### (0) Obligation:

JBC Problem based on JBC Program:
Manifest-Version: 1.0 Created-By: 1.6.0_16 (Sun Microsystems Inc.) Main-Class: LogIterative
`public class LogIterative {  public static int log(int x, int y) {    int res = 0;    while (x >= y && y > 1) {      res++;      x = x/y;    }    return res;  }   public static void main(String[] args) {    Random.args = args;    int x = Random.random();    int y = Random.random();    log(x, y);  }}public class Random {  static String[] args;  static int index = 0;  public static int random() {    String string = args[index];    index++;    return string.length();  }}`

### (1) JBC2FIG (SOUND transformation)

Constructed FIGraph.

### (2) Obligation:

FIGraph based on JBC Program:
Graph of 206 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:
Load1335(i349, i315, i349, i317) → Cond_Load1335(i349 > 1 && i315 >= i349 && i317 + 1 > 0, i349, i315, i349, i317)
Cond_Load1335(TRUE, i349, i315, i349, i317) → Load1335(i349, i315 / i349, i349, i317 + 1)
The set Q consists of the following terms:

### (5) DuplicateArgsRemoverProof (EQUIVALENT transformation)

Some arguments are removed because they only appear as duplicates.
We removed arguments according to the following replacements:

### (6) 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:
Load1335(i315, i349, i317) → Cond_Load1335(i349 > 1 && i315 >= i349 && i317 + 1 > 0, i315, i349, i317)
Cond_Load1335(TRUE, i315, i349, i317) → Load1335(i315 / i349, i349, i317 + 1)
The set Q consists of the following terms:

### (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

The ITRS R consists of the following rules:
Load1335(i315, i349, i317) → Cond_Load1335(i349 > 1 && i315 >= i349 && i317 + 1 > 0, i315, i349, i317)
Cond_Load1335(TRUE, i315, i349, i317) → Load1335(i315 / i349, i349, i317 + 1)

The integer pair graph contains the following rules and edges:
(0): LOAD1335(i315[0], i349[0], i317[0]) → COND_LOAD1335(i349[0] > 1 && i315[0] >= i349[0] && i317[0] + 1 > 0, i315[0], i349[0], i317[0])
(1): COND_LOAD1335(TRUE, i315[1], i349[1], i317[1]) → LOAD1335(i315[1] / i349[1], i349[1], i317[1] + 1)

(0) -> (1), if ((i315[0]* i315[1])∧(i349[0] > 1 && i315[0] >= i349[0] && i317[0] + 1 > 0* TRUE)∧(i317[0]* i317[1])∧(i349[0]* i349[1]))

(1) -> (0), if ((i315[1] / i349[1]* i315[0])∧(i317[1] + 1* i317[0])∧(i349[1]* i349[0]))

The set Q consists of the following terms:

### (9) 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.

### (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): LOAD1335(i315[0], i349[0], i317[0]) → COND_LOAD1335(i349[0] > 1 && i315[0] >= i349[0] && i317[0] + 1 > 0, i315[0], i349[0], i317[0])
(1): COND_LOAD1335(TRUE, i315[1], i349[1], i317[1]) → LOAD1335(i315[1] / i349[1], i349[1], i317[1] + 1)

(0) -> (1), if ((i315[0]* i315[1])∧(i349[0] > 1 && i315[0] >= i349[0] && i317[0] + 1 > 0* TRUE)∧(i317[0]* i317[1])∧(i349[0]* i349[1]))

(1) -> (0), if ((i315[1] / i349[1]* i315[0])∧(i317[1] + 1* i317[0])∧(i349[1]* i349[0]))

The set Q consists of the following terms:

### (11) 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 LOAD1335(i315, i349, i317) → COND_LOAD1335(&&(&&(>(i349, 1), >=(i315, i349)), >(+(i317, 1), 0)), i315, i349, i317) the following chains were created:
• We consider the chain LOAD1335(i315[0], i349[0], i317[0]) → COND_LOAD1335(&&(&&(>(i349[0], 1), >=(i315[0], i349[0])), >(+(i317[0], 1), 0)), i315[0], i349[0], i317[0]), COND_LOAD1335(TRUE, i315[1], i349[1], i317[1]) → LOAD1335(/(i315[1], i349[1]), i349[1], +(i317[1], 1)) which results in the following constraint:

(1)    (i315[0]=i315[1]&&(&&(>(i349[0], 1), >=(i315[0], i349[0])), >(+(i317[0], 1), 0))=TRUEi317[0]=i317[1]i349[0]=i349[1]LOAD1335(i315[0], i349[0], i317[0])≥NonInfC∧LOAD1335(i315[0], i349[0], i317[0])≥COND_LOAD1335(&&(&&(>(i349[0], 1), >=(i315[0], i349[0])), >(+(i317[0], 1), 0)), i315[0], i349[0], i317[0])∧(UIncreasing(COND_LOAD1335(&&(&&(>(i349[0], 1), >=(i315[0], i349[0])), >(+(i317[0], 1), 0)), i315[0], i349[0], i317[0])), ≥))

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

(2)    (>(+(i317[0], 1), 0)=TRUE>(i349[0], 1)=TRUE>=(i315[0], i349[0])=TRUELOAD1335(i315[0], i349[0], i317[0])≥NonInfC∧LOAD1335(i315[0], i349[0], i317[0])≥COND_LOAD1335(&&(&&(>(i349[0], 1), >=(i315[0], i349[0])), >(+(i317[0], 1), 0)), i315[0], i349[0], i317[0])∧(UIncreasing(COND_LOAD1335(&&(&&(>(i349[0], 1), >=(i315[0], i349[0])), >(+(i317[0], 1), 0)), i315[0], i349[0], i317[0])), ≥))

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

(3)    (i317[0] ≥ 0∧i349[0] + [-2] ≥ 0∧i315[0] + [-1]i349[0] ≥ 0 ⇒ (UIncreasing(COND_LOAD1335(&&(&&(>(i349[0], 1), >=(i315[0], i349[0])), >(+(i317[0], 1), 0)), i315[0], i349[0], i317[0])), ≥)∧[(-1)bni_15 + (-1)Bound*bni_15] + [bni_15]i315[0] ≥ 0∧[(-1)bso_16] ≥ 0)

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

(4)    (i317[0] ≥ 0∧i349[0] + [-2] ≥ 0∧i315[0] + [-1]i349[0] ≥ 0 ⇒ (UIncreasing(COND_LOAD1335(&&(&&(>(i349[0], 1), >=(i315[0], i349[0])), >(+(i317[0], 1), 0)), i315[0], i349[0], i317[0])), ≥)∧[(-1)bni_15 + (-1)Bound*bni_15] + [bni_15]i315[0] ≥ 0∧[(-1)bso_16] ≥ 0)

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

(5)    (i317[0] ≥ 0∧i349[0] + [-2] ≥ 0∧i315[0] + [-1]i349[0] ≥ 0 ⇒ (UIncreasing(COND_LOAD1335(&&(&&(>(i349[0], 1), >=(i315[0], i349[0])), >(+(i317[0], 1), 0)), i315[0], i349[0], i317[0])), ≥)∧[(-1)bni_15 + (-1)Bound*bni_15] + [bni_15]i315[0] ≥ 0∧[(-1)bso_16] ≥ 0)

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

(6)    (i317[0] ≥ 0∧i349[0] ≥ 0∧i315[0] + [-2] + [-1]i349[0] ≥ 0 ⇒ (UIncreasing(COND_LOAD1335(&&(&&(>(i349[0], 1), >=(i315[0], i349[0])), >(+(i317[0], 1), 0)), i315[0], i349[0], i317[0])), ≥)∧[(-1)bni_15 + (-1)Bound*bni_15] + [bni_15]i315[0] ≥ 0∧[(-1)bso_16] ≥ 0)

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

(7)    (i317[0] ≥ 0∧i349[0] ≥ 0∧i315[0] ≥ 0 ⇒ (UIncreasing(COND_LOAD1335(&&(&&(>(i349[0], 1), >=(i315[0], i349[0])), >(+(i317[0], 1), 0)), i315[0], i349[0], i317[0])), ≥)∧[bni_15 + (-1)Bound*bni_15] + [bni_15]i349[0] + [bni_15]i315[0] ≥ 0∧[(-1)bso_16] ≥ 0)

For Pair COND_LOAD1335(TRUE, i315, i349, i317) → LOAD1335(/(i315, i349), i349, +(i317, 1)) the following chains were created:
• We consider the chain LOAD1335(i315[0], i349[0], i317[0]) → COND_LOAD1335(&&(&&(>(i349[0], 1), >=(i315[0], i349[0])), >(+(i317[0], 1), 0)), i315[0], i349[0], i317[0]), COND_LOAD1335(TRUE, i315[1], i349[1], i317[1]) → LOAD1335(/(i315[1], i349[1]), i349[1], +(i317[1], 1)), LOAD1335(i315[0], i349[0], i317[0]) → COND_LOAD1335(&&(&&(>(i349[0], 1), >=(i315[0], i349[0])), >(+(i317[0], 1), 0)), i315[0], i349[0], i317[0]) which results in the following constraint:

(8)    (i315[0]=i315[1]&&(&&(>(i349[0], 1), >=(i315[0], i349[0])), >(+(i317[0], 1), 0))=TRUEi317[0]=i317[1]i349[0]=i349[1]/(i315[1], i349[1])=i315[0]1+(i317[1], 1)=i317[0]1i349[1]=i349[0]1COND_LOAD1335(TRUE, i315[1], i349[1], i317[1])≥NonInfC∧COND_LOAD1335(TRUE, i315[1], i349[1], i317[1])≥LOAD1335(/(i315[1], i349[1]), i349[1], +(i317[1], 1))∧(UIncreasing(LOAD1335(/(i315[1], i349[1]), i349[1], +(i317[1], 1))), ≥))

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

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

(10)    (i317[0] ≥ 0∧i349[0] + [-2] ≥ 0∧i315[0] + [-1]i349[0] ≥ 0 ⇒ (UIncreasing(LOAD1335(/(i315[1], i349[1]), i349[1], +(i317[1], 1))), ≥)∧[(-1)bni_17 + (-1)Bound*bni_17] + [bni_17]i315[0] ≥ 0∧[(-1)bso_21] + i315[0] + [-1]max{i315[0], [-1]i315[0]} + min{max{i349[0], [-1]i349[0]} + [-1], max{i315[0], [-1]i315[0]}} ≥ 0)

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

(11)    (i317[0] ≥ 0∧i349[0] + [-2] ≥ 0∧i315[0] + [-1]i349[0] ≥ 0 ⇒ (UIncreasing(LOAD1335(/(i315[1], i349[1]), i349[1], +(i317[1], 1))), ≥)∧[(-1)bni_17 + (-1)Bound*bni_17] + [bni_17]i315[0] ≥ 0∧[(-1)bso_21] + i315[0] + [-1]max{i315[0], [-1]i315[0]} + min{max{i349[0], [-1]i349[0]} + [-1], max{i315[0], [-1]i315[0]}} ≥ 0)

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

(12)    (i317[0] ≥ 0∧i349[0] + [-2] ≥ 0∧i315[0] + [-1]i349[0] ≥ 0∧[2]i315[0] ≥ 0∧[2]i349[0] ≥ 0 ⇒ (UIncreasing(LOAD1335(/(i315[1], i349[1]), i349[1], +(i317[1], 1))), ≥)∧[(-1)bni_17 + (-1)Bound*bni_17] + [bni_17]i315[0] ≥ 0∧[-1 + (-1)bso_21] + i349[0] ≥ 0)

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

(13)    (i317[0] ≥ 0∧i349[0] ≥ 0∧i315[0] + [-2] + [-1]i349[0] ≥ 0∧[2]i315[0] ≥ 0∧[4] + [2]i349[0] ≥ 0 ⇒ (UIncreasing(LOAD1335(/(i315[1], i349[1]), i349[1], +(i317[1], 1))), ≥)∧[(-1)bni_17 + (-1)Bound*bni_17] + [bni_17]i315[0] ≥ 0∧[1 + (-1)bso_21] + i349[0] ≥ 0)

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

(14)    (i317[0] ≥ 0∧i349[0] ≥ 0∧i315[0] ≥ 0∧[4] + [2]i349[0] + [2]i315[0] ≥ 0∧[4] + [2]i349[0] ≥ 0 ⇒ (UIncreasing(LOAD1335(/(i315[1], i349[1]), i349[1], +(i317[1], 1))), ≥)∧[bni_17 + (-1)Bound*bni_17] + [bni_17]i349[0] + [bni_17]i315[0] ≥ 0∧[1 + (-1)bso_21] + i349[0] ≥ 0)

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

(15)    (i317[0] ≥ 0∧i349[0] ≥ 0∧i315[0] ≥ 0∧[2] + i349[0] + i315[0] ≥ 0∧[2] + i349[0] ≥ 0 ⇒ (UIncreasing(LOAD1335(/(i315[1], i349[1]), i349[1], +(i317[1], 1))), ≥)∧[bni_17 + (-1)Bound*bni_17] + [bni_17]i349[0] + [bni_17]i315[0] ≥ 0∧[1 + (-1)bso_21] + i349[0] ≥ 0)

To summarize, we get the following constraints P for the following pairs.
• LOAD1335(i315, i349, i317) → COND_LOAD1335(&&(&&(>(i349, 1), >=(i315, i349)), >(+(i317, 1), 0)), i315, i349, i317)
• (i317[0] ≥ 0∧i349[0] ≥ 0∧i315[0] ≥ 0 ⇒ (UIncreasing(COND_LOAD1335(&&(&&(>(i349[0], 1), >=(i315[0], i349[0])), >(+(i317[0], 1), 0)), i315[0], i349[0], i317[0])), ≥)∧[bni_15 + (-1)Bound*bni_15] + [bni_15]i349[0] + [bni_15]i315[0] ≥ 0∧[(-1)bso_16] ≥ 0)

• (i317[0] ≥ 0∧i349[0] ≥ 0∧i315[0] ≥ 0∧[2] + i349[0] + i315[0] ≥ 0∧[2] + i349[0] ≥ 0 ⇒ (UIncreasing(LOAD1335(/(i315[1], i349[1]), i349[1], +(i317[1], 1))), ≥)∧[bni_17 + (-1)Bound*bni_17] + [bni_17]i349[0] + [bni_17]i315[0] ≥ 0∧[1 + (-1)bso_21] + i349[0] ≥ 0)

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

POL(TRUE) = 0
POL(FALSE) = 0
POL(LOAD1335(x1, x2, x3)) = [-1] + x1
POL(COND_LOAD1335(x1, x2, x3, x4)) = [-1] + x2 + [-1]x1
POL(&&(x1, x2)) = 0
POL(>(x1, x2)) = [-1]
POL(1) = [1]
POL(>=(x1, x2)) = [-1]
POL(+(x1, x2)) = x1 + x2
POL(0) = 0

Polynomial Interpretations with Context Sensitive Arithemetic Replacement
POL(TermCSAR-Mode @ Context)

POL(/(x1, i349[0])1 @ {LOAD1335_3/0}) = max{x1, [-1]x1} + [-1]min{max{x2, [-1]x2} + [-1], max{x1, [-1]x1}}

The following pairs are in P>:

The following pairs are in Pbound:

LOAD1335(i315[0], i349[0], i317[0]) → COND_LOAD1335(&&(&&(>(i349[0], 1), >=(i315[0], i349[0])), >(+(i317[0], 1), 0)), i315[0], i349[0], i317[0])

The following pairs are in P:

LOAD1335(i315[0], i349[0], i317[0]) → COND_LOAD1335(&&(&&(>(i349[0], 1), >=(i315[0], i349[0])), >(+(i317[0], 1), 0)), i315[0], i349[0], i317[0])

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
/1

### (13) Obligation:

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

The following domains are used:

Boolean, Integer

R is empty.

The integer pair graph contains the following rules and edges:
(0): LOAD1335(i315[0], i349[0], i317[0]) → COND_LOAD1335(i349[0] > 1 && i315[0] >= i349[0] && i317[0] + 1 > 0, i315[0], i349[0], i317[0])

The set Q consists of the following terms:

### (14) IDependencyGraphProof (EQUIVALENT transformation)

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

### (16) Obligation:

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

The following domains are used:
none

R is empty.

The integer pair graph is empty.

The set Q consists of the following terms: