(-1)bni_38 + (-1)Bound*bni_38] ≥ 0∧0 ≥ 0∧0 ≥ 0∧[(-1)bso_39] ≥ 0)

For Pair 215_0_ACK_GT(x1, x0) → COND_215_0_ACK_GT1(&&(>(x1, 0), >(x0, 0)), x1, x0) the following chains were created:
• We consider the chain 215_0_ACK_GT(x1[2], x0[2]) → COND_215_0_ACK_GT1(&&(>(x1[2], 0), >(x0[2], 0)), x1[2], x0[2]), COND_215_0_ACK_GT1(TRUE, x1[3], x0[3]) → 277_1_ACK_INVOKEMETHOD(215_0_ack_GT(-(x1[3], 1), x0[3]), -(x0[3], 1), x0[3], -(x1[3], 1)) which results in the following constraint:

(19)    (&&(>(x1[2], 0), >(x0[2], 0))=TRUEx1[2]=x1[3]x0[2]=x0[3]215_0_ACK_GT(x1[2], x0[2])≥NonInfC∧215_0_ACK_GT(x1[2], x0[2])≥COND_215_0_ACK_GT1(&&(>(x1[2], 0), >(x0[2], 0)), x1[2], x0[2])∧(UIncreasing(COND_215_0_ACK_GT1(&&(>(x1[2], 0), >(x0[2], 0)), x1[2], x0[2])), ≥))

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

(20)    (&&(>(x1[2], 0), >(x0[2], 0))=TRUE215_0_ACK_GT(x1[2], x0[2])≥NonInfC∧215_0_ACK_GT(x1[2], x0[2])≥COND_215_0_ACK_GT1(&&(>(x1[2], 0), >(x0[2], 0)), x1[2], x0[2])∧(UIncreasing(COND_215_0_ACK_GT1(&&(>(x1[2], 0), >(x0[2], 0)), x1[2], x0[2])), ≥))

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

(21)    (0 ≥ 0 ⇒ (UIncreasing(COND_215_0_ACK_GT1(&&(>(x1[2], 0), >(x0[2], 0)), x1[2], x0[2])), ≥)∧[(-1)bni_40 + (-1)Bound*bni_40] ≥ 0∧[(-1)bso_41] ≥ 0)

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

(22)    (0 ≥ 0 ⇒ (UIncreasing(COND_215_0_ACK_GT1(&&(>(x1[2], 0), >(x0[2], 0)), x1[2], x0[2])), ≥)∧[(-1)bni_40 + (-1)Bound*bni_40] ≥ 0∧[(-1)bso_41] ≥ 0)

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

(23)    (0 ≥ 0 ⇒ (UIncreasing(COND_215_0_ACK_GT1(&&(>(x1[2], 0), >(x0[2], 0)), x1[2], x0[2])), ≥)∧[(-1)bni_40 + (-1)Bound*bni_40] ≥ 0∧[(-1)bso_41] ≥ 0)

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

(24)    (0 ≥ 0 ⇒ (UIncreasing(COND_215_0_ACK_GT1(&&(>(x1[2], 0), >(x0[2], 0)), x1[2], x0[2])), ≥)∧0 ≥ 0∧0 ≥ 0∧[(-1)bni_40 + (-1)Bound*bni_40] ≥ 0∧0 ≥ 0∧0 ≥ 0∧[(-1)bso_41] ≥ 0)

• We consider the chain 215_0_ACK_GT(x1[2], x0[2]) → COND_215_0_ACK_GT1(&&(>(x1[2], 0), >(x0[2], 0)), x1[2], x0[2]), COND_215_0_ACK_GT1(TRUE, x1[4], x0[4]) → 215_0_ACK_GT(-(x1[4], 1), x0[4]) which results in the following constraint:

(25)    (&&(>(x1[2], 0), >(x0[2], 0))=TRUEx1[2]=x1[4]x0[2]=x0[4]215_0_ACK_GT(x1[2], x0[2])≥NonInfC∧215_0_ACK_GT(x1[2], x0[2])≥COND_215_0_ACK_GT1(&&(>(x1[2], 0), >(x0[2], 0)), x1[2], x0[2])∧(UIncreasing(COND_215_0_ACK_GT1(&&(>(x1[2], 0), >(x0[2], 0)), x1[2], x0[2])), ≥))

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

(26)    (&&(>(x1[2], 0), >(x0[2], 0))=TRUE215_0_ACK_GT(x1[2], x0[2])≥NonInfC∧215_0_ACK_GT(x1[2], x0[2])≥COND_215_0_ACK_GT1(&&(>(x1[2], 0), >(x0[2], 0)), x1[2], x0[2])∧(UIncreasing(COND_215_0_ACK_GT1(&&(>(x1[2], 0), >(x0[2], 0)), x1[2], x0[2])), ≥))

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

(27)    (0 ≥ 0 ⇒ (UIncreasing(COND_215_0_ACK_GT1(&&(>(x1[2], 0), >(x0[2], 0)), x1[2], x0[2])), ≥)∧[(-1)bni_40 + (-1)Bound*bni_40] ≥ 0∧[(-1)bso_41] ≥ 0)

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

(28)    (0 ≥ 0 ⇒ (UIncreasing(COND_215_0_ACK_GT1(&&(>(x1[2], 0), >(x0[2], 0)), x1[2], x0[2])), ≥)∧[(-1)bni_40 + (-1)Bound*bni_40] ≥ 0∧[(-1)bso_41] ≥ 0)

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

(29)    (0 ≥ 0 ⇒ (UIncreasing(COND_215_0_ACK_GT1(&&(>(x1[2], 0), >(x0[2], 0)), x1[2], x0[2])), ≥)∧[(-1)bni_40 + (-1)Bound*bni_40] ≥ 0∧[(-1)bso_41] ≥ 0)

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

(30)    (0 ≥ 0 ⇒ (UIncreasing(COND_215_0_ACK_GT1(&&(>(x1[2], 0), >(x0[2], 0)), x1[2], x0[2])), ≥)∧0 ≥ 0∧0 ≥ 0∧[(-1)bni_40 + (-1)Bound*bni_40] ≥ 0∧0 ≥ 0∧0 ≥ 0∧[(-1)bso_41] ≥ 0)

For Pair COND_215_0_ACK_GT1(TRUE, x1, x0) → 277_1_ACK_INVOKEMETHOD(215_0_ack_GT(-(x1, 1), x0), -(x0, 1), x0, -(x1, 1)) the following chains were created:
• We consider the chain 215_0_ACK_GT(x1[2], x0[2]) → COND_215_0_ACK_GT1(&&(>(x1[2], 0), >(x0[2], 0)), x1[2], x0[2]), COND_215_0_ACK_GT1(TRUE, x1[3], x0[3]) → 277_1_ACK_INVOKEMETHOD(215_0_ack_GT(-(x1[3], 1), x0[3]), -(x0[3], 1), x0[3], -(x1[3], 1)), 277_1_ACK_INVOKEMETHOD(266_0_ack_Return(x0[5], 0, x2[5]), x3[5], x0[5], 0) → 215_0_ACK_GT(x2[5], x3[5]) which results in the following constraint:

(31)    (&&(>(x1[2], 0), >(x0[2], 0))=TRUEx1[2]=x1[3]x0[2]=x0[3]215_0_ack_GT(-(x1[3], 1), x0[3])=266_0_ack_Return(x0[5], 0, x2[5])∧-(x0[3], 1)=x3[5]x0[3]=x0[5]-(x1[3], 1)=0COND_215_0_ACK_GT1(TRUE, x1[3], x0[3])≥NonInfC∧COND_215_0_ACK_GT1(TRUE, x1[3], x0[3])≥277_1_ACK_INVOKEMETHOD(215_0_ack_GT(-(x1[3], 1), x0[3]), -(x0[3], 1), x0[3], -(x1[3], 1))∧(UIncreasing(277_1_ACK_INVOKEMETHOD(215_0_ack_GT(-(x1[3], 1), x0[3]), -(x0[3], 1), x0[3], -(x1[3], 1))), ≥))

We simplified constraint (31) using rules (III), (IV) which results in the following new constraint:

(32)    (&&(>(x1[2], 0), >(x0[2], 0))=TRUE215_0_ack_GT(-(x1[2], 1), x0[2])=266_0_ack_Return(x0[2], 0, x2[5])∧-(x1[2], 1)=0COND_215_0_ACK_GT1(TRUE, x1[2], x0[2])≥NonInfC∧COND_215_0_ACK_GT1(TRUE, x1[2], x0[2])≥277_1_ACK_INVOKEMETHOD(215_0_ack_GT(-(x1[2], 1), x0[2]), -(x0[2], 1), x0[2], -(x1[2], 1))∧(UIncreasing(277_1_ACK_INVOKEMETHOD(215_0_ack_GT(-(x1[3], 1), x0[3]), -(x0[3], 1), x0[3], -(x1[3], 1))), ≥))

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

(33)    (0 ≥ 0∧0 ≥ 0 ⇒ (UIncreasing(277_1_ACK_INVOKEMETHOD(215_0_ack_GT(-(x1[3], 1), x0[3]), -(x0[3], 1), x0[3], -(x1[3], 1))), ≥)∧[(-1)bni_42 + (-1)Bound*bni_42] ≥ 0∧[1 + (-1)bso_43] + x0[2] ≥ 0)

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

(34)    (0 ≥ 0∧0 ≥ 0 ⇒ (UIncreasing(277_1_ACK_INVOKEMETHOD(215_0_ack_GT(-(x1[3], 1), x0[3]), -(x0[3], 1), x0[3], -(x1[3], 1))), ≥)∧[(-1)bni_42 + (-1)Bound*bni_42] ≥ 0∧[1 + (-1)bso_43] + x0[2] ≥ 0)

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

(35)    (0 ≥ 0∧0 ≥ 0 ⇒ (UIncreasing(277_1_ACK_INVOKEMETHOD(215_0_ack_GT(-(x1[3], 1), x0[3]), -(x0[3], 1), x0[3], -(x1[3], 1))), ≥)∧[(-1)bni_42 + (-1)Bound*bni_42] ≥ 0∧[1 + (-1)bso_43] + x0[2] ≥ 0)

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

(36)    (0 ≥ 0∧0 ≥ 0 ⇒ (UIncreasing(277_1_ACK_INVOKEMETHOD(215_0_ack_GT(-(x1[3], 1), x0[3]), -(x0[3], 1), x0[3], -(x1[3], 1))), ≥)∧0 ≥ 0∧0 ≥ 0∧[(-1)bni_42 + (-1)Bound*bni_42] ≥ 0∧[1] ≥ 0∧0 ≥ 0∧[1 + (-1)bso_43] ≥ 0)

• We consider the chain 215_0_ACK_GT(x1[2], x0[2]) → COND_215_0_ACK_GT1(&&(>(x1[2], 0), >(x0[2], 0)), x1[2], x0[2]), COND_215_0_ACK_GT1(TRUE, x1[3], x0[3]) → 277_1_ACK_INVOKEMETHOD(215_0_ack_GT(-(x1[3], 1), x0[3]), -(x0[3], 1), x0[3], -(x1[3], 1)), 277_1_ACK_INVOKEMETHOD(295_0_ack_Return(x0[6]), x1[6], x2[6], x3[6]) → 215_0_ACK_GT(x0[6], x1[6]) which results in the following constraint:

(37)    (&&(>(x1[2], 0), >(x0[2], 0))=TRUEx1[2]=x1[3]x0[2]=x0[3]215_0_ack_GT(-(x1[3], 1), x0[3])=295_0_ack_Return(x0[6])∧-(x0[3], 1)=x1[6]x0[3]=x2[6]-(x1[3], 1)=x3[6]COND_215_0_ACK_GT1(TRUE, x1[3], x0[3])≥NonInfC∧COND_215_0_ACK_GT1(TRUE, x1[3], x0[3])≥277_1_ACK_INVOKEMETHOD(215_0_ack_GT(-(x1[3], 1), x0[3]), -(x0[3], 1), x0[3], -(x1[3], 1))∧(UIncreasing(277_1_ACK_INVOKEMETHOD(215_0_ack_GT(-(x1[3], 1), x0[3]), -(x0[3], 1), x0[3], -(x1[3], 1))), ≥))

We simplified constraint (37) using rules (III), (IV) which results in the following new constraint:

(38)    (&&(>(x1[2], 0), >(x0[2], 0))=TRUE215_0_ack_GT(-(x1[2], 1), x0[2])=295_0_ack_Return(x0[6]) ⇒ COND_215_0_ACK_GT1(TRUE, x1[2], x0[2])≥NonInfC∧COND_215_0_ACK_GT1(TRUE, x1[2], x0[2])≥277_1_ACK_INVOKEMETHOD(215_0_ack_GT(-(x1[2], 1), x0[2]), -(x0[2], 1), x0[2], -(x1[2], 1))∧(UIncreasing(277_1_ACK_INVOKEMETHOD(215_0_ack_GT(-(x1[3], 1), x0[3]), -(x0[3], 1), x0[3], -(x1[3], 1))), ≥))

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

(39)    (0 ≥ 0 ⇒ (UIncreasing(277_1_ACK_INVOKEMETHOD(215_0_ack_GT(-(x1[3], 1), x0[3]), -(x0[3], 1), x0[3], -(x1[3], 1))), ≥)∧[(-1)bni_42 + (-1)Bound*bni_42] ≥ 0∧[1 + (-1)bso_43] + x0[2] ≥ 0)

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

(40)    (0 ≥ 0 ⇒ (UIncreasing(277_1_ACK_INVOKEMETHOD(215_0_ack_GT(-(x1[3], 1), x0[3]), -(x0[3], 1), x0[3], -(x1[3], 1))), ≥)∧[(-1)bni_42 + (-1)Bound*bni_42] ≥ 0∧[1 + (-1)bso_43] + x0[2] ≥ 0)

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

(41)    (0 ≥ 0 ⇒ (UIncreasing(277_1_ACK_INVOKEMETHOD(215_0_ack_GT(-(x1[3], 1), x0[3]), -(x0[3], 1), x0[3], -(x1[3], 1))), ≥)∧[(-1)bni_42 + (-1)Bound*bni_42] ≥ 0∧[1 + (-1)bso_43] + x0[2] ≥ 0)

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

(42)    (0 ≥ 0 ⇒ (UIncreasing(277_1_ACK_INVOKEMETHOD(215_0_ack_GT(-(x1[3], 1), x0[3]), -(x0[3], 1), x0[3], -(x1[3], 1))), ≥)∧0 ≥ 0∧0 ≥ 0∧[(-1)bni_42 + (-1)Bound*bni_42] ≥ 0∧[1] ≥ 0∧0 ≥ 0∧[1 + (-1)bso_43] ≥ 0)

For Pair COND_215_0_ACK_GT1(TRUE, x1, x0) → 215_0_ACK_GT(-(x1, 1), x0) the following chains were created:
• We consider the chain 215_0_ACK_GT(x1[2], x0[2]) → COND_215_0_ACK_GT1(&&(>(x1[2], 0), >(x0[2], 0)), x1[2], x0[2]), COND_215_0_ACK_GT1(TRUE, x1[4], x0[4]) → 215_0_ACK_GT(-(x1[4], 1), x0[4]), 215_0_ACK_GT(x1[0], x0[0]) → COND_215_0_ACK_GT(&&(<=(x1[0], 0), >(x0[0], 0)), x1[0], x0[0]) which results in the following constraint:

(43)    (&&(>(x1[2], 0), >(x0[2], 0))=TRUEx1[2]=x1[4]x0[2]=x0[4]-(x1[4], 1)=x1[0]x0[4]=x0[0]COND_215_0_ACK_GT1(TRUE, x1[4], x0[4])≥NonInfC∧COND_215_0_ACK_GT1(TRUE, x1[4], x0[4])≥215_0_ACK_GT(-(x1[4], 1), x0[4])∧(UIncreasing(215_0_ACK_GT(-(x1[4], 1), x0[4])), ≥))

We simplified constraint (43) using rules (III), (IV) which results in the following new constraint:

(44)    (&&(>(x1[2], 0), >(x0[2], 0))=TRUECOND_215_0_ACK_GT1(TRUE, x1[2], x0[2])≥NonInfC∧COND_215_0_ACK_GT1(TRUE, x1[2], x0[2])≥215_0_ACK_GT(-(x1[2], 1), x0[2])∧(UIncreasing(215_0_ACK_GT(-(x1[4], 1), x0[4])), ≥))

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

(45)    (0 ≥ 0 ⇒ (UIncreasing(215_0_ACK_GT(-(x1[4], 1), x0[4])), ≥)∧[(-1)bni_44 + (-1)Bound*bni_44] ≥ 0∧[(-1)bso_45] ≥ 0)

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

(46)    (0 ≥ 0 ⇒ (UIncreasing(215_0_ACK_GT(-(x1[4], 1), x0[4])), ≥)∧[(-1)bni_44 + (-1)Bound*bni_44] ≥ 0∧[(-1)bso_45] ≥ 0)

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

(47)    (0 ≥ 0 ⇒ (UIncreasing(215_0_ACK_GT(-(x1[4], 1), x0[4])), ≥)∧[(-1)bni_44 + (-1)Bound*bni_44] ≥ 0∧[(-1)bso_45] ≥ 0)

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

(48)    (0 ≥ 0 ⇒ (UIncreasing(215_0_ACK_GT(-(x1[4], 1), x0[4])), ≥)∧0 ≥ 0∧0 ≥ 0∧[(-1)bni_44 + (-1)Bound*bni_44] ≥ 0∧0 ≥ 0∧0 ≥ 0∧[(-1)bso_45] ≥ 0)

• We consider the chain 215_0_ACK_GT(x1[2], x0[2]) → COND_215_0_ACK_GT1(&&(>(x1[2], 0), >(x0[2], 0)), x1[2], x0[2]), COND_215_0_ACK_GT1(TRUE, x1[4], x0[4]) → 215_0_ACK_GT(-(x1[4], 1), x0[4]), 215_0_ACK_GT(x1[2], x0[2]) → COND_215_0_ACK_GT1(&&(>(x1[2], 0), >(x0[2], 0)), x1[2], x0[2]) which results in the following constraint:

(49)    (&&(>(x1[2], 0), >(x0[2], 0))=TRUEx1[2]=x1[4]x0[2]=x0[4]-(x1[4], 1)=x1[2]1x0[4]=x0[2]1COND_215_0_ACK_GT1(TRUE, x1[4], x0[4])≥NonInfC∧COND_215_0_ACK_GT1(TRUE, x1[4], x0[4])≥215_0_ACK_GT(-(x1[4], 1), x0[4])∧(UIncreasing(215_0_ACK_GT(-(x1[4], 1), x0[4])), ≥))

We simplified constraint (49) using rules (III), (IV) which results in the following new constraint:

(50)    (&&(>(x1[2], 0), >(x0[2], 0))=TRUECOND_215_0_ACK_GT1(TRUE, x1[2], x0[2])≥NonInfC∧COND_215_0_ACK_GT1(TRUE, x1[2], x0[2])≥215_0_ACK_GT(-(x1[2], 1), x0[2])∧(UIncreasing(215_0_ACK_GT(-(x1[4], 1), x0[4])), ≥))

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

(51)    (0 ≥ 0 ⇒ (UIncreasing(215_0_ACK_GT(-(x1[4], 1), x0[4])), ≥)∧[(-1)bni_44 + (-1)Bound*bni_44] ≥ 0∧[(-1)bso_45] ≥ 0)

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

(52)    (0 ≥ 0 ⇒ (UIncreasing(215_0_ACK_GT(-(x1[4], 1), x0[4])), ≥)∧[(-1)bni_44 + (-1)Bound*bni_44] ≥ 0∧[(-1)bso_45] ≥ 0)

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

(53)    (0 ≥ 0 ⇒ (UIncreasing(215_0_ACK_GT(-(x1[4], 1), x0[4])), ≥)∧[(-1)bni_44 + (-1)Bound*bni_44] ≥ 0∧[(-1)bso_45] ≥ 0)

We simplified constraint (53) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint:

(54)    (0 ≥ 0 ⇒ (UIncreasing(215_0_ACK_GT(-(x1[4], 1), x0[4])), ≥)∧0 ≥ 0∧0 ≥ 0∧[(-1)bni_44 + (-1)Bound*bni_44] ≥ 0∧0 ≥ 0∧0 ≥ 0∧[(-1)bso_45] ≥ 0)

For Pair 277_1_ACK_INVOKEMETHOD(266_0_ack_Return(x0, 0, x2), x3, x0, 0) → 215_0_ACK_GT(x2, x3) the following chains were created:
• We consider the chain COND_215_0_ACK_GT1(TRUE, x1[3], x0[3]) → 277_1_ACK_INVOKEMETHOD(215_0_ack_GT(-(x1[3], 1), x0[3]), -(x0[3], 1), x0[3], -(x1[3], 1)), 277_1_ACK_INVOKEMETHOD(266_0_ack_Return(x0[5], 0, x2[5]), x3[5], x0[5], 0) → 215_0_ACK_GT(x2[5], x3[5]), 215_0_ACK_GT(x1[0], x0[0]) → COND_215_0_ACK_GT(&&(<=(x1[0], 0), >(x0[0], 0)), x1[0], x0[0]) which results in the following constraint:

(55)    (215_0_ack_GT(-(x1[3], 1), x0[3])=266_0_ack_Return(x0[5], 0, x2[5])∧-(x0[3], 1)=x3[5]x0[3]=x0[5]-(x1[3], 1)=0x2[5]=x1[0]x3[5]=x0[0]277_1_ACK_INVOKEMETHOD(266_0_ack_Return(x0[5], 0, x2[5]), x3[5], x0[5], 0)≥NonInfC∧277_1_ACK_INVOKEMETHOD(266_0_ack_Return(x0[5], 0, x2[5]), x3[5], x0[5], 0)≥215_0_ACK_GT(x2[5], x3[5])∧(UIncreasing(215_0_ACK_GT(x2[5], x3[5])), ≥))

We simplified constraint (55) using rules (III), (IV), (VII) which results in the following new constraint:

(56)    (-(x1[3], 1)=x0215_0_ack_GT(x0, x0[3])=266_0_ack_Return(x0[3], 0, x2[5])∧-(x1[3], 1)=0277_1_ACK_INVOKEMETHOD(266_0_ack_Return(x0[3], 0, x2[5]), -(x0[3], 1), x0[3], 0)≥NonInfC∧277_1_ACK_INVOKEMETHOD(266_0_ack_Return(x0[3], 0, x2[5]), -(x0[3], 1), x0[3], 0)≥215_0_ACK_GT(x2[5], -(x0[3], 1))∧(UIncreasing(215_0_ACK_GT(x2[5], x3[5])), ≥))

We solved constraint (56) using rule (V) (with possible (I) afterwards).
• We consider the chain COND_215_0_ACK_GT1(TRUE, x1[3], x0[3]) → 277_1_ACK_INVOKEMETHOD(215_0_ack_GT(-(x1[3], 1), x0[3]), -(x0[3], 1), x0[3], -(x1[3], 1)), 277_1_ACK_INVOKEMETHOD(266_0_ack_Return(x0[5], 0, x2[5]), x3[5], x0[5], 0) → 215_0_ACK_GT(x2[5], x3[5]), 215_0_ACK_GT(x1[2], x0[2]) → COND_215_0_ACK_GT1(&&(>(x1[2], 0), >(x0[2], 0)), x1[2], x0[2]) which results in the following constraint:

(57)    (215_0_ack_GT(-(x1[3], 1), x0[3])=266_0_ack_Return(x0[5], 0, x2[5])∧-(x0[3], 1)=x3[5]x0[3]=x0[5]-(x1[3], 1)=0x2[5]=x1[2]x3[5]=x0[2]277_1_ACK_INVOKEMETHOD(266_0_ack_Return(x0[5], 0, x2[5]), x3[5], x0[5], 0)≥NonInfC∧277_1_ACK_INVOKEMETHOD(266_0_ack_Return(x0[5], 0, x2[5]), x3[5], x0[5], 0)≥215_0_ACK_GT(x2[5], x3[5])∧(UIncreasing(215_0_ACK_GT(x2[5], x3[5])), ≥))

We simplified constraint (57) using rules (III), (IV), (VII) which results in the following new constraint:

(58)    (-(x1[3], 1)=x2215_0_ack_GT(x2, x0[3])=266_0_ack_Return(x0[3], 0, x2[5])∧-(x1[3], 1)=0277_1_ACK_INVOKEMETHOD(266_0_ack_Return(x0[3], 0, x2[5]), -(x0[3], 1), x0[3], 0)≥NonInfC∧277_1_ACK_INVOKEMETHOD(266_0_ack_Return(x0[3], 0, x2[5]), -(x0[3], 1), x0[3], 0)≥215_0_ACK_GT(x2[5], -(x0[3], 1))∧(UIncreasing(215_0_ACK_GT(x2[5], x3[5])), ≥))

We solved constraint (58) using rule (V) (with possible (I) afterwards).

For Pair 277_1_ACK_INVOKEMETHOD(295_0_ack_Return(x0), x1, x2, x3) → 215_0_ACK_GT(x0, x1) the following chains were created:
• We consider the chain COND_215_0_ACK_GT1(TRUE, x1[3], x0[3]) → 277_1_ACK_INVOKEMETHOD(215_0_ack_GT(-(x1[3], 1), x0[3]), -(x0[3], 1), x0[3], -(x1[3], 1)), 277_1_ACK_INVOKEMETHOD(295_0_ack_Return(x0[6]), x1[6], x2[6], x3[6]) → 215_0_ACK_GT(x0[6], x1[6]), 215_0_ACK_GT(x1[0], x0[0]) → COND_215_0_ACK_GT(&&(<=(x1[0], 0), >(x0[0], 0)), x1[0], x0[0]) which results in the following constraint:

(59)    (215_0_ack_GT(-(x1[3], 1), x0[3])=295_0_ack_Return(x0[6])∧-(x0[3], 1)=x1[6]x0[3]=x2[6]-(x1[3], 1)=x3[6]x0[6]=x1[0]x1[6]=x0[0]277_1_ACK_INVOKEMETHOD(295_0_ack_Return(x0[6]), x1[6], x2[6], x3[6])≥NonInfC∧277_1_ACK_INVOKEMETHOD(295_0_ack_Return(x0[6]), x1[6], x2[6], x3[6])≥215_0_ACK_GT(x0[6], x1[6])∧(UIncreasing(215_0_ACK_GT(x0[6], x1[6])), ≥))

We simplified constraint (59) using rules (III), (IV), (VII) which results in the following new constraint:

(60)    (-(x1[3], 1)=x4215_0_ack_GT(x4, x0[3])=295_0_ack_Return(x0[6]) ⇒ 277_1_ACK_INVOKEMETHOD(295_0_ack_Return(x0[6]), -(x0[3], 1), x0[3], -(x1[3], 1))≥NonInfC∧277_1_ACK_INVOKEMETHOD(295_0_ack_Return(x0[6]), -(x0[3], 1), x0[3], -(x1[3], 1))≥215_0_ACK_GT(x0[6], -(x0[3], 1))∧(UIncreasing(215_0_ACK_GT(x0[6], x1[6])), ≥))

We solved constraint (60) using rule (V) (with possible (I) afterwards).
• We consider the chain COND_215_0_ACK_GT1(TRUE, x1[3], x0[3]) → 277_1_ACK_INVOKEMETHOD(215_0_ack_GT(-(x1[3], 1), x0[3]), -(x0[3], 1), x0[3], -(x1[3], 1)), 277_1_ACK_INVOKEMETHOD(295_0_ack_Return(x0[6]), x1[6], x2[6], x3[6]) → 215_0_ACK_GT(x0[6], x1[6]), 215_0_ACK_GT(x1[2], x0[2]) → COND_215_0_ACK_GT1(&&(>(x1[2], 0), >(x0[2], 0)), x1[2], x0[2]) which results in the following constraint:

(61)    (215_0_ack_GT(-(x1[3], 1), x0[3])=295_0_ack_Return(x0[6])∧-(x0[3], 1)=x1[6]x0[3]=x2[6]-(x1[3], 1)=x3[6]x0[6]=x1[2]x1[6]=x0[2]277_1_ACK_INVOKEMETHOD(295_0_ack_Return(x0[6]), x1[6], x2[6], x3[6])≥NonInfC∧277_1_ACK_INVOKEMETHOD(295_0_ack_Return(x0[6]), x1[6], x2[6], x3[6])≥215_0_ACK_GT(x0[6], x1[6])∧(UIncreasing(215_0_ACK_GT(x0[6], x1[6])), ≥))

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

(62)    (-(x1[3], 1)=x6215_0_ack_GT(x6, x0[3])=295_0_ack_Return(x0[6]) ⇒ 277_1_ACK_INVOKEMETHOD(295_0_ack_Return(x0[6]), -(x0[3], 1), x0[3], -(x1[3], 1))≥NonInfC∧277_1_ACK_INVOKEMETHOD(295_0_ack_Return(x0[6]), -(x0[3], 1), x0[3], -(x1[3], 1))≥215_0_ACK_GT(x0[6], -(x0[3], 1))∧(UIncreasing(215_0_ACK_GT(x0[6], x1[6])), ≥))

We solved constraint (62) using rule (V) (with possible (I) afterwards).

To summarize, we get the following constraints P for the following pairs.
• 215_0_ACK_GT(x1, x0) → COND_215_0_ACK_GT(&&(<=(x1, 0), >(x0, 0)), x1, x0)
• (0 ≥ 0 ⇒ (UIncreasing(COND_215_0_ACK_GT(&&(<=(x1[0], 0), >(x0[0], 0)), x1[0], x0[0])), ≥)∧0 ≥ 0∧0 ≥ 0∧[(-1)bni_36 + (-1)Bound*bni_36] ≥ 0∧0 ≥ 0∧0 ≥ 0∧[(-1)bso_37] ≥ 0)

• COND_215_0_ACK_GT(TRUE, x1, x0) → 215_0_ACK_GT(1, -(x0, 1))
• (0 ≥ 0 ⇒ (UIncreasing(215_0_ACK_GT(1, -(x0[1], 1))), ≥)∧0 ≥ 0∧0 ≥ 0∧[(-1)bni_38 + (-1)Bound*bni_38] ≥ 0∧0 ≥ 0∧0 ≥ 0∧[(-1)bso_39] ≥ 0)
• (0 ≥ 0 ⇒ (UIncreasing(215_0_ACK_GT(1, -(x0[1], 1))), ≥)∧0 ≥ 0∧0 ≥ 0∧[(-1)bni_38 + (-1)Bound*bni_38] ≥ 0∧0 ≥ 0∧0 ≥ 0∧[(-1)bso_39] ≥ 0)

• 215_0_ACK_GT(x1, x0) → COND_215_0_ACK_GT1(&&(>(x1, 0), >(x0, 0)), x1, x0)
• (0 ≥ 0 ⇒ (UIncreasing(COND_215_0_ACK_GT1(&&(>(x1[2], 0), >(x0[2], 0)), x1[2], x0[2])), ≥)∧0 ≥ 0∧0 ≥ 0∧[(-1)bni_40 + (-1)Bound*bni_40] ≥ 0∧0 ≥ 0∧0 ≥ 0∧[(-1)bso_41] ≥ 0)
• (0 ≥ 0 ⇒ (UIncreasing(COND_215_0_ACK_GT1(&&(>(x1[2], 0), >(x0[2], 0)), x1[2], x0[2])), ≥)∧0 ≥ 0∧0 ≥ 0∧[(-1)bni_40 + (-1)Bound*bni_40] ≥ 0∧0 ≥ 0∧0 ≥ 0∧[(-1)bso_41] ≥ 0)

• COND_215_0_ACK_GT1(TRUE, x1, x0) → 277_1_ACK_INVOKEMETHOD(215_0_ack_GT(-(x1, 1), x0), -(x0, 1), x0, -(x1, 1))
• (0 ≥ 0∧0 ≥ 0 ⇒ (UIncreasing(277_1_ACK_INVOKEMETHOD(215_0_ack_GT(-(x1[3], 1), x0[3]), -(x0[3], 1), x0[3], -(x1[3], 1))), ≥)∧0 ≥ 0∧0 ≥ 0∧[(-1)bni_42 + (-1)Bound*bni_42] ≥ 0∧[1] ≥ 0∧0 ≥ 0∧[1 + (-1)bso_43] ≥ 0)
• (0 ≥ 0 ⇒ (UIncreasing(277_1_ACK_INVOKEMETHOD(215_0_ack_GT(-(x1[3], 1), x0[3]), -(x0[3], 1), x0[3], -(x1[3], 1))), ≥)∧0 ≥ 0∧0 ≥ 0∧[(-1)bni_42 + (-1)Bound*bni_42] ≥ 0∧[1] ≥ 0∧0 ≥ 0∧[1 + (-1)bso_43] ≥ 0)

• COND_215_0_ACK_GT1(TRUE, x1, x0) → 215_0_ACK_GT(-(x1, 1), x0)
• (0 ≥ 0 ⇒ (UIncreasing(215_0_ACK_GT(-(x1[4], 1), x0[4])), ≥)∧0 ≥ 0∧0 ≥ 0∧[(-1)bni_44 + (-1)Bound*bni_44] ≥ 0∧0 ≥ 0∧0 ≥ 0∧[(-1)bso_45] ≥ 0)
• (0 ≥ 0 ⇒ (UIncreasing(215_0_ACK_GT(-(x1[4], 1), x0[4])), ≥)∧0 ≥ 0∧0 ≥ 0∧[(-1)bni_44 + (-1)Bound*bni_44] ≥ 0∧0 ≥ 0∧0 ≥ 0∧[(-1)bso_45] ≥ 0)

• 277_1_ACK_INVOKEMETHOD(266_0_ack_Return(x0, 0, x2), x3, x0, 0) → 215_0_ACK_GT(x2, x3)

• 277_1_ACK_INVOKEMETHOD(295_0_ack_Return(x0), x1, x2, x3) → 215_0_ACK_GT(x0, x1)

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 with natural coefficients for non-tuple symbols [NONINF][POLO]:

POL(TRUE) = 0
POL(FALSE) = 0
POL(215_0_ack_GT(x1, x2)) = [2] + [3]x2 + [3]x1
POL(0) = 0
POL(231_0_ack_Return(x1, x2)) = [3] + [3]x1
POL(+(x1, x2)) = 0
POL(1) = 0
POL(258_1_ack_InvokeMethod(x1, x2)) = 0
POL(266_0_ack_Return(x1, x2, x3)) = 0
POL(295_0_ack_Return(x1)) = 0
POL(287_1_ack_InvokeMethod(x1, x2, x3)) = 0
POL(215_0_ACK_GT(x1, x2)) = [-1]
POL(COND_215_0_ACK_GT(x1, x2, x3)) = [-1]
POL(&&(x1, x2)) = 0
POL(<=(x1, x2)) = 0
POL(>(x1, x2)) = 0
POL(-(x1, x2)) = 0
POL(COND_215_0_ACK_GT1(x1, x2, x3)) = [-1] + [2]x1
POL(277_1_ACK_INVOKEMETHOD(x1, x2, x3, x4)) = [-1]x4 + [2]x3 + [-1]x2 + [-1]x1

The following pairs are in P>:

COND_215_0_ACK_GT1(TRUE, x1[3], x0[3]) → 277_1_ACK_INVOKEMETHOD(215_0_ack_GT(-(x1[3], 1), x0[3]), -(x0[3], 1), x0[3], -(x1[3], 1))
277_1_ACK_INVOKEMETHOD(266_0_ack_Return(x0[5], 0, x2[5]), x3[5], x0[5], 0) → 215_0_ACK_GT(x2[5], x3[5])
277_1_ACK_INVOKEMETHOD(295_0_ack_Return(x0[6]), x1[6], x2[6], x3[6]) → 215_0_ACK_GT(x0[6], x1[6])

The following pairs are in Pbound:

215_0_ACK_GT(x1[0], x0[0]) → COND_215_0_ACK_GT(&&(<=(x1[0], 0), >(x0[0], 0)), x1[0], x0[0])
COND_215_0_ACK_GT(TRUE, x1[1], x0[1]) → 215_0_ACK_GT(1, -(x0[1], 1))
215_0_ACK_GT(x1[2], x0[2]) → COND_215_0_ACK_GT1(&&(>(x1[2], 0), >(x0[2], 0)), x1[2], x0[2])
COND_215_0_ACK_GT1(TRUE, x1[3], x0[3]) → 277_1_ACK_INVOKEMETHOD(215_0_ack_GT(-(x1[3], 1), x0[3]), -(x0[3], 1), x0[3], -(x1[3], 1))
COND_215_0_ACK_GT1(TRUE, x1[4], x0[4]) → 215_0_ACK_GT(-(x1[4], 1), x0[4])
277_1_ACK_INVOKEMETHOD(266_0_ack_Return(x0[5], 0, x2[5]), x3[5], x0[5], 0) → 215_0_ACK_GT(x2[5], x3[5])
277_1_ACK_INVOKEMETHOD(295_0_ack_Return(x0[6]), x1[6], x2[6], x3[6]) → 215_0_ACK_GT(x0[6], x1[6])

The following pairs are in P:

215_0_ACK_GT(x1[0], x0[0]) → COND_215_0_ACK_GT(&&(<=(x1[0], 0), >(x0[0], 0)), x1[0], x0[0])
COND_215_0_ACK_GT(TRUE, x1[1], x0[1]) → 215_0_ACK_GT(1, -(x0[1], 1))
215_0_ACK_GT(x1[2], x0[2]) → COND_215_0_ACK_GT1(&&(>(x1[2], 0), >(x0[2], 0)), x1[2], x0[2])
COND_215_0_ACK_GT1(TRUE, x1[4], x0[4]) → 215_0_ACK_GT(-(x1[4], 1), x0[4])

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

&&(TRUE, TRUE)1TRUE1
&&(TRUE, FALSE)1FALSE1
&&(FALSE, TRUE)1FALSE1
&&(FALSE, FALSE)1FALSE1
231_0_ack_Return(x1, +(x1, 1))1215_0_ack_GT(x1, 0)1

### (6) Obligation:

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

The following domains are used:

Integer, Boolean

The ITRS R consists of the following rules:
215_0_ack_GT(x1, 0) → 231_0_ack_Return(x1, x1 + 1)
258_1_ack_InvokeMethod(231_0_ack_Return(1, x2), 0) → 266_0_ack_Return(x3, x4, x2)
258_1_ack_InvokeMethod(295_0_ack_Return(x0), x3) → 266_0_ack_Return(x1, x2, x0)
287_1_ack_InvokeMethod(231_0_ack_Return(x1, x2), 0, x1) → 295_0_ack_Return(x2)
287_1_ack_InvokeMethod(266_0_ack_Return(x0, x1, x2), x0, x1) → 295_0_ack_Return(x2)
287_1_ack_InvokeMethod(295_0_ack_Return(x0), x1, x2) → 295_0_ack_Return(x0)

The integer pair graph contains the following rules and edges:
(0): 215_0_ACK_GT(x1[0], x0[0]) → COND_215_0_ACK_GT(x1[0] <= 0 && x0[0] > 0, x1[0], x0[0])
(1): COND_215_0_ACK_GT(TRUE, x1[1], x0[1]) → 215_0_ACK_GT(1, x0[1] - 1)
(2): 215_0_ACK_GT(x1[2], x0[2]) → COND_215_0_ACK_GT1(x1[2] > 0 && x0[2] > 0, x1[2], x0[2])
(4): COND_215_0_ACK_GT1(TRUE, x1[4], x0[4]) → 215_0_ACK_GT(x1[4] - 1, x0[4])

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

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

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

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

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

(2) -> (4), if ((x1[2] > 0 && x0[2] > 0* TRUE)∧(x1[2]* x1[4])∧(x0[2]* x0[4]))

The set Q consists of the following terms:
215_0_ack_GT(x0, 0)
258_1_ack_InvokeMethod(231_0_ack_Return(1, x0), 0)
258_1_ack_InvokeMethod(295_0_ack_Return(x0), x1)
287_1_ack_InvokeMethod(231_0_ack_Return(x0, x1), 0, x0)
287_1_ack_InvokeMethod(266_0_ack_Return(x0, x1, x2), x0, x1)
287_1_ack_InvokeMethod(295_0_ack_Return(x0), x1, x2)

### (7) 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 215_0_ACK_GT(x1[0], x0[0]) → COND_215_0_ACK_GT(&&(<=(x1[0], 0), >(x0[0], 0)), x1[0], x0[0]) the following chains were created:
• We consider the chain 215_0_ACK_GT(x1[0], x0[0]) → COND_215_0_ACK_GT(&&(<=(x1[0], 0), >(x0[0], 0)), x1[0], x0[0]), COND_215_0_ACK_GT(TRUE, x1[1], x0[1]) → 215_0_ACK_GT(1, -(x0[1], 1)) which results in the following constraint:

(1)    (&&(<=(x1[0], 0), >(x0[0], 0))=TRUEx1[0]=x1[1]x0[0]=x0[1]215_0_ACK_GT(x1[0], x0[0])≥NonInfC∧215_0_ACK_GT(x1[0], x0[0])≥COND_215_0_ACK_GT(&&(<=(x1[0], 0), >(x0[0], 0)), x1[0], x0[0])∧(UIncreasing(COND_215_0_ACK_GT(&&(<=(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)    (<=(x1[0], 0)=TRUE>(x0[0], 0)=TRUE215_0_ACK_GT(x1[0], x0[0])≥NonInfC∧215_0_ACK_GT(x1[0], x0[0])≥COND_215_0_ACK_GT(&&(<=(x1[0], 0), >(x0[0], 0)), x1[0], x0[0])∧(UIncreasing(COND_215_0_ACK_GT(&&(<=(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)    ([-1]x1[0] ≥ 0∧x0[0] + [-1] ≥ 0 ⇒ (UIncreasing(COND_215_0_ACK_GT(&&(<=(x1[0], 0), >(x0[0], 0)), x1[0], x0[0])), ≥)∧[(-1)Bound*bni_36] + [(2)bni_36]x0[0] ≥ 0∧[1 + (-1)bso_37] ≥ 0)

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

(4)    ([-1]x1[0] ≥ 0∧x0[0] + [-1] ≥ 0 ⇒ (UIncreasing(COND_215_0_ACK_GT(&&(<=(x1[0], 0), >(x0[0], 0)), x1[0], x0[0])), ≥)∧[(-1)Bound*bni_36] + [(2)bni_36]x0[0] ≥ 0∧[1 + (-1)bso_37] ≥ 0)

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

(5)    ([-1]x1[0] ≥ 0∧x0[0] + [-1] ≥ 0 ⇒ (UIncreasing(COND_215_0_ACK_GT(&&(<=(x1[0], 0), >(x0[0], 0)), x1[0], x0[0])), ≥)∧[(-1)Bound*bni_36] + [(2)bni_36]x0[0] ≥ 0∧[1 + (-1)bso_37] ≥ 0)

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

(6)    (x1[0] ≥ 0∧x0[0] + [-1] ≥ 0 ⇒ (UIncreasing(COND_215_0_ACK_GT(&&(<=(x1[0], 0), >(x0[0], 0)), x1[0], x0[0])), ≥)∧[(-1)Bound*bni_36] + [(2)bni_36]x0[0] ≥ 0∧[1 + (-1)bso_37] ≥ 0)

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

(7)    (x1[0] ≥ 0∧x0[0] ≥ 0 ⇒ (UIncreasing(COND_215_0_ACK_GT(&&(<=(x1[0], 0), >(x0[0], 0)), x1[0], x0[0])), ≥)∧[(-1)Bound*bni_36 + (2)bni_36] + [(2)bni_36]x0[0] ≥ 0∧[1 + (-1)bso_37] ≥ 0)

For Pair COND_215_0_ACK_GT(TRUE, x1[1], x0[1]) → 215_0_ACK_GT(1, -(x0[1], 1)) the following chains were created:
• We consider the chain 215_0_ACK_GT(x1[0], x0[0]) → COND_215_0_ACK_GT(&&(<=(x1[0], 0), >(x0[0], 0)), x1[0], x0[0]), COND_215_0_ACK_GT(TRUE, x1[1], x0[1]) → 215_0_ACK_GT(1, -(x0[1], 1)), 215_0_ACK_GT(x1[0], x0[0]) → COND_215_0_ACK_GT(&&(<=(x1[0], 0), >(x0[0], 0)), x1[0], x0[0]) which results in the following constraint:

(8)    (&&(<=(x1[0], 0), >(x0[0], 0))=TRUEx1[0]=x1[1]x0[0]=x0[1]1=x1[0]1-(x0[1], 1)=x0[0]1COND_215_0_ACK_GT(TRUE, x1[1], x0[1])≥NonInfC∧COND_215_0_ACK_GT(TRUE, x1[1], x0[1])≥215_0_ACK_GT(1, -(x0[1], 1))∧(UIncreasing(215_0_ACK_GT(1, -(x0[1], 1))), ≥))

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

(9)    (<=(x1[0], 0)=TRUE>(x0[0], 0)=TRUECOND_215_0_ACK_GT(TRUE, x1[0], x0[0])≥NonInfC∧COND_215_0_ACK_GT(TRUE, x1[0], x0[0])≥215_0_ACK_GT(1, -(x0[0], 1))∧(UIncreasing(215_0_ACK_GT(1, -(x0[1], 1))), ≥))

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

(10)    ([-1]x1[0] ≥ 0∧x0[0] + [-1] ≥ 0 ⇒ (UIncreasing(215_0_ACK_GT(1, -(x0[1], 1))), ≥)∧[(-1)bni_38 + (-1)Bound*bni_38] + [(2)bni_38]x0[0] ≥ 0∧[1 + (-1)bso_39] ≥ 0)

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

(11)    ([-1]x1[0] ≥ 0∧x0[0] + [-1] ≥ 0 ⇒ (UIncreasing(215_0_ACK_GT(1, -(x0[1], 1))), ≥)∧[(-1)bni_38 + (-1)Bound*bni_38] + [(2)bni_38]x0[0] ≥ 0∧[1 + (-1)bso_39] ≥ 0)

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

(12)    ([-1]x1[0] ≥ 0∧x0[0] + [-1] ≥ 0 ⇒ (UIncreasing(215_0_ACK_GT(1, -(x0[1], 1))), ≥)∧[(-1)bni_38 + (-1)Bound*bni_38] + [(2)bni_38]x0[0] ≥ 0∧[1 + (-1)bso_39] ≥ 0)

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

(13)    (x1[0] ≥ 0∧x0[0] + [-1] ≥ 0 ⇒ (UIncreasing(215_0_ACK_GT(1, -(x0[1], 1))), ≥)∧[(-1)bni_38 + (-1)Bound*bni_38] + [(2)bni_38]x0[0] ≥ 0∧[1 + (-1)bso_39] ≥ 0)

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

(14)    (x1[0] ≥ 0∧x0[0] ≥ 0 ⇒ (UIncreasing(215_0_ACK_GT(1, -(x0[1], 1))), ≥)∧[bni_38 + (-1)Bound*bni_38] + [(2)bni_38]x0[0] ≥ 0∧[1 + (-1)bso_39] ≥ 0)

• We consider the chain 215_0_ACK_GT(x1[0], x0[0]) → COND_215_0_ACK_GT(&&(<=(x1[0], 0), >(x0[0], 0)), x1[0], x0[0]), COND_215_0_ACK_GT(TRUE, x1[1], x0[1]) → 215_0_ACK_GT(1, -(x0[1], 1)), 215_0_ACK_GT(x1[2], x0[2]) → COND_215_0_ACK_GT1(&&(>(x1[2], 0), >(x0[2], 0)), x1[2], x0[2]) which results in the following constraint:

(15)    (&&(<=(x1[0], 0), >(x0[0], 0))=TRUEx1[0]=x1[1]x0[0]=x0[1]1=x1[2]-(x0[1], 1)=x0[2]COND_215_0_ACK_GT(TRUE, x1[1], x0[1])≥NonInfC∧COND_215_0_ACK_GT(TRUE, x1[1], x0[1])≥215_0_ACK_GT(1, -(x0[1], 1))∧(UIncreasing(215_0_ACK_GT(1, -(x0[1], 1))), ≥))

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

(16)    (<=(x1[0], 0)=TRUE>(x0[0], 0)=TRUECOND_215_0_ACK_GT(TRUE, x1[0], x0[0])≥NonInfC∧COND_215_0_ACK_GT(TRUE, x1[0], x0[0])≥215_0_ACK_GT(1, -(x0[0], 1))∧(UIncreasing(215_0_ACK_GT(1, -(x0[1], 1))), ≥))

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

(17)    ([-1]x1[0] ≥ 0∧x0[0] + [-1] ≥ 0 ⇒ (UIncreasing(215_0_ACK_GT(1, -(x0[1], 1))), ≥)∧[(-1)bni_38 + (-1)Bound*bni_38] + [(2)bni_38]x0[0] ≥ 0∧[1 + (-1)bso_39] ≥ 0)

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

(18)    ([-1]x1[0] ≥ 0∧x0[0] + [-1] ≥ 0 ⇒ (UIncreasing(215_0_ACK_GT(1, -(x0[1], 1))), ≥)∧[(-1)bni_38 + (-1)Bound*bni_38] + [(2)bni_38]x0[0] ≥ 0∧[1 + (-1)bso_39] ≥ 0)

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

(19)    ([-1]x1[0] ≥ 0∧x0[0] + [-1] ≥ 0 ⇒ (UIncreasing(215_0_ACK_GT(1, -(x0[1], 1))), ≥)∧[(-1)bni_38 + (-1)Bound*bni_38] + [(2)bni_38]x0[0] ≥ 0∧[1 + (-1)bso_39] ≥ 0)

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

(20)    (x1[0] ≥ 0∧x0[0] + [-1] ≥ 0 ⇒ (UIncreasing(215_0_ACK_GT(1, -(x0[1], 1))), ≥)∧[(-1)bni_38 + (-1)Bound*bni_38] + [(2)bni_38]x0[0] ≥ 0∧[1 + (-1)bso_39] ≥ 0)

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

(21)    (x1[0] ≥ 0∧x0[0] ≥ 0 ⇒ (UIncreasing(215_0_ACK_GT(1, -(x0[1], 1))), ≥)∧[bni_38 + (-1)Bound*bni_38] + [(2)bni_38]x0[0] ≥ 0∧[1 + (-1)bso_39] ≥ 0)

For Pair 215_0_ACK_GT(x1[2], x0[2]) → COND_215_0_ACK_GT1(&&(>(x1[2], 0), >(x0[2], 0)), x1[2], x0[2]) the following chains were created:
• We consider the chain 215_0_ACK_GT(x1[2], x0[2]) → COND_215_0_ACK_GT1(&&(>(x1[2], 0), >(x0[2], 0)), x1[2], x0[2]), COND_215_0_ACK_GT1(TRUE, x1[4], x0[4]) → 215_0_ACK_GT(-(x1[4], 1), x0[4]) which results in the following constraint:

(22)    (&&(>(x1[2], 0), >(x0[2], 0))=TRUEx1[2]=x1[4]x0[2]=x0[4]215_0_ACK_GT(x1[2], x0[2])≥NonInfC∧215_0_ACK_GT(x1[2], x0[2])≥COND_215_0_ACK_GT1(&&(>(x1[2], 0), >(x0[2], 0)), x1[2], x0[2])∧(UIncreasing(COND_215_0_ACK_GT1(&&(>(x1[2], 0), >(x0[2], 0)), x1[2], x0[2])), ≥))

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

(23)    (>(x1[2], 0)=TRUE>(x0[2], 0)=TRUE215_0_ACK_GT(x1[2], x0[2])≥NonInfC∧215_0_ACK_GT(x1[2], x0[2])≥COND_215_0_ACK_GT1(&&(>(x1[2], 0), >(x0[2], 0)), x1[2], x0[2])∧(UIncreasing(COND_215_0_ACK_GT1(&&(>(x1[2], 0), >(x0[2], 0)), x1[2], x0[2])), ≥))

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

(24)    (x1[2] + [-1] ≥ 0∧x0[2] + [-1] ≥ 0 ⇒ (UIncreasing(COND_215_0_ACK_GT1(&&(>(x1[2], 0), >(x0[2], 0)), x1[2], x0[2])), ≥)∧[(-1)Bound*bni_40] + [(2)bni_40]x0[2] ≥ 0∧[(-1)bso_41] ≥ 0)

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

(25)    (x1[2] + [-1] ≥ 0∧x0[2] + [-1] ≥ 0 ⇒ (UIncreasing(COND_215_0_ACK_GT1(&&(>(x1[2], 0), >(x0[2], 0)), x1[2], x0[2])), ≥)∧[(-1)Bound*bni_40] + [(2)bni_40]x0[2] ≥ 0∧[(-1)bso_41] ≥ 0)

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

(26)    (x1[2] + [-1] ≥ 0∧x0[2] + [-1] ≥ 0 ⇒ (UIncreasing(COND_215_0_ACK_GT1(&&(>(x1[2], 0), >(x0[2], 0)), x1[2], x0[2])), ≥)∧[(-1)Bound*bni_40] + [(2)bni_40]x0[2] ≥ 0∧[(-1)bso_41] ≥ 0)

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

(27)    (x1[2] ≥ 0∧x0[2] + [-1] ≥ 0 ⇒ (UIncreasing(COND_215_0_ACK_GT1(&&(>(x1[2], 0), >(x0[2], 0)), x1[2], x0[2])), ≥)∧[(-1)Bound*bni_40] + [(2)bni_40]x0[2] ≥ 0∧[(-1)bso_41] ≥ 0)

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

(28)    (x1[2] ≥ 0∧x0[2] ≥ 0 ⇒ (UIncreasing(COND_215_0_ACK_GT1(&&(>(x1[2], 0), >(x0[2], 0)), x1[2], x0[2])), ≥)∧[(-1)Bound*bni_40 + (2)bni_40] + [(2)bni_40]x0[2] ≥ 0∧[(-1)bso_41] ≥ 0)

For Pair COND_215_0_ACK_GT1(TRUE, x1[4], x0[4]) → 215_0_ACK_GT(-(x1[4], 1), x0[4]) the following chains were created:
• We consider the chain 215_0_ACK_GT(x1[2], x0[2]) → COND_215_0_ACK_GT1(&&(>(x1[2], 0), >(x0[2], 0)), x1[2], x0[2]), COND_215_0_ACK_GT1(TRUE, x1[4], x0[4]) → 215_0_ACK_GT(-(x1[4], 1), x0[4]), 215_0_ACK_GT(x1[0], x0[0]) → COND_215_0_ACK_GT(&&(<=(x1[0], 0), >(x0[0], 0)), x1[0], x0[0]) which results in the following constraint:

(29)    (&&(>(x1[2], 0), >(x0[2], 0))=TRUEx1[2]=x1[4]x0[2]=x0[4]-(x1[4], 1)=x1[0]x0[4]=x0[0]COND_215_0_ACK_GT1(TRUE, x1[4], x0[4])≥NonInfC∧COND_215_0_ACK_GT1(TRUE, x1[4], x0[4])≥215_0_ACK_GT(-(x1[4], 1), x0[4])∧(UIncreasing(215_0_ACK_GT(-(x1[4], 1), x0[4])), ≥))

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

(30)    (>(x1[2], 0)=TRUE>(x0[2], 0)=TRUECOND_215_0_ACK_GT1(TRUE, x1[2], x0[2])≥NonInfC∧COND_215_0_ACK_GT1(TRUE, x1[2], x0[2])≥215_0_ACK_GT(-(x1[2], 1), x0[2])∧(UIncreasing(215_0_ACK_GT(-(x1[4], 1), x0[4])), ≥))

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

(31)    (x1[2] + [-1] ≥ 0∧x0[2] + [-1] ≥ 0 ⇒ (UIncreasing(215_0_ACK_GT(-(x1[4], 1), x0[4])), ≥)∧[(-1)Bound*bni_42] + [(2)bni_42]x0[2] ≥ 0∧[(-1)bso_43] ≥ 0)

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

(32)    (x1[2] + [-1] ≥ 0∧x0[2] + [-1] ≥ 0 ⇒ (UIncreasing(215_0_ACK_GT(-(x1[4], 1), x0[4])), ≥)∧[(-1)Bound*bni_42] + [(2)bni_42]x0[2] ≥ 0∧[(-1)bso_43] ≥ 0)

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

(33)    (x1[2] + [-1] ≥ 0∧x0[2] + [-1] ≥ 0 ⇒ (UIncreasing(215_0_ACK_GT(-(x1[4], 1), x0[4])), ≥)∧[(-1)Bound*bni_42] + [(2)bni_42]x0[2] ≥ 0∧[(-1)bso_43] ≥ 0)

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

(34)    (x1[2] ≥ 0∧x0[2] + [-1] ≥ 0 ⇒ (UIncreasing(215_0_ACK_GT(-(x1[4], 1), x0[4])), ≥)∧[(-1)Bound*bni_42] + [(2)bni_42]x0[2] ≥ 0∧[(-1)bso_43] ≥ 0)

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

(35)    (x1[2] ≥ 0∧x0[2] ≥ 0 ⇒ (UIncreasing(215_0_ACK_GT(-(x1[4], 1), x0[4])), ≥)∧[(-1)Bound*bni_42 + (2)bni_42] + [(2)bni_42]x0[2] ≥ 0∧[(-1)bso_43] ≥ 0)

• We consider the chain 215_0_ACK_GT(x1[2], x0[2]) → COND_215_0_ACK_GT1(&&(>(x1[2], 0), >(x0[2], 0)), x1[2], x0[2]), COND_215_0_ACK_GT1(TRUE, x1[4], x0[4]) → 215_0_ACK_GT(-(x1[4], 1), x0[4]), 215_0_ACK_GT(x1[2], x0[2]) → COND_215_0_ACK_GT1(&&(>(x1[2], 0), >(x0[2], 0)), x1[2], x0[2]) which results in the following constraint:

(36)    (&&(>(x1[2], 0), >(x0[2], 0))=TRUEx1[2]=x1[4]x0[2]=x0[4]-(x1[4], 1)=x1[2]1x0[4]=x0[2]1COND_215_0_ACK_GT1(TRUE, x1[4], x0[4])≥NonInfC∧COND_215_0_ACK_GT1(TRUE, x1[4], x0[4])≥215_0_ACK_GT(-(x1[4], 1), x0[4])∧(UIncreasing(215_0_ACK_GT(-(x1[4], 1), x0[4])), ≥))

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

(37)    (>(x1[2], 0)=TRUE>(x0[2], 0)=TRUECOND_215_0_ACK_GT1(TRUE, x1[2], x0[2])≥NonInfC∧COND_215_0_ACK_GT1(TRUE, x1[2], x0[2])≥215_0_ACK_GT(-(x1[2], 1), x0[2])∧(UIncreasing(215_0_ACK_GT(-(x1[4], 1), x0[4])), ≥))

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

(38)    (x1[2] + [-1] ≥ 0∧x0[2] + [-1] ≥ 0 ⇒ (UIncreasing(215_0_ACK_GT(-(x1[4], 1), x0[4])), ≥)∧[(-1)Bound*bni_42] + [(2)bni_42]x0[2] ≥ 0∧[(-1)bso_43] ≥ 0)

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

(39)    (x1[2] + [-1] ≥ 0∧x0[2] + [-1] ≥ 0 ⇒ (UIncreasing(215_0_ACK_GT(-(x1[4], 1), x0[4])), ≥)∧[(-1)Bound*bni_42] + [(2)bni_42]x0[2] ≥ 0∧[(-1)bso_43] ≥ 0)

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

(40)    (x1[2] + [-1] ≥ 0∧x0[2] + [-1] ≥ 0 ⇒ (UIncreasing(215_0_ACK_GT(-(x1[4], 1), x0[4])), ≥)∧[(-1)Bound*bni_42] + [(2)bni_42]x0[2] ≥ 0∧[(-1)bso_43] ≥ 0)

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

(41)    (x1[2] ≥ 0∧x0[2] + [-1] ≥ 0 ⇒ (UIncreasing(215_0_ACK_GT(-(x1[4], 1), x0[4])), ≥)∧[(-1)Bound*bni_42] + [(2)bni_42]x0[2] ≥ 0∧[(-1)bso_43] ≥ 0)

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

(42)    (x1[2] ≥ 0∧x0[2] ≥ 0 ⇒ (UIncreasing(215_0_ACK_GT(-(x1[4], 1), x0[4])), ≥)∧[(-1)Bound*bni_42 + (2)bni_42] + [(2)bni_42]x0[2] ≥ 0∧[(-1)bso_43] ≥ 0)

To summarize, we get the following constraints P for the following pairs.
• 215_0_ACK_GT(x1[0], x0[0]) → COND_215_0_ACK_GT(&&(<=(x1[0], 0), >(x0[0], 0)), x1[0], x0[0])
• (x1[0] ≥ 0∧x0[0] ≥ 0 ⇒ (UIncreasing(COND_215_0_ACK_GT(&&(<=(x1[0], 0), >(x0[0], 0)), x1[0], x0[0])), ≥)∧[(-1)Bound*bni_36 + (2)bni_36] + [(2)bni_36]x0[0] ≥ 0∧[1 + (-1)bso_37] ≥ 0)

• COND_215_0_ACK_GT(TRUE, x1[1], x0[1]) → 215_0_ACK_GT(1, -(x0[1], 1))
• (x1[0] ≥ 0∧x0[0] ≥ 0 ⇒ (UIncreasing(215_0_ACK_GT(1, -(x0[1], 1))), ≥)∧[bni_38 + (-1)Bound*bni_38] + [(2)bni_38]x0[0] ≥ 0∧[1 + (-1)bso_39] ≥ 0)
• (x1[0] ≥ 0∧x0[0] ≥ 0 ⇒ (UIncreasing(215_0_ACK_GT(1, -(x0[1], 1))), ≥)∧[bni_38 + (-1)Bound*bni_38] + [(2)bni_38]x0[0] ≥ 0∧[1 + (-1)bso_39] ≥ 0)

• 215_0_ACK_GT(x1[2], x0[2]) → COND_215_0_ACK_GT1(&&(>(x1[2], 0), >(x0[2], 0)), x1[2], x0[2])
• (x1[2] ≥ 0∧x0[2] ≥ 0 ⇒ (UIncreasing(COND_215_0_ACK_GT1(&&(>(x1[2], 0), >(x0[2], 0)), x1[2], x0[2])), ≥)∧[(-1)Bound*bni_40 + (2)bni_40] + [(2)bni_40]x0[2] ≥ 0∧[(-1)bso_41] ≥ 0)

• COND_215_0_ACK_GT1(TRUE, x1[4], x0[4]) → 215_0_ACK_GT(-(x1[4], 1), x0[4])
• (x1[2] ≥ 0∧x0[2] ≥ 0 ⇒ (UIncreasing(215_0_ACK_GT(-(x1[4], 1), x0[4])), ≥)∧[(-1)Bound*bni_42 + (2)bni_42] + [(2)bni_42]x0[2] ≥ 0∧[(-1)bso_43] ≥ 0)
• (x1[2] ≥ 0∧x0[2] ≥ 0 ⇒ (UIncreasing(215_0_ACK_GT(-(x1[4], 1), x0[4])), ≥)∧[(-1)Bound*bni_42 + (2)bni_42] + [(2)bni_42]x0[2] ≥ 0∧[(-1)bso_43] ≥ 0)

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

POL(TRUE) = 0
POL(FALSE) = [2]
POL(215_0_ack_GT(x1, x2)) = [-1] + [-1]x2 + [-1]x1
POL(0) = 0
POL(231_0_ack_Return(x1, x2)) = [-1] + [-1]x2 + [-1]x1
POL(+(x1, x2)) = x1 + x2
POL(1) = [1]
POL(258_1_ack_InvokeMethod(x1, x2)) = [-1] + [-1]x2 + [-1]x1
POL(266_0_ack_Return(x1, x2, x3)) = [-1] + [-1]x3 + [-1]x2 + [-1]x1
POL(295_0_ack_Return(x1)) = [-1] + [-1]x1
POL(287_1_ack_InvokeMethod(x1, x2, x3)) = [-1] + [-1]x3 + [-1]x2 + [-1]x1
POL(215_0_ACK_GT(x1, x2)) = [2]x2
POL(COND_215_0_ACK_GT(x1, x2, x3)) = [-1] + [2]x3
POL(&&(x1, x2)) = [-1]
POL(<=(x1, x2)) = [-1]
POL(>(x1, x2)) = [-1]
POL(-(x1, x2)) = x1 + [-1]x2
POL(COND_215_0_ACK_GT1(x1, x2, x3)) = [2]x3

The following pairs are in P>:

215_0_ACK_GT(x1[0], x0[0]) → COND_215_0_ACK_GT(&&(<=(x1[0], 0), >(x0[0], 0)), x1[0], x0[0])
COND_215_0_ACK_GT(TRUE, x1[1], x0[1]) → 215_0_ACK_GT(1, -(x0[1], 1))

The following pairs are in Pbound:

215_0_ACK_GT(x1[0], x0[0]) → COND_215_0_ACK_GT(&&(<=(x1[0], 0), >(x0[0], 0)), x1[0], x0[0])
COND_215_0_ACK_GT(TRUE, x1[1], x0[1]) → 215_0_ACK_GT(1, -(x0[1], 1))
215_0_ACK_GT(x1[2], x0[2]) → COND_215_0_ACK_GT1(&&(>(x1[2], 0), >(x0[2], 0)), x1[2], x0[2])
COND_215_0_ACK_GT1(TRUE, x1[4], x0[4]) → 215_0_ACK_GT(-(x1[4], 1), x0[4])

The following pairs are in P:

215_0_ACK_GT(x1[2], x0[2]) → COND_215_0_ACK_GT1(&&(>(x1[2], 0), >(x0[2], 0)), x1[2], x0[2])
COND_215_0_ACK_GT1(TRUE, x1[4], x0[4]) → 215_0_ACK_GT(-(x1[4], 1), x0[4])

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

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

### (8) Obligation:

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

The following domains are used:

Integer, Boolean

The ITRS R consists of the following rules:
215_0_ack_GT(x1, 0) → 231_0_ack_Return(x1, x1 + 1)
258_1_ack_InvokeMethod(231_0_ack_Return(1, x2), 0) → 266_0_ack_Return(x3, x4, x2)
258_1_ack_InvokeMethod(295_0_ack_Return(x0), x3) → 266_0_ack_Return(x1, x2, x0)
287_1_ack_InvokeMethod(231_0_ack_Return(x1, x2), 0, x1) → 295_0_ack_Return(x2)
287_1_ack_InvokeMethod(266_0_ack_Return(x0, x1, x2), x0, x1) → 295_0_ack_Return(x2)
287_1_ack_InvokeMethod(295_0_ack_Return(x0), x1, x2) → 295_0_ack_Return(x0)

The integer pair graph contains the following rules and edges:
(2): 215_0_ACK_GT(x1[2], x0[2]) → COND_215_0_ACK_GT1(x1[2] > 0 && x0[2] > 0, x1[2], x0[2])
(4): COND_215_0_ACK_GT1(TRUE, x1[4], x0[4]) → 215_0_ACK_GT(x1[4] - 1, x0[4])

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

(2) -> (4), if ((x1[2] > 0 && x0[2] > 0* TRUE)∧(x1[2]* x1[4])∧(x0[2]* x0[4]))

The set Q consists of the following terms:
215_0_ack_GT(x0, 0)
258_1_ack_InvokeMethod(231_0_ack_Return(1, x0), 0)
258_1_ack_InvokeMethod(295_0_ack_Return(x0), x1)
287_1_ack_InvokeMethod(231_0_ack_Return(x0, x1), 0, x0)
287_1_ack_InvokeMethod(266_0_ack_Return(x0, x1, x2), x0, x1)
287_1_ack_InvokeMethod(295_0_ack_Return(x0), x1, x2)

### (9) IDPNonInfProof (SOUND transformation)

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

For Pair 215_0_ACK_GT(x1[2], x0[2]) → COND_215_0_ACK_GT1(&&(>(x1[2], 0), >(x0[2], 0)), x1[2], x0[2]) the following chains were created:
• We consider the chain 215_0_ACK_GT(x1[2], x0[2]) → COND_215_0_ACK_GT1(&&(>(x1[2], 0), >(x0[2], 0)), x1[2], x0[2]), COND_215_0_ACK_GT1(TRUE, x1[4], x0[4]) → 215_0_ACK_GT(-(x1[4], 1), x0[4]) which results in the following constraint:

(1)    (&&(>(x1[2], 0), >(x0[2], 0))=TRUEx1[2]=x1[4]x0[2]=x0[4]215_0_ACK_GT(x1[2], x0[2])≥NonInfC∧215_0_ACK_GT(x1[2], x0[2])≥COND_215_0_ACK_GT1(&&(>(x1[2], 0), >(x0[2], 0)), x1[2], x0[2])∧(UIncreasing(COND_215_0_ACK_GT1(&&(>(x1[2], 0), >(x0[2], 0)), x1[2], x0[2])), ≥))

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

(2)    (>(x1[2], 0)=TRUE>(x0[2], 0)=TRUE215_0_ACK_GT(x1[2], x0[2])≥NonInfC∧215_0_ACK_GT(x1[2], x0[2])≥COND_215_0_ACK_GT1(&&(>(x1[2], 0), >(x0[2], 0)), x1[2], x0[2])∧(UIncreasing(COND_215_0_ACK_GT1(&&(>(x1[2], 0), >(x0[2], 0)), x1[2], x0[2])), ≥))

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

(3)    (x1[2] + [-1] ≥ 0∧x0[2] + [-1] ≥ 0 ⇒ (UIncreasing(COND_215_0_ACK_GT1(&&(>(x1[2], 0), >(x0[2], 0)), x1[2], x0[2])), ≥)∧[bni_31 + (-1)Bound*bni_31] + [(2)bni_31]x1[2] ≥ 0∧[(-1)bso_32] ≥ 0)

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

(4)    (x1[2] + [-1] ≥ 0∧x0[2] + [-1] ≥ 0 ⇒ (UIncreasing(COND_215_0_ACK_GT1(&&(>(x1[2], 0), >(x0[2], 0)), x1[2], x0[2])), ≥)∧[bni_31 + (-1)Bound*bni_31] + [(2)bni_31]x1[2] ≥ 0∧[(-1)bso_32] ≥ 0)

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

(5)    (x1[2] + [-1] ≥ 0∧x0[2] + [-1] ≥ 0 ⇒ (UIncreasing(COND_215_0_ACK_GT1(&&(>(x1[2], 0), >(x0[2], 0)), x1[2], x0[2])), ≥)∧[bni_31 + (-1)Bound*bni_31] + [(2)bni_31]x1[2] ≥ 0∧[(-1)bso_32] ≥ 0)

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

(6)    (x1[2] ≥ 0∧x0[2] + [-1] ≥ 0 ⇒ (UIncreasing(COND_215_0_ACK_GT1(&&(>(x1[2], 0), >(x0[2], 0)), x1[2], x0[2])), ≥)∧[(3)bni_31 + (-1)Bound*bni_31] + [(2)bni_31]x1[2] ≥ 0∧[(-1)bso_32] ≥ 0)

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

(7)    (x1[2] ≥ 0∧x0[2] ≥ 0 ⇒ (UIncreasing(COND_215_0_ACK_GT1(&&(>(x1[2], 0), >(x0[2], 0)), x1[2], x0[2])), ≥)∧[(3)bni_31 + (-1)Bound*bni_31] + [(2)bni_31]x1[2] ≥ 0∧[(-1)bso_32] ≥ 0)

For Pair COND_215_0_ACK_GT1(TRUE, x1[4], x0[4]) → 215_0_ACK_GT(-(x1[4], 1), x0[4]) the following chains were created:
• We consider the chain 215_0_ACK_GT(x1[2], x0[2]) → COND_215_0_ACK_GT1(&&(>(x1[2], 0), >(x0[2], 0)), x1[2], x0[2]), COND_215_0_ACK_GT1(TRUE, x1[4], x0[4]) → 215_0_ACK_GT(-(x1[4], 1), x0[4]), 215_0_ACK_GT(x1[2], x0[2]) → COND_215_0_ACK_GT1(&&(>(x1[2], 0), >(x0[2], 0)), x1[2], x0[2]) which results in the following constraint:

(8)    (&&(>(x1[2], 0), >(x0[2], 0))=TRUEx1[2]=x1[4]x0[2]=x0[4]-(x1[4], 1)=x1[2]1x0[4]=x0[2]1COND_215_0_ACK_GT1(TRUE, x1[4], x0[4])≥NonInfC∧COND_215_0_ACK_GT1(TRUE, x1[4], x0[4])≥215_0_ACK_GT(-(x1[4], 1), x0[4])∧(UIncreasing(215_0_ACK_GT(-(x1[4], 1), x0[4])), ≥))

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

(9)    (>(x1[2], 0)=TRUE>(x0[2], 0)=TRUECOND_215_0_ACK_GT1(TRUE, x1[2], x0[2])≥NonInfC∧COND_215_0_ACK_GT1(TRUE, x1[2], x0[2])≥215_0_ACK_GT(-(x1[2], 1), x0[2])∧(UIncreasing(215_0_ACK_GT(-(x1[4], 1), x0[4])), ≥))

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

(10)    (x1[2] + [-1] ≥ 0∧x0[2] + [-1] ≥ 0 ⇒ (UIncreasing(215_0_ACK_GT(-(x1[4], 1), x0[4])), ≥)∧[(-1)Bound*bni_33] + [(2)bni_33]x1[2] ≥ 0∧[1 + (-1)bso_34] ≥ 0)

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

(11)    (x1[2] + [-1] ≥ 0∧x0[2] + [-1] ≥ 0 ⇒ (UIncreasing(215_0_ACK_GT(-(x1[4], 1), x0[4])), ≥)∧[(-1)Bound*bni_33] + [(2)bni_33]x1[2] ≥ 0∧[1 + (-1)bso_34] ≥ 0)

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

(12)    (x1[2] + [-1] ≥ 0∧x0[2] + [-1] ≥ 0 ⇒ (UIncreasing(215_0_ACK_GT(-(x1[4], 1), x0[4])), ≥)∧[(-1)Bound*bni_33] + [(2)bni_33]x1[2] ≥ 0∧[1 + (-1)bso_34] ≥ 0)

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

(13)    (x1[2] ≥ 0∧x0[2] + [-1] ≥ 0 ⇒ (UIncreasing(215_0_ACK_GT(-(x1[4], 1), x0[4])), ≥)∧[(-1)Bound*bni_33 + (2)bni_33] + [(2)bni_33]x1[2] ≥ 0∧[1 + (-1)bso_34] ≥ 0)

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

(14)    (x1[2] ≥ 0∧x0[2] ≥ 0 ⇒ (UIncreasing(215_0_ACK_GT(-(x1[4], 1), x0[4])), ≥)∧[(-1)Bound*bni_33 + (2)bni_33] + [(2)bni_33]x1[2] ≥ 0∧[1 + (-1)bso_34] ≥ 0)

To summarize, we get the following constraints P for the following pairs.
• 215_0_ACK_GT(x1[2], x0[2]) → COND_215_0_ACK_GT1(&&(>(x1[2], 0), >(x0[2], 0)), x1[2], x0[2])
• (x1[2] ≥ 0∧x0[2] ≥ 0 ⇒ (UIncreasing(COND_215_0_ACK_GT1(&&(>(x1[2], 0), >(x0[2], 0)), x1[2], x0[2])), ≥)∧[(3)bni_31 + (-1)Bound*bni_31] + [(2)bni_31]x1[2] ≥ 0∧[(-1)bso_32] ≥ 0)

• COND_215_0_ACK_GT1(TRUE, x1[4], x0[4]) → 215_0_ACK_GT(-(x1[4], 1), x0[4])
• (x1[2] ≥ 0∧x0[2] ≥ 0 ⇒ (UIncreasing(215_0_ACK_GT(-(x1[4], 1), x0[4])), ≥)∧[(-1)Bound*bni_33 + (2)bni_33] + [(2)bni_33]x1[2] ≥ 0∧[1 + (-1)bso_34] ≥ 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) = [1]
POL(FALSE) = [1]
POL(215_0_ack_GT(x1, x2)) = [-1] + [-1]x2 + [-1]x1
POL(0) = 0
POL(231_0_ack_Return(x1, x2)) = [-1] + [-1]x2 + [-1]x1
POL(+(x1, x2)) = x1 + x2
POL(1) = [1]
POL(258_1_ack_InvokeMethod(x1, x2)) = [-1] + [-1]x2 + [-1]x1
POL(266_0_ack_Return(x1, x2, x3)) = [-1] + [-1]x3 + [-1]x2 + [-1]x1
POL(295_0_ack_Return(x1)) = [-1] + [-1]x1
POL(287_1_ack_InvokeMethod(x1, x2, x3)) = [-1] + [-1]x3 + [-1]x2 + [-1]x1
POL(215_0_ACK_GT(x1, x2)) = [1] + [2]x1
POL(COND_215_0_ACK_GT1(x1, x2, x3)) = [1] + [2]x2 + [-1]x1
POL(&&(x1, x2)) = 0
POL(>(x1, x2)) = [-1]
POL(-(x1, x2)) = x1 + [-1]x2

The following pairs are in P>:

COND_215_0_ACK_GT1(TRUE, x1[4], x0[4]) → 215_0_ACK_GT(-(x1[4], 1), x0[4])

The following pairs are in Pbound:

215_0_ACK_GT(x1[2], x0[2]) → COND_215_0_ACK_GT1(&&(>(x1[2], 0), >(x0[2], 0)), x1[2], x0[2])
COND_215_0_ACK_GT1(TRUE, x1[4], x0[4]) → 215_0_ACK_GT(-(x1[4], 1), x0[4])

The following pairs are in P:

215_0_ACK_GT(x1[2], x0[2]) → COND_215_0_ACK_GT1(&&(>(x1[2], 0), >(x0[2], 0)), x1[2], x0[2])

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

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

### (10) Obligation:

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

The following domains are used:

Integer, Boolean

The ITRS R consists of the following rules:
215_0_ack_GT(x1, 0) → 231_0_ack_Return(x1, x1 + 1)
258_1_ack_InvokeMethod(231_0_ack_Return(1, x2), 0) → 266_0_ack_Return(x3, x4, x2)
258_1_ack_InvokeMethod(295_0_ack_Return(x0), x3) → 266_0_ack_Return(x1, x2, x0)
287_1_ack_InvokeMethod(231_0_ack_Return(x1, x2), 0, x1) → 295_0_ack_Return(x2)
287_1_ack_InvokeMethod(266_0_ack_Return(x0, x1, x2), x0, x1) → 295_0_ack_Return(x2)
287_1_ack_InvokeMethod(295_0_ack_Return(x0), x1, x2) → 295_0_ack_Return(x0)

The integer pair graph contains the following rules and edges:
(2): 215_0_ACK_GT(x1[2], x0[2]) → COND_215_0_ACK_GT1(x1[2] > 0 && x0[2] > 0, x1[2], x0[2])

The set Q consists of the following terms:
215_0_ack_GT(x0, 0)
258_1_ack_InvokeMethod(231_0_ack_Return(1, x0), 0)
258_1_ack_InvokeMethod(295_0_ack_Return(x0), x1)
287_1_ack_InvokeMethod(231_0_ack_Return(x0, x1), 0, x0)
287_1_ack_InvokeMethod(266_0_ack_Return(x0, x1, x2), x0, x1)
287_1_ack_InvokeMethod(295_0_ack_Return(x0), x1, x2)

### (11) IDependencyGraphProof (EQUIVALENT transformation)

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