### (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:
PastaA1.main([Ljava/lang/String;)V: Graph of 98 nodes with 1 SCC.

### (3) FIGtoITRSProof (SOUND transformation)

Transformed FIGraph SCCs to IDPs. Logs:

Log for SCC 0:

Generated 20 rules for P and 2 rules for R.

Combined rules. Obtained 2 rules for P and 0 rules for R.

Filtered ground terms:

348_0_main_GE(x1, x2, x3, x4, x5) → 348_0_main_GE(x2, x3, x4, x5)
Cond_348_0_main_GE1(x1, x2, x3, x4, x5, x6) → Cond_348_0_main_GE1(x1, x3, x4, x5, x6)
Cond_348_0_main_GE(x1, x2, x3, x4, x5, x6) → Cond_348_0_main_GE(x1, x3, x4, x5, x6)

Filtered duplicate args:

348_0_main_GE(x1, x2, x3, x4) → 348_0_main_GE(x3, x4)
Cond_348_0_main_GE1(x1, x2, x3, x4, x5) → Cond_348_0_main_GE1(x1, x4, x5)
Cond_348_0_main_GE(x1, x2, x3, x4, x5) → Cond_348_0_main_GE(x1, x4, x5)

Filtered unneeded arguments:

Cond_348_0_main_GE(x1, x2, x3) → Cond_348_0_main_GE(x1, x3)

Combined rules. Obtained 2 rules for P and 0 rules for R.

Finished conversion. Obtained 2 rules for P and 0 rules for R. System has predefined symbols.

### (4) Obligation:

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

The following domains are used:

Boolean, Integer

R is empty.

The integer pair graph contains the following rules and edges:
(0): 348_0_MAIN_GE(x1[0], x0[0]) → COND_348_0_MAIN_GE(x1[0] >= x0[0] && x0[0] > 0 && 0 < x0[0] + -1, x1[0], x0[0])
(1): COND_348_0_MAIN_GE(TRUE, x1[1], x0[1]) → 348_0_MAIN_GE(0, x0[1] + -1)
(2): 348_0_MAIN_GE(x1[2], x0[2]) → COND_348_0_MAIN_GE1(x1[2] >= 0 && x1[2] < x0[2], x1[2], x0[2])
(3): COND_348_0_MAIN_GE1(TRUE, x1[3], x0[3]) → 348_0_MAIN_GE(x1[3] + 1, x0[3])

(0) -> (1), if ((x1[0] >= x0[0] && x0[0] > 0 && 0 < x0[0] + -1* TRUE)∧(x1[0]* x1[1])∧(x0[0]* x0[1]))

(1) -> (0), if ((0* x1[0])∧(x0[1] + -1* x0[0]))

(1) -> (2), if ((0* x1[2])∧(x0[1] + -1* x0[2]))

(2) -> (3), if ((x1[2] >= 0 && x1[2] < x0[2]* TRUE)∧(x1[2]* x1[3])∧(x0[2]* x0[3]))

(3) -> (0), if ((x1[3] + 1* x1[0])∧(x0[3]* x0[0]))

(3) -> (2), if ((x1[3] + 1* x1[2])∧(x0[3]* x0[2]))

The set Q is empty.

### (5) 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 348_0_MAIN_GE(x1, x0) → COND_348_0_MAIN_GE(&&(&&(>=(x1, x0), >(x0, 0)), <(0, +(x0, -1))), x1, x0) the following chains were created:
• We consider the chain 348_0_MAIN_GE(x1[0], x0[0]) → COND_348_0_MAIN_GE(&&(&&(>=(x1[0], x0[0]), >(x0[0], 0)), <(0, +(x0[0], -1))), x1[0], x0[0]), COND_348_0_MAIN_GE(TRUE, x1[1], x0[1]) → 348_0_MAIN_GE(0, +(x0[1], -1)) which results in the following constraint:

(1)    (&&(&&(>=(x1[0], x0[0]), >(x0[0], 0)), <(0, +(x0[0], -1)))=TRUEx1[0]=x1[1]x0[0]=x0[1]348_0_MAIN_GE(x1[0], x0[0])≥NonInfC∧348_0_MAIN_GE(x1[0], x0[0])≥COND_348_0_MAIN_GE(&&(&&(>=(x1[0], x0[0]), >(x0[0], 0)), <(0, +(x0[0], -1))), x1[0], x0[0])∧(UIncreasing(COND_348_0_MAIN_GE(&&(&&(>=(x1[0], x0[0]), >(x0[0], 0)), <(0, +(x0[0], -1))), x1[0], x0[0])), ≥))

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

(2)    (<(0, +(x0[0], -1))=TRUE>=(x1[0], x0[0])=TRUE>(x0[0], 0)=TRUE348_0_MAIN_GE(x1[0], x0[0])≥NonInfC∧348_0_MAIN_GE(x1[0], x0[0])≥COND_348_0_MAIN_GE(&&(&&(>=(x1[0], x0[0]), >(x0[0], 0)), <(0, +(x0[0], -1))), x1[0], x0[0])∧(UIncreasing(COND_348_0_MAIN_GE(&&(&&(>=(x1[0], x0[0]), >(x0[0], 0)), <(0, +(x0[0], -1))), x1[0], x0[0])), ≥))

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

(3)    (x0[0] + [-2] ≥ 0∧x1[0] + [-1]x0[0] ≥ 0∧x0[0] + [-1] ≥ 0 ⇒ (UIncreasing(COND_348_0_MAIN_GE(&&(&&(>=(x1[0], x0[0]), >(x0[0], 0)), <(0, +(x0[0], -1))), x1[0], x0[0])), ≥)∧[(-1)bni_12 + (-1)Bound*bni_12] + [bni_12]x0[0] ≥ 0∧[(-1)bso_13] ≥ 0)

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

(4)    (x0[0] + [-2] ≥ 0∧x1[0] + [-1]x0[0] ≥ 0∧x0[0] + [-1] ≥ 0 ⇒ (UIncreasing(COND_348_0_MAIN_GE(&&(&&(>=(x1[0], x0[0]), >(x0[0], 0)), <(0, +(x0[0], -1))), x1[0], x0[0])), ≥)∧[(-1)bni_12 + (-1)Bound*bni_12] + [bni_12]x0[0] ≥ 0∧[(-1)bso_13] ≥ 0)

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

(5)    (x0[0] + [-2] ≥ 0∧x1[0] + [-1]x0[0] ≥ 0∧x0[0] + [-1] ≥ 0 ⇒ (UIncreasing(COND_348_0_MAIN_GE(&&(&&(>=(x1[0], x0[0]), >(x0[0], 0)), <(0, +(x0[0], -1))), x1[0], x0[0])), ≥)∧[(-1)bni_12 + (-1)Bound*bni_12] + [bni_12]x0[0] ≥ 0∧[(-1)bso_13] ≥ 0)

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

(6)    (x0[0] ≥ 0∧x1[0] + [-2] + [-1]x0[0] ≥ 0∧[1] + x0[0] ≥ 0 ⇒ (UIncreasing(COND_348_0_MAIN_GE(&&(&&(>=(x1[0], x0[0]), >(x0[0], 0)), <(0, +(x0[0], -1))), x1[0], x0[0])), ≥)∧[bni_12 + (-1)Bound*bni_12] + [bni_12]x0[0] ≥ 0∧[(-1)bso_13] ≥ 0)

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

(7)    (x0[0] ≥ 0∧x1[0] ≥ 0∧[1] + x0[0] ≥ 0 ⇒ (UIncreasing(COND_348_0_MAIN_GE(&&(&&(>=(x1[0], x0[0]), >(x0[0], 0)), <(0, +(x0[0], -1))), x1[0], x0[0])), ≥)∧[bni_12 + (-1)Bound*bni_12] + [bni_12]x0[0] ≥ 0∧[(-1)bso_13] ≥ 0)

For Pair COND_348_0_MAIN_GE(TRUE, x1, x0) → 348_0_MAIN_GE(0, +(x0, -1)) the following chains were created:
• We consider the chain COND_348_0_MAIN_GE(TRUE, x1[1], x0[1]) → 348_0_MAIN_GE(0, +(x0[1], -1)) which results in the following constraint:

(8)    (COND_348_0_MAIN_GE(TRUE, x1[1], x0[1])≥NonInfC∧COND_348_0_MAIN_GE(TRUE, x1[1], x0[1])≥348_0_MAIN_GE(0, +(x0[1], -1))∧(UIncreasing(348_0_MAIN_GE(0, +(x0[1], -1))), ≥))

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

(9)    ((UIncreasing(348_0_MAIN_GE(0, +(x0[1], -1))), ≥)∧[1 + (-1)bso_15] ≥ 0)

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

(10)    ((UIncreasing(348_0_MAIN_GE(0, +(x0[1], -1))), ≥)∧[1 + (-1)bso_15] ≥ 0)

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

(11)    ((UIncreasing(348_0_MAIN_GE(0, +(x0[1], -1))), ≥)∧[1 + (-1)bso_15] ≥ 0)

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

(12)    ((UIncreasing(348_0_MAIN_GE(0, +(x0[1], -1))), ≥)∧0 = 0∧0 = 0∧[1 + (-1)bso_15] ≥ 0)

For Pair 348_0_MAIN_GE(x1, x0) → COND_348_0_MAIN_GE1(&&(>=(x1, 0), <(x1, x0)), x1, x0) the following chains were created:
• We consider the chain 348_0_MAIN_GE(x1[2], x0[2]) → COND_348_0_MAIN_GE1(&&(>=(x1[2], 0), <(x1[2], x0[2])), x1[2], x0[2]), COND_348_0_MAIN_GE1(TRUE, x1[3], x0[3]) → 348_0_MAIN_GE(+(x1[3], 1), x0[3]) which results in the following constraint:

(13)    (&&(>=(x1[2], 0), <(x1[2], x0[2]))=TRUEx1[2]=x1[3]x0[2]=x0[3]348_0_MAIN_GE(x1[2], x0[2])≥NonInfC∧348_0_MAIN_GE(x1[2], x0[2])≥COND_348_0_MAIN_GE1(&&(>=(x1[2], 0), <(x1[2], x0[2])), x1[2], x0[2])∧(UIncreasing(COND_348_0_MAIN_GE1(&&(>=(x1[2], 0), <(x1[2], x0[2])), x1[2], x0[2])), ≥))

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

(14)    (>=(x1[2], 0)=TRUE<(x1[2], x0[2])=TRUE348_0_MAIN_GE(x1[2], x0[2])≥NonInfC∧348_0_MAIN_GE(x1[2], x0[2])≥COND_348_0_MAIN_GE1(&&(>=(x1[2], 0), <(x1[2], x0[2])), x1[2], x0[2])∧(UIncreasing(COND_348_0_MAIN_GE1(&&(>=(x1[2], 0), <(x1[2], x0[2])), x1[2], x0[2])), ≥))

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

(15)    (x1[2] ≥ 0∧x0[2] + [-1] + [-1]x1[2] ≥ 0 ⇒ (UIncreasing(COND_348_0_MAIN_GE1(&&(>=(x1[2], 0), <(x1[2], x0[2])), x1[2], x0[2])), ≥)∧[(-1)bni_16 + (-1)Bound*bni_16] + [bni_16]x0[2] ≥ 0∧[(-1)bso_17] ≥ 0)

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

(16)    (x1[2] ≥ 0∧x0[2] + [-1] + [-1]x1[2] ≥ 0 ⇒ (UIncreasing(COND_348_0_MAIN_GE1(&&(>=(x1[2], 0), <(x1[2], x0[2])), x1[2], x0[2])), ≥)∧[(-1)bni_16 + (-1)Bound*bni_16] + [bni_16]x0[2] ≥ 0∧[(-1)bso_17] ≥ 0)

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

(17)    (x1[2] ≥ 0∧x0[2] + [-1] + [-1]x1[2] ≥ 0 ⇒ (UIncreasing(COND_348_0_MAIN_GE1(&&(>=(x1[2], 0), <(x1[2], x0[2])), x1[2], x0[2])), ≥)∧[(-1)bni_16 + (-1)Bound*bni_16] + [bni_16]x0[2] ≥ 0∧[(-1)bso_17] ≥ 0)

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

(18)    (x1[2] ≥ 0∧x0[2] ≥ 0 ⇒ (UIncreasing(COND_348_0_MAIN_GE1(&&(>=(x1[2], 0), <(x1[2], x0[2])), x1[2], x0[2])), ≥)∧[(-1)Bound*bni_16] + [bni_16]x1[2] + [bni_16]x0[2] ≥ 0∧[(-1)bso_17] ≥ 0)

For Pair COND_348_0_MAIN_GE1(TRUE, x1, x0) → 348_0_MAIN_GE(+(x1, 1), x0) the following chains were created:
• We consider the chain COND_348_0_MAIN_GE1(TRUE, x1[3], x0[3]) → 348_0_MAIN_GE(+(x1[3], 1), x0[3]) which results in the following constraint:

(19)    (COND_348_0_MAIN_GE1(TRUE, x1[3], x0[3])≥NonInfC∧COND_348_0_MAIN_GE1(TRUE, x1[3], x0[3])≥348_0_MAIN_GE(+(x1[3], 1), x0[3])∧(UIncreasing(348_0_MAIN_GE(+(x1[3], 1), x0[3])), ≥))

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

(20)    ((UIncreasing(348_0_MAIN_GE(+(x1[3], 1), x0[3])), ≥)∧[(-1)bso_19] ≥ 0)

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

(21)    ((UIncreasing(348_0_MAIN_GE(+(x1[3], 1), x0[3])), ≥)∧[(-1)bso_19] ≥ 0)

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

(22)    ((UIncreasing(348_0_MAIN_GE(+(x1[3], 1), x0[3])), ≥)∧[(-1)bso_19] ≥ 0)

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

(23)    ((UIncreasing(348_0_MAIN_GE(+(x1[3], 1), x0[3])), ≥)∧0 = 0∧0 = 0∧[(-1)bso_19] ≥ 0)

To summarize, we get the following constraints P for the following pairs.
• 348_0_MAIN_GE(x1, x0) → COND_348_0_MAIN_GE(&&(&&(>=(x1, x0), >(x0, 0)), <(0, +(x0, -1))), x1, x0)
• (x0[0] ≥ 0∧x1[0] ≥ 0∧[1] + x0[0] ≥ 0 ⇒ (UIncreasing(COND_348_0_MAIN_GE(&&(&&(>=(x1[0], x0[0]), >(x0[0], 0)), <(0, +(x0[0], -1))), x1[0], x0[0])), ≥)∧[bni_12 + (-1)Bound*bni_12] + [bni_12]x0[0] ≥ 0∧[(-1)bso_13] ≥ 0)

• COND_348_0_MAIN_GE(TRUE, x1, x0) → 348_0_MAIN_GE(0, +(x0, -1))
• ((UIncreasing(348_0_MAIN_GE(0, +(x0[1], -1))), ≥)∧0 = 0∧0 = 0∧[1 + (-1)bso_15] ≥ 0)

• 348_0_MAIN_GE(x1, x0) → COND_348_0_MAIN_GE1(&&(>=(x1, 0), <(x1, x0)), x1, x0)
• (x1[2] ≥ 0∧x0[2] ≥ 0 ⇒ (UIncreasing(COND_348_0_MAIN_GE1(&&(>=(x1[2], 0), <(x1[2], x0[2])), x1[2], x0[2])), ≥)∧[(-1)Bound*bni_16] + [bni_16]x1[2] + [bni_16]x0[2] ≥ 0∧[(-1)bso_17] ≥ 0)

• COND_348_0_MAIN_GE1(TRUE, x1, x0) → 348_0_MAIN_GE(+(x1, 1), x0)
• ((UIncreasing(348_0_MAIN_GE(+(x1[3], 1), x0[3])), ≥)∧0 = 0∧0 = 0∧[(-1)bso_19] ≥ 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(348_0_MAIN_GE(x1, x2)) = [-1] + x2
POL(COND_348_0_MAIN_GE(x1, x2, x3)) = [-1] + x3
POL(&&(x1, x2)) = [-1]
POL(>=(x1, x2)) = [-1]
POL(>(x1, x2)) = [-1]
POL(0) = 0
POL(<(x1, x2)) = [-1]
POL(+(x1, x2)) = x1 + x2
POL(-1) = [-1]
POL(COND_348_0_MAIN_GE1(x1, x2, x3)) = [-1] + x3
POL(1) = [1]

The following pairs are in P>:

COND_348_0_MAIN_GE(TRUE, x1[1], x0[1]) → 348_0_MAIN_GE(0, +(x0[1], -1))

The following pairs are in Pbound:

348_0_MAIN_GE(x1[0], x0[0]) → COND_348_0_MAIN_GE(&&(&&(>=(x1[0], x0[0]), >(x0[0], 0)), <(0, +(x0[0], -1))), x1[0], x0[0])
348_0_MAIN_GE(x1[2], x0[2]) → COND_348_0_MAIN_GE1(&&(>=(x1[2], 0), <(x1[2], x0[2])), x1[2], x0[2])

The following pairs are in P:

348_0_MAIN_GE(x1[0], x0[0]) → COND_348_0_MAIN_GE(&&(&&(>=(x1[0], x0[0]), >(x0[0], 0)), <(0, +(x0[0], -1))), x1[0], x0[0])
348_0_MAIN_GE(x1[2], x0[2]) → COND_348_0_MAIN_GE1(&&(>=(x1[2], 0), <(x1[2], x0[2])), x1[2], x0[2])
COND_348_0_MAIN_GE1(TRUE, x1[3], x0[3]) → 348_0_MAIN_GE(+(x1[3], 1), x0[3])

There are no usable rules.

### (7) Obligation:

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

The following domains are used:

Boolean, Integer

R is empty.

The integer pair graph contains the following rules and edges:
(0): 348_0_MAIN_GE(x1[0], x0[0]) → COND_348_0_MAIN_GE(x1[0] >= x0[0] && x0[0] > 0 && 0 < x0[0] + -1, x1[0], x0[0])
(2): 348_0_MAIN_GE(x1[2], x0[2]) → COND_348_0_MAIN_GE1(x1[2] >= 0 && x1[2] < x0[2], x1[2], x0[2])
(3): COND_348_0_MAIN_GE1(TRUE, x1[3], x0[3]) → 348_0_MAIN_GE(x1[3] + 1, x0[3])

(3) -> (0), if ((x1[3] + 1* x1[0])∧(x0[3]* x0[0]))

(3) -> (2), if ((x1[3] + 1* x1[2])∧(x0[3]* x0[2]))

(2) -> (3), if ((x1[2] >= 0 && x1[2] < x0[2]* TRUE)∧(x1[2]* x1[3])∧(x0[2]* x0[3]))

The set Q is empty.

### (8) IDependencyGraphProof (EQUIVALENT transformation)

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

### (9) 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:
(3): COND_348_0_MAIN_GE1(TRUE, x1[3], x0[3]) → 348_0_MAIN_GE(x1[3] + 1, x0[3])
(2): 348_0_MAIN_GE(x1[2], x0[2]) → COND_348_0_MAIN_GE1(x1[2] >= 0 && x1[2] < x0[2], x1[2], x0[2])

(3) -> (2), if ((x1[3] + 1* x1[2])∧(x0[3]* x0[2]))

(2) -> (3), if ((x1[2] >= 0 && x1[2] < x0[2]* TRUE)∧(x1[2]* x1[3])∧(x0[2]* x0[3]))

The set Q is empty.

### (10) 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_348_0_MAIN_GE1(TRUE, x1[3], x0[3]) → 348_0_MAIN_GE(+(x1[3], 1), x0[3]) the following chains were created:
• We consider the chain COND_348_0_MAIN_GE1(TRUE, x1[3], x0[3]) → 348_0_MAIN_GE(+(x1[3], 1), x0[3]) which results in the following constraint:

(1)    (COND_348_0_MAIN_GE1(TRUE, x1[3], x0[3])≥NonInfC∧COND_348_0_MAIN_GE1(TRUE, x1[3], x0[3])≥348_0_MAIN_GE(+(x1[3], 1), x0[3])∧(UIncreasing(348_0_MAIN_GE(+(x1[3], 1), x0[3])), ≥))

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

(2)    ((UIncreasing(348_0_MAIN_GE(+(x1[3], 1), x0[3])), ≥)∧[(-1)bso_11] ≥ 0)

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

(3)    ((UIncreasing(348_0_MAIN_GE(+(x1[3], 1), x0[3])), ≥)∧[(-1)bso_11] ≥ 0)

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

(4)    ((UIncreasing(348_0_MAIN_GE(+(x1[3], 1), x0[3])), ≥)∧[(-1)bso_11] ≥ 0)

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

(5)    ((UIncreasing(348_0_MAIN_GE(+(x1[3], 1), x0[3])), ≥)∧0 = 0∧0 = 0∧[(-1)bso_11] ≥ 0)

For Pair 348_0_MAIN_GE(x1[2], x0[2]) → COND_348_0_MAIN_GE1(&&(>=(x1[2], 0), <(x1[2], x0[2])), x1[2], x0[2]) the following chains were created:
• We consider the chain 348_0_MAIN_GE(x1[2], x0[2]) → COND_348_0_MAIN_GE1(&&(>=(x1[2], 0), <(x1[2], x0[2])), x1[2], x0[2]), COND_348_0_MAIN_GE1(TRUE, x1[3], x0[3]) → 348_0_MAIN_GE(+(x1[3], 1), x0[3]) which results in the following constraint:

(6)    (&&(>=(x1[2], 0), <(x1[2], x0[2]))=TRUEx1[2]=x1[3]x0[2]=x0[3]348_0_MAIN_GE(x1[2], x0[2])≥NonInfC∧348_0_MAIN_GE(x1[2], x0[2])≥COND_348_0_MAIN_GE1(&&(>=(x1[2], 0), <(x1[2], x0[2])), x1[2], x0[2])∧(UIncreasing(COND_348_0_MAIN_GE1(&&(>=(x1[2], 0), <(x1[2], x0[2])), x1[2], x0[2])), ≥))

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

(7)    (>=(x1[2], 0)=TRUE<(x1[2], x0[2])=TRUE348_0_MAIN_GE(x1[2], x0[2])≥NonInfC∧348_0_MAIN_GE(x1[2], x0[2])≥COND_348_0_MAIN_GE1(&&(>=(x1[2], 0), <(x1[2], x0[2])), x1[2], x0[2])∧(UIncreasing(COND_348_0_MAIN_GE1(&&(>=(x1[2], 0), <(x1[2], x0[2])), x1[2], x0[2])), ≥))

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

(8)    (x1[2] ≥ 0∧x0[2] + [-1] + [-1]x1[2] ≥ 0 ⇒ (UIncreasing(COND_348_0_MAIN_GE1(&&(>=(x1[2], 0), <(x1[2], x0[2])), x1[2], x0[2])), ≥)∧[(-1)Bound*bni_12] + [(-1)bni_12]x1[2] + [(2)bni_12]x0[2] ≥ 0∧[1 + (-1)bso_13] ≥ 0)

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

(9)    (x1[2] ≥ 0∧x0[2] + [-1] + [-1]x1[2] ≥ 0 ⇒ (UIncreasing(COND_348_0_MAIN_GE1(&&(>=(x1[2], 0), <(x1[2], x0[2])), x1[2], x0[2])), ≥)∧[(-1)Bound*bni_12] + [(-1)bni_12]x1[2] + [(2)bni_12]x0[2] ≥ 0∧[1 + (-1)bso_13] ≥ 0)

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

(10)    (x1[2] ≥ 0∧x0[2] + [-1] + [-1]x1[2] ≥ 0 ⇒ (UIncreasing(COND_348_0_MAIN_GE1(&&(>=(x1[2], 0), <(x1[2], x0[2])), x1[2], x0[2])), ≥)∧[(-1)Bound*bni_12] + [(-1)bni_12]x1[2] + [(2)bni_12]x0[2] ≥ 0∧[1 + (-1)bso_13] ≥ 0)

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

(11)    (x1[2] ≥ 0∧x0[2] ≥ 0 ⇒ (UIncreasing(COND_348_0_MAIN_GE1(&&(>=(x1[2], 0), <(x1[2], x0[2])), x1[2], x0[2])), ≥)∧[(-1)Bound*bni_12 + (2)bni_12] + [bni_12]x1[2] + [(2)bni_12]x0[2] ≥ 0∧[1 + (-1)bso_13] ≥ 0)

To summarize, we get the following constraints P for the following pairs.
• COND_348_0_MAIN_GE1(TRUE, x1[3], x0[3]) → 348_0_MAIN_GE(+(x1[3], 1), x0[3])
• ((UIncreasing(348_0_MAIN_GE(+(x1[3], 1), x0[3])), ≥)∧0 = 0∧0 = 0∧[(-1)bso_11] ≥ 0)

• 348_0_MAIN_GE(x1[2], x0[2]) → COND_348_0_MAIN_GE1(&&(>=(x1[2], 0), <(x1[2], x0[2])), x1[2], x0[2])
• (x1[2] ≥ 0∧x0[2] ≥ 0 ⇒ (UIncreasing(COND_348_0_MAIN_GE1(&&(>=(x1[2], 0), <(x1[2], x0[2])), x1[2], x0[2])), ≥)∧[(-1)Bound*bni_12 + (2)bni_12] + [bni_12]x1[2] + [(2)bni_12]x0[2] ≥ 0∧[1 + (-1)bso_13] ≥ 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_348_0_MAIN_GE1(x1, x2, x3)) = [-1] + [2]x3 + [-1]x2
POL(348_0_MAIN_GE(x1, x2)) = [-1]x1 + [2]x2
POL(+(x1, x2)) = x1 + x2
POL(1) = [1]
POL(&&(x1, x2)) = [-1]
POL(>=(x1, x2)) = [-1]
POL(0) = 0
POL(<(x1, x2)) = [-1]

The following pairs are in P>:

348_0_MAIN_GE(x1[2], x0[2]) → COND_348_0_MAIN_GE1(&&(>=(x1[2], 0), <(x1[2], x0[2])), x1[2], x0[2])

The following pairs are in Pbound:

348_0_MAIN_GE(x1[2], x0[2]) → COND_348_0_MAIN_GE1(&&(>=(x1[2], 0), <(x1[2], x0[2])), x1[2], x0[2])

The following pairs are in P:

COND_348_0_MAIN_GE1(TRUE, x1[3], x0[3]) → 348_0_MAIN_GE(+(x1[3], 1), x0[3])

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

R is empty.

The integer pair graph contains the following rules and edges:
(3): COND_348_0_MAIN_GE1(TRUE, x1[3], x0[3]) → 348_0_MAIN_GE(x1[3] + 1, x0[3])

The set Q is empty.

### (12) IDependencyGraphProof (EQUIVALENT transformation)

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

### (14) 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): COND_348_0_MAIN_GE(TRUE, x1[1], x0[1]) → 348_0_MAIN_GE(0, x0[1] + -1)
(3): COND_348_0_MAIN_GE1(TRUE, x1[3], x0[3]) → 348_0_MAIN_GE(x1[3] + 1, x0[3])

The set Q is empty.

### (15) IDependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 2 less nodes.