### (0) Obligation:

JBC Problem based on JBC Program:
Manifest-Version: 1.0 Created-By: 1.6.0_16 (Sun Microsystems Inc.) Main-Class: PastaA1
`/** * Example taken from "A Term Rewriting Approach to the Automated Termination * Analysis of Imperative Programs" (http://www.cs.unm.edu/~spf/papers/2009-02.pdf) * and converted to Java. */public class PastaA1 {    public static void main(String[] args) {        Random.args = args;        int x = Random.random();        while (x > 0) {            int y = 0;            while (y < x) {                y++;            }            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 133 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:
Load597(i20, i31) → Cond_Load597(i31 >= 0 && i31 < i20 && i31 + 1 > 0, i20, i31)
Load597(i20, i31) → Cond_Load5971(i20 > 0 && i31 >= i20, i20, i31)
The set Q consists of the following terms:

### (6) Obligation:

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

The following domains are used:

Integer, Boolean

The ITRS R consists of the following rules:
Load597(i20, i31) → Cond_Load597(i31 >= 0 && i31 < i20 && i31 + 1 > 0, i20, i31)
Load597(i20, i31) → Cond_Load5971(i20 > 0 && i31 >= i20, i20, i31)

The integer pair graph contains the following rules and edges:
(2): LOAD597(i20[2], i31[2]) → COND_LOAD597(i31[2] >= 0 && i31[2] < i20[2] && i31[2] + 1 > 0, i20[2], i31[2])
(4): LOAD597(i20[4], i31[4]) → COND_LOAD5971(i20[4] > 0 && i31[4] >= i20[4], i20[4], i31[4])

(0) -> (1), if ((i20[0] > 0* TRUE)∧(i20[0]* i20[1]))

(1) -> (2), if ((i20[1]* i20[2])∧(0* i31[2]))

(1) -> (4), if ((i20[1]* i20[4])∧(0* i31[4]))

(2) -> (3), if ((i20[2]* i20[3])∧(i31[2]* i31[3])∧(i31[2] >= 0 && i31[2] < i20[2] && i31[2] + 1 > 0* TRUE))

(3) -> (2), if ((i31[3] + 1* i31[2])∧(i20[3]* i20[2]))

(3) -> (4), if ((i20[3]* i20[4])∧(i31[3] + 1* i31[4]))

(4) -> (5), if ((i20[4] > 0 && i31[4] >= i20[4]* TRUE)∧(i20[4]* i20[5])∧(i31[4]* i31[5]))

(5) -> (0), if ((i20[5] + -1* i20[0]))

The set Q consists of the following terms:

### (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:

Integer, Boolean

R is empty.

The integer pair graph contains the following rules and edges:
(2): LOAD597(i20[2], i31[2]) → COND_LOAD597(i31[2] >= 0 && i31[2] < i20[2] && i31[2] + 1 > 0, i20[2], i31[2])
(4): LOAD597(i20[4], i31[4]) → COND_LOAD5971(i20[4] > 0 && i31[4] >= i20[4], i20[4], i31[4])

(0) -> (1), if ((i20[0] > 0* TRUE)∧(i20[0]* i20[1]))

(1) -> (2), if ((i20[1]* i20[2])∧(0* i31[2]))

(1) -> (4), if ((i20[1]* i20[4])∧(0* i31[4]))

(2) -> (3), if ((i20[2]* i20[3])∧(i31[2]* i31[3])∧(i31[2] >= 0 && i31[2] < i20[2] && i31[2] + 1 > 0* TRUE))

(3) -> (2), if ((i31[3] + 1* i31[2])∧(i20[3]* i20[2]))

(3) -> (4), if ((i20[3]* i20[4])∧(i31[3] + 1* i31[4]))

(4) -> (5), if ((i20[4] > 0 && i31[4] >= i20[4]* TRUE)∧(i20[4]* i20[5])∧(i31[4]* i31[5]))

(5) -> (0), if ((i20[5] + -1* i20[0]))

The set Q consists of the following terms:

### (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 LOAD323(i20) → COND_LOAD323(>(i20, 0), i20) the following chains were created:

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

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

(3)    (i20[0] + [-1] ≥ 0 ⇒ (UIncreasing(COND_LOAD323(>(i20[0], 0), i20[0])), ≥)∧[bni_23 + (-1)Bound*bni_23] + [bni_23]i20[0] ≥ 0∧[1 + (-1)bso_24] ≥ 0)

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

(4)    (i20[0] + [-1] ≥ 0 ⇒ (UIncreasing(COND_LOAD323(>(i20[0], 0), i20[0])), ≥)∧[bni_23 + (-1)Bound*bni_23] + [bni_23]i20[0] ≥ 0∧[1 + (-1)bso_24] ≥ 0)

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

(5)    (i20[0] + [-1] ≥ 0 ⇒ (UIncreasing(COND_LOAD323(>(i20[0], 0), i20[0])), ≥)∧[bni_23 + (-1)Bound*bni_23] + [bni_23]i20[0] ≥ 0∧[1 + (-1)bso_24] ≥ 0)

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

(6)    (i20[0] ≥ 0 ⇒ (UIncreasing(COND_LOAD323(>(i20[0], 0), i20[0])), ≥)∧[(2)bni_23 + (-1)Bound*bni_23] + [bni_23]i20[0] ≥ 0∧[1 + (-1)bso_24] ≥ 0)

For Pair COND_LOAD323(TRUE, i20) → LOAD597(i20, 0) the following chains were created:

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

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

(9)    (i20[0] + [-1] ≥ 0 ⇒ (UIncreasing(LOAD597(i20[1], 0)), ≥)∧[(-1)Bound*bni_25] + [bni_25]i20[0] ≥ 0∧[(-1)bso_26] ≥ 0)

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

(10)    (i20[0] + [-1] ≥ 0 ⇒ (UIncreasing(LOAD597(i20[1], 0)), ≥)∧[(-1)Bound*bni_25] + [bni_25]i20[0] ≥ 0∧[(-1)bso_26] ≥ 0)

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

(11)    (i20[0] + [-1] ≥ 0 ⇒ (UIncreasing(LOAD597(i20[1], 0)), ≥)∧[(-1)Bound*bni_25] + [bni_25]i20[0] ≥ 0∧[(-1)bso_26] ≥ 0)

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

(12)    (i20[0] ≥ 0 ⇒ (UIncreasing(LOAD597(i20[1], 0)), ≥)∧[(-1)Bound*bni_25 + bni_25] + [bni_25]i20[0] ≥ 0∧[(-1)bso_26] ≥ 0)

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

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

(15)    (i20[0] + [-1] ≥ 0 ⇒ (UIncreasing(LOAD597(i20[1], 0)), ≥)∧[(-1)Bound*bni_25] + [bni_25]i20[0] ≥ 0∧[(-1)bso_26] ≥ 0)

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

(16)    (i20[0] + [-1] ≥ 0 ⇒ (UIncreasing(LOAD597(i20[1], 0)), ≥)∧[(-1)Bound*bni_25] + [bni_25]i20[0] ≥ 0∧[(-1)bso_26] ≥ 0)

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

(17)    (i20[0] + [-1] ≥ 0 ⇒ (UIncreasing(LOAD597(i20[1], 0)), ≥)∧[(-1)Bound*bni_25] + [bni_25]i20[0] ≥ 0∧[(-1)bso_26] ≥ 0)

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

(18)    (i20[0] ≥ 0 ⇒ (UIncreasing(LOAD597(i20[1], 0)), ≥)∧[(-1)Bound*bni_25 + bni_25] + [bni_25]i20[0] ≥ 0∧[(-1)bso_26] ≥ 0)

For Pair LOAD597(i20, i31) → COND_LOAD597(&&(&&(>=(i31, 0), <(i31, i20)), >(+(i31, 1), 0)), i20, i31) the following chains were created:
• We consider the chain LOAD597(i20[2], i31[2]) → COND_LOAD597(&&(&&(>=(i31[2], 0), <(i31[2], i20[2])), >(+(i31[2], 1), 0)), i20[2], i31[2]), COND_LOAD597(TRUE, i20[3], i31[3]) → LOAD597(i20[3], +(i31[3], 1)) which results in the following constraint:

(19)    (i20[2]=i20[3]i31[2]=i31[3]&&(&&(>=(i31[2], 0), <(i31[2], i20[2])), >(+(i31[2], 1), 0))=TRUELOAD597(i20[2], i31[2])≥NonInfC∧LOAD597(i20[2], i31[2])≥COND_LOAD597(&&(&&(>=(i31[2], 0), <(i31[2], i20[2])), >(+(i31[2], 1), 0)), i20[2], i31[2])∧(UIncreasing(COND_LOAD597(&&(&&(>=(i31[2], 0), <(i31[2], i20[2])), >(+(i31[2], 1), 0)), i20[2], i31[2])), ≥))

We simplified constraint (19) using rules (IV), (IDP_BOOLEAN) which results in the following new constraint:

(20)    (>(+(i31[2], 1), 0)=TRUE>=(i31[2], 0)=TRUE<(i31[2], i20[2])=TRUELOAD597(i20[2], i31[2])≥NonInfC∧LOAD597(i20[2], i31[2])≥COND_LOAD597(&&(&&(>=(i31[2], 0), <(i31[2], i20[2])), >(+(i31[2], 1), 0)), i20[2], i31[2])∧(UIncreasing(COND_LOAD597(&&(&&(>=(i31[2], 0), <(i31[2], i20[2])), >(+(i31[2], 1), 0)), i20[2], i31[2])), ≥))

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

(21)    (i31[2] ≥ 0∧i31[2] ≥ 0∧i20[2] + [-1] + [-1]i31[2] ≥ 0 ⇒ (UIncreasing(COND_LOAD597(&&(&&(>=(i31[2], 0), <(i31[2], i20[2])), >(+(i31[2], 1), 0)), i20[2], i31[2])), ≥)∧[(-1)Bound*bni_27] + [bni_27]i20[2] ≥ 0∧[(-1)bso_28] ≥ 0)

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

(22)    (i31[2] ≥ 0∧i31[2] ≥ 0∧i20[2] + [-1] + [-1]i31[2] ≥ 0 ⇒ (UIncreasing(COND_LOAD597(&&(&&(>=(i31[2], 0), <(i31[2], i20[2])), >(+(i31[2], 1), 0)), i20[2], i31[2])), ≥)∧[(-1)Bound*bni_27] + [bni_27]i20[2] ≥ 0∧[(-1)bso_28] ≥ 0)

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

(23)    (i31[2] ≥ 0∧i31[2] ≥ 0∧i20[2] + [-1] + [-1]i31[2] ≥ 0 ⇒ (UIncreasing(COND_LOAD597(&&(&&(>=(i31[2], 0), <(i31[2], i20[2])), >(+(i31[2], 1), 0)), i20[2], i31[2])), ≥)∧[(-1)Bound*bni_27] + [bni_27]i20[2] ≥ 0∧[(-1)bso_28] ≥ 0)

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

(24)    (i31[2] ≥ 0∧i31[2] ≥ 0∧i20[2] ≥ 0 ⇒ (UIncreasing(COND_LOAD597(&&(&&(>=(i31[2], 0), <(i31[2], i20[2])), >(+(i31[2], 1), 0)), i20[2], i31[2])), ≥)∧[(-1)Bound*bni_27 + bni_27] + [bni_27]i31[2] + [bni_27]i20[2] ≥ 0∧[(-1)bso_28] ≥ 0)

For Pair COND_LOAD597(TRUE, i20, i31) → LOAD597(i20, +(i31, 1)) the following chains were created:
• We consider the chain LOAD597(i20[2], i31[2]) → COND_LOAD597(&&(&&(>=(i31[2], 0), <(i31[2], i20[2])), >(+(i31[2], 1), 0)), i20[2], i31[2]), COND_LOAD597(TRUE, i20[3], i31[3]) → LOAD597(i20[3], +(i31[3], 1)), LOAD597(i20[2], i31[2]) → COND_LOAD597(&&(&&(>=(i31[2], 0), <(i31[2], i20[2])), >(+(i31[2], 1), 0)), i20[2], i31[2]) which results in the following constraint:

We simplified constraint (25) using rules (III), (IV), (IDP_BOOLEAN) which results in the following new constraint:

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

(27)    (i31[2] ≥ 0∧i31[2] ≥ 0∧i20[2] + [-1] + [-1]i31[2] ≥ 0 ⇒ (UIncreasing(LOAD597(i20[3], +(i31[3], 1))), ≥)∧[(-1)Bound*bni_29] + [bni_29]i20[2] ≥ 0∧[(-1)bso_30] ≥ 0)

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

(28)    (i31[2] ≥ 0∧i31[2] ≥ 0∧i20[2] + [-1] + [-1]i31[2] ≥ 0 ⇒ (UIncreasing(LOAD597(i20[3], +(i31[3], 1))), ≥)∧[(-1)Bound*bni_29] + [bni_29]i20[2] ≥ 0∧[(-1)bso_30] ≥ 0)

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

(29)    (i31[2] ≥ 0∧i31[2] ≥ 0∧i20[2] + [-1] + [-1]i31[2] ≥ 0 ⇒ (UIncreasing(LOAD597(i20[3], +(i31[3], 1))), ≥)∧[(-1)Bound*bni_29] + [bni_29]i20[2] ≥ 0∧[(-1)bso_30] ≥ 0)

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

(30)    (i31[2] ≥ 0∧i31[2] ≥ 0∧i20[2] ≥ 0 ⇒ (UIncreasing(LOAD597(i20[3], +(i31[3], 1))), ≥)∧[(-1)Bound*bni_29 + bni_29] + [bni_29]i31[2] + [bni_29]i20[2] ≥ 0∧[(-1)bso_30] ≥ 0)

• We consider the chain LOAD597(i20[2], i31[2]) → COND_LOAD597(&&(&&(>=(i31[2], 0), <(i31[2], i20[2])), >(+(i31[2], 1), 0)), i20[2], i31[2]), COND_LOAD597(TRUE, i20[3], i31[3]) → LOAD597(i20[3], +(i31[3], 1)), LOAD597(i20[4], i31[4]) → COND_LOAD5971(&&(>(i20[4], 0), >=(i31[4], i20[4])), i20[4], i31[4]) which results in the following constraint:

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

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

(33)    (i31[2] ≥ 0∧i31[2] ≥ 0∧i20[2] + [-1] + [-1]i31[2] ≥ 0 ⇒ (UIncreasing(LOAD597(i20[3], +(i31[3], 1))), ≥)∧[(-1)Bound*bni_29] + [bni_29]i20[2] ≥ 0∧[(-1)bso_30] ≥ 0)

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

(34)    (i31[2] ≥ 0∧i31[2] ≥ 0∧i20[2] + [-1] + [-1]i31[2] ≥ 0 ⇒ (UIncreasing(LOAD597(i20[3], +(i31[3], 1))), ≥)∧[(-1)Bound*bni_29] + [bni_29]i20[2] ≥ 0∧[(-1)bso_30] ≥ 0)

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

(35)    (i31[2] ≥ 0∧i31[2] ≥ 0∧i20[2] + [-1] + [-1]i31[2] ≥ 0 ⇒ (UIncreasing(LOAD597(i20[3], +(i31[3], 1))), ≥)∧[(-1)Bound*bni_29] + [bni_29]i20[2] ≥ 0∧[(-1)bso_30] ≥ 0)

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

(36)    (i31[2] ≥ 0∧i31[2] ≥ 0∧i20[2] ≥ 0 ⇒ (UIncreasing(LOAD597(i20[3], +(i31[3], 1))), ≥)∧[(-1)Bound*bni_29 + bni_29] + [bni_29]i31[2] + [bni_29]i20[2] ≥ 0∧[(-1)bso_30] ≥ 0)

For Pair LOAD597(i20, i31) → COND_LOAD5971(&&(>(i20, 0), >=(i31, i20)), i20, i31) the following chains were created:
• We consider the chain LOAD597(i20[4], i31[4]) → COND_LOAD5971(&&(>(i20[4], 0), >=(i31[4], i20[4])), i20[4], i31[4]), COND_LOAD5971(TRUE, i20[5], i31[5]) → LOAD323(+(i20[5], -1)) which results in the following constraint:

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

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

(39)    (i20[4] + [-1] ≥ 0∧i31[4] + [-1]i20[4] ≥ 0 ⇒ (UIncreasing(COND_LOAD5971(&&(>(i20[4], 0), >=(i31[4], i20[4])), i20[4], i31[4])), ≥)∧[(-1)Bound*bni_31] + [bni_31]i20[4] ≥ 0∧[(-1)bso_32] ≥ 0)

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

(40)    (i20[4] + [-1] ≥ 0∧i31[4] + [-1]i20[4] ≥ 0 ⇒ (UIncreasing(COND_LOAD5971(&&(>(i20[4], 0), >=(i31[4], i20[4])), i20[4], i31[4])), ≥)∧[(-1)Bound*bni_31] + [bni_31]i20[4] ≥ 0∧[(-1)bso_32] ≥ 0)

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

(41)    (i20[4] + [-1] ≥ 0∧i31[4] + [-1]i20[4] ≥ 0 ⇒ (UIncreasing(COND_LOAD5971(&&(>(i20[4], 0), >=(i31[4], i20[4])), i20[4], i31[4])), ≥)∧[(-1)Bound*bni_31] + [bni_31]i20[4] ≥ 0∧[(-1)bso_32] ≥ 0)

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

(42)    (i20[4] ≥ 0∧i31[4] + [-1] + [-1]i20[4] ≥ 0 ⇒ (UIncreasing(COND_LOAD5971(&&(>(i20[4], 0), >=(i31[4], i20[4])), i20[4], i31[4])), ≥)∧[(-1)Bound*bni_31 + bni_31] + [bni_31]i20[4] ≥ 0∧[(-1)bso_32] ≥ 0)

We simplified constraint (42) using rule (IDP_SMT_SPLIT) which results in the following new constraint:

(43)    (i20[4] ≥ 0∧i31[4] ≥ 0 ⇒ (UIncreasing(COND_LOAD5971(&&(>(i20[4], 0), >=(i31[4], i20[4])), i20[4], i31[4])), ≥)∧[(-1)Bound*bni_31 + bni_31] + [bni_31]i20[4] ≥ 0∧[(-1)bso_32] ≥ 0)

For Pair COND_LOAD5971(TRUE, i20, i31) → LOAD323(+(i20, -1)) the following chains were created:

We simplified constraint (44) using rules (III), (IV), (IDP_BOOLEAN) which results in the following new constraint:

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

(46)    (i20[4] + [-1] ≥ 0∧i31[4] + [-1]i20[4] ≥ 0 ⇒ (UIncreasing(LOAD323(+(i20[5], -1))), ≥)∧[(-1)Bound*bni_33] + [bni_33]i20[4] ≥ 0∧[(-1)bso_34] ≥ 0)

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

(47)    (i20[4] + [-1] ≥ 0∧i31[4] + [-1]i20[4] ≥ 0 ⇒ (UIncreasing(LOAD323(+(i20[5], -1))), ≥)∧[(-1)Bound*bni_33] + [bni_33]i20[4] ≥ 0∧[(-1)bso_34] ≥ 0)

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

(48)    (i20[4] + [-1] ≥ 0∧i31[4] + [-1]i20[4] ≥ 0 ⇒ (UIncreasing(LOAD323(+(i20[5], -1))), ≥)∧[(-1)Bound*bni_33] + [bni_33]i20[4] ≥ 0∧[(-1)bso_34] ≥ 0)

We simplified constraint (48) using rule (IDP_SMT_SPLIT) which results in the following new constraint:

(49)    (i20[4] ≥ 0∧i31[4] + [-1] + [-1]i20[4] ≥ 0 ⇒ (UIncreasing(LOAD323(+(i20[5], -1))), ≥)∧[(-1)Bound*bni_33 + bni_33] + [bni_33]i20[4] ≥ 0∧[(-1)bso_34] ≥ 0)

We simplified constraint (49) using rule (IDP_SMT_SPLIT) which results in the following new constraint:

(50)    (i20[4] ≥ 0∧i31[4] ≥ 0 ⇒ (UIncreasing(LOAD323(+(i20[5], -1))), ≥)∧[(-1)Bound*bni_33 + bni_33] + [bni_33]i20[4] ≥ 0∧[(-1)bso_34] ≥ 0)

To summarize, we get the following constraints P for the following pairs.
• (i20[0] ≥ 0 ⇒ (UIncreasing(COND_LOAD323(>(i20[0], 0), i20[0])), ≥)∧[(2)bni_23 + (-1)Bound*bni_23] + [bni_23]i20[0] ≥ 0∧[1 + (-1)bso_24] ≥ 0)

• (i20[0] ≥ 0 ⇒ (UIncreasing(LOAD597(i20[1], 0)), ≥)∧[(-1)Bound*bni_25 + bni_25] + [bni_25]i20[0] ≥ 0∧[(-1)bso_26] ≥ 0)
• (i20[0] ≥ 0 ⇒ (UIncreasing(LOAD597(i20[1], 0)), ≥)∧[(-1)Bound*bni_25 + bni_25] + [bni_25]i20[0] ≥ 0∧[(-1)bso_26] ≥ 0)

• LOAD597(i20, i31) → COND_LOAD597(&&(&&(>=(i31, 0), <(i31, i20)), >(+(i31, 1), 0)), i20, i31)
• (i31[2] ≥ 0∧i31[2] ≥ 0∧i20[2] ≥ 0 ⇒ (UIncreasing(COND_LOAD597(&&(&&(>=(i31[2], 0), <(i31[2], i20[2])), >(+(i31[2], 1), 0)), i20[2], i31[2])), ≥)∧[(-1)Bound*bni_27 + bni_27] + [bni_27]i31[2] + [bni_27]i20[2] ≥ 0∧[(-1)bso_28] ≥ 0)

• (i31[2] ≥ 0∧i31[2] ≥ 0∧i20[2] ≥ 0 ⇒ (UIncreasing(LOAD597(i20[3], +(i31[3], 1))), ≥)∧[(-1)Bound*bni_29 + bni_29] + [bni_29]i31[2] + [bni_29]i20[2] ≥ 0∧[(-1)bso_30] ≥ 0)
• (i31[2] ≥ 0∧i31[2] ≥ 0∧i20[2] ≥ 0 ⇒ (UIncreasing(LOAD597(i20[3], +(i31[3], 1))), ≥)∧[(-1)Bound*bni_29 + bni_29] + [bni_29]i31[2] + [bni_29]i20[2] ≥ 0∧[(-1)bso_30] ≥ 0)

• (i20[4] ≥ 0∧i31[4] ≥ 0 ⇒ (UIncreasing(COND_LOAD5971(&&(>(i20[4], 0), >=(i31[4], i20[4])), i20[4], i31[4])), ≥)∧[(-1)Bound*bni_31 + bni_31] + [bni_31]i20[4] ≥ 0∧[(-1)bso_32] ≥ 0)

• (i20[4] ≥ 0∧i31[4] ≥ 0 ⇒ (UIncreasing(LOAD323(+(i20[5], -1))), ≥)∧[(-1)Bound*bni_33 + bni_33] + [bni_33]i20[4] ≥ 0∧[(-1)bso_34] ≥ 0)

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

POL(TRUE) = 0
POL(FALSE) = [2]
POL(>(x1, x2)) = 0
POL(0) = 0
POL(COND_LOAD597(x1, x2, x3)) = x2 + [-1]x1
POL(&&(x1, x2)) = 0
POL(>=(x1, x2)) = [-1]
POL(<(x1, x2)) = [-1]
POL(+(x1, x2)) = x1 + x2
POL(1) = [1]
POL(COND_LOAD5971(x1, x2, x3)) = x2 + [-1]x1
POL(-1) = [-1]

The following pairs are in P>:

The following pairs are in Pbound:

LOAD597(i20[2], i31[2]) → COND_LOAD597(&&(&&(>=(i31[2], 0), <(i31[2], i20[2])), >(+(i31[2], 1), 0)), i20[2], i31[2])

The following pairs are in P:

LOAD597(i20[2], i31[2]) → COND_LOAD597(&&(&&(>=(i31[2], 0), <(i31[2], i20[2])), >(+(i31[2], 1), 0)), i20[2], i31[2])

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

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

### (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:
(2): LOAD597(i20[2], i31[2]) → COND_LOAD597(i31[2] >= 0 && i31[2] < i20[2] && i31[2] + 1 > 0, i20[2], i31[2])
(4): LOAD597(i20[4], i31[4]) → COND_LOAD5971(i20[4] > 0 && i31[4] >= i20[4], i20[4], i31[4])

(1) -> (2), if ((i20[1]* i20[2])∧(0* i31[2]))

(3) -> (2), if ((i31[3] + 1* i31[2])∧(i20[3]* i20[2]))

(2) -> (3), if ((i20[2]* i20[3])∧(i31[2]* i31[3])∧(i31[2] >= 0 && i31[2] < i20[2] && i31[2] + 1 > 0* TRUE))

(1) -> (4), if ((i20[1]* i20[4])∧(0* i31[4]))

(3) -> (4), if ((i20[3]* i20[4])∧(i31[3] + 1* i31[4]))

(4) -> (5), if ((i20[4] > 0 && i31[4] >= i20[4]* TRUE)∧(i20[4]* i20[5])∧(i31[4]* i31[5]))

The set Q consists of the following terms:

### (12) IDependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC with 3 less nodes.

### (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:

Integer, Boolean

R is empty.

The integer pair graph contains the following rules and edges:
(2): LOAD597(i20[2], i31[2]) → COND_LOAD597(i31[2] >= 0 && i31[2] < i20[2] && i31[2] + 1 > 0, i20[2], i31[2])

(3) -> (2), if ((i31[3] + 1* i31[2])∧(i20[3]* i20[2]))

(2) -> (3), if ((i20[2]* i20[3])∧(i31[2]* i31[3])∧(i31[2] >= 0 && i31[2] < i20[2] && i31[2] + 1 > 0* TRUE))

The set Q consists of the following terms:

### (14) 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 COND_LOAD597(TRUE, i20[3], i31[3]) → LOAD597(i20[3], +(i31[3], 1)) the following chains were created:
• We consider the chain LOAD597(i20[2], i31[2]) → COND_LOAD597(&&(&&(>=(i31[2], 0), <(i31[2], i20[2])), >(+(i31[2], 1), 0)), i20[2], i31[2]), COND_LOAD597(TRUE, i20[3], i31[3]) → LOAD597(i20[3], +(i31[3], 1)), LOAD597(i20[2], i31[2]) → COND_LOAD597(&&(&&(>=(i31[2], 0), <(i31[2], i20[2])), >(+(i31[2], 1), 0)), i20[2], i31[2]) which results in the following constraint:

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

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

(3)    (i31[2] ≥ 0∧i31[2] ≥ 0∧i20[2] + [-1] + [-1]i31[2] ≥ 0 ⇒ (UIncreasing(LOAD597(i20[3], +(i31[3], 1))), ≥)∧[(-1)bni_14 + (-1)Bound*bni_14] + [(-1)bni_14]i31[2] + [bni_14]i20[2] ≥ 0∧[1 + (-1)bso_15] ≥ 0)

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

(4)    (i31[2] ≥ 0∧i31[2] ≥ 0∧i20[2] + [-1] + [-1]i31[2] ≥ 0 ⇒ (UIncreasing(LOAD597(i20[3], +(i31[3], 1))), ≥)∧[(-1)bni_14 + (-1)Bound*bni_14] + [(-1)bni_14]i31[2] + [bni_14]i20[2] ≥ 0∧[1 + (-1)bso_15] ≥ 0)

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

(5)    (i31[2] ≥ 0∧i31[2] ≥ 0∧i20[2] + [-1] + [-1]i31[2] ≥ 0 ⇒ (UIncreasing(LOAD597(i20[3], +(i31[3], 1))), ≥)∧[(-1)bni_14 + (-1)Bound*bni_14] + [(-1)bni_14]i31[2] + [bni_14]i20[2] ≥ 0∧[1 + (-1)bso_15] ≥ 0)

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

(6)    (i31[2] ≥ 0∧i31[2] ≥ 0∧i20[2] ≥ 0 ⇒ (UIncreasing(LOAD597(i20[3], +(i31[3], 1))), ≥)∧[(-1)Bound*bni_14] + [bni_14]i20[2] ≥ 0∧[1 + (-1)bso_15] ≥ 0)

For Pair LOAD597(i20[2], i31[2]) → COND_LOAD597(&&(&&(>=(i31[2], 0), <(i31[2], i20[2])), >(+(i31[2], 1), 0)), i20[2], i31[2]) the following chains were created:
• We consider the chain LOAD597(i20[2], i31[2]) → COND_LOAD597(&&(&&(>=(i31[2], 0), <(i31[2], i20[2])), >(+(i31[2], 1), 0)), i20[2], i31[2]), COND_LOAD597(TRUE, i20[3], i31[3]) → LOAD597(i20[3], +(i31[3], 1)) which results in the following constraint:

(7)    (i20[2]=i20[3]i31[2]=i31[3]&&(&&(>=(i31[2], 0), <(i31[2], i20[2])), >(+(i31[2], 1), 0))=TRUELOAD597(i20[2], i31[2])≥NonInfC∧LOAD597(i20[2], i31[2])≥COND_LOAD597(&&(&&(>=(i31[2], 0), <(i31[2], i20[2])), >(+(i31[2], 1), 0)), i20[2], i31[2])∧(UIncreasing(COND_LOAD597(&&(&&(>=(i31[2], 0), <(i31[2], i20[2])), >(+(i31[2], 1), 0)), i20[2], i31[2])), ≥))

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

(8)    (>(+(i31[2], 1), 0)=TRUE>=(i31[2], 0)=TRUE<(i31[2], i20[2])=TRUELOAD597(i20[2], i31[2])≥NonInfC∧LOAD597(i20[2], i31[2])≥COND_LOAD597(&&(&&(>=(i31[2], 0), <(i31[2], i20[2])), >(+(i31[2], 1), 0)), i20[2], i31[2])∧(UIncreasing(COND_LOAD597(&&(&&(>=(i31[2], 0), <(i31[2], i20[2])), >(+(i31[2], 1), 0)), i20[2], i31[2])), ≥))

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

(9)    (i31[2] ≥ 0∧i31[2] ≥ 0∧i20[2] + [-1] + [-1]i31[2] ≥ 0 ⇒ (UIncreasing(COND_LOAD597(&&(&&(>=(i31[2], 0), <(i31[2], i20[2])), >(+(i31[2], 1), 0)), i20[2], i31[2])), ≥)∧[(-1)bni_16 + (-1)Bound*bni_16] + [(-1)bni_16]i31[2] + [bni_16]i20[2] ≥ 0∧[(-1)bso_17] ≥ 0)

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

(10)    (i31[2] ≥ 0∧i31[2] ≥ 0∧i20[2] + [-1] + [-1]i31[2] ≥ 0 ⇒ (UIncreasing(COND_LOAD597(&&(&&(>=(i31[2], 0), <(i31[2], i20[2])), >(+(i31[2], 1), 0)), i20[2], i31[2])), ≥)∧[(-1)bni_16 + (-1)Bound*bni_16] + [(-1)bni_16]i31[2] + [bni_16]i20[2] ≥ 0∧[(-1)bso_17] ≥ 0)

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

(11)    (i31[2] ≥ 0∧i31[2] ≥ 0∧i20[2] + [-1] + [-1]i31[2] ≥ 0 ⇒ (UIncreasing(COND_LOAD597(&&(&&(>=(i31[2], 0), <(i31[2], i20[2])), >(+(i31[2], 1), 0)), i20[2], i31[2])), ≥)∧[(-1)bni_16 + (-1)Bound*bni_16] + [(-1)bni_16]i31[2] + [bni_16]i20[2] ≥ 0∧[(-1)bso_17] ≥ 0)

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

(12)    (i31[2] ≥ 0∧i31[2] ≥ 0∧i20[2] ≥ 0 ⇒ (UIncreasing(COND_LOAD597(&&(&&(>=(i31[2], 0), <(i31[2], i20[2])), >(+(i31[2], 1), 0)), i20[2], i31[2])), ≥)∧[(-1)Bound*bni_16] + [bni_16]i20[2] ≥ 0∧[(-1)bso_17] ≥ 0)

To summarize, we get the following constraints P for the following pairs.
• (i31[2] ≥ 0∧i31[2] ≥ 0∧i20[2] ≥ 0 ⇒ (UIncreasing(LOAD597(i20[3], +(i31[3], 1))), ≥)∧[(-1)Bound*bni_14] + [bni_14]i20[2] ≥ 0∧[1 + (-1)bso_15] ≥ 0)

• LOAD597(i20[2], i31[2]) → COND_LOAD597(&&(&&(>=(i31[2], 0), <(i31[2], i20[2])), >(+(i31[2], 1), 0)), i20[2], i31[2])
• (i31[2] ≥ 0∧i31[2] ≥ 0∧i20[2] ≥ 0 ⇒ (UIncreasing(COND_LOAD597(&&(&&(>=(i31[2], 0), <(i31[2], i20[2])), >(+(i31[2], 1), 0)), i20[2], i31[2])), ≥)∧[(-1)Bound*bni_16] + [bni_16]i20[2] ≥ 0∧[(-1)bso_17] ≥ 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(COND_LOAD597(x1, x2, x3)) = [-1] + [-1]x3 + x2
POL(LOAD597(x1, x2)) = [-1] + [-1]x2 + x1
POL(+(x1, x2)) = x1 + x2
POL(1) = [1]
POL(&&(x1, x2)) = [-1]
POL(>=(x1, x2)) = [-1]
POL(0) = 0
POL(<(x1, x2)) = [-1]
POL(>(x1, x2)) = [-1]

The following pairs are in P>:

The following pairs are in Pbound:

LOAD597(i20[2], i31[2]) → COND_LOAD597(&&(&&(>=(i31[2], 0), <(i31[2], i20[2])), >(+(i31[2], 1), 0)), i20[2], i31[2])

The following pairs are in P:

LOAD597(i20[2], i31[2]) → COND_LOAD597(&&(&&(>=(i31[2], 0), <(i31[2], i20[2])), >(+(i31[2], 1), 0)), i20[2], i31[2])

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

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

### (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:

Boolean, Integer

R is empty.

The integer pair graph contains the following rules and edges:
(2): LOAD597(i20[2], i31[2]) → COND_LOAD597(i31[2] >= 0 && i31[2] < i20[2] && i31[2] + 1 > 0, i20[2], i31[2])

The set Q consists of the following terms:

### (17) IDependencyGraphProof (EQUIVALENT transformation)

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

### (19) 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:

### (20) IDependencyGraphProof (EQUIVALENT transformation)

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

### (22) 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: