Termination proof of /tmp/tmpxcu7wg/PlusSwap.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 IDP

↳11 IDependencyGraphProof (⇔)

↳12 TRUE

(0) Obligation:

JBC Problem based on JBC Program:
Manifest-Version: 1.0 Created-By: 1.6.0_16 (Sun Microsystems Inc.) Main-Class: PlusSwap

public class PlusSwap{
  public static void main(String[] args) {
    Random.args = args;
    int x = Random.random();
    int y = Random.random();
    int z;
    int res = 0;

    while (y > 0) {

      z = x;
      x = y-1;
      y = z;
      res++;

    }

    res = res + x;
  }
}


public class Random {
  static String[] args;
  static int index = 0;

  public static int random() {
    String string = args[index];
    index++;
    return string.length();
  }
}

(1) JBC2FIG (SOUND transformation)

Constructed FIGraph.

(2) Obligation:

FIGraph based on JBC Program:
Graph of 198 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:
Load1127(i218, i237, i220) → Cond_Load1127(i237 > 0 && i220 + 1 > 0, i218, i237, i220)
Cond_Load1127(TRUE, i218, i237, i220) → Load1127(i237 - 1, i218, i220 + 1)
The set Q consists of the following terms:
Load1127(x0, x1, x2)
Cond_Load1127(TRUE, x0, x1, x2)

(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:
Load1127(i218, i237, i220) → Cond_Load1127(i237 > 0 && i220 + 1 > 0, i218, i237, i220)
Cond_Load1127(TRUE, i218, i237, i220) → Load1127(i237 - 1, i218, i220 + 1)

The integer pair graph contains the following rules and edges:
(0): LOAD1127(i218[0], i237[0], i220[0]) → COND_LOAD1127(i237[0] > 0 && i220[0] + 1 > 0, i218[0], i237[0], i220[0])
(1): COND_LOAD1127(TRUE, i218[1], i237[1], i220[1]) → LOAD1127(i237[1] - 1, i218[1], i220[1] + 1)

(0) -> (1), if ((i237[0] > 0 && i220[0] + 1 > 0 →^* TRUE)∧(i218[0] →^* i218[1])∧(i220[0] →^* i220[1])∧(i237[0] →^* i237[1]))

(1) -> (0), if ((i220[1] + 1 →^* i220[0])∧(i237[1] - 1 →^* i218[0])∧(i218[1] →^* i237[0]))

The set Q consists of the following terms:
Load1127(x0, x1, x2)
Cond_Load1127(TRUE, x0, x1, x2)

(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): LOAD1127(i218[0], i237[0], i220[0]) → COND_LOAD1127(i237[0] > 0 && i220[0] + 1 > 0, i218[0], i237[0], i220[0])
(1): COND_LOAD1127(TRUE, i218[1], i237[1], i220[1]) → LOAD1127(i237[1] - 1, i218[1], i220[1] + 1)

(0) -> (1), if ((i237[0] > 0 && i220[0] + 1 > 0 →^* TRUE)∧(i218[0] →^* i218[1])∧(i220[0] →^* i220[1])∧(i237[0] →^* i237[1]))

(1) -> (0), if ((i220[1] + 1 →^* i220[0])∧(i237[1] - 1 →^* i218[0])∧(i218[1] →^* i237[0]))

The set Q consists of the following terms:
Load1127(x0, x1, x2)
Cond_Load1127(TRUE, 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 LOAD1127(i218, i237, i220) → COND_LOAD1127(&&(>(i237, 0), >(+(i220, 1), 0)), i218, i237, i220) the following chains were created:

We consider the chain LOAD1127(i218[0], i237[0], i220[0]) → COND_LOAD1127(&&(>(i237[0], 0), >(+(i220[0], 1), 0)), i218[0], i237[0], i220[0]), COND_LOAD1127(TRUE, i218[1], i237[1], i220[1]) → LOAD1127(-(i237[1], 1), i218[1], +(i220[1], 1)) which results in the following constraint:
(1)    (&&(>(i237[0], 0), >(+(i220[0], 1), 0))=TRUE∧i218[0]=i218[1]∧i220[0]=i220[1]∧i237[0]=i237[1] ⇒ LOAD1127(i218[0], i237[0], i220[0])≥NonInfC∧LOAD1127(i218[0], i237[0], i220[0])≥COND_LOAD1127(&&(>(i237[0], 0), >(+(i220[0], 1), 0)), i218[0], i237[0], i220[0])∧(U^Increasing(COND_LOAD1127(&&(>(i237[0], 0), >(+(i220[0], 1), 0)), i218[0], i237[0], i220[0])), ≥))

We simplified constraint (1) using rules (IV), (IDP_BOOLEAN) which results in the following new constraint:
(2)    (>(i237[0], 0)=TRUE∧>(+(i220[0], 1), 0)=TRUE ⇒ LOAD1127(i218[0], i237[0], i220[0])≥NonInfC∧LOAD1127(i218[0], i237[0], i220[0])≥COND_LOAD1127(&&(>(i237[0], 0), >(+(i220[0], 1), 0)), i218[0], i237[0], i220[0])∧(U^Increasing(COND_LOAD1127(&&(>(i237[0], 0), >(+(i220[0], 1), 0)), i218[0], i237[0], i220[0])), ≥))

We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(3)    (i237[0] + [-1] ≥ 0∧i220[0] ≥ 0 ⇒ (U^Increasing(COND_LOAD1127(&&(>(i237[0], 0), >(+(i220[0], 1), 0)), i218[0], i237[0], i220[0])), ≥)∧[(-1)bni_14 + (-1)Bound*bni_14] + [bni_14]i220[0] + [(2)bni_14]i237[0] + [(2)bni_14]i218[0] ≥ 0∧[(-1)bso_15] ≥ 0)

We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(4)    (i237[0] + [-1] ≥ 0∧i220[0] ≥ 0 ⇒ (U^Increasing(COND_LOAD1127(&&(>(i237[0], 0), >(+(i220[0], 1), 0)), i218[0], i237[0], i220[0])), ≥)∧[(-1)bni_14 + (-1)Bound*bni_14] + [bni_14]i220[0] + [(2)bni_14]i237[0] + [(2)bni_14]i218[0] ≥ 0∧[(-1)bso_15] ≥ 0)

We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(5)    (i237[0] + [-1] ≥ 0∧i220[0] ≥ 0 ⇒ (U^Increasing(COND_LOAD1127(&&(>(i237[0], 0), >(+(i220[0], 1), 0)), i218[0], i237[0], i220[0])), ≥)∧[(-1)bni_14 + (-1)Bound*bni_14] + [bni_14]i220[0] + [(2)bni_14]i237[0] + [(2)bni_14]i218[0] ≥ 0∧[(-1)bso_15] ≥ 0)

We simplified constraint (5) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint:
(6)    (i237[0] + [-1] ≥ 0∧i220[0] ≥ 0 ⇒ (U^Increasing(COND_LOAD1127(&&(>(i237[0], 0), >(+(i220[0], 1), 0)), i218[0], i237[0], i220[0])), ≥)∧[(2)bni_14] = 0∧[(-1)bni_14 + (-1)Bound*bni_14] + [bni_14]i220[0] + [(2)bni_14]i237[0] ≥ 0∧0 = 0∧[(-1)bso_15] ≥ 0)

We simplified constraint (6) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(7)    (i237[0] ≥ 0∧i220[0] ≥ 0 ⇒ (U^Increasing(COND_LOAD1127(&&(>(i237[0], 0), >(+(i220[0], 1), 0)), i218[0], i237[0], i220[0])), ≥)∧[(2)bni_14] = 0∧[bni_14 + (-1)Bound*bni_14] + [bni_14]i220[0] + [(2)bni_14]i237[0] ≥ 0∧0 = 0∧[(-1)bso_15] ≥ 0)

For Pair COND_LOAD1127(TRUE, i218, i237, i220) → LOAD1127(-(i237, 1), i218, +(i220, 1)) the following chains were created:

We consider the chain LOAD1127(i218[0], i237[0], i220[0]) → COND_LOAD1127(&&(>(i237[0], 0), >(+(i220[0], 1), 0)), i218[0], i237[0], i220[0]), COND_LOAD1127(TRUE, i218[1], i237[1], i220[1]) → LOAD1127(-(i237[1], 1), i218[1], +(i220[1], 1)), LOAD1127(i218[0], i237[0], i220[0]) → COND_LOAD1127(&&(>(i237[0], 0), >(+(i220[0], 1), 0)), i218[0], i237[0], i220[0]), COND_LOAD1127(TRUE, i218[1], i237[1], i220[1]) → LOAD1127(-(i237[1], 1), i218[1], +(i220[1], 1)), LOAD1127(i218[0], i237[0], i220[0]) → COND_LOAD1127(&&(>(i237[0], 0), >(+(i220[0], 1), 0)), i218[0], i237[0], i220[0]), COND_LOAD1127(TRUE, i218[1], i237[1], i220[1]) → LOAD1127(-(i237[1], 1), i218[1], +(i220[1], 1)) which results in the following constraint:
(8)    (&&(>(i237[0], 0), >(+(i220[0], 1), 0))=TRUE∧i218[0]=i218[1]∧i220[0]=i220[1]∧i237[0]=i237[1]∧+(i220[1], 1)=i220[0]1∧-(i237[1], 1)=i218[0]1∧i218[1]=i237[0]1∧&&(>(i237[0]1, 0), >(+(i220[0]1, 1), 0))=TRUE∧i218[0]1=i218[1]1∧i220[0]1=i220[1]1∧i237[0]1=i237[1]1∧+(i220[1]1, 1)=i220[0]2∧-(i237[1]1, 1)=i218[0]2∧i218[1]1=i237[0]2∧&&(>(i237[0]2, 0), >(+(i220[0]2, 1), 0))=TRUE∧i218[0]2=i218[1]2∧i220[0]2=i220[1]2∧i237[0]2=i237[1]2 ⇒ COND_LOAD1127(TRUE, i218[1]1, i237[1]1, i220[1]1)≥NonInfC∧COND_LOAD1127(TRUE, i218[1]1, i237[1]1, i220[1]1)≥LOAD1127(-(i237[1]1, 1), i218[1]1, +(i220[1]1, 1))∧(U^Increasing(LOAD1127(-(i237[1]1, 1), i218[1]1, +(i220[1]1, 1))), ≥))

We simplified constraint (8) using rules (III), (IV), (IDP_BOOLEAN) which results in the following new constraint:
(9)    (>(i237[0], 0)=TRUE∧>(+(i220[0], 1), 0)=TRUE∧>(i237[0]1, 0)=TRUE∧>(+(+(i220[0], 1), 1), 0)=TRUE∧>(-(i237[0], 1), 0)=TRUE∧>(+(+(+(i220[0], 1), 1), 1), 0)=TRUE ⇒ COND_LOAD1127(TRUE, -(i237[0], 1), i237[0]1, +(i220[0], 1))≥NonInfC∧COND_LOAD1127(TRUE, -(i237[0], 1), i237[0]1, +(i220[0], 1))≥LOAD1127(-(i237[0]1, 1), -(i237[0], 1), +(+(i220[0], 1), 1))∧(U^Increasing(LOAD1127(-(i237[1]1, 1), i218[1]1, +(i220[1]1, 1))), ≥))

We simplified constraint (9) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(10)    (i237[0] + [-1] ≥ 0∧i220[0] ≥ 0∧i237[0]1 + [-1] ≥ 0∧i220[0] + [1] ≥ 0∧i237[0] + [-2] ≥ 0∧i220[0] + [2] ≥ 0 ⇒ (U^Increasing(LOAD1127(-(i237[1]1, 1), i218[1]1, +(i220[1]1, 1))), ≥)∧[(-2)bni_16 + (-1)Bound*bni_16] + [bni_16]i220[0] + [(2)bni_16]i237[0]1 + [(2)bni_16]i237[0] ≥ 0∧[1 + (-1)bso_17] ≥ 0)

We simplified constraint (10) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(11)    (i237[0] + [-1] ≥ 0∧i220[0] ≥ 0∧i237[0]1 + [-1] ≥ 0∧i220[0] + [1] ≥ 0∧i237[0] + [-2] ≥ 0∧i220[0] + [2] ≥ 0 ⇒ (U^Increasing(LOAD1127(-(i237[1]1, 1), i218[1]1, +(i220[1]1, 1))), ≥)∧[(-2)bni_16 + (-1)Bound*bni_16] + [bni_16]i220[0] + [(2)bni_16]i237[0]1 + [(2)bni_16]i237[0] ≥ 0∧[1 + (-1)bso_17] ≥ 0)

We simplified constraint (11) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(12)    (i237[0] + [-1] ≥ 0∧i220[0] ≥ 0∧i237[0]1 + [-1] ≥ 0∧i220[0] + [1] ≥ 0∧i237[0] + [-2] ≥ 0∧i220[0] + [2] ≥ 0 ⇒ (U^Increasing(LOAD1127(-(i237[1]1, 1), i218[1]1, +(i220[1]1, 1))), ≥)∧[(-2)bni_16 + (-1)Bound*bni_16] + [bni_16]i220[0] + [(2)bni_16]i237[0]1 + [(2)bni_16]i237[0] ≥ 0∧[1 + (-1)bso_17] ≥ 0)

We simplified constraint (12) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(13)    (i237[0] ≥ 0∧i220[0] ≥ 0∧i237[0]1 + [-1] ≥ 0∧i220[0] + [1] ≥ 0∧[-1] + i237[0] ≥ 0∧i220[0] + [2] ≥ 0 ⇒ (U^Increasing(LOAD1127(-(i237[1]1, 1), i218[1]1, +(i220[1]1, 1))), ≥)∧[(-1)Bound*bni_16] + [bni_16]i220[0] + [(2)bni_16]i237[0]1 + [(2)bni_16]i237[0] ≥ 0∧[1 + (-1)bso_17] ≥ 0)

We simplified constraint (13) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(14)    ([1] + i237[0] ≥ 0∧i220[0] ≥ 0∧i237[0]1 + [-1] ≥ 0∧i220[0] + [1] ≥ 0∧i237[0] ≥ 0∧i220[0] + [2] ≥ 0 ⇒ (U^Increasing(LOAD1127(-(i237[1]1, 1), i218[1]1, +(i220[1]1, 1))), ≥)∧[(2)bni_16 + (-1)Bound*bni_16] + [bni_16]i220[0] + [(2)bni_16]i237[0]1 + [(2)bni_16]i237[0] ≥ 0∧[1 + (-1)bso_17] ≥ 0)

We simplified constraint (14) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(15)    ([1] + i237[0] ≥ 0∧i220[0] ≥ 0∧i237[0]1 ≥ 0∧i220[0] + [1] ≥ 0∧i237[0] ≥ 0∧i220[0] + [2] ≥ 0 ⇒ (U^Increasing(LOAD1127(-(i237[1]1, 1), i218[1]1, +(i220[1]1, 1))), ≥)∧[(-1)Bound*bni_16 + (4)bni_16] + [bni_16]i220[0] + [(2)bni_16]i237[0]1 + [(2)bni_16]i237[0] ≥ 0∧[1 + (-1)bso_17] ≥ 0)

We simplified constraint (15) using rule (IDP_POLY_GCD) which results in the following new constraint:
(16)    ([1] + i237[0] ≥ 0∧i220[0] ≥ 0∧i237[0]1 ≥ 0∧i220[0] + [1] ≥ 0∧i237[0] ≥ 0∧[1] ≥ 0 ⇒ (U^Increasing(LOAD1127(-(i237[1]1, 1), i218[1]1, +(i220[1]1, 1))), ≥)∧[(-1)Bound*bni_16 + (4)bni_16] + [bni_16]i220[0] + [(2)bni_16]i237[0]1 + [(2)bni_16]i237[0] ≥ 0∧[1 + (-1)bso_17] ≥ 0)

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

LOAD1127(i218, i237, i220) → COND_LOAD1127(&&(>(i237, 0), >(+(i220, 1), 0)), i218, i237, i220)
- (i237[0] ≥ 0∧i220[0] ≥ 0 ⇒ (U^Increasing(COND_LOAD1127(&&(>(i237[0], 0), >(+(i220[0], 1), 0)), i218[0], i237[0], i220[0])), ≥)∧[(2)bni_14] = 0∧[bni_14 + (-1)Bound*bni_14] + [bni_14]i220[0] + [(2)bni_14]i237[0] ≥ 0∧0 = 0∧[(-1)bso_15] ≥ 0)
COND_LOAD1127(TRUE, i218, i237, i220) → LOAD1127(-(i237, 1), i218, +(i220, 1))
- ([1] + i237[0] ≥ 0∧i220[0] ≥ 0∧i237[0]1 ≥ 0∧i220[0] + [1] ≥ 0∧i237[0] ≥ 0∧[1] ≥ 0 ⇒ (U^Increasing(LOAD1127(-(i237[1]1, 1), i218[1]1, +(i220[1]1, 1))), ≥)∧[(-1)Bound*bni_16 + (4)bni_16] + [bni_16]i220[0] + [(2)bni_16]i237[0]1 + [(2)bni_16]i237[0] ≥ 0∧[1 + (-1)bso_17] ≥ 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(LOAD1127(x₁, x₂, x₃)) = [-1] + x₃ + [2]x₂ + [2]x₁
POL(COND_LOAD1127(x₁, x₂, x₃, x₄)) = [-1] + x₄ + [2]x₃ + [2]x₂ + [-1]x₁
POL(&&(x₁, x₂)) = 0
POL(>(x₁, x₂)) = [-1]
POL(0) = 0
POL(+(x₁, x₂)) = x₁ + x₂
POL(1) = [1]
POL(-(x₁, x₂)) = x₁ + [-1]x₂

The following pairs are in P_>:

COND_LOAD1127(TRUE, i218[1], i237[1], i220[1]) → LOAD1127(-(i237[1], 1), i218[1], +(i220[1], 1))

The following pairs are in P_bound:

COND_LOAD1127(TRUE, i218[1], i237[1], i220[1]) → LOAD1127(-(i237[1], 1), i218[1], +(i220[1], 1))

The following pairs are in P_≥:

LOAD1127(i218[0], i237[0], i220[0]) → COND_LOAD1127(&&(>(i237[0], 0), >(+(i220[0], 1), 0)), i218[0], i237[0], i220[0])

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

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

(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

R is empty.

The integer pair graph contains the following rules and edges:
(0): LOAD1127(i218[0], i237[0], i220[0]) → COND_LOAD1127(i237[0] > 0 && i220[0] + 1 > 0, i218[0], i237[0], i220[0])

The set Q consists of the following terms:
Load1127(x0, x1, x2)
Cond_Load1127(TRUE, x0, x1, x2)

(11) IDependencyGraphProof (EQUIVALENT transformation)

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

(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) Obligation:

(11) IDependencyGraphProof (EQUIVALENT transformation)

(12) TRUE