(0) Obligation:

JBC Problem based on JBC Program:
Manifest-Version: 1.0 Created-By: 1.6.0_16 (Sun Microsystems Inc.) Main-Class: PastaB12
`/** * 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 PastaB12 {    public static void main(String[] args) {        Random.args = args;        int x = Random.random();        int y = Random.random();        while (x > 0 || y > 0) {            if (x > 0) {                x--;            } else if (y > 0) {                y--;            } else {                continue;            }        }    }}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 196 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:
Load635(i50, i57) → Cond_Load6351(i57 > 0 && i50 <= 0, i50, i57)
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:
Load635(i50, i57) → Cond_Load6351(i57 > 0 && i50 <= 0, i50, i57)

The integer pair graph contains the following rules and edges:
(2): LOAD635(i50[2], i57[2]) → COND_LOAD6351(i57[2] > 0 && i50[2] <= 0, i50[2], i57[2])

(0) -> (1), if ((i36[0]* i36[1])∧(i49[0]* i49[1])∧(i49[0] > 0* TRUE))

(1) -> (0), if ((i49[1] + -1* i49[0])∧(i36[1]* i36[0]))

(1) -> (2), if ((i49[1] + -1* i50[2])∧(i36[1]* i57[2]))

(2) -> (3), if ((i57[2] > 0 && i50[2] <= 0* TRUE)∧(i50[2]* i50[3])∧(i57[2]* i57[3]))

(3) -> (0), if ((i57[3] + -1* i36[0])∧(i50[3]* i49[0]))

(3) -> (2), if ((i50[3]* i50[2])∧(i57[3] + -1* i57[2]))

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): LOAD635(i50[2], i57[2]) → COND_LOAD6351(i57[2] > 0 && i50[2] <= 0, i50[2], i57[2])

(0) -> (1), if ((i36[0]* i36[1])∧(i49[0]* i49[1])∧(i49[0] > 0* TRUE))

(1) -> (0), if ((i49[1] + -1* i49[0])∧(i36[1]* i36[0]))

(1) -> (2), if ((i49[1] + -1* i50[2])∧(i36[1]* i57[2]))

(2) -> (3), if ((i57[2] > 0 && i50[2] <= 0* TRUE)∧(i50[2]* i50[3])∧(i57[2]* i57[3]))

(3) -> (0), if ((i57[3] + -1* i36[0])∧(i50[3]* i49[0]))

(3) -> (2), if ((i50[3]* i50[2])∧(i57[3] + -1* i57[2]))

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 LOAD635(i49, i36) → COND_LOAD635(>(i49, 0), i49, i36) the following chains were created:
• We consider the chain LOAD635(i49[0], i36[0]) → COND_LOAD635(>(i49[0], 0), i49[0], i36[0]), COND_LOAD635(TRUE, i49[1], i36[1]) → LOAD635(+(i49[1], -1), i36[1]) which results in the following constraint:

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)    (i49[0] + [-1] ≥ 0 ⇒ (UIncreasing(COND_LOAD635(>(i49[0], 0), i49[0], i36[0])), ≥)∧[(-1)bni_11 + (-1)Bound*bni_11] + [bni_11]i36[0] ≥ 0∧[(-1)bso_12] ≥ 0)

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

(4)    (i49[0] + [-1] ≥ 0 ⇒ (UIncreasing(COND_LOAD635(>(i49[0], 0), i49[0], i36[0])), ≥)∧[(-1)bni_11 + (-1)Bound*bni_11] + [bni_11]i36[0] ≥ 0∧[(-1)bso_12] ≥ 0)

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

(5)    (i49[0] + [-1] ≥ 0 ⇒ (UIncreasing(COND_LOAD635(>(i49[0], 0), i49[0], i36[0])), ≥)∧[(-1)bni_11 + (-1)Bound*bni_11] + [bni_11]i36[0] ≥ 0∧[(-1)bso_12] ≥ 0)

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

(6)    (i49[0] + [-1] ≥ 0 ⇒ (UIncreasing(COND_LOAD635(>(i49[0], 0), i49[0], i36[0])), ≥)∧[bni_11] = 0∧[(-1)bni_11 + (-1)Bound*bni_11] ≥ 0∧0 = 0∧[(-1)bso_12] ≥ 0)

