### (0) Obligation:

JBC Problem based on JBC Program:
Manifest-Version: 1.0 Created-By: 1.6.0_16 (Sun Microsystems Inc.) Main-Class: PastaC1
`/** * 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 PastaC1 {    public static void main(String[] args) {        Random.args = args;        int x = Random.random();		while (x >= 0) {			int y = 1;			while (x > y) {				y = 2*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 136 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:
Load671(1, i47, i48) → Cond_Load671(i48 > 0 && i47 > i48, 1, i47, i48)
Load671(1, i47, i48) → Cond_Load6711(i47 >= 0 && i48 > 0 && i47 <= i48, 1, i47, i48)
The set Q consists of the following terms:

### (5) GroundTermsRemoverProof (EQUIVALENT transformation)

Some arguments are removed because they always contain the same ground term.
We removed the following ground terms:
• 1

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:
Load671(i47, i48) → Cond_Load671(i48 > 0 && i47 > i48, i47, i48)
Load671(i47, i48) → Cond_Load6711(i47 >= 0 && i48 > 0 && i47 <= i48, i47, i48)
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:

Integer, Boolean

The ITRS R consists of the following rules:
Load671(i47, i48) → Cond_Load671(i48 > 0 && i47 > i48, i47, i48)
Load671(i47, i48) → Cond_Load6711(i47 >= 0 && i48 > 0 && i47 <= i48, i47, i48)

The integer pair graph contains the following rules and edges:
(2): LOAD671(i47[2], i48[2]) → COND_LOAD671(i48[2] > 0 && i47[2] > i48[2], i47[2], i48[2])
(4): LOAD671(i47[4], i48[4]) → COND_LOAD6711(i47[4] >= 0 && i48[4] > 0 && i47[4] <= i48[4], i47[4], i48[4])

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

(1) -> (2), if ((i20[1]* i47[2])∧(1* i48[2]))

(1) -> (4), if ((1* i48[4])∧(i20[1]* i47[4]))

(2) -> (3), if ((i48[2]* i48[3])∧(i48[2] > 0 && i47[2] > i48[2]* TRUE)∧(i47[2]* i47[3]))

(3) -> (2), if ((i47[3]* i47[2])∧(2 * i48[3]* i48[2]))

(3) -> (4), if ((i47[3]* i47[4])∧(2 * i48[3]* i48[4]))

(4) -> (5), if ((i47[4] >= 0 && i48[4] > 0 && i47[4] <= i48[4]* TRUE)∧(i48[4]* i48[5])∧(i47[4]* i47[5]))

(5) -> (0), if ((i47[5] + -1* i20[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:

Integer, Boolean

R is empty.

The integer pair graph contains the following rules and edges:
(2): LOAD671(i47[2], i48[2]) → COND_LOAD671(i48[2] > 0 && i47[2] > i48[2], i47[2], i48[2])
(4): LOAD671(i47[4], i48[4]) → COND_LOAD6711(i47[4] >= 0 && i48[4] > 0 && i47[4] <= i48[4], i47[4], i48[4])

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

(1) -> (2), if ((i20[1]* i47[2])∧(1* i48[2]))

(1) -> (4), if ((1* i48[4])∧(i20[1]* i47[4]))

(2) -> (3), if ((i48[2]* i48[3])∧(i48[2] > 0 && i47[2] > i48[2]* TRUE)∧(i47[2]* i47[3]))

(3) -> (2), if ((i47[3]* i47[2])∧(2 * i48[3]* i48[2]))

(3) -> (4), if ((i47[3]* i47[4])∧(2 * i48[3]* i48[4]))

(4) -> (5), if ((i47[4] >= 0 && i48[4] > 0 && i47[4] <= i48[4]* TRUE)∧(i48[4]* i48[5])∧(i47[4]* i47[5]))

(5) -> (0), if ((i47[5] + -1* i20[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 LOAD288(i20) → COND_LOAD288(>=(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] ≥ 0 ⇒ (UIncreasing(COND_LOAD288(>=(i20[0], 0), i20[0])), ≥)∧[bni_23 + (-1)Bound*bni_23] + [(2)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] ≥ 0 ⇒ (UIncreasing(COND_LOAD288(>=(i20[0], 0), i20[0])), ≥)∧[bni_23 + (-1)Bound*bni_23] + [(2)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] ≥ 0 ⇒ (UIncreasing(COND_LOAD288(>=(i20[0], 0), i20[0])), ≥)∧[bni_23 + (-1)Bound*bni_23] + [(2)bni_23]i20[0] ≥ 0∧[1 + (-1)bso_24] ≥ 0)

For Pair COND_LOAD288(TRUE, i20) → LOAD671(i20, 1) the following chains were created:

We simplified constraint (6) using rules (III), (IV) which results in the following new constraint:

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

(8)    (i20[0] ≥ 0 ⇒ (UIncreasing(LOAD671(i20[1], 1)), ≥)∧[(-1)Bound*bni_25] + [(2)bni_25]i20[0] ≥ 0∧[(-1)bso_26] ≥ 0)

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

(9)    (i20[0] ≥ 0 ⇒ (UIncreasing(LOAD671(i20[1], 1)), ≥)∧[(-1)Bound*bni_25] + [(2)bni_25]i20[0] ≥ 0∧[(-1)bso_26] ≥ 0)

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

(10)    (i20[0] ≥ 0 ⇒ (UIncreasing(LOAD671(i20[1], 1)), ≥)∧[(-1)Bound*bni_25] + [(2)bni_25]i20[0] ≥ 0∧[(-1)bso_26] ≥ 0)

We simplified constraint (11) using rules (III), (IV) which results in the following new constraint:

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

(13)    (i20[0] ≥ 0 ⇒ (UIncreasing(LOAD671(i20[1], 1)), ≥)∧[(-1)Bound*bni_25] + [(2)bni_25]i20[0] ≥ 0∧[(-1)bso_26] ≥ 0)

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

(14)    (i20[0] ≥ 0 ⇒ (UIncreasing(LOAD671(i20[1], 1)), ≥)∧[(-1)Bound*bni_25] + [(2)bni_25]i20[0] ≥ 0∧[(-1)bso_26] ≥ 0)

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

(15)    (i20[0] ≥ 0 ⇒ (UIncreasing(LOAD671(i20[1], 1)), ≥)∧[(-1)Bound*bni_25] + [(2)bni_25]i20[0] ≥ 0∧[(-1)bso_26] ≥ 0)

For Pair LOAD671(i47, i48) → COND_LOAD671(&&(>(i48, 0), >(i47, i48)), i47, i48) the following chains were created:
• We consider the chain LOAD671(i47[2], i48[2]) → COND_LOAD671(&&(>(i48[2], 0), >(i47[2], i48[2])), i47[2], i48[2]), COND_LOAD671(TRUE, i47[3], i48[3]) → LOAD671(i47[3], *(2, i48[3])) which results in the following constraint:

We simplified constraint (16) using rules (IV), (IDP_BOOLEAN) which results in the following new constraint:

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

(18)    (i48[2] + [-1] ≥ 0∧i47[2] + [-1] + [-1]i48[2] ≥ 0 ⇒ (UIncreasing(COND_LOAD671(&&(>(i48[2], 0), >(i47[2], i48[2])), i47[2], i48[2])), ≥)∧[(-1)Bound*bni_27] + [(2)bni_27]i47[2] ≥ 0∧[(-1)bso_28] ≥ 0)

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

(19)    (i48[2] + [-1] ≥ 0∧i47[2] + [-1] + [-1]i48[2] ≥ 0 ⇒ (UIncreasing(COND_LOAD671(&&(>(i48[2], 0), >(i47[2], i48[2])), i47[2], i48[2])), ≥)∧[(-1)Bound*bni_27] + [(2)bni_27]i47[2] ≥ 0∧[(-1)bso_28] ≥ 0)

We simplified constraint (19) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:

(20)    (i48[2] + [-1] ≥ 0∧i47[2] + [-1] + [-1]i48[2] ≥ 0 ⇒ (UIncreasing(COND_LOAD671(&&(>(i48[2], 0), >(i47[2], i48[2])), i47[2], i48[2])), ≥)∧[(-1)Bound*bni_27] + [(2)bni_27]i47[2] ≥ 0∧[(-1)bso_28] ≥ 0)

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

(21)    (i48[2] ≥ 0∧i47[2] + [-2] + [-1]i48[2] ≥ 0 ⇒ (UIncreasing(COND_LOAD671(&&(>(i48[2], 0), >(i47[2], i48[2])), i47[2], i48[2])), ≥)∧[(-1)Bound*bni_27] + [(2)bni_27]i47[2] ≥ 0∧[(-1)bso_28] ≥ 0)

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

(22)    (i48[2] ≥ 0∧i47[2] ≥ 0 ⇒ (UIncreasing(COND_LOAD671(&&(>(i48[2], 0), >(i47[2], i48[2])), i47[2], i48[2])), ≥)∧[(-1)Bound*bni_27 + (4)bni_27] + [(2)bni_27]i48[2] + [(2)bni_27]i47[2] ≥ 0∧[(-1)bso_28] ≥ 0)

For Pair COND_LOAD671(TRUE, i47, i48) → LOAD671(i47, *(2, i48)) the following chains were created:
• We consider the chain LOAD671(i47[2], i48[2]) → COND_LOAD671(&&(>(i48[2], 0), >(i47[2], i48[2])), i47[2], i48[2]), COND_LOAD671(TRUE, i47[3], i48[3]) → LOAD671(i47[3], *(2, i48[3])), LOAD671(i47[2], i48[2]) → COND_LOAD671(&&(>(i48[2], 0), >(i47[2], i48[2])), i47[2], i48[2]) which results in the following constraint:

We simplified constraint (23) using rules (III), (IV), (IDP_BOOLEAN) which results in the following new constraint:

We simplified constraint (24) using rule (POLY_CONSTRAINTS) which results in the following new constraint:

(25)    (i48[2] + [-1] ≥ 0∧i47[2] + [-1] + [-1]i48[2] ≥ 0 ⇒ (UIncreasing(LOAD671(i47[3], *(2, i48[3]))), ≥)∧[(-1)Bound*bni_29] + [(2)bni_29]i47[2] ≥ 0∧[(-1)bso_30] ≥ 0)

We simplified constraint (25) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:

(26)    (i48[2] + [-1] ≥ 0∧i47[2] + [-1] + [-1]i48[2] ≥ 0 ⇒ (UIncreasing(LOAD671(i47[3], *(2, i48[3]))), ≥)∧[(-1)Bound*bni_29] + [(2)bni_29]i47[2] ≥ 0∧[(-1)bso_30] ≥ 0)

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

(27)    (i48[2] + [-1] ≥ 0∧i47[2] + [-1] + [-1]i48[2] ≥ 0 ⇒ (UIncreasing(LOAD671(i47[3], *(2, i48[3]))), ≥)∧[(-1)Bound*bni_29] + [(2)bni_29]i47[2] ≥ 0∧[(-1)bso_30] ≥ 0)

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

(28)    (i48[2] ≥ 0∧i47[2] + [-2] + [-1]i48[2] ≥ 0 ⇒ (UIncreasing(LOAD671(i47[3], *(2, i48[3]))), ≥)∧[(-1)Bound*bni_29] + [(2)bni_29]i47[2] ≥ 0∧[(-1)bso_30] ≥ 0)

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

(29)    (i48[2] ≥ 0∧i47[2] ≥ 0 ⇒ (UIncreasing(LOAD671(i47[3], *(2, i48[3]))), ≥)∧[(-1)Bound*bni_29 + (4)bni_29] + [(2)bni_29]i48[2] + [(2)bni_29]i47[2] ≥ 0∧[(-1)bso_30] ≥ 0)

• We consider the chain LOAD671(i47[2], i48[2]) → COND_LOAD671(&&(>(i48[2], 0), >(i47[2], i48[2])), i47[2], i48[2]), COND_LOAD671(TRUE, i47[3], i48[3]) → LOAD671(i47[3], *(2, i48[3])), LOAD671(i47[4], i48[4]) → COND_LOAD6711(&&(&&(>=(i47[4], 0), >(i48[4], 0)), <=(i47[4], i48[4])), i47[4], i48[4]) which results in the following constraint:

We simplified constraint (30) using rules (III), (IV), (IDP_BOOLEAN) which results in the following new constraint:

We simplified constraint (31) using rule (POLY_CONSTRAINTS) which results in the following new constraint:

(32)    (i48[2] + [-1] ≥ 0∧i47[2] + [-1] + [-1]i48[2] ≥ 0 ⇒ (UIncreasing(LOAD671(i47[3], *(2, i48[3]))), ≥)∧[(-1)Bound*bni_29] + [(2)bni_29]i47[2] ≥ 0∧[(-1)bso_30] ≥ 0)

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

(33)    (i48[2] + [-1] ≥ 0∧i47[2] + [-1] + [-1]i48[2] ≥ 0 ⇒ (UIncreasing(LOAD671(i47[3], *(2, i48[3]))), ≥)∧[(-1)Bound*bni_29] + [(2)bni_29]i47[2] ≥ 0∧[(-1)bso_30] ≥ 0)

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

(34)    (i48[2] + [-1] ≥ 0∧i47[2] + [-1] + [-1]i48[2] ≥ 0 ⇒ (UIncreasing(LOAD671(i47[3], *(2, i48[3]))), ≥)∧[(-1)Bound*bni_29] + [(2)bni_29]i47[2] ≥ 0∧[(-1)bso_30] ≥ 0)

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

(35)    (i48[2] ≥ 0∧i47[2] + [-2] + [-1]i48[2] ≥ 0 ⇒ (UIncreasing(LOAD671(i47[3], *(2, i48[3]))), ≥)∧[(-1)Bound*bni_29] + [(2)bni_29]i47[2] ≥ 0∧[(-1)bso_30] ≥ 0)

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

(36)    (i48[2] ≥ 0∧i47[2] ≥ 0 ⇒ (UIncreasing(LOAD671(i47[3], *(2, i48[3]))), ≥)∧[(-1)Bound*bni_29 + (4)bni_29] + [(2)bni_29]i48[2] + [(2)bni_29]i47[2] ≥ 0∧[(-1)bso_30] ≥ 0)

For Pair LOAD671(i47, i48) → COND_LOAD6711(&&(&&(>=(i47, 0), >(i48, 0)), <=(i47, i48)), i47, i48) the following chains were created:
• We consider the chain LOAD671(i47[4], i48[4]) → COND_LOAD6711(&&(&&(>=(i47[4], 0), >(i48[4], 0)), <=(i47[4], i48[4])), i47[4], i48[4]), COND_LOAD6711(TRUE, i47[5], i48[5]) → LOAD288(+(i47[5], -1)) which results in the following constraint:

(37)    (&&(&&(>=(i47[4], 0), >(i48[4], 0)), <=(i47[4], i48[4]))=TRUEi48[4]=i48[5]i47[4]=i47[5]LOAD671(i47[4], i48[4])≥NonInfC∧LOAD671(i47[4], i48[4])≥COND_LOAD6711(&&(&&(>=(i47[4], 0), >(i48[4], 0)), <=(i47[4], i48[4])), i47[4], i48[4])∧(UIncreasing(COND_LOAD6711(&&(&&(>=(i47[4], 0), >(i48[4], 0)), <=(i47[4], i48[4])), i47[4], i48[4])), ≥))

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)    (i48[4] + [-1]i47[4] ≥ 0∧i47[4] ≥ 0∧i48[4] + [-1] ≥ 0 ⇒ (UIncreasing(COND_LOAD6711(&&(&&(>=(i47[4], 0), >(i48[4], 0)), <=(i47[4], i48[4])), i47[4], i48[4])), ≥)∧[(-1)Bound*bni_31] + [(2)bni_31]i47[4] ≥ 0∧[(-1)bso_32] ≥ 0)

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

(40)    (i48[4] + [-1]i47[4] ≥ 0∧i47[4] ≥ 0∧i48[4] + [-1] ≥ 0 ⇒ (UIncreasing(COND_LOAD6711(&&(&&(>=(i47[4], 0), >(i48[4], 0)), <=(i47[4], i48[4])), i47[4], i48[4])), ≥)∧[(-1)Bound*bni_31] + [(2)bni_31]i47[4] ≥ 0∧[(-1)bso_32] ≥ 0)

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

(41)    (i48[4] + [-1]i47[4] ≥ 0∧i47[4] ≥ 0∧i48[4] + [-1] ≥ 0 ⇒ (UIncreasing(COND_LOAD6711(&&(&&(>=(i47[4], 0), >(i48[4], 0)), <=(i47[4], i48[4])), i47[4], i48[4])), ≥)∧[(-1)Bound*bni_31] + [(2)bni_31]i47[4] ≥ 0∧[(-1)bso_32] ≥ 0)

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

(42)    (i48[4] ≥ 0∧i47[4] ≥ 0∧i47[4] + [-1] + i48[4] ≥ 0 ⇒ (UIncreasing(COND_LOAD6711(&&(&&(>=(i47[4], 0), >(i48[4], 0)), <=(i47[4], i48[4])), i47[4], i48[4])), ≥)∧[(-1)Bound*bni_31] + [(2)bni_31]i47[4] ≥ 0∧[(-1)bso_32] ≥ 0)

For Pair COND_LOAD6711(TRUE, i47, i48) → LOAD288(+(i47, -1)) the following chains were created:

We simplified constraint (43) using rules (III), (IV), (IDP_BOOLEAN) which results in the following new constraint:

We simplified constraint (44) using rule (POLY_CONSTRAINTS) which results in the following new constraint:

(45)    (i48[4] + [-1]i47[4] ≥ 0∧i47[4] ≥ 0∧i48[4] + [-1] ≥ 0 ⇒ (UIncreasing(LOAD288(+(i47[5], -1))), ≥)∧[(-1)bni_33 + (-1)Bound*bni_33] + [(2)bni_33]i47[4] ≥ 0∧[(-1)bso_34] ≥ 0)

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

(46)    (i48[4] + [-1]i47[4] ≥ 0∧i47[4] ≥ 0∧i48[4] + [-1] ≥ 0 ⇒ (UIncreasing(LOAD288(+(i47[5], -1))), ≥)∧[(-1)bni_33 + (-1)Bound*bni_33] + [(2)bni_33]i47[4] ≥ 0∧[(-1)bso_34] ≥ 0)

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

(47)    (i48[4] + [-1]i47[4] ≥ 0∧i47[4] ≥ 0∧i48[4] + [-1] ≥ 0 ⇒ (UIncreasing(LOAD288(+(i47[5], -1))), ≥)∧[(-1)bni_33 + (-1)Bound*bni_33] + [(2)bni_33]i47[4] ≥ 0∧[(-1)bso_34] ≥ 0)

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

(48)    (i48[4] ≥ 0∧i47[4] ≥ 0∧i47[4] + [-1] + i48[4] ≥ 0 ⇒ (UIncreasing(LOAD288(+(i47[5], -1))), ≥)∧[(-1)bni_33 + (-1)Bound*bni_33] + [(2)bni_33]i47[4] ≥ 0∧[(-1)bso_34] ≥ 0)

To summarize, we get the following constraints P for the following pairs.
• (i20[0] ≥ 0 ⇒ (UIncreasing(COND_LOAD288(>=(i20[0], 0), i20[0])), ≥)∧[bni_23 + (-1)Bound*bni_23] + [(2)bni_23]i20[0] ≥ 0∧[1 + (-1)bso_24] ≥ 0)

• (i20[0] ≥ 0 ⇒ (UIncreasing(LOAD671(i20[1], 1)), ≥)∧[(-1)Bound*bni_25] + [(2)bni_25]i20[0] ≥ 0∧[(-1)bso_26] ≥ 0)
• (i20[0] ≥ 0 ⇒ (UIncreasing(LOAD671(i20[1], 1)), ≥)∧[(-1)Bound*bni_25] + [(2)bni_25]i20[0] ≥ 0∧[(-1)bso_26] ≥ 0)

• (i48[2] ≥ 0∧i47[2] ≥ 0 ⇒ (UIncreasing(COND_LOAD671(&&(>(i48[2], 0), >(i47[2], i48[2])), i47[2], i48[2])), ≥)∧[(-1)Bound*bni_27 + (4)bni_27] + [(2)bni_27]i48[2] + [(2)bni_27]i47[2] ≥ 0∧[(-1)bso_28] ≥ 0)

• (i48[2] ≥ 0∧i47[2] ≥ 0 ⇒ (UIncreasing(LOAD671(i47[3], *(2, i48[3]))), ≥)∧[(-1)Bound*bni_29 + (4)bni_29] + [(2)bni_29]i48[2] + [(2)bni_29]i47[2] ≥ 0∧[(-1)bso_30] ≥ 0)
• (i48[2] ≥ 0∧i47[2] ≥ 0 ⇒ (UIncreasing(LOAD671(i47[3], *(2, i48[3]))), ≥)∧[(-1)Bound*bni_29 + (4)bni_29] + [(2)bni_29]i48[2] + [(2)bni_29]i47[2] ≥ 0∧[(-1)bso_30] ≥ 0)

• LOAD671(i47, i48) → COND_LOAD6711(&&(&&(>=(i47, 0), >(i48, 0)), <=(i47, i48)), i47, i48)
• (i48[4] ≥ 0∧i47[4] ≥ 0∧i47[4] + [-1] + i48[4] ≥ 0 ⇒ (UIncreasing(COND_LOAD6711(&&(&&(>=(i47[4], 0), >(i48[4], 0)), <=(i47[4], i48[4])), i47[4], i48[4])), ≥)∧[(-1)Bound*bni_31] + [(2)bni_31]i47[4] ≥ 0∧[(-1)bso_32] ≥ 0)

• (i48[4] ≥ 0∧i47[4] ≥ 0∧i47[4] + [-1] + i48[4] ≥ 0 ⇒ (UIncreasing(LOAD288(+(i47[5], -1))), ≥)∧[(-1)bni_33 + (-1)Bound*bni_33] + [(2)bni_33]i47[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) = [3]
POL(>=(x1, x2)) = [-1]
POL(0) = 0
POL(1) = [1]
POL(&&(x1, x2)) = [-1]
POL(>(x1, x2)) = [-1]
POL(*(x1, x2)) = x1·x2
POL(2) = [2]
POL(COND_LOAD6711(x1, x2, x3)) = [-1] + [2]x2 + [-1]x1
POL(<=(x1, x2)) = [-1]
POL(+(x1, x2)) = x1 + x2
POL(-1) = [-1]

The following pairs are in P>:

The following pairs are in Pbound:

The following pairs are in P:

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

### (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:
(2): LOAD671(i47[2], i48[2]) → COND_LOAD671(i48[2] > 0 && i47[2] > i48[2], i47[2], i48[2])
(4): LOAD671(i47[4], i48[4]) → COND_LOAD6711(i47[4] >= 0 && i48[4] > 0 && i47[4] <= i48[4], i47[4], i48[4])

(1) -> (2), if ((i20[1]* i47[2])∧(1* i48[2]))

(3) -> (2), if ((i47[3]* i47[2])∧(2 * i48[3]* i48[2]))

(2) -> (3), if ((i48[2]* i48[3])∧(i48[2] > 0 && i47[2] > i48[2]* TRUE)∧(i47[2]* i47[3]))

(1) -> (4), if ((1* i48[4])∧(i20[1]* i47[4]))

(3) -> (4), if ((i47[3]* i47[4])∧(2 * i48[3]* i48[4]))

(4) -> (5), if ((i47[4] >= 0 && i48[4] > 0 && i47[4] <= i48[4]* TRUE)∧(i48[4]* i48[5])∧(i47[4]* i47[5]))

The set Q consists of the following terms:

### (14) IDependencyGraphProof (EQUIVALENT transformation)

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

### (15) 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): LOAD671(i47[2], i48[2]) → COND_LOAD671(i48[2] > 0 && i47[2] > i48[2], i47[2], i48[2])

(3) -> (2), if ((i47[3]* i47[2])∧(2 * i48[3]* i48[2]))

(2) -> (3), if ((i48[2]* i48[3])∧(i48[2] > 0 && i47[2] > i48[2]* TRUE)∧(i47[2]* i47[3]))

The set Q consists of the following terms:

### (16) 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_LOAD671(TRUE, i47[3], i48[3]) → LOAD671(i47[3], *(2, i48[3])) the following chains were created:
• We consider the chain LOAD671(i47[2], i48[2]) → COND_LOAD671(&&(>(i48[2], 0), >(i47[2], i48[2])), i47[2], i48[2]), COND_LOAD671(TRUE, i47[3], i48[3]) → LOAD671(i47[3], *(2, i48[3])), LOAD671(i47[2], i48[2]) → COND_LOAD671(&&(>(i48[2], 0), >(i47[2], i48[2])), i47[2], i48[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)    (i48[2] + [-1] ≥ 0∧i47[2] + [-1] + [-1]i48[2] ≥ 0 ⇒ (UIncreasing(LOAD671(i47[3], *(2, i48[3]))), ≥)∧[(-1)bni_12 + (-1)Bound*bni_12] + [(-1)bni_12]i48[2] + [bni_12]i47[2] ≥ 0∧[(-1)bso_13] + i48[2] ≥ 0)

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

(4)    (i48[2] + [-1] ≥ 0∧i47[2] + [-1] + [-1]i48[2] ≥ 0 ⇒ (UIncreasing(LOAD671(i47[3], *(2, i48[3]))), ≥)∧[(-1)bni_12 + (-1)Bound*bni_12] + [(-1)bni_12]i48[2] + [bni_12]i47[2] ≥ 0∧[(-1)bso_13] + i48[2] ≥ 0)

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

(5)    (i48[2] + [-1] ≥ 0∧i47[2] + [-1] + [-1]i48[2] ≥ 0 ⇒ (UIncreasing(LOAD671(i47[3], *(2, i48[3]))), ≥)∧[(-1)bni_12 + (-1)Bound*bni_12] + [(-1)bni_12]i48[2] + [bni_12]i47[2] ≥ 0∧[(-1)bso_13] + i48[2] ≥ 0)

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

(6)    (i48[2] ≥ 0∧i47[2] + [-2] + [-1]i48[2] ≥ 0 ⇒ (UIncreasing(LOAD671(i47[3], *(2, i48[3]))), ≥)∧[(-2)bni_12 + (-1)Bound*bni_12] + [(-1)bni_12]i48[2] + [bni_12]i47[2] ≥ 0∧[1 + (-1)bso_13] + i48[2] ≥ 0)

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

(7)    (i48[2] ≥ 0∧i47[2] ≥ 0 ⇒ (UIncreasing(LOAD671(i47[3], *(2, i48[3]))), ≥)∧[(-1)Bound*bni_12] + [bni_12]i47[2] ≥ 0∧[1 + (-1)bso_13] + i48[2] ≥ 0)

For Pair LOAD671(i47[2], i48[2]) → COND_LOAD671(&&(>(i48[2], 0), >(i47[2], i48[2])), i47[2], i48[2]) the following chains were created:
• We consider the chain LOAD671(i47[2], i48[2]) → COND_LOAD671(&&(>(i48[2], 0), >(i47[2], i48[2])), i47[2], i48[2]), COND_LOAD671(TRUE, i47[3], i48[3]) → LOAD671(i47[3], *(2, i48[3])) which results in the following constraint:

We simplified constraint (8) using rules (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)    (i48[2] + [-1] ≥ 0∧i47[2] + [-1] + [-1]i48[2] ≥ 0 ⇒ (UIncreasing(COND_LOAD671(&&(>(i48[2], 0), >(i47[2], i48[2])), i47[2], i48[2])), ≥)∧[(-1)bni_14 + (-1)Bound*bni_14] + [(-1)bni_14]i48[2] + [bni_14]i47[2] ≥ 0∧[(-1)bso_15] ≥ 0)

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

(11)    (i48[2] + [-1] ≥ 0∧i47[2] + [-1] + [-1]i48[2] ≥ 0 ⇒ (UIncreasing(COND_LOAD671(&&(>(i48[2], 0), >(i47[2], i48[2])), i47[2], i48[2])), ≥)∧[(-1)bni_14 + (-1)Bound*bni_14] + [(-1)bni_14]i48[2] + [bni_14]i47[2] ≥ 0∧[(-1)bso_15] ≥ 0)

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

(12)    (i48[2] + [-1] ≥ 0∧i47[2] + [-1] + [-1]i48[2] ≥ 0 ⇒ (UIncreasing(COND_LOAD671(&&(>(i48[2], 0), >(i47[2], i48[2])), i47[2], i48[2])), ≥)∧[(-1)bni_14 + (-1)Bound*bni_14] + [(-1)bni_14]i48[2] + [bni_14]i47[2] ≥ 0∧[(-1)bso_15] ≥ 0)

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

(13)    (i48[2] ≥ 0∧i47[2] + [-2] + [-1]i48[2] ≥ 0 ⇒ (UIncreasing(COND_LOAD671(&&(>(i48[2], 0), >(i47[2], i48[2])), i47[2], i48[2])), ≥)∧[(-2)bni_14 + (-1)Bound*bni_14] + [(-1)bni_14]i48[2] + [bni_14]i47[2] ≥ 0∧[(-1)bso_15] ≥ 0)

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

(14)    (i48[2] ≥ 0∧i47[2] ≥ 0 ⇒ (UIncreasing(COND_LOAD671(&&(>(i48[2], 0), >(i47[2], i48[2])), i47[2], i48[2])), ≥)∧[(-1)Bound*bni_14] + [bni_14]i47[2] ≥ 0∧[(-1)bso_15] ≥ 0)

To summarize, we get the following constraints P for the following pairs.
• (i48[2] ≥ 0∧i47[2] ≥ 0 ⇒ (UIncreasing(LOAD671(i47[3], *(2, i48[3]))), ≥)∧[(-1)Bound*bni_12] + [bni_12]i47[2] ≥ 0∧[1 + (-1)bso_13] + i48[2] ≥ 0)

• (i48[2] ≥ 0∧i47[2] ≥ 0 ⇒ (UIncreasing(COND_LOAD671(&&(>(i48[2], 0), >(i47[2], i48[2])), i47[2], i48[2])), ≥)∧[(-1)Bound*bni_14] + [bni_14]i47[2] ≥ 0∧[(-1)bso_15] ≥ 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) = [1]
POL(COND_LOAD671(x1, x2, x3)) = [-1] + [-1]x3 + x2
POL(LOAD671(x1, x2)) = [-1] + [-1]x2 + x1
POL(*(x1, x2)) = x1·x2
POL(2) = [2]
POL(&&(x1, x2)) = [-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:

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

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

### (18) 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): LOAD671(i47[2], i48[2]) → COND_LOAD671(i48[2] > 0 && i47[2] > i48[2], i47[2], i48[2])

The set Q consists of the following terms:

### (19) IDependencyGraphProof (EQUIVALENT transformation)

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

### (21) 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:

### (22) IDependencyGraphProof (EQUIVALENT transformation)

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

### (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:
none

R is empty.

The integer pair graph is empty.

The set Q consists of the following terms: