Termination proof of /tmp/tmpV8NKpd/Test1.jar

Termination of the given JBC could be proven:

0 JBC

↳1 JBC2FIG (⇒)

↳2 JBCTerminationGraph

↳3 FIGtoITRSProof (⇒)

↳4 IDP

↳5 IDPNonInfProof (⇒)

↳6 AND

    ↳7 IDP

        ↳8 IDependencyGraphProof (⇔)

        ↳9 TRUE

    ↳10 IDP

        ↳11 IDependencyGraphProof (⇔)

        ↳12 IDP

        ↳13 UsableRulesProof (⇔)

        ↳14 IDP

        ↳15 IDPNonInfProof (⇒)

        ↳16 AND

            ↳17 IDP

                ↳18 IDependencyGraphProof (⇔)

                ↳19 IDP

                ↳20 IDPNonInfProof (⇒)

                ↳21 IDP

                ↳22 IDependencyGraphProof (⇔)

                ↳23 TRUE

            ↳24 IDP

                ↳25 IDependencyGraphProof (⇔)

                ↳26 IDP

                ↳27 IDPNonInfProof (⇒)

                ↳28 IDP

                ↳29 IDependencyGraphProof (⇔)

                ↳30 TRUE

(0) Obligation:

JBC Problem based on JBC Program:
Manifest-Version: 1.0 Created-By: 1.6.0_20 (Apple Inc.) Main-Class: Test1

public class Test1 {
    public static void main(String[] args) {
	rec(args.length, args.length % 5, args.length % 4);
    }

    private static void rec(int x, int y, int z) {
	if (x + y + 3 * z < 0)
	    return;
	else if (x > y)
	    rec(x - 1, y, z);
	else if (y > z)
	    rec (x, y - 2, z);
	else
	    rec (x, y, z - 1);
    }
}

(1) JBC2FIG (SOUND transformation)

Constructed FIGraph.

(2) Obligation:

FIGraph based on JBC Program:
Test1.main([Ljava/lang/String;)V: Graph of 22 nodes with 0 SCCs.

Test1.rec(III)V: Graph of 61 nodes with 0 SCCs.

(3) FIGtoITRSProof (SOUND transformation)

Transformed FIGraph SCCs to IDPs. Logs:

Log for SCC 0:

Generated 45 rules for P and 21 rules for R.

Combined rules. Obtained 3 rules for P and 6 rules for R.

Filtered ground terms:

1062_0_rec_Load(x1, x2, x3, x4, x5) → 1062_0_rec_Load(x2, x3, x4, x5)
Cond_1062_0_rec_Load2(x1, x2, x3, x4, x5, x6) → Cond_1062_0_rec_Load2(x1, x3, x4, x5, x6)
Cond_1062_0_rec_Load1(x1, x2, x3, x4, x5, x6) → Cond_1062_0_rec_Load1(x1, x3, x4, x5, x6)
Cond_1062_0_rec_Load(x1, x2, x3, x4, x5, x6) → Cond_1062_0_rec_Load(x1, x3, x4, x5, x6)
1150_0_rec_Return(x1) → 1150_0_rec_Return
1081_0_rec_Return(x1, x2, x3, x4) → 1081_0_rec_Return(x2, x3, x4)

Filtered duplicate args:

1062_0_rec_Load(x1, x2, x3, x4) → 1062_0_rec_Load(x2, x3, x4)
Cond_1062_0_rec_Load2(x1, x2, x3, x4, x5) → Cond_1062_0_rec_Load2(x1, x3, x4, x5)
Cond_1062_0_rec_Load1(x1, x2, x3, x4, x5) → Cond_1062_0_rec_Load1(x1, x3, x4, x5)
Cond_1062_0_rec_Load(x1, x2, x3, x4, x5) → Cond_1062_0_rec_Load(x1, x3, x4, x5)

Combined rules. Obtained 3 rules for P and 6 rules for R.

Finished conversion. Obtained 3 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:

Boolean, Integer

The ITRS R consists of the following rules:
1162_1_rec_InvokeMethod(1081_0_rec_Return(x0, x1, x2), x0, x1, x2) → 1150_0_rec_Return
1162_1_rec_InvokeMethod(1150_0_rec_Return, x0, x1, x2) → 1150_0_rec_Return
1170_1_rec_InvokeMethod(1081_0_rec_Return(x0, x1, x2), x0, x1, x2) → 1150_0_rec_Return
1170_1_rec_InvokeMethod(1150_0_rec_Return, x0, x1, x2) → 1150_0_rec_Return
1133_1_rec_InvokeMethod(1081_0_rec_Return(x0, x1, x2), x0, x1, x2) → 1150_0_rec_Return
1133_1_rec_InvokeMethod(1150_0_rec_Return, x0, x1, x2) → 1150_0_rec_Return

The integer pair graph contains the following rules and edges:
(0): 1062_0_REC_LOAD(x1[0], x2[0], x0[0]) → COND_1062_0_REC_LOAD(x2[0] >= x1[0] && x1[0] >= x0[0] && 0 <= x0[0] + x1[0] + 3 * x2[0], x1[0], x2[0], x0[0])
(1): COND_1062_0_REC_LOAD(TRUE, x1[1], x2[1], x0[1]) → 1062_0_REC_LOAD(x1[1], x2[1] - 1, x0[1])
(2): 1062_0_REC_LOAD(x1[2], x2[2], x0[2]) → COND_1062_0_REC_LOAD1(x2[2] < x1[2] && x1[2] >= x0[2] && 0 <= x0[2] + x1[2] + 3 * x2[2], x1[2], x2[2], x0[2])
(3): COND_1062_0_REC_LOAD1(TRUE, x1[3], x2[3], x0[3]) → 1062_0_REC_LOAD(x1[3] - 2, x2[3], x0[3])
(4): 1062_0_REC_LOAD(x1[4], x2[4], x0[4]) → COND_1062_0_REC_LOAD2(x1[4] < x0[4] && 0 <= x0[4] + x1[4] + 3 * x2[4], x1[4], x2[4], x0[4])
(5): COND_1062_0_REC_LOAD2(TRUE, x1[5], x2[5], x0[5]) → 1062_0_REC_LOAD(x1[5], x2[5], x0[5] - 1)

(0) -> (1), if ((x2[0] >= x1[0] && x1[0] >= x0[0] && 0 <= x0[0] + x1[0] + 3 * x2[0] →^* TRUE)∧(x1[0] →^* x1[1])∧(x2[0] →^* x2[1])∧(x0[0] →^* x0[1]))

(1) -> (0), if ((x1[1] →^* x1[0])∧(x2[1] - 1 →^* x2[0])∧(x0[1] →^* x0[0]))

(1) -> (2), if ((x1[1] →^* x1[2])∧(x2[1] - 1 →^* x2[2])∧(x0[1] →^* x0[2]))

(1) -> (4), if ((x1[1] →^* x1[4])∧(x2[1] - 1 →^* x2[4])∧(x0[1] →^* x0[4]))

(2) -> (3), if ((x2[2] < x1[2] && x1[2] >= x0[2] && 0 <= x0[2] + x1[2] + 3 * x2[2] →^* TRUE)∧(x1[2] →^* x1[3])∧(x2[2] →^* x2[3])∧(x0[2] →^* x0[3]))

(3) -> (0), if ((x1[3] - 2 →^* x1[0])∧(x2[3] →^* x2[0])∧(x0[3] →^* x0[0]))

(3) -> (2), if ((x1[3] - 2 →^* x1[2])∧(x2[3] →^* x2[2])∧(x0[3] →^* x0[2]))

(3) -> (4), if ((x1[3] - 2 →^* x1[4])∧(x2[3] →^* x2[4])∧(x0[3] →^* x0[4]))

(4) -> (5), if ((x1[4] < x0[4] && 0 <= x0[4] + x1[4] + 3 * x2[4] →^* TRUE)∧(x1[4] →^* x1[5])∧(x2[4] →^* x2[5])∧(x0[4] →^* x0[5]))

(5) -> (0), if ((x1[5] →^* x1[0])∧(x2[5] →^* x2[0])∧(x0[5] - 1 →^* x0[0]))

(5) -> (2), if ((x1[5] →^* x1[2])∧(x2[5] →^* x2[2])∧(x0[5] - 1 →^* x0[2]))

(5) -> (4), if ((x1[5] →^* x1[4])∧(x2[5] →^* x2[4])∧(x0[5] - 1 →^* x0[4]))

The set Q consists of the following terms:
1162_1_rec_InvokeMethod(1081_0_rec_Return(x0, x1, x2), x0, x1, x2)
1162_1_rec_InvokeMethod(1150_0_rec_Return, x0, x1, x2)
1170_1_rec_InvokeMethod(1081_0_rec_Return(x0, x1, x2), x0, x1, x2)
1170_1_rec_InvokeMethod(1150_0_rec_Return, x0, x1, x2)
1133_1_rec_InvokeMethod(1081_0_rec_Return(x0, x1, x2), x0, x1, x2)
1133_1_rec_InvokeMethod(1150_0_rec_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 1062_0_REC_LOAD(x1, x2, x0) → COND_1062_0_REC_LOAD(&&(&&(>=(x2, x1), >=(x1, x0)), <=(0, +(+(x0, x1), *(3, x2)))), x1, x2, x0) the following chains were created:

We consider the chain 1062_0_REC_LOAD(x1[0], x2[0], x0[0]) → COND_1062_0_REC_LOAD(&&(&&(>=(x2[0], x1[0]), >=(x1[0], x0[0])), <=(0, +(+(x0[0], x1[0]), *(3, x2[0])))), x1[0], x2[0], x0[0]), COND_1062_0_REC_LOAD(TRUE, x1[1], x2[1], x0[1]) → 1062_0_REC_LOAD(x1[1], -(x2[1], 1), x0[1]) which results in the following constraint:
(1)    (&&(&&(>=(x2[0], x1[0]), >=(x1[0], x0[0])), <=(0, +(+(x0[0], x1[0]), *(3, x2[0]))))=TRUE∧x1[0]=x1[1]∧x2[0]=x2[1]∧x0[0]=x0[1] ⇒ 1062_0_REC_LOAD(x1[0], x2[0], x0[0])≥NonInfC∧1062_0_REC_LOAD(x1[0], x2[0], x0[0])≥COND_1062_0_REC_LOAD(&&(&&(>=(x2[0], x1[0]), >=(x1[0], x0[0])), <=(0, +(+(x0[0], x1[0]), *(3, x2[0])))), x1[0], x2[0], x0[0])∧(U^Increasing(COND_1062_0_REC_LOAD(&&(&&(>=(x2[0], x1[0]), >=(x1[0], x0[0])), <=(0, +(+(x0[0], x1[0]), *(3, x2[0])))), x1[0], x2[0], x0[0])), ≥))

We simplified constraint (1) using rules (IV), (IDP_BOOLEAN) which results in the following new constraint:
(2)    (<=(0, +(+(x0[0], x1[0]), *(3, x2[0])))=TRUE∧>=(x2[0], x1[0])=TRUE∧>=(x1[0], x0[0])=TRUE ⇒ 1062_0_REC_LOAD(x1[0], x2[0], x0[0])≥NonInfC∧1062_0_REC_LOAD(x1[0], x2[0], x0[0])≥COND_1062_0_REC_LOAD(&&(&&(>=(x2[0], x1[0]), >=(x1[0], x0[0])), <=(0, +(+(x0[0], x1[0]), *(3, x2[0])))), x1[0], x2[0], x0[0])∧(U^Increasing(COND_1062_0_REC_LOAD(&&(&&(>=(x2[0], x1[0]), >=(x1[0], x0[0])), <=(0, +(+(x0[0], x1[0]), *(3, x2[0])))), x1[0], x2[0], x0[0])), ≥))

We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(3)    (x0[0] + x1[0] + [3]x2[0] ≥ 0∧x2[0] + [-1]x1[0] ≥ 0∧x1[0] + [-1]x0[0] ≥ 0 ⇒ (U^Increasing(COND_1062_0_REC_LOAD(&&(&&(>=(x2[0], x1[0]), >=(x1[0], x0[0])), <=(0, +(+(x0[0], x1[0]), *(3, x2[0])))), x1[0], x2[0], x0[0])), ≥)∧[bni_19 + (-1)Bound*bni_19] + [bni_19]x0[0] + [(2)bni_19]x2[0] + [(2)bni_19]x1[0] ≥ 0∧[1 + (-1)bso_20] ≥ 0)

We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(4)    (x0[0] + x1[0] + [3]x2[0] ≥ 0∧x2[0] + [-1]x1[0] ≥ 0∧x1[0] + [-1]x0[0] ≥ 0 ⇒ (U^Increasing(COND_1062_0_REC_LOAD(&&(&&(>=(x2[0], x1[0]), >=(x1[0], x0[0])), <=(0, +(+(x0[0], x1[0]), *(3, x2[0])))), x1[0], x2[0], x0[0])), ≥)∧[bni_19 + (-1)Bound*bni_19] + [bni_19]x0[0] + [(2)bni_19]x2[0] + [(2)bni_19]x1[0] ≥ 0∧[1 + (-1)bso_20] ≥ 0)

We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(5)    (x0[0] + x1[0] + [3]x2[0] ≥ 0∧x2[0] + [-1]x1[0] ≥ 0∧x1[0] + [-1]x0[0] ≥ 0 ⇒ (U^Increasing(COND_1062_0_REC_LOAD(&&(&&(>=(x2[0], x1[0]), >=(x1[0], x0[0])), <=(0, +(+(x0[0], x1[0]), *(3, x2[0])))), x1[0], x2[0], x0[0])), ≥)∧[bni_19 + (-1)Bound*bni_19] + [bni_19]x0[0] + [(2)bni_19]x2[0] + [(2)bni_19]x1[0] ≥ 0∧[1 + (-1)bso_20] ≥ 0)

We simplified constraint (5) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(6)    (x0[0] ≥ 0∧x2[0] + [-1]x1[0] ≥ 0∧[2]x1[0] + [3]x2[0] + [-1]x0[0] ≥ 0 ⇒ (U^Increasing(COND_1062_0_REC_LOAD(&&(&&(>=(x2[0], x1[0]), >=(x1[0], x0[0])), <=(0, +(+(x0[0], x1[0]), *(3, x2[0])))), x1[0], x2[0], x0[0])), ≥)∧[bni_19 + (-1)Bound*bni_19] + [bni_19]x1[0] + [(-1)bni_19]x2[0] + [bni_19]x0[0] ≥ 0∧[1 + (-1)bso_20] ≥ 0)

We simplified constraint (6) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(7)    (x0[0] ≥ 0∧x1[0] ≥ 0∧[5]x2[0] + [-2]x1[0] + [-1]x0[0] ≥ 0 ⇒ (U^Increasing(COND_1062_0_REC_LOAD(&&(&&(>=(x2[0], x1[0]), >=(x1[0], x0[0])), <=(0, +(+(x0[0], x1[0]), *(3, x2[0])))), x1[0], x2[0], x0[0])), ≥)∧[bni_19 + (-1)Bound*bni_19] + [(-1)bni_19]x1[0] + [bni_19]x0[0] ≥ 0∧[1 + (-1)bso_20] ≥ 0)

For Pair COND_1062_0_REC_LOAD(TRUE, x1, x2, x0) → 1062_0_REC_LOAD(x1, -(x2, 1), x0) the following chains were created:

We consider the chain COND_1062_0_REC_LOAD(TRUE, x1[1], x2[1], x0[1]) → 1062_0_REC_LOAD(x1[1], -(x2[1], 1), x0[1]) which results in the following constraint:
(8)    (COND_1062_0_REC_LOAD(TRUE, x1[1], x2[1], x0[1])≥NonInfC∧COND_1062_0_REC_LOAD(TRUE, x1[1], x2[1], x0[1])≥1062_0_REC_LOAD(x1[1], -(x2[1], 1), x0[1])∧(U^Increasing(1062_0_REC_LOAD(x1[1], -(x2[1], 1), x0[1])), ≥))

We simplified constraint (8) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(9)    ((U^Increasing(1062_0_REC_LOAD(x1[1], -(x2[1], 1), x0[1])), ≥)∧[1 + (-1)bso_22] ≥ 0)

We simplified constraint (9) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(10)    ((U^Increasing(1062_0_REC_LOAD(x1[1], -(x2[1], 1), x0[1])), ≥)∧[1 + (-1)bso_22] ≥ 0)

We simplified constraint (10) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(11)    ((U^Increasing(1062_0_REC_LOAD(x1[1], -(x2[1], 1), x0[1])), ≥)∧[1 + (-1)bso_22] ≥ 0)

We simplified constraint (11) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint:
(12)    ((U^Increasing(1062_0_REC_LOAD(x1[1], -(x2[1], 1), x0[1])), ≥)∧0 = 0∧0 = 0∧0 = 0∧[1 + (-1)bso_22] ≥ 0)

For Pair 1062_0_REC_LOAD(x1, x2, x0) → COND_1062_0_REC_LOAD1(&&(&&(<(x2, x1), >=(x1, x0)), <=(0, +(+(x0, x1), *(3, x2)))), x1, x2, x0) the following chains were created:

We consider the chain 1062_0_REC_LOAD(x1[2], x2[2], x0[2]) → COND_1062_0_REC_LOAD1(&&(&&(<(x2[2], x1[2]), >=(x1[2], x0[2])), <=(0, +(+(x0[2], x1[2]), *(3, x2[2])))), x1[2], x2[2], x0[2]), COND_1062_0_REC_LOAD1(TRUE, x1[3], x2[3], x0[3]) → 1062_0_REC_LOAD(-(x1[3], 2), x2[3], x0[3]) which results in the following constraint:
(13)    (&&(&&(<(x2[2], x1[2]), >=(x1[2], x0[2])), <=(0, +(+(x0[2], x1[2]), *(3, x2[2]))))=TRUE∧x1[2]=x1[3]∧x2[2]=x2[3]∧x0[2]=x0[3] ⇒ 1062_0_REC_LOAD(x1[2], x2[2], x0[2])≥NonInfC∧1062_0_REC_LOAD(x1[2], x2[2], x0[2])≥COND_1062_0_REC_LOAD1(&&(&&(<(x2[2], x1[2]), >=(x1[2], x0[2])), <=(0, +(+(x0[2], x1[2]), *(3, x2[2])))), x1[2], x2[2], x0[2])∧(U^Increasing(COND_1062_0_REC_LOAD1(&&(&&(<(x2[2], x1[2]), >=(x1[2], x0[2])), <=(0, +(+(x0[2], x1[2]), *(3, x2[2])))), x1[2], x2[2], x0[2])), ≥))

We simplified constraint (13) using rules (IV), (IDP_BOOLEAN) which results in the following new constraint:
(14)    (<=(0, +(+(x0[2], x1[2]), *(3, x2[2])))=TRUE∧<(x2[2], x1[2])=TRUE∧>=(x1[2], x0[2])=TRUE ⇒ 1062_0_REC_LOAD(x1[2], x2[2], x0[2])≥NonInfC∧1062_0_REC_LOAD(x1[2], x2[2], x0[2])≥COND_1062_0_REC_LOAD1(&&(&&(<(x2[2], x1[2]), >=(x1[2], x0[2])), <=(0, +(+(x0[2], x1[2]), *(3, x2[2])))), x1[2], x2[2], x0[2])∧(U^Increasing(COND_1062_0_REC_LOAD1(&&(&&(<(x2[2], x1[2]), >=(x1[2], x0[2])), <=(0, +(+(x0[2], x1[2]), *(3, x2[2])))), x1[2], x2[2], x0[2])), ≥))

We simplified constraint (14) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(15)    (x0[2] + x1[2] + [3]x2[2] ≥ 0∧x1[2] + [-1] + [-1]x2[2] ≥ 0∧x1[2] + [-1]x0[2] ≥ 0 ⇒ (U^Increasing(COND_1062_0_REC_LOAD1(&&(&&(<(x2[2], x1[2]), >=(x1[2], x0[2])), <=(0, +(+(x0[2], x1[2]), *(3, x2[2])))), x1[2], x2[2], x0[2])), ≥)∧[bni_23 + (-1)Bound*bni_23] + [bni_23]x0[2] + [(2)bni_23]x2[2] + [(2)bni_23]x1[2] ≥ 0∧[(-1)bso_24] ≥ 0)

We simplified constraint (15) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(16)    (x0[2] + x1[2] + [3]x2[2] ≥ 0∧x1[2] + [-1] + [-1]x2[2] ≥ 0∧x1[2] + [-1]x0[2] ≥ 0 ⇒ (U^Increasing(COND_1062_0_REC_LOAD1(&&(&&(<(x2[2], x1[2]), >=(x1[2], x0[2])), <=(0, +(+(x0[2], x1[2]), *(3, x2[2])))), x1[2], x2[2], x0[2])), ≥)∧[bni_23 + (-1)Bound*bni_23] + [bni_23]x0[2] + [(2)bni_23]x2[2] + [(2)bni_23]x1[2] ≥ 0∧[(-1)bso_24] ≥ 0)

We simplified constraint (16) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(17)    (x0[2] + x1[2] + [3]x2[2] ≥ 0∧x1[2] + [-1] + [-1]x2[2] ≥ 0∧x1[2] + [-1]x0[2] ≥ 0 ⇒ (U^Increasing(COND_1062_0_REC_LOAD1(&&(&&(<(x2[2], x1[2]), >=(x1[2], x0[2])), <=(0, +(+(x0[2], x1[2]), *(3, x2[2])))), x1[2], x2[2], x0[2])), ≥)∧[bni_23 + (-1)Bound*bni_23] + [bni_23]x0[2] + [(2)bni_23]x2[2] + [(2)bni_23]x1[2] ≥ 0∧[(-1)bso_24] ≥ 0)

We simplified constraint (17) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(18)    (x0[2] ≥ 0∧x1[2] + [-1] + [-1]x2[2] ≥ 0∧[2]x1[2] + [3]x2[2] + [-1]x0[2] ≥ 0 ⇒ (U^Increasing(COND_1062_0_REC_LOAD1(&&(&&(<(x2[2], x1[2]), >=(x1[2], x0[2])), <=(0, +(+(x0[2], x1[2]), *(3, x2[2])))), x1[2], x2[2], x0[2])), ≥)∧[bni_23 + (-1)Bound*bni_23] + [bni_23]x1[2] + [(-1)bni_23]x2[2] + [bni_23]x0[2] ≥ 0∧[(-1)bso_24] ≥ 0)

We simplified constraint (18) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(19)    (x0[2] ≥ 0∧x1[2] ≥ 0∧[2] + [5]x2[2] + [2]x1[2] + [-1]x0[2] ≥ 0 ⇒ (U^Increasing(COND_1062_0_REC_LOAD1(&&(&&(<(x2[2], x1[2]), >=(x1[2], x0[2])), <=(0, +(+(x0[2], x1[2]), *(3, x2[2])))), x1[2], x2[2], x0[2])), ≥)∧[(-1)Bound*bni_23 + (2)bni_23] + [bni_23]x1[2] + [bni_23]x0[2] ≥ 0∧[(-1)bso_24] ≥ 0)

We simplified constraint (19) using rule (IDP_SMT_SPLIT) which results in the following new constraints:
(20)    (x0[2] ≥ 0∧x1[2] ≥ 0∧[2] + [5]x2[2] + [2]x1[2] + [-1]x0[2] ≥ 0∧x2[2] ≥ 0 ⇒ (U^Increasing(COND_1062_0_REC_LOAD1(&&(&&(<(x2[2], x1[2]), >=(x1[2], x0[2])), <=(0, +(+(x0[2], x1[2]), *(3, x2[2])))), x1[2], x2[2], x0[2])), ≥)∧[(-1)Bound*bni_23 + (2)bni_23] + [bni_23]x1[2] + [bni_23]x0[2] ≥ 0∧[(-1)bso_24] ≥ 0)

(21)    (x0[2] ≥ 0∧x1[2] ≥ 0∧[2] + [-5]x2[2] + [2]x1[2] + [-1]x0[2] ≥ 0∧x2[2] ≥ 0 ⇒ (U^Increasing(COND_1062_0_REC_LOAD1(&&(&&(<(x2[2], x1[2]), >=(x1[2], x0[2])), <=(0, +(+(x0[2], x1[2]), *(3, x2[2])))), x1[2], x2[2], x0[2])), ≥)∧[(-1)Bound*bni_23 + (2)bni_23] + [bni_23]x1[2] + [bni_23]x0[2] ≥ 0∧[(-1)bso_24] ≥ 0)

For Pair COND_1062_0_REC_LOAD1(TRUE, x1, x2, x0) → 1062_0_REC_LOAD(-(x1, 2), x2, x0) the following chains were created:

We consider the chain COND_1062_0_REC_LOAD1(TRUE, x1[3], x2[3], x0[3]) → 1062_0_REC_LOAD(-(x1[3], 2), x2[3], x0[3]) which results in the following constraint:
(22)    (COND_1062_0_REC_LOAD1(TRUE, x1[3], x2[3], x0[3])≥NonInfC∧COND_1062_0_REC_LOAD1(TRUE, x1[3], x2[3], x0[3])≥1062_0_REC_LOAD(-(x1[3], 2), x2[3], x0[3])∧(U^Increasing(1062_0_REC_LOAD(-(x1[3], 2), x2[3], x0[3])), ≥))

We simplified constraint (22) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(23)    ((U^Increasing(1062_0_REC_LOAD(-(x1[3], 2), x2[3], x0[3])), ≥)∧[4 + (-1)bso_26] ≥ 0)

We simplified constraint (23) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(24)    ((U^Increasing(1062_0_REC_LOAD(-(x1[3], 2), x2[3], x0[3])), ≥)∧[4 + (-1)bso_26] ≥ 0)

We simplified constraint (24) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(25)    ((U^Increasing(1062_0_REC_LOAD(-(x1[3], 2), x2[3], x0[3])), ≥)∧[4 + (-1)bso_26] ≥ 0)

We simplified constraint (25) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint:
(26)    ((U^Increasing(1062_0_REC_LOAD(-(x1[3], 2), x2[3], x0[3])), ≥)∧0 = 0∧0 = 0∧0 = 0∧[4 + (-1)bso_26] ≥ 0)

For Pair 1062_0_REC_LOAD(x1, x2, x0) → COND_1062_0_REC_LOAD2(&&(<(x1, x0), <=(0, +(+(x0, x1), *(3, x2)))), x1, x2, x0) the following chains were created:

We consider the chain 1062_0_REC_LOAD(x1[4], x2[4], x0[4]) → COND_1062_0_REC_LOAD2(&&(<(x1[4], x0[4]), <=(0, +(+(x0[4], x1[4]), *(3, x2[4])))), x1[4], x2[4], x0[4]), COND_1062_0_REC_LOAD2(TRUE, x1[5], x2[5], x0[5]) → 1062_0_REC_LOAD(x1[5], x2[5], -(x0[5], 1)) which results in the following constraint:
(27)    (&&(<(x1[4], x0[4]), <=(0, +(+(x0[4], x1[4]), *(3, x2[4]))))=TRUE∧x1[4]=x1[5]∧x2[4]=x2[5]∧x0[4]=x0[5] ⇒ 1062_0_REC_LOAD(x1[4], x2[4], x0[4])≥NonInfC∧1062_0_REC_LOAD(x1[4], x2[4], x0[4])≥COND_1062_0_REC_LOAD2(&&(<(x1[4], x0[4]), <=(0, +(+(x0[4], x1[4]), *(3, x2[4])))), x1[4], x2[4], x0[4])∧(U^Increasing(COND_1062_0_REC_LOAD2(&&(<(x1[4], x0[4]), <=(0, +(+(x0[4], x1[4]), *(3, x2[4])))), x1[4], x2[4], x0[4])), ≥))

We simplified constraint (27) using rules (IV), (IDP_BOOLEAN) which results in the following new constraint:
(28)    (<(x1[4], x0[4])=TRUE∧<=(0, +(+(x0[4], x1[4]), *(3, x2[4])))=TRUE ⇒ 1062_0_REC_LOAD(x1[4], x2[4], x0[4])≥NonInfC∧1062_0_REC_LOAD(x1[4], x2[4], x0[4])≥COND_1062_0_REC_LOAD2(&&(<(x1[4], x0[4]), <=(0, +(+(x0[4], x1[4]), *(3, x2[4])))), x1[4], x2[4], x0[4])∧(U^Increasing(COND_1062_0_REC_LOAD2(&&(<(x1[4], x0[4]), <=(0, +(+(x0[4], x1[4]), *(3, x2[4])))), x1[4], x2[4], x0[4])), ≥))

We simplified constraint (28) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(29)    (x0[4] + [-1] + [-1]x1[4] ≥ 0∧x0[4] + x1[4] + [3]x2[4] ≥ 0 ⇒ (U^Increasing(COND_1062_0_REC_LOAD2(&&(<(x1[4], x0[4]), <=(0, +(+(x0[4], x1[4]), *(3, x2[4])))), x1[4], x2[4], x0[4])), ≥)∧[bni_27 + (-1)Bound*bni_27] + [bni_27]x0[4] + [(2)bni_27]x2[4] + [(2)bni_27]x1[4] ≥ 0∧[(-1)bso_28] ≥ 0)

We simplified constraint (29) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(30)    (x0[4] + [-1] + [-1]x1[4] ≥ 0∧x0[4] + x1[4] + [3]x2[4] ≥ 0 ⇒ (U^Increasing(COND_1062_0_REC_LOAD2(&&(<(x1[4], x0[4]), <=(0, +(+(x0[4], x1[4]), *(3, x2[4])))), x1[4], x2[4], x0[4])), ≥)∧[bni_27 + (-1)Bound*bni_27] + [bni_27]x0[4] + [(2)bni_27]x2[4] + [(2)bni_27]x1[4] ≥ 0∧[(-1)bso_28] ≥ 0)

We simplified constraint (30) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(31)    (x0[4] + [-1] + [-1]x1[4] ≥ 0∧x0[4] + x1[4] + [3]x2[4] ≥ 0 ⇒ (U^Increasing(COND_1062_0_REC_LOAD2(&&(<(x1[4], x0[4]), <=(0, +(+(x0[4], x1[4]), *(3, x2[4])))), x1[4], x2[4], x0[4])), ≥)∧[bni_27 + (-1)Bound*bni_27] + [bni_27]x0[4] + [(2)bni_27]x2[4] + [(2)bni_27]x1[4] ≥ 0∧[(-1)bso_28] ≥ 0)

We simplified constraint (31) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(32)    (x0[4] ≥ 0∧[1] + [2]x1[4] + x0[4] + [3]x2[4] ≥ 0 ⇒ (U^Increasing(COND_1062_0_REC_LOAD2(&&(<(x1[4], x0[4]), <=(0, +(+(x0[4], x1[4]), *(3, x2[4])))), x1[4], x2[4], x0[4])), ≥)∧[(2)bni_27 + (-1)Bound*bni_27] + [(3)bni_27]x1[4] + [bni_27]x0[4] + [(2)bni_27]x2[4] ≥ 0∧[(-1)bso_28] ≥ 0)

We simplified constraint (32) using rule (IDP_SMT_SPLIT) which results in the following new constraints:
(33)    (x0[4] ≥ 0∧[1] + [2]x1[4] + x0[4] + [3]x2[4] ≥ 0∧x1[4] ≥ 0 ⇒ (U^Increasing(COND_1062_0_REC_LOAD2(&&(<(x1[4], x0[4]), <=(0, +(+(x0[4], x1[4]), *(3, x2[4])))), x1[4], x2[4], x0[4])), ≥)∧[(2)bni_27 + (-1)Bound*bni_27] + [(3)bni_27]x1[4] + [bni_27]x0[4] + [(2)bni_27]x2[4] ≥ 0∧[(-1)bso_28] ≥ 0)

(34)    (x0[4] ≥ 0∧[1] + [-2]x1[4] + x0[4] + [3]x2[4] ≥ 0∧x1[4] ≥ 0 ⇒ (U^Increasing(COND_1062_0_REC_LOAD2(&&(<(x1[4], x0[4]), <=(0, +(+(x0[4], x1[4]), *(3, x2[4])))), x1[4], x2[4], x0[4])), ≥)∧[(2)bni_27 + (-1)Bound*bni_27] + [(-3)bni_27]x1[4] + [bni_27]x0[4] + [(2)bni_27]x2[4] ≥ 0∧[(-1)bso_28] ≥ 0)

We simplified constraint (33) using rule (IDP_SMT_SPLIT) which results in the following new constraints:
(35)    (x0[4] ≥ 0∧[1] + [2]x1[4] + x0[4] + [3]x2[4] ≥ 0∧x1[4] ≥ 0∧x2[4] ≥ 0 ⇒ (U^Increasing(COND_1062_0_REC_LOAD2(&&(<(x1[4], x0[4]), <=(0, +(+(x0[4], x1[4]), *(3, x2[4])))), x1[4], x2[4], x0[4])), ≥)∧[(2)bni_27 + (-1)Bound*bni_27] + [(3)bni_27]x1[4] + [bni_27]x0[4] + [(2)bni_27]x2[4] ≥ 0∧[(-1)bso_28] ≥ 0)

(36)    (x0[4] ≥ 0∧[1] + [2]x1[4] + x0[4] + [-3]x2[4] ≥ 0∧x1[4] ≥ 0∧x2[4] ≥ 0 ⇒ (U^Increasing(COND_1062_0_REC_LOAD2(&&(<(x1[4], x0[4]), <=(0, +(+(x0[4], x1[4]), *(3, x2[4])))), x1[4], x2[4], x0[4])), ≥)∧[(2)bni_27 + (-1)Bound*bni_27] + [(3)bni_27]x1[4] + [bni_27]x0[4] + [(-2)bni_27]x2[4] ≥ 0∧[(-1)bso_28] ≥ 0)

We simplified constraint (34) using rule (IDP_SMT_SPLIT) which results in the following new constraints:
(37)    (x0[4] ≥ 0∧[1] + [-2]x1[4] + x0[4] + [3]x2[4] ≥ 0∧x1[4] ≥ 0∧x2[4] ≥ 0 ⇒ (U^Increasing(COND_1062_0_REC_LOAD2(&&(<(x1[4], x0[4]), <=(0, +(+(x0[4], x1[4]), *(3, x2[4])))), x1[4], x2[4], x0[4])), ≥)∧[(2)bni_27 + (-1)Bound*bni_27] + [(-3)bni_27]x1[4] + [bni_27]x0[4] + [(2)bni_27]x2[4] ≥ 0∧[(-1)bso_28] ≥ 0)

(38)    (x0[4] ≥ 0∧[1] + [-2]x1[4] + x0[4] + [-3]x2[4] ≥ 0∧x1[4] ≥ 0∧x2[4] ≥ 0 ⇒ (U^Increasing(COND_1062_0_REC_LOAD2(&&(<(x1[4], x0[4]), <=(0, +(+(x0[4], x1[4]), *(3, x2[4])))), x1[4], x2[4], x0[4])), ≥)∧[(2)bni_27 + (-1)Bound*bni_27] + [(-3)bni_27]x1[4] + [bni_27]x0[4] + [(-2)bni_27]x2[4] ≥ 0∧[(-1)bso_28] ≥ 0)

We simplified constraint (35) using rule (IDP_POLY_GCD) which results in the following new constraint:
(39)    (x0[4] ≥ 0∧x1[4] ≥ 0∧x2[4] ≥ 0∧x2[4] ≥ 0 ⇒ (U^Increasing(COND_1062_0_REC_LOAD2(&&(<(x1[4], x0[4]), <=(0, +(+(x0[4], x1[4]), *(3, x2[4])))), x1[4], x2[4], x0[4])), ≥)∧[(2)bni_27 + (-1)Bound*bni_27] + [(3)bni_27]x1[4] + [bni_27]x0[4] + [(2)bni_27]x2[4] ≥ 0∧[(-1)bso_28] ≥ 0)

We simplified constraint (36) using rule (IDP_POLY_GCD) which results in the following new constraint:
(40)    (x0[4] ≥ 0∧x1[4] ≥ 0∧x2[4] ≥ 0∧[-1]x2[4] ≥ 0 ⇒ (U^Increasing(COND_1062_0_REC_LOAD2(&&(<(x1[4], x0[4]), <=(0, +(+(x0[4], x1[4]), *(3, x2[4])))), x1[4], x2[4], x0[4])), ≥)∧[(2)bni_27 + (-1)Bound*bni_27] + [(3)bni_27]x1[4] + [bni_27]x0[4] + [(-2)bni_27]x2[4] ≥ 0∧[(-1)bso_28] ≥ 0)

We simplified constraint (37) using rule (IDP_POLY_GCD) which results in the following new constraint:
(41)    (x0[4] ≥ 0∧x1[4] ≥ 0∧x2[4] ≥ 0∧x2[4] ≥ 0 ⇒ (U^Increasing(COND_1062_0_REC_LOAD2(&&(<(x1[4], x0[4]), <=(0, +(+(x0[4], x1[4]), *(3, x2[4])))), x1[4], x2[4], x0[4])), ≥)∧[(2)bni_27 + (-1)Bound*bni_27] + [(-3)bni_27]x1[4] + [bni_27]x0[4] + [(2)bni_27]x2[4] ≥ 0∧[(-1)bso_28] ≥ 0)

We simplified constraint (38) using rule (IDP_POLY_GCD) which results in the following new constraint:
(42)    (x0[4] ≥ 0∧x1[4] ≥ 0∧x2[4] ≥ 0∧[-1]x2[4] ≥ 0 ⇒ (U^Increasing(COND_1062_0_REC_LOAD2(&&(<(x1[4], x0[4]), <=(0, +(+(x0[4], x1[4]), *(3, x2[4])))), x1[4], x2[4], x0[4])), ≥)∧[(2)bni_27 + (-1)Bound*bni_27] + [(-3)bni_27]x1[4] + [bni_27]x0[4] + [(-2)bni_27]x2[4] ≥ 0∧[(-1)bso_28] ≥ 0)

For Pair COND_1062_0_REC_LOAD2(TRUE, x1, x2, x0) → 1062_0_REC_LOAD(x1, x2, -(x0, 1)) the following chains were created:

We consider the chain COND_1062_0_REC_LOAD2(TRUE, x1[5], x2[5], x0[5]) → 1062_0_REC_LOAD(x1[5], x2[5], -(x0[5], 1)) which results in the following constraint:
(43)    (COND_1062_0_REC_LOAD2(TRUE, x1[5], x2[5], x0[5])≥NonInfC∧COND_1062_0_REC_LOAD2(TRUE, x1[5], x2[5], x0[5])≥1062_0_REC_LOAD(x1[5], x2[5], -(x0[5], 1))∧(U^Increasing(1062_0_REC_LOAD(x1[5], x2[5], -(x0[5], 1))), ≥))

We simplified constraint (43) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(44)    ((U^Increasing(1062_0_REC_LOAD(x1[5], x2[5], -(x0[5], 1))), ≥)∧[1 + (-1)bso_30] ≥ 0)

We simplified constraint (44) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(45)    ((U^Increasing(1062_0_REC_LOAD(x1[5], x2[5], -(x0[5], 1))), ≥)∧[1 + (-1)bso_30] ≥ 0)

We simplified constraint (45) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(46)    ((U^Increasing(1062_0_REC_LOAD(x1[5], x2[5], -(x0[5], 1))), ≥)∧[1 + (-1)bso_30] ≥ 0)

We simplified constraint (46) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint:
(47)    ((U^Increasing(1062_0_REC_LOAD(x1[5], x2[5], -(x0[5], 1))), ≥)∧0 = 0∧0 = 0∧0 = 0∧[1 + (-1)bso_30] ≥ 0)

To summarize, we get the following constraints P_≥ for the following pairs.

1062_0_REC_LOAD(x1, x2, x0) → COND_1062_0_REC_LOAD(&&(&&(>=(x2, x1), >=(x1, x0)), <=(0, +(+(x0, x1), *(3, x2)))), x1, x2, x0)
- (x0[0] ≥ 0∧x1[0] ≥ 0∧[5]x2[0] + [-2]x1[0] + [-1]x0[0] ≥ 0 ⇒ (U^Increasing(COND_1062_0_REC_LOAD(&&(&&(>=(x2[0], x1[0]), >=(x1[0], x0[0])), <=(0, +(+(x0[0], x1[0]), *(3, x2[0])))), x1[0], x2[0], x0[0])), ≥)∧[bni_19 + (-1)Bound*bni_19] + [(-1)bni_19]x1[0] + [bni_19]x0[0] ≥ 0∧[1 + (-1)bso_20] ≥ 0)
COND_1062_0_REC_LOAD(TRUE, x1, x2, x0) → 1062_0_REC_LOAD(x1, -(x2, 1), x0)
- ((U^Increasing(1062_0_REC_LOAD(x1[1], -(x2[1], 1), x0[1])), ≥)∧0 = 0∧0 = 0∧0 = 0∧[1 + (-1)bso_22] ≥ 0)
1062_0_REC_LOAD(x1, x2, x0) → COND_1062_0_REC_LOAD1(&&(&&(<(x2, x1), >=(x1, x0)), <=(0, +(+(x0, x1), *(3, x2)))), x1, x2, x0)
- (x0[2] ≥ 0∧x1[2] ≥ 0∧[2] + [5]x2[2] + [2]x1[2] + [-1]x0[2] ≥ 0∧x2[2] ≥ 0 ⇒ (U^Increasing(COND_1062_0_REC_LOAD1(&&(&&(<(x2[2], x1[2]), >=(x1[2], x0[2])), <=(0, +(+(x0[2], x1[2]), *(3, x2[2])))), x1[2], x2[2], x0[2])), ≥)∧[(-1)Bound*bni_23 + (2)bni_23] + [bni_23]x1[2] + [bni_23]x0[2] ≥ 0∧[(-1)bso_24] ≥ 0)
- (x0[2] ≥ 0∧x1[2] ≥ 0∧[2] + [-5]x2[2] + [2]x1[2] + [-1]x0[2] ≥ 0∧x2[2] ≥ 0 ⇒ (U^Increasing(COND_1062_0_REC_LOAD1(&&(&&(<(x2[2], x1[2]), >=(x1[2], x0[2])), <=(0, +(+(x0[2], x1[2]), *(3, x2[2])))), x1[2], x2[2], x0[2])), ≥)∧[(-1)Bound*bni_23 + (2)bni_23] + [bni_23]x1[2] + [bni_23]x0[2] ≥ 0∧[(-1)bso_24] ≥ 0)
COND_1062_0_REC_LOAD1(TRUE, x1, x2, x0) → 1062_0_REC_LOAD(-(x1, 2), x2, x0)
- ((U^Increasing(1062_0_REC_LOAD(-(x1[3], 2), x2[3], x0[3])), ≥)∧0 = 0∧0 = 0∧0 = 0∧[4 + (-1)bso_26] ≥ 0)
1062_0_REC_LOAD(x1, x2, x0) → COND_1062_0_REC_LOAD2(&&(<(x1, x0), <=(0, +(+(x0, x1), *(3, x2)))), x1, x2, x0)
- (x0[4] ≥ 0∧x1[4] ≥ 0∧x2[4] ≥ 0∧x2[4] ≥ 0 ⇒ (U^Increasing(COND_1062_0_REC_LOAD2(&&(<(x1[4], x0[4]), <=(0, +(+(x0[4], x1[4]), *(3, x2[4])))), x1[4], x2[4], x0[4])), ≥)∧[(2)bni_27 + (-1)Bound*bni_27] + [(3)bni_27]x1[4] + [bni_27]x0[4] + [(2)bni_27]x2[4] ≥ 0∧[(-1)bso_28] ≥ 0)
- (x0[4] ≥ 0∧x1[4] ≥ 0∧x2[4] ≥ 0∧[-1]x2[4] ≥ 0 ⇒ (U^Increasing(COND_1062_0_REC_LOAD2(&&(<(x1[4], x0[4]), <=(0, +(+(x0[4], x1[4]), *(3, x2[4])))), x1[4], x2[4], x0[4])), ≥)∧[(2)bni_27 + (-1)Bound*bni_27] + [(3)bni_27]x1[4] + [bni_27]x0[4] + [(-2)bni_27]x2[4] ≥ 0∧[(-1)bso_28] ≥ 0)
- (x0[4] ≥ 0∧x1[4] ≥ 0∧x2[4] ≥ 0∧x2[4] ≥ 0 ⇒ (U^Increasing(COND_1062_0_REC_LOAD2(&&(<(x1[4], x0[4]), <=(0, +(+(x0[4], x1[4]), *(3, x2[4])))), x1[4], x2[4], x0[4])), ≥)∧[(2)bni_27 + (-1)Bound*bni_27] + [(-3)bni_27]x1[4] + [bni_27]x0[4] + [(2)bni_27]x2[4] ≥ 0∧[(-1)bso_28] ≥ 0)
- (x0[4] ≥ 0∧x1[4] ≥ 0∧x2[4] ≥ 0∧[-1]x2[4] ≥ 0 ⇒ (U^Increasing(COND_1062_0_REC_LOAD2(&&(<(x1[4], x0[4]), <=(0, +(+(x0[4], x1[4]), *(3, x2[4])))), x1[4], x2[4], x0[4])), ≥)∧[(2)bni_27 + (-1)Bound*bni_27] + [(-3)bni_27]x1[4] + [bni_27]x0[4] + [(-2)bni_27]x2[4] ≥ 0∧[(-1)bso_28] ≥ 0)
COND_1062_0_REC_LOAD2(TRUE, x1, x2, x0) → 1062_0_REC_LOAD(x1, x2, -(x0, 1))
- ((U^Increasing(1062_0_REC_LOAD(x1[5], x2[5], -(x0[5], 1))), ≥)∧0 = 0∧0 = 0∧0 = 0∧[1 + (-1)bso_30] ≥ 0)

The constraints for P_> respective P_bound 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 P_bound.
Using the following integer polynomial ordering the resulting constraints can be solved
Polynomial interpretation over integers[POLO]:

POL(TRUE) = 0
POL(FALSE) = 0
POL(1162_1_rec_InvokeMethod(x₁, x₂, x₃, x₄)) = [-1]
POL(1081_0_rec_Return(x₁, x₂, x₃)) = [-1]
POL(1150_0_rec_Return) = [-1]
POL(1170_1_rec_InvokeMethod(x₁, x₂, x₃, x₄)) = [-1]
POL(1133_1_rec_InvokeMethod(x₁, x₂, x₃, x₄)) = [-1]
POL(1062_0_REC_LOAD(x₁, x₂, x₃)) = [1] + x₃ + [2]x₂ + [2]x₁
POL(COND_1062_0_REC_LOAD(x₁, x₂, x₃, x₄)) = x₄ + [2]x₃ + [2]x₂
POL(&&(x₁, x₂)) = [-1]
POL(>=(x₁, x₂)) = [-1]
POL(<=(x₁, x₂)) = [-1]
POL(0) = 0
POL(+(x₁, x₂)) = x₁ + x₂
POL(*(x₁, x₂)) = x₁·x₂
POL(3) = [3]
POL(-(x₁, x₂)) = x₁ + [-1]x₂
POL(1) = [1]
POL(COND_1062_0_REC_LOAD1(x₁, x₂, x₃, x₄)) = [1] + x₄ + [2]x₃ + [2]x₂
POL(<(x₁, x₂)) = [-1]
POL(2) = [2]
POL(COND_1062_0_REC_LOAD2(x₁, x₂, x₃, x₄)) = [1] + x₄ + [2]x₃ + [2]x₂

The following pairs are in P_>:

1062_0_REC_LOAD(x1[0], x2[0], x0[0]) → COND_1062_0_REC_LOAD(&&(&&(>=(x2[0], x1[0]), >=(x1[0], x0[0])), <=(0, +(+(x0[0], x1[0]), *(3, x2[0])))), x1[0], x2[0], x0[0])
COND_1062_0_REC_LOAD(TRUE, x1[1], x2[1], x0[1]) → 1062_0_REC_LOAD(x1[1], -(x2[1], 1), x0[1])
COND_1062_0_REC_LOAD1(TRUE, x1[3], x2[3], x0[3]) → 1062_0_REC_LOAD(-(x1[3], 2), x2[3], x0[3])
COND_1062_0_REC_LOAD2(TRUE, x1[5], x2[5], x0[5]) → 1062_0_REC_LOAD(x1[5], x2[5], -(x0[5], 1))

The following pairs are in P_bound:

1062_0_REC_LOAD(x1[2], x2[2], x0[2]) → COND_1062_0_REC_LOAD1(&&(&&(<(x2[2], x1[2]), >=(x1[2], x0[2])), <=(0, +(+(x0[2], x1[2]), *(3, x2[2])))), x1[2], x2[2], x0[2])

The following pairs are in P_≥:

1062_0_REC_LOAD(x1[2], x2[2], x0[2]) → COND_1062_0_REC_LOAD1(&&(&&(<(x2[2], x1[2]), >=(x1[2], x0[2])), <=(0, +(+(x0[2], x1[2]), *(3, x2[2])))), x1[2], x2[2], x0[2])
1062_0_REC_LOAD(x1[4], x2[4], x0[4]) → COND_1062_0_REC_LOAD2(&&(<(x1[4], x0[4]), <=(0, +(+(x0[4], x1[4]), *(3, x2[4])))), x1[4], x2[4], x0[4])

There are no usable rules.

(6) Complex Obligation (AND)

(7) Obligation:

IDP problem:
The following function symbols are pre-defined:

!=	~	Neq: (Integer, Integer) -> Boolean
*	~	Mul: (Integer, Integer) -> Integer
>=	~	Ge: (Integer, Integer) -> Boolean
-1	~	UnaryMinus: (Integer) -> Integer
\|	~	Bwor: (Integer, Integer) -> Integer
/	~	Div: (Integer, Integer) -> Integer
=	~	Eq: (Integer, Integer) -> Boolean
	~	Bwxor: (Integer, Integer) -> Integer
\|\|	~	Lor: (Boolean, Boolean) -> Boolean
!	~	Lnot: (Boolean) -> Boolean
<	~	Lt: (Integer, Integer) -> Boolean
-	~	Sub: (Integer, Integer) -> Integer
<=	~	Le: (Integer, Integer) -> Boolean
>	~	Gt: (Integer, Integer) -> Boolean
~	~	Bwnot: (Integer) -> Integer
%	~	Mod: (Integer, Integer) -> Integer
&	~	Bwand: (Integer, Integer) -> Integer
+	~	Add: (Integer, Integer) -> Integer
&&	~	Land: (Boolean, Boolean) -> Boolean

The following domains are used:

Boolean, Integer

The ITRS R consists of the following rules:
1162_1_rec_InvokeMethod(1081_0_rec_Return(x0, x1, x2), x0, x1, x2) → 1150_0_rec_Return
1162_1_rec_InvokeMethod(1150_0_rec_Return, x0, x1, x2) → 1150_0_rec_Return
1170_1_rec_InvokeMethod(1081_0_rec_Return(x0, x1, x2), x0, x1, x2) → 1150_0_rec_Return
1170_1_rec_InvokeMethod(1150_0_rec_Return, x0, x1, x2) → 1150_0_rec_Return
1133_1_rec_InvokeMethod(1081_0_rec_Return(x0, x1, x2), x0, x1, x2) → 1150_0_rec_Return
1133_1_rec_InvokeMethod(1150_0_rec_Return, x0, x1, x2) → 1150_0_rec_Return

The integer pair graph contains the following rules and edges:
(2): 1062_0_REC_LOAD(x1[2], x2[2], x0[2]) → COND_1062_0_REC_LOAD1(x2[2] < x1[2] && x1[2] >= x0[2] && 0 <= x0[2] + x1[2] + 3 * x2[2], x1[2], x2[2], x0[2])
(4): 1062_0_REC_LOAD(x1[4], x2[4], x0[4]) → COND_1062_0_REC_LOAD2(x1[4] < x0[4] && 0 <= x0[4] + x1[4] + 3 * x2[4], x1[4], x2[4], x0[4])

The set Q consists of the following terms:
1162_1_rec_InvokeMethod(1081_0_rec_Return(x0, x1, x2), x0, x1, x2)
1162_1_rec_InvokeMethod(1150_0_rec_Return, x0, x1, x2)
1170_1_rec_InvokeMethod(1081_0_rec_Return(x0, x1, x2), x0, x1, x2)
1170_1_rec_InvokeMethod(1150_0_rec_Return, x0, x1, x2)
1133_1_rec_InvokeMethod(1081_0_rec_Return(x0, x1, x2), x0, x1, x2)
1133_1_rec_InvokeMethod(1150_0_rec_Return, x0, x1, x2)

(8) IDependencyGraphProof (EQUIVALENT transformation)

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

(9) TRUE

(10) Obligation:

IDP problem:
The following function symbols are pre-defined:

!=	~	Neq: (Integer, Integer) -> Boolean
*	~	Mul: (Integer, Integer) -> Integer
>=	~	Ge: (Integer, Integer) -> Boolean
-1	~	UnaryMinus: (Integer) -> Integer
\|	~	Bwor: (Integer, Integer) -> Integer
/	~	Div: (Integer, Integer) -> Integer
=	~	Eq: (Integer, Integer) -> Boolean
	~	Bwxor: (Integer, Integer) -> Integer
\|\|	~	Lor: (Boolean, Boolean) -> Boolean
!	~	Lnot: (Boolean) -> Boolean
<	~	Lt: (Integer, Integer) -> Boolean
-	~	Sub: (Integer, Integer) -> Integer
<=	~	Le: (Integer, Integer) -> Boolean
>	~	Gt: (Integer, Integer) -> Boolean
~	~	Bwnot: (Integer) -> Integer
%	~	Mod: (Integer, Integer) -> Integer
&	~	Bwand: (Integer, Integer) -> Integer
+	~	Add: (Integer, Integer) -> Integer
&&	~	Land: (Boolean, Boolean) -> Boolean

The following domains are used:

Boolean, Integer

The ITRS R consists of the following rules:
1162_1_rec_InvokeMethod(1081_0_rec_Return(x0, x1, x2), x0, x1, x2) → 1150_0_rec_Return
1162_1_rec_InvokeMethod(1150_0_rec_Return, x0, x1, x2) → 1150_0_rec_Return
1170_1_rec_InvokeMethod(1081_0_rec_Return(x0, x1, x2), x0, x1, x2) → 1150_0_rec_Return
1170_1_rec_InvokeMethod(1150_0_rec_Return, x0, x1, x2) → 1150_0_rec_Return
1133_1_rec_InvokeMethod(1081_0_rec_Return(x0, x1, x2), x0, x1, x2) → 1150_0_rec_Return
1133_1_rec_InvokeMethod(1150_0_rec_Return, x0, x1, x2) → 1150_0_rec_Return

The integer pair graph contains the following rules and edges:
(0): 1062_0_REC_LOAD(x1[0], x2[0], x0[0]) → COND_1062_0_REC_LOAD(x2[0] >= x1[0] && x1[0] >= x0[0] && 0 <= x0[0] + x1[0] + 3 * x2[0], x1[0], x2[0], x0[0])
(1): COND_1062_0_REC_LOAD(TRUE, x1[1], x2[1], x0[1]) → 1062_0_REC_LOAD(x1[1], x2[1] - 1, x0[1])
(3): COND_1062_0_REC_LOAD1(TRUE, x1[3], x2[3], x0[3]) → 1062_0_REC_LOAD(x1[3] - 2, x2[3], x0[3])
(4): 1062_0_REC_LOAD(x1[4], x2[4], x0[4]) → COND_1062_0_REC_LOAD2(x1[4] < x0[4] && 0 <= x0[4] + x1[4] + 3 * x2[4], x1[4], x2[4], x0[4])
(5): COND_1062_0_REC_LOAD2(TRUE, x1[5], x2[5], x0[5]) → 1062_0_REC_LOAD(x1[5], x2[5], x0[5] - 1)