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

(7)    (i49[0] ≥ 0 ⇒ (UIncreasing(COND_LOAD635(>(i49[0], 0), i49[0], i36[0])), ≥)∧[bni_11] = 0∧[(-1)bni_11 + (-1)Bound*bni_11] ≥ 0∧0 = 0∧[(-1)bso_12] ≥ 0)

For Pair COND_LOAD635(TRUE, i49, i36) → LOAD635(+(i49, -1), i36) the following chains were created:
• We consider the chain COND_LOAD635(TRUE, i49[1], i36[1]) → LOAD635(+(i49[1], -1), i36[1]) which results in the following constraint:

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

(9)    ((UIncreasing(LOAD635(+(i49[1], -1), i36[1])), ≥)∧[(-1)bso_14] ≥ 0)

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

(10)    ((UIncreasing(LOAD635(+(i49[1], -1), i36[1])), ≥)∧[(-1)bso_14] ≥ 0)

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

(11)    ((UIncreasing(LOAD635(+(i49[1], -1), i36[1])), ≥)∧[(-1)bso_14] ≥ 0)

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

(12)    ((UIncreasing(LOAD635(+(i49[1], -1), i36[1])), ≥)∧0 = 0∧0 = 0∧[(-1)bso_14] ≥ 0)

For Pair LOAD635(i50, i57) → COND_LOAD6351(&&(>(i57, 0), <=(i50, 0)), i50, i57) the following chains were created:
• We consider the chain LOAD635(i50[2], i57[2]) → COND_LOAD6351(&&(>(i57[2], 0), <=(i50[2], 0)), i50[2], i57[2]), COND_LOAD6351(TRUE, i50[3], i57[3]) → LOAD635(i50[3], +(i57[3], -1)) which results in the following constraint:

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

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

(15)    (i57[2] + [-1] ≥ 0∧[-1]i50[2] ≥ 0 ⇒ (UIncreasing(COND_LOAD6351(&&(>(i57[2], 0), <=(i50[2], 0)), i50[2], i57[2])), ≥)∧[(-1)bni_15 + (-1)Bound*bni_15] + [bni_15]i57[2] ≥ 0∧[(-1)bso_16] ≥ 0)

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

(16)    (i57[2] + [-1] ≥ 0∧[-1]i50[2] ≥ 0 ⇒ (UIncreasing(COND_LOAD6351(&&(>(i57[2], 0), <=(i50[2], 0)), i50[2], i57[2])), ≥)∧[(-1)bni_15 + (-1)Bound*bni_15] + [bni_15]i57[2] ≥ 0∧[(-1)bso_16] ≥ 0)

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

(17)    (i57[2] + [-1] ≥ 0∧[-1]i50[2] ≥ 0 ⇒ (UIncreasing(COND_LOAD6351(&&(>(i57[2], 0), <=(i50[2], 0)), i50[2], i57[2])), ≥)∧[(-1)bni_15 + (-1)Bound*bni_15] + [bni_15]i57[2] ≥ 0∧[(-1)bso_16] ≥ 0)

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

(18)    (i57[2] ≥ 0∧[-1]i50[2] ≥ 0 ⇒ (UIncreasing(COND_LOAD6351(&&(>(i57[2], 0), <=(i50[2], 0)), i50[2], i57[2])), ≥)∧[(-1)Bound*bni_15] + [bni_15]i57[2] ≥ 0∧[(-1)bso_16] ≥ 0)

We simplified constraint (18) using rule (IDP_SMT_SPLIT) which results in the following new constraint:

(19)    (i57[2] ≥ 0∧i50[2] ≥ 0 ⇒ (UIncreasing(COND_LOAD6351(&&(>(i57[2], 0), <=(i50[2], 0)), i50[2], i57[2])), ≥)∧[(-1)Bound*bni_15] + [bni_15]i57[2] ≥ 0∧[(-1)bso_16] ≥ 0)

