Termination proof of /tmp/tmpNkdKVL/PastaB8.jar

Termination of the given JBC could be proven:

0 JBC

↳1 JBC2FIG (⇐)

↳2 FIGraph

↳3 FIGtoITRSProof (⇐)

↳4 ITRS

↳5 ITRStoIDPProof (⇔)

↳6 IDP

↳7 UsableRulesProof (⇔)

↳8 IDP

↳9 IDPNonInfProof (⇐)

↳10 AND

    ↳11 IDP

        ↳12 IDependencyGraphProof (⇔)

        ↳13 TRUE

    ↳14 IDP

        ↳15 IDependencyGraphProof (⇔)

        ↳16 TRUE

(0) Obligation:

JBC Problem based on JBC Program:

No human-readable program information known.

Manifest-Version: 1.0 Created-By: 1.6.0_16 (Sun Microsystems Inc.) Main-Class: PastaB8

(1) JBC2FIG (SOUND transformation)

Constructed FIGraph.

(2) Obligation:

FIGraph based on JBC Program:
Graph of 117 nodes with 1 SCC.

(3) FIGtoITRSProof (SOUND transformation)

Transformed FIGraph to ITRS rules

(4) Obligation:

ITRS 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 TRS R consists of the following rules:
Load292(i41) → Cond_Load292(i41 > 0 && 0 = i41 % 2, i41)
Cond_Load292(TRUE, i41) → Load292(i41 / 2)
Load292(i41) → Cond_Load2921(i41 % 2 > 0 && i41 > 0, i41)
Cond_Load2921(TRUE, i41) → Load292(i41 + -1)
The set Q consists of the following terms:
Load292(x0)
Cond_Load292(TRUE, x0)
Cond_Load2921(TRUE, x0)

(5) ITRStoIDPProof (EQUIVALENT transformation)

Added dependency pairs

(6) Obligation:

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

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

The following domains are used:

Boolean, Integer

The ITRS R consists of the following rules:
Load292(i41) → Cond_Load292(i41 > 0 && 0 = i41 % 2, i41)
Cond_Load292(TRUE, i41) → Load292(i41 / 2)
Load292(i41) → Cond_Load2921(i41 % 2 > 0 && i41 > 0, i41)
Cond_Load2921(TRUE, i41) → Load292(i41 + -1)

The integer pair graph contains the following rules and edges:
(0): LOAD292(i41[0]) → COND_LOAD292(i41[0] > 0 && 0 = i41[0] % 2, i41[0])
(1): COND_LOAD292(TRUE, i41[1]) → LOAD292(i41[1] / 2)
(2): LOAD292(i41[2]) → COND_LOAD2921(i41[2] % 2 > 0 && i41[2] > 0, i41[2])
(3): COND_LOAD2921(TRUE, i41[3]) → LOAD292(i41[3] + -1)

(0) -> (1), if ((i41[0] →^* i41[1])∧(i41[0] > 0 && 0 = i41[0] % 2 →^* TRUE))

(1) -> (0), if ((i41[1] / 2 →^* i41[0]))

(1) -> (2), if ((i41[1] / 2 →^* i41[2]))

(2) -> (3), if ((i41[2] % 2 > 0 && i41[2] > 0 →^* TRUE)∧(i41[2] →^* i41[3]))

(3) -> (0), if ((i41[3] + -1 →^* i41[0]))

(3) -> (2), if ((i41[3] + -1 →^* i41[2]))

The set Q consists of the following terms:
Load292(x0)
Cond_Load292(TRUE, x0)
Cond_Load2921(TRUE, x0)

(7) UsableRulesProof (EQUIVALENT transformation)

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

(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:

Boolean, Integer

R is empty.

The integer pair graph contains the following rules and edges:
(0): LOAD292(i41[0]) → COND_LOAD292(i41[0] > 0 && 0 = i41[0] % 2, i41[0])
(1): COND_LOAD292(TRUE, i41[1]) → LOAD292(i41[1] / 2)
(2): LOAD292(i41[2]) → COND_LOAD2921(i41[2] % 2 > 0 && i41[2] > 0, i41[2])
(3): COND_LOAD2921(TRUE, i41[3]) → LOAD292(i41[3] + -1)

(0) -> (1), if ((i41[0] →^* i41[1])∧(i41[0] > 0 && 0 = i41[0] % 2 →^* TRUE))

(1) -> (0), if ((i41[1] / 2 →^* i41[0]))

(1) -> (2), if ((i41[1] / 2 →^* i41[2]))

(2) -> (3), if ((i41[2] % 2 > 0 && i41[2] > 0 →^* TRUE)∧(i41[2] →^* i41[3]))

(3) -> (0), if ((i41[3] + -1 →^* i41[0]))

(3) -> (2), if ((i41[3] + -1 →^* i41[2]))

The set Q consists of the following terms:
Load292(x0)
Cond_Load292(TRUE, x0)
Cond_Load2921(TRUE, x0)

(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 LOAD292(i41) → COND_LOAD292(&&(>(i41, 0), =(0, %(i41, 2))), i41) the following chains were created:

We consider the chain LOAD292(i41[0]) → COND_LOAD292(&&(>(i41[0], 0), =(0, %(i41[0], 2))), i41[0]), COND_LOAD292(TRUE, i41[1]) → LOAD292(/(i41[1], 2)) which results in the following constraint:
(1)    (i41[0]=i41[1]∧&&(>(i41[0], 0), =(0, %(i41[0], 2)))=TRUE ⇒ LOAD292(i41[0])≥NonInfC∧LOAD292(i41[0])≥COND_LOAD292(&&(>(i41[0], 0), =(0, %(i41[0], 2))), i41[0])∧(U^Increasing(COND_LOAD292(&&(>(i41[0], 0), =(0, %(i41[0], 2))), i41[0])), ≥))

We simplified constraint (1) using rules (IV), (IDP_BOOLEAN) which results in the following new constraint:
(2)    (>(i41[0], 0)=TRUE∧>=(0, %(i41[0], 2))=TRUE∧<=(0, %(i41[0], 2))=TRUE ⇒ LOAD292(i41[0])≥NonInfC∧LOAD292(i41[0])≥COND_LOAD292(&&(>(i41[0], 0), =(0, %(i41[0], 2))), i41[0])∧(U^Increasing(COND_LOAD292(&&(>(i41[0], 0), =(0, %(i41[0], 2))), i41[0])), ≥))

We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(3)    (i41[0] + [-1] ≥ 0∧[-1]min{[2], [-2]} ≥ 0∧max{[2], [-2]} ≥ 0 ⇒ (U^Increasing(COND_LOAD292(&&(>(i41[0], 0), =(0, %(i41[0], 2))), i41[0])), ≥)∧[(-1)bni_14 + (-1)Bound*bni_14] + [bni_14]i41[0] ≥ 0∧[(-1)bso_15] ≥ 0)

We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(4)    (i41[0] + [-1] ≥ 0∧[-1]min{[2], [-2]} ≥ 0∧max{[2], [-2]} ≥ 0 ⇒ (U^Increasing(COND_LOAD292(&&(>(i41[0], 0), =(0, %(i41[0], 2))), i41[0])), ≥)∧[(-1)bni_14 + (-1)Bound*bni_14] + [bni_14]i41[0] ≥ 0∧[(-1)bso_15] ≥ 0)

We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(5)    (i41[0] + [-1] ≥ 0∧[4] ≥ 0∧[2] ≥ 0∧[2] ≥ 0 ⇒ (U^Increasing(COND_LOAD292(&&(>(i41[0], 0), =(0, %(i41[0], 2))), i41[0])), ≥)∧[(-1)bni_14 + (-1)Bound*bni_14] + [bni_14]i41[0] ≥ 0∧[(-1)bso_15] ≥ 0)

We simplified constraint (5) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(6)    (i41[0] ≥ 0∧[4] ≥ 0∧[2] ≥ 0∧[2] ≥ 0 ⇒ (U^Increasing(COND_LOAD292(&&(>(i41[0], 0), =(0, %(i41[0], 2))), i41[0])), ≥)∧[(-1)Bound*bni_14] + [bni_14]i41[0] ≥ 0∧[(-1)bso_15] ≥ 0)

We simplified constraint (6) using rule (IDP_POLY_GCD) which results in the following new constraint:
(7)    (i41[0] ≥ 0∧[1] ≥ 0∧[1] ≥ 0∧[1] ≥ 0 ⇒ (U^Increasing(COND_LOAD292(&&(>(i41[0], 0), =(0, %(i41[0], 2))), i41[0])), ≥)∧[(-1)Bound*bni_14] + [bni_14]i41[0] ≥ 0∧[(-1)bso_15] ≥ 0)

For Pair COND_LOAD292(TRUE, i41) → LOAD292(/(i41, 2)) the following chains were created:

We consider the chain LOAD292(i41[0]) → COND_LOAD292(&&(>(i41[0], 0), =(0, %(i41[0], 2))), i41[0]), COND_LOAD292(TRUE, i41[1]) → LOAD292(/(i41[1], 2)) which results in the following constraint:
(8)    (i41[0]=i41[1]∧&&(>(i41[0], 0), =(0, %(i41[0], 2)))=TRUE ⇒ COND_LOAD292(TRUE, i41[1])≥NonInfC∧COND_LOAD292(TRUE, i41[1])≥LOAD292(/(i41[1], 2))∧(U^Increasing(LOAD292(/(i41[1], 2))), ≥))

We simplified constraint (8) using rules (III), (IDP_BOOLEAN) which results in the following new constraint:
(9)    (>(i41[0], 0)=TRUE∧>=(0, %(i41[0], 2))=TRUE∧<=(0, %(i41[0], 2))=TRUE ⇒ COND_LOAD292(TRUE, i41[0])≥NonInfC∧COND_LOAD292(TRUE, i41[0])≥LOAD292(/(i41[0], 2))∧(U^Increasing(LOAD292(/(i41[1], 2))), ≥))

We simplified constraint (9) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(10)    (i41[0] + [-1] ≥ 0∧[-1]min{[2], [-2]} ≥ 0∧max{[2], [-2]} ≥ 0 ⇒ (U^Increasing(LOAD292(/(i41[1], 2))), ≥)∧[(-1)bni_16 + (-1)Bound*bni_16] + [bni_16]i41[0] ≥ 0∧[1 + (-1)bso_20] + i41[0] + [-1]max{i41[0], [-1]i41[0]} ≥ 0)

We simplified constraint (10) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(11)    (i41[0] + [-1] ≥ 0∧[-1]min{[2], [-2]} ≥ 0∧max{[2], [-2]} ≥ 0 ⇒ (U^Increasing(LOAD292(/(i41[1], 2))), ≥)∧[(-1)bni_16 + (-1)Bound*bni_16] + [bni_16]i41[0] ≥ 0∧[1 + (-1)bso_20] + i41[0] + [-1]max{i41[0], [-1]i41[0]} ≥ 0)

We simplified constraint (11) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(12)    (i41[0] + [-1] ≥ 0∧[4] ≥ 0∧[2] ≥ 0∧[2] ≥ 0∧[2]i41[0] ≥ 0 ⇒ (U^Increasing(LOAD292(/(i41[1], 2))), ≥)∧[(-1)bni_16 + (-1)Bound*bni_16] + [bni_16]i41[0] ≥ 0∧[1 + (-1)bso_20] ≥ 0)

We simplified constraint (12) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(13)    (i41[0] ≥ 0∧[4] ≥ 0∧[2] ≥ 0∧[2] ≥ 0∧[2] + [2]i41[0] ≥ 0 ⇒ (U^Increasing(LOAD292(/(i41[1], 2))), ≥)∧[(-1)Bound*bni_16] + [bni_16]i41[0] ≥ 0∧[1 + (-1)bso_20] ≥ 0)

We simplified constraint (13) using rule (IDP_POLY_GCD) which results in the following new constraint:
(14)    (i41[0] ≥ 0∧[1] ≥ 0∧[1] ≥ 0∧[1] ≥ 0∧[1] + i41[0] ≥ 0 ⇒ (U^Increasing(LOAD292(/(i41[1], 2))), ≥)∧[(-1)Bound*bni_16] + [bni_16]i41[0] ≥ 0∧[1 + (-1)bso_20] ≥ 0)

For Pair LOAD292(i41) → COND_LOAD2921(&&(>(%(i41, 2), 0), >(i41, 0)), i41) the following chains were created:

We consider the chain LOAD292(i41[2]) → COND_LOAD2921(&&(>(%(i41[2], 2), 0), >(i41[2], 0)), i41[2]), COND_LOAD2921(TRUE, i41[3]) → LOAD292(+(i41[3], -1)) which results in the following constraint:
(15)    (&&(>(%(i41[2], 2), 0), >(i41[2], 0))=TRUE∧i41[2]=i41[3] ⇒ LOAD292(i41[2])≥NonInfC∧LOAD292(i41[2])≥COND_LOAD2921(&&(>(%(i41[2], 2), 0), >(i41[2], 0)), i41[2])∧(U^Increasing(COND_LOAD2921(&&(>(%(i41[2], 2), 0), >(i41[2], 0)), i41[2])), ≥))

We simplified constraint (15) using rules (IV), (IDP_BOOLEAN) which results in the following new constraint:
(16)    (>(%(i41[2], 2), 0)=TRUE∧>(i41[2], 0)=TRUE ⇒ LOAD292(i41[2])≥NonInfC∧LOAD292(i41[2])≥COND_LOAD2921(&&(>(%(i41[2], 2), 0), >(i41[2], 0)), i41[2])∧(U^Increasing(COND_LOAD2921(&&(>(%(i41[2], 2), 0), >(i41[2], 0)), i41[2])), ≥))

We simplified constraint (16) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(17)    (max{[2], [-2]} + [-1] ≥ 0∧i41[2] + [-1] ≥ 0 ⇒ (U^Increasing(COND_LOAD2921(&&(>(%(i41[2], 2), 0), >(i41[2], 0)), i41[2])), ≥)∧[(-1)bni_21 + (-1)Bound*bni_21] + [bni_21]i41[2] ≥ 0∧[(-1)bso_22] ≥ 0)

We simplified constraint (17) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(18)    (max{[2], [-2]} + [-1] ≥ 0∧i41[2] + [-1] ≥ 0 ⇒ (U^Increasing(COND_LOAD2921(&&(>(%(i41[2], 2), 0), >(i41[2], 0)), i41[2])), ≥)∧[(-1)bni_21 + (-1)Bound*bni_21] + [bni_21]i41[2] ≥ 0∧[(-1)bso_22] ≥ 0)

We simplified constraint (18) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(19)    (i41[2] + [-1] ≥ 0∧[4] ≥ 0∧[1] ≥ 0 ⇒ (U^Increasing(COND_LOAD2921(&&(>(%(i41[2], 2), 0), >(i41[2], 0)), i41[2])), ≥)∧[(-1)bni_21 + (-1)Bound*bni_21] + [bni_21]i41[2] ≥ 0∧[(-1)bso_22] ≥ 0)

We simplified constraint (19) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(20)    (i41[2] ≥ 0∧[4] ≥ 0∧[1] ≥ 0 ⇒ (U^Increasing(COND_LOAD2921(&&(>(%(i41[2], 2), 0), >(i41[2], 0)), i41[2])), ≥)∧[(-1)Bound*bni_21] + [bni_21]i41[2] ≥ 0∧[(-1)bso_22] ≥ 0)

We simplified constraint (20) using rule (IDP_POLY_GCD) which results in the following new constraint:
(21)    (i41[2] ≥ 0∧[1] ≥ 0∧[1] ≥ 0 ⇒ (U^Increasing(COND_LOAD2921(&&(>(%(i41[2], 2), 0), >(i41[2], 0)), i41[2])), ≥)∧[(-1)Bound*bni_21] + [bni_21]i41[2] ≥ 0∧[(-1)bso_22] ≥ 0)

For Pair COND_LOAD2921(TRUE, i41) → LOAD292(+(i41, -1)) the following chains were created:

We consider the chain LOAD292(i41[2]) → COND_LOAD2921(&&(>(%(i41[2], 2), 0), >(i41[2], 0)), i41[2]), COND_LOAD2921(TRUE, i41[3]) → LOAD292(+(i41[3], -1)), LOAD292(i41[0]) → COND_LOAD292(&&(>(i41[0], 0), =(0, %(i41[0], 2))), i41[0]) which results in the following constraint:
(22)    (&&(>(%(i41[2], 2), 0), >(i41[2], 0))=TRUE∧i41[2]=i41[3]∧+(i41[3], -1)=i41[0] ⇒ COND_LOAD2921(TRUE, i41[3])≥NonInfC∧COND_LOAD2921(TRUE, i41[3])≥LOAD292(+(i41[3], -1))∧(U^Increasing(LOAD292(+(i41[3], -1))), ≥))

We simplified constraint (22) using rules (III), (IV), (IDP_BOOLEAN) which results in the following new constraint:
(23)    (>(%(i41[2], 2), 0)=TRUE∧>(i41[2], 0)=TRUE ⇒ COND_LOAD2921(TRUE, i41[2])≥NonInfC∧COND_LOAD2921(TRUE, i41[2])≥LOAD292(+(i41[2], -1))∧(U^Increasing(LOAD292(+(i41[3], -1))), ≥))

We simplified constraint (23) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(24)    (max{[2], [-2]} + [-1] ≥ 0∧i41[2] + [-1] ≥ 0 ⇒ (U^Increasing(LOAD292(+(i41[3], -1))), ≥)∧[(-1)bni_23 + (-1)Bound*bni_23] + [bni_23]i41[2] ≥ 0∧[1 + (-1)bso_24] ≥ 0)

We simplified constraint (24) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(25)    (max{[2], [-2]} + [-1] ≥ 0∧i41[2] + [-1] ≥ 0 ⇒ (U^Increasing(LOAD292(+(i41[3], -1))), ≥)∧[(-1)bni_23 + (-1)Bound*bni_23] + [bni_23]i41[2] ≥ 0∧[1 + (-1)bso_24] ≥ 0)

We simplified constraint (25) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(26)    (i41[2] + [-1] ≥ 0∧[4] ≥ 0∧[1] ≥ 0 ⇒ (U^Increasing(LOAD292(+(i41[3], -1))), ≥)∧[(-1)bni_23 + (-1)Bound*bni_23] + [bni_23]i41[2] ≥ 0∧[1 + (-1)bso_24] ≥ 0)

We simplified constraint (26) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(27)    (i41[2] ≥ 0∧[4] ≥ 0∧[1] ≥ 0 ⇒ (U^Increasing(LOAD292(+(i41[3], -1))), ≥)∧[(-1)Bound*bni_23] + [bni_23]i41[2] ≥ 0∧[1 + (-1)bso_24] ≥ 0)

We simplified constraint (27) using rule (IDP_POLY_GCD) which results in the following new constraint:
(28)    (i41[2] ≥ 0∧[1] ≥ 0∧[1] ≥ 0 ⇒ (U^Increasing(LOAD292(+(i41[3], -1))), ≥)∧[(-1)Bound*bni_23] + [bni_23]i41[2] ≥ 0∧[1 + (-1)bso_24] ≥ 0)
We consider the chain LOAD292(i41[2]) → COND_LOAD2921(&&(>(%(i41[2], 2), 0), >(i41[2], 0)), i41[2]), COND_LOAD2921(TRUE, i41[3]) → LOAD292(+(i41[3], -1)), LOAD292(i41[2]) → COND_LOAD2921(&&(>(%(i41[2], 2), 0), >(i41[2], 0)), i41[2]) which results in the following constraint:
(29)    (&&(>(%(i41[2], 2), 0), >(i41[2], 0))=TRUE∧i41[2]=i41[3]∧+(i41[3], -1)=i41[2]1 ⇒ COND_LOAD2921(TRUE, i41[3])≥NonInfC∧COND_LOAD2921(TRUE, i41[3])≥LOAD292(+(i41[3], -1))∧(U^Increasing(LOAD292(+(i41[3], -1))), ≥))

We simplified constraint (29) using rules (III), (IV), (IDP_BOOLEAN) which results in the following new constraint:
(30)    (>(%(i41[2], 2), 0)=TRUE∧>(i41[2], 0)=TRUE ⇒ COND_LOAD2921(TRUE, i41[2])≥NonInfC∧COND_LOAD2921(TRUE, i41[2])≥LOAD292(+(i41[2], -1))∧(U^Increasing(LOAD292(+(i41[3], -1))), ≥))

We simplified constraint (30) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(31)    (max{[2], [-2]} + [-1] ≥ 0∧i41[2] + [-1] ≥ 0 ⇒ (U^Increasing(LOAD292(+(i41[3], -1))), ≥)∧[(-1)bni_23 + (-1)Bound*bni_23] + [bni_23]i41[2] ≥ 0∧[1 + (-1)bso_24] ≥ 0)

We simplified constraint (31) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(32)    (max{[2], [-2]} + [-1] ≥ 0∧i41[2] + [-1] ≥ 0 ⇒ (U^Increasing(LOAD292(+(i41[3], -1))), ≥)∧[(-1)bni_23 + (-1)Bound*bni_23] + [bni_23]i41[2] ≥ 0∧[1 + (-1)bso_24] ≥ 0)

We simplified constraint (32) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(33)    (i41[2] + [-1] ≥ 0∧[4] ≥ 0∧[1] ≥ 0 ⇒ (U^Increasing(LOAD292(+(i41[3], -1))), ≥)∧[(-1)bni_23 + (-1)Bound*bni_23] + [bni_23]i41[2] ≥ 0∧[1 + (-1)bso_24] ≥ 0)

We simplified constraint (33) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(34)    (i41[2] ≥ 0∧[4] ≥ 0∧[1] ≥ 0 ⇒ (U^Increasing(LOAD292(+(i41[3], -1))), ≥)∧[(-1)Bound*bni_23] + [bni_23]i41[2] ≥ 0∧[1 + (-1)bso_24] ≥ 0)

We simplified constraint (34) using rule (IDP_POLY_GCD) which results in the following new constraint:
(35)    (i41[2] ≥ 0∧[1] ≥ 0∧[1] ≥ 0 ⇒ (U^Increasing(LOAD292(+(i41[3], -1))), ≥)∧[(-1)Bound*bni_23] + [bni_23]i41[2] ≥ 0∧[1 + (-1)bso_24] ≥ 0)

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

LOAD292(i41) → COND_LOAD292(&&(>(i41, 0), =(0, %(i41, 2))), i41)
- (i41[0] ≥ 0∧[1] ≥ 0∧[1] ≥ 0∧[1] ≥ 0 ⇒ (U^Increasing(COND_LOAD292(&&(>(i41[0], 0), =(0, %(i41[0], 2))), i41[0])), ≥)∧[(-1)Bound*bni_14] + [bni_14]i41[0] ≥ 0∧[(-1)bso_15] ≥ 0)
COND_LOAD292(TRUE, i41) → LOAD292(/(i41, 2))
- (i41[0] ≥ 0∧[1] ≥ 0∧[1] ≥ 0∧[1] ≥ 0∧[1] + i41[0] ≥ 0 ⇒ (U^Increasing(LOAD292(/(i41[1], 2))), ≥)∧[(-1)Bound*bni_16] + [bni_16]i41[0] ≥ 0∧[1 + (-1)bso_20] ≥ 0)
LOAD292(i41) → COND_LOAD2921(&&(>(%(i41, 2), 0), >(i41, 0)), i41)
- (i41[2] ≥ 0∧[1] ≥ 0∧[1] ≥ 0 ⇒ (U^Increasing(COND_LOAD2921(&&(>(%(i41[2], 2), 0), >(i41[2], 0)), i41[2])), ≥)∧[(-1)Bound*bni_21] + [bni_21]i41[2] ≥ 0∧[(-1)bso_22] ≥ 0)
COND_LOAD2921(TRUE, i41) → LOAD292(+(i41, -1))
- (i41[2] ≥ 0∧[1] ≥ 0∧[1] ≥ 0 ⇒ (U^Increasing(LOAD292(+(i41[3], -1))), ≥)∧[(-1)Bound*bni_23] + [bni_23]i41[2] ≥ 0∧[1 + (-1)bso_24] ≥ 0)
- (i41[2] ≥ 0∧[1] ≥ 0∧[1] ≥ 0 ⇒ (U^Increasing(LOAD292(+(i41[3], -1))), ≥)∧[(-1)Bound*bni_23] + [bni_23]i41[2] ≥ 0∧[1 + (-1)bso_24] ≥ 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(LOAD292(x₁)) = [-1] + x₁
POL(COND_LOAD292(x₁, x₂)) = [-1] + x₂
POL(&&(x₁, x₂)) = [-1]
POL(>(x₁, x₂)) = [-1]
POL(0) = 0
POL(=(x₁, x₂)) = [-1]
POL(2) = [2]
POL(COND_LOAD2921(x₁, x₂)) = [-1] + x₂
POL(+(x₁, x₂)) = x₁ + x₂
POL(-1) = [-1]

Polynomial Interpretations with Context Sensitive Arithemetic Replacement
POL(Term^CSAR-Mode @ Context)

POL(%(x₁, 2)^-1 @ {}) = min{x₂, [-1]x₂}
POL(%(x₁, 2)¹ @ {}) = max{x₂, [-1]x₂}
POL(/(x₁, 2)¹ @ {LOAD292_1/0}) = max{x₁, [-1]x₁} + [-1]

The following pairs are in P_>:

COND_LOAD292(TRUE, i41[1]) → LOAD292(/(i41[1], 2))
COND_LOAD2921(TRUE, i41[3]) → LOAD292(+(i41[3], -1))

The following pairs are in P_bound:

LOAD292(i41[0]) → COND_LOAD292(&&(>(i41[0], 0), =(0, %(i41[0], 2))), i41[0])
COND_LOAD292(TRUE, i41[1]) → LOAD292(/(i41[1], 2))
LOAD292(i41[2]) → COND_LOAD2921(&&(>(%(i41[2], 2), 0), >(i41[2], 0)), i41[2])
COND_LOAD2921(TRUE, i41[3]) → LOAD292(+(i41[3], -1))

The following pairs are in P_≥:

LOAD292(i41[0]) → COND_LOAD292(&&(>(i41[0], 0), =(0, %(i41[0], 2))), i41[0])
LOAD292(i41[2]) → COND_LOAD2921(&&(>(%(i41[2], 2), 0), >(i41[2], 0)), i41[2])

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

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

(10) Complex Obligation (AND)

(11) Obligation:

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

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

The following domains are used:

Boolean, Integer

R is empty.

The integer pair graph contains the following rules and edges:
(0): LOAD292(i41[0]) → COND_LOAD292(i41[0] > 0 && 0 = i41[0] % 2, i41[0])
(2): LOAD292(i41[2]) → COND_LOAD2921(i41[2] % 2 > 0 && i41[2] > 0, i41[2])

The set Q consists of the following terms:
Load292(x0)
Cond_Load292(TRUE, x0)
Cond_Load2921(TRUE, x0)

(12) IDependencyGraphProof (EQUIVALENT transformation)

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

(13) TRUE

(14) 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:
none

R is empty.

The integer pair graph is empty.

The set Q consists of the following terms:
Load292(x0)
Cond_Load292(TRUE, x0)
Cond_Load2921(TRUE, x0)

(15) IDependencyGraphProof (EQUIVALENT transformation)

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

(0) Obligation:

(1) JBC2FIG (SOUND transformation)

(2) Obligation:

(3) FIGtoITRSProof (SOUND transformation)

(4) Obligation:

(5) ITRStoIDPProof (EQUIVALENT transformation)

(6) Obligation:

(7) UsableRulesProof (EQUIVALENT transformation)

(8) Obligation:

(9) IDPNonInfProof (SOUND transformation)

(10) Complex Obligation (AND)

(11) Obligation:

(12) IDependencyGraphProof (EQUIVALENT transformation)

(13) TRUE

(14) Obligation:

(15) IDependencyGraphProof (EQUIVALENT transformation)

(16) TRUE