For Pair COND_LOAD6351(TRUE, i50, i57) → LOAD635(i50, +(i57, -1)) the following chains were created:
• We consider the chain COND_LOAD6351(TRUE, i50[3], i57[3]) → LOAD635(i50[3], +(i57[3], -1)) which results in the following constraint:

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

(21)    ((UIncreasing(LOAD635(i50[3], +(i57[3], -1))), ≥)∧[1 + (-1)bso_18] ≥ 0)

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

(22)    ((UIncreasing(LOAD635(i50[3], +(i57[3], -1))), ≥)∧[1 + (-1)bso_18] ≥ 0)

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

(23)    ((UIncreasing(LOAD635(i50[3], +(i57[3], -1))), ≥)∧[1 + (-1)bso_18] ≥ 0)

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

(24)    ((UIncreasing(LOAD635(i50[3], +(i57[3], -1))), ≥)∧0 = 0∧0 = 0∧[1 + (-1)bso_18] ≥ 0)

To summarize, we get the following constraints P for the following pairs.
• (i49[0] ≥ 0 ⇒ (UIncreasing(COND_LOAD635(>(i49[0], 0), i49[0], i36[0])), ≥)∧[bni_11] = 0∧[(-1)bni_11 + (-1)Bound*bni_11] ≥ 0∧0 = 0∧[(-1)bso_12] ≥ 0)

• ((UIncreasing(LOAD635(+(i49[1], -1), i36[1])), ≥)∧0 = 0∧0 = 0∧[(-1)bso_14] ≥ 0)

• (i57[2] ≥ 0∧i50[2] ≥ 0 ⇒ (UIncreasing(COND_LOAD6351(&&(>(i57[2], 0), <=(i50[2], 0)), i50[2], i57[2])), ≥)∧[(-1)Bound*bni_15] + [bni_15]i57[2] ≥ 0∧[(-1)bso_16] ≥ 0)

• ((UIncreasing(LOAD635(i50[3], +(i57[3], -1))), ≥)∧0 = 0∧0 = 0∧[1 + (-1)bso_18] ≥ 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(LOAD635(x1, x2)) = [-1] + x2
POL(COND_LOAD635(x1, x2, x3)) = [-1] + x3
POL(>(x1, x2)) = [-1]
POL(0) = 0
POL(+(x1, x2)) = x1 + x2
POL(-1) = [-1]
POL(COND_LOAD6351(x1, x2, x3)) = [-1] + x3
POL(&&(x1, x2)) = [-1]
POL(<=(x1, x2)) = [-1]

The following pairs are in P>:

The following pairs are in Pbound:

The following pairs are in P:

There are no usable rules.

(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:

Integer, Boolean

R is empty.

The integer pair graph contains the following rules and edges:
(2): LOAD635(i50[2], i57[2]) → COND_LOAD6351(i57[2] > 0 && i50[2] <= 0, i50[2], i57[2])

(1) -> (0), if ((i49[1] + -1* i49[0])∧(i36[1]* i36[0]))

(0) -> (1), if ((i36[0]* i36[1])∧(i49[0]* i49[1])∧(i49[0] > 0* TRUE))

(1) -> (2), if ((i49[1] + -1* i50[2])∧(i36[1]* i57[2]))

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 1 less node.

(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

R is empty.

The integer pair graph contains the following rules and edges:

(1) -> (0), if ((i49[1] + -1* i49[0])∧(i36[1]* i36[0]))

(0) -> (1), if ((i36[0]* i36[1])∧(i49[0]* i49[1])∧(i49[0] > 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_LOAD635(TRUE, i49[1], i36[1]) → LOAD635(+(i49[1], -1), i36[1]) the following chains were created:
• We consider the chain COND_LOAD635(TRUE, i49[1], i36[1]) → LOAD635(+(i49[1], -1), i36[1]) which results in the following constraint:

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

(2)    ((UIncreasing(LOAD635(+(i49[1], -1), i36[1])), ≥)∧[1 + (-1)bso_7] ≥ 0)

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

(3)    ((UIncreasing(LOAD635(+(i49[1], -1), i36[1])), ≥)∧[1 + (-1)bso_7] ≥ 0)

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

(4)    ((UIncreasing(LOAD635(+(i49[1], -1), i36[1])), ≥)∧[1 + (-1)bso_7] ≥ 0)

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

(5)    ((UIncreasing(LOAD635(+(i49[1], -1), i36[1])), ≥)∧0 = 0∧[1 + (-1)bso_7] ≥ 0)

For Pair LOAD635(i49[0], i36[0]) → COND_LOAD635(>(i49[0], 0), i49[0], i36[0]) the following chains were created:
• We consider the chain LOAD635(i49[0], i36[0]) → COND_LOAD635(>(i49[0], 0), i49[0], i36[0]), COND_LOAD635(TRUE, i49[1], i36[1]) → LOAD635(+(i49[1], -1), i36[1]) which results in the following constraint:

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

We simplified constraint (7) using rule (POLY_CONSTRAINTS) which results in the following new constraint:

(8)    (i49[0] + [-1] ≥ 0 ⇒ (UIncreasing(COND_LOAD635(>(i49[0], 0), i49[0], i36[0])), ≥)∧[bni_8 + (-1)Bound*bni_8] + [bni_8]i49[0] ≥ 0∧[(-1)bso_9] ≥ 0)

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

(9)    (i49[0] + [-1] ≥ 0 ⇒ (UIncreasing(COND_LOAD635(>(i49[0], 0), i49[0], i36[0])), ≥)∧[bni_8 + (-1)Bound*bni_8] + [bni_8]i49[0] ≥ 0∧[(-1)bso_9] ≥ 0)

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

(10)    (i49[0] + [-1] ≥ 0 ⇒ (UIncreasing(COND_LOAD635(>(i49[0], 0), i49[0], i36[0])), ≥)∧[bni_8 + (-1)Bound*bni_8] + [bni_8]i49[0] ≥ 0∧[(-1)bso_9] ≥ 0)

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

(11)    (i49[0] ≥ 0 ⇒ (UIncreasing(COND_LOAD635(>(i49[0], 0), i49[0], i36[0])), ≥)∧[(2)bni_8 + (-1)Bound*bni_8] + [bni_8]i49[0] ≥ 0∧[(-1)bso_9] ≥ 0)

To summarize, we get the following constraints P for the following pairs.
• ((UIncreasing(LOAD635(+(i49[1], -1), i36[1])), ≥)∧0 = 0∧[1 + (-1)bso_7] ≥ 0)

• (i49[0] ≥ 0 ⇒ (UIncreasing(COND_LOAD635(>(i49[0], 0), i49[0], i36[0])), ≥)∧[(2)bni_8 + (-1)Bound*bni_8] + [bni_8]i49[0] ≥ 0∧[(-1)bso_9] ≥ 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_LOAD635(x1, x2, x3)) = [1] + x2
POL(LOAD635(x1, x2)) = [1] + x1
POL(+(x1, x2)) = x1 + x2
POL(-1) = [-1]
POL(>(x1, x2)) = [1]
POL(0) = 0

The following pairs are in P>:

The following pairs are in Pbound:

The following pairs are in P:

There are no usable rules.

(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:

Integer

R is empty.

The integer pair graph contains the following rules and edges:

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:

Integer

R is empty.

The integer pair graph contains the following rules and edges:

The set Q consists of the following terms:

(20) IDependencyGraphProof (EQUIVALENT transformation)

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

(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:

Integer

R is empty.

The integer pair graph contains the following rules and edges:

(1) -> (0), if ((i49[1] + -1* i49[0])∧(i36[1]* i36[0]))

(3) -> (0), if ((i57[3] + -1* i36[0])∧(i50[3]* i49[0]))

(0) -> (1), if ((i36[0]* i36[1])∧(i49[0]* i49[1])∧(i49[0] > 0* TRUE))

The set Q consists of the following terms:

(23) IDependencyGraphProof (EQUIVALENT transformation)

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

(24) 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

R is empty.

The integer pair graph contains the following rules and edges:

(1) -> (0), if ((i49[1] + -1* i49[0])∧(i36[1]* i36[0]))

(0) -> (1), if ((i36[0]* i36[1])∧(i49[0]* i49[1])∧(i49[0] > 0* TRUE))

The set Q consists of the following terms:

(25) 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_LOAD635(TRUE, i49[1], i36[1]) → LOAD635(+(i49[1], -1), i36[1]) the following chains were created:
• We consider the chain COND_LOAD635(TRUE, i49[1], i36[1]) → LOAD635(+(i49[1], -1), i36[1]) which results in the following constraint:

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

(2)    ((UIncreasing(LOAD635(+(i49[1], -1), i36[1])), ≥)∧[1 + (-1)bso_8] ≥ 0)

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

(3)    ((UIncreasing(LOAD635(+(i49[1], -1), i36[1])), ≥)∧[1 + (-1)bso_8] ≥ 0)

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

(4)    ((UIncreasing(LOAD635(+(i49[1], -1), i36[1])), ≥)∧[1 + (-1)bso_8] ≥ 0)

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

(5)    ((UIncreasing(LOAD635(+(i49[1], -1), i36[1])), ≥)∧0 = 0∧[1 + (-1)bso_8] ≥ 0)

For Pair LOAD635(i49[0], i36[0]) → COND_LOAD635(>(i49[0], 0), i49[0], i36[0]) the following chains were created:
• We consider the chain LOAD635(i49[0], i36[0]) → COND_LOAD635(>(i49[0], 0), i49[0], i36[0]), COND_LOAD635(TRUE, i49[1], i36[1]) → LOAD635(+(i49[1], -1), i36[1]) which results in the following constraint:

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

We simplified constraint (7) using rule (POLY_CONSTRAINTS) which results in the following new constraint:

(8)    (i49[0] + [-1] ≥ 0 ⇒ (UIncreasing(COND_LOAD635(>(i49[0], 0), i49[0], i36[0])), ≥)∧[bni_9 + (-1)Bound*bni_9] + [bni_9]i49[0] ≥ 0∧[(-1)bso_10] ≥ 0)

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

(9)    (i49[0] + [-1] ≥ 0 ⇒ (UIncreasing(COND_LOAD635(>(i49[0], 0), i49[0], i36[0])), ≥)∧[bni_9 + (-1)Bound*bni_9] + [bni_9]i49[0] ≥ 0∧[(-1)bso_10] ≥ 0)

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

(10)    (i49[0] + [-1] ≥ 0 ⇒ (UIncreasing(COND_LOAD635(>(i49[0], 0), i49[0], i36[0])), ≥)∧[bni_9 + (-1)Bound*bni_9] + [bni_9]i49[0] ≥ 0∧[(-1)bso_10] ≥ 0)

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

(11)    (i49[0] ≥ 0 ⇒ (UIncreasing(COND_LOAD635(>(i49[0], 0), i49[0], i36[0])), ≥)∧[(2)bni_9 + (-1)Bound*bni_9] + [bni_9]i49[0] ≥ 0∧[(-1)bso_10] ≥ 0)

To summarize, we get the following constraints P for the following pairs.
• ((UIncreasing(LOAD635(+(i49[1], -1), i36[1])), ≥)∧0 = 0∧[1 + (-1)bso_8] ≥ 0)

• (i49[0] ≥ 0 ⇒ (UIncreasing(COND_LOAD635(>(i49[0], 0), i49[0], i36[0])), ≥)∧[(2)bni_9 + (-1)Bound*bni_9] + [bni_9]i49[0] ≥ 0∧[(-1)bso_10] ≥ 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_LOAD635(x1, x2, x3)) = [1] + x2
POL(LOAD635(x1, x2)) = [1] + x1
POL(+(x1, x2)) = x1 + x2
POL(-1) = [-1]
POL(>(x1, x2)) = [-1]
POL(0) = 0

The following pairs are in P>:

The following pairs are in Pbound:

The following pairs are in P:

There are no usable rules.

(27) 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

R is empty.

The integer pair graph contains the following rules and edges:

The set Q consists of the following terms:

(28) IDependencyGraphProof (EQUIVALENT transformation)

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

(30) 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

R is empty.

The integer pair graph contains the following rules and edges: