### (0) Obligation:

JBC Problem based on JBC Program:
Manifest-Version: 1.0 Created-By: 1.6.0_16 (Sun Microsystems Inc.) Main-Class: GCD3
`public class GCD3 {  public static int mod(int a, int b) {    if(b == 0) {      return b;    }    if(b < 0) {      a = -a;    }    if(a > 0) {      while(a>=b) {        a -= b;      }      return a;    } else {      while(a < 0) {        a -= b;      }      return a;    }  }  public static int gcd(int a, int b) {    int tmp;    while(b > 0 && a > 0) {      tmp = b;      b = mod(a, b);      a = tmp;    }    return a;  }  public static void main(String[] args) {    Random.args = args;    int x = Random.random();    int y = Random.random();    gcd(x, y);  }}public class Random {  static String[] args;  static int index = 0;  public static int random() {    String string = args[index];    index++;    return string.length();  }}`

### (1) JBC2FIG (SOUND transformation)

Constructed FIGraph.

### (2) Obligation:

FIGraph based on JBC Program:
GCD3.main([Ljava/lang/String;)V: Graph of 191 nodes with 1 SCC.

### (3) FIGtoITRSProof (SOUND transformation)

Transformed FIGraph SCCs to IDPs. Logs:

Log for SCC 0:

Generated 37 rules for P and 8 rules for R.

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

Filtered ground terms:

1129_0_mod_LT(x1, x2, x3, x4, x5) → 1129_0_mod_LT(x2, x3, x4, x5)

Filtered duplicate args:

1129_0_mod_LT(x1, x2, x3, x4) → 1129_0_mod_LT(x3, x4)
1129_1_gcd_InvokeMethod(x1, x2, x3) → 1129_1_gcd_InvokeMethod(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): 1129_2_MAIN_INVOKEMETHOD(1129_1_gcd_InvokeMethod(1129_0_mod_LT(x0[0], x1[0]), x1[0])) → COND_1129_2_MAIN_INVOKEMETHOD(x1[0] > x0[0] && x1[0] > 0 && x0[0] > 0, 1129_1_gcd_InvokeMethod(1129_0_mod_LT(x0[0], x1[0]), x1[0]))
(1): COND_1129_2_MAIN_INVOKEMETHOD(TRUE, 1129_1_gcd_InvokeMethod(1129_0_mod_LT(x0[1], x1[1]), x1[1])) → 1129_2_MAIN_INVOKEMETHOD(1129_1_gcd_InvokeMethod(1129_0_mod_LT(x1[1], x0[1]), x0[1]))
(2): 1129_2_MAIN_INVOKEMETHOD(1129_1_gcd_InvokeMethod(1129_0_mod_LT(x0[2], x1[2]), x1[2])) → COND_1129_2_MAIN_INVOKEMETHOD1(x1[2] > 0 && x1[2] <= x0[2], 1129_1_gcd_InvokeMethod(1129_0_mod_LT(x0[2], x1[2]), x1[2]))
(3): COND_1129_2_MAIN_INVOKEMETHOD1(TRUE, 1129_1_gcd_InvokeMethod(1129_0_mod_LT(x0[3], x1[3]), x1[3])) → 1129_2_MAIN_INVOKEMETHOD(1129_1_gcd_InvokeMethod(1129_0_mod_LT(x0[3] - x1[3], x1[3]), x1[3]))

(0) -> (1), if ((x1[0] > x0[0] && x1[0] > 0 && x0[0] > 0* TRUE)∧(1129_1_gcd_InvokeMethod(1129_0_mod_LT(x0[0], x1[0]), x1[0]) →* 1129_1_gcd_InvokeMethod(1129_0_mod_LT(x0[1], x1[1]), x1[1])))

(1) -> (0), if ((1129_1_gcd_InvokeMethod(1129_0_mod_LT(x1[1], x0[1]), x0[1]) →* 1129_1_gcd_InvokeMethod(1129_0_mod_LT(x0[0], x1[0]), x1[0])))

(1) -> (2), if ((1129_1_gcd_InvokeMethod(1129_0_mod_LT(x1[1], x0[1]), x0[1]) →* 1129_1_gcd_InvokeMethod(1129_0_mod_LT(x0[2], x1[2]), x1[2])))

(2) -> (3), if ((x1[2] > 0 && x1[2] <= x0[2]* TRUE)∧(1129_1_gcd_InvokeMethod(1129_0_mod_LT(x0[2], x1[2]), x1[2]) →* 1129_1_gcd_InvokeMethod(1129_0_mod_LT(x0[3], x1[3]), x1[3])))

(3) -> (0), if ((1129_1_gcd_InvokeMethod(1129_0_mod_LT(x0[3] - x1[3], x1[3]), x1[3]) →* 1129_1_gcd_InvokeMethod(1129_0_mod_LT(x0[0], x1[0]), x1[0])))

(3) -> (2), if ((1129_1_gcd_InvokeMethod(1129_0_mod_LT(x0[3] - x1[3], x1[3]), x1[3]) →* 1129_1_gcd_InvokeMethod(1129_0_mod_LT(x0[2], x1[2]), x1[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 1129_2_MAIN_INVOKEMETHOD(1129_1_gcd_InvokeMethod(1129_0_mod_LT(x0, x1), x1)) → COND_1129_2_MAIN_INVOKEMETHOD(&&(&&(>(x1, x0), >(x1, 0)), >(x0, 0)), 1129_1_gcd_InvokeMethod(1129_0_mod_LT(x0, x1), x1)) the following chains were created:
• We consider the chain 1129_2_MAIN_INVOKEMETHOD(1129_1_gcd_InvokeMethod(1129_0_mod_LT(x0[0], x1[0]), x1[0])) → COND_1129_2_MAIN_INVOKEMETHOD(&&(&&(>(x1[0], x0[0]), >(x1[0], 0)), >(x0[0], 0)), 1129_1_gcd_InvokeMethod(1129_0_mod_LT(x0[0], x1[0]), x1[0])), COND_1129_2_MAIN_INVOKEMETHOD(TRUE, 1129_1_gcd_InvokeMethod(1129_0_mod_LT(x0[1], x1[1]), x1[1])) → 1129_2_MAIN_INVOKEMETHOD(1129_1_gcd_InvokeMethod(1129_0_mod_LT(x1[1], x0[1]), x0[1])) which results in the following constraint:

(1)    (&&(&&(>(x1[0], x0[0]), >(x1[0], 0)), >(x0[0], 0))=TRUE1129_1_gcd_InvokeMethod(1129_0_mod_LT(x0[0], x1[0]), x1[0])=1129_1_gcd_InvokeMethod(1129_0_mod_LT(x0[1], x1[1]), x1[1]) ⇒ 1129_2_MAIN_INVOKEMETHOD(1129_1_gcd_InvokeMethod(1129_0_mod_LT(x0[0], x1[0]), x1[0]))≥NonInfC∧1129_2_MAIN_INVOKEMETHOD(1129_1_gcd_InvokeMethod(1129_0_mod_LT(x0[0], x1[0]), x1[0]))≥COND_1129_2_MAIN_INVOKEMETHOD(&&(&&(>(x1[0], x0[0]), >(x1[0], 0)), >(x0[0], 0)), 1129_1_gcd_InvokeMethod(1129_0_mod_LT(x0[0], x1[0]), x1[0]))∧(UIncreasing(COND_1129_2_MAIN_INVOKEMETHOD(&&(&&(>(x1[0], x0[0]), >(x1[0], 0)), >(x0[0], 0)), 1129_1_gcd_InvokeMethod(1129_0_mod_LT(x0[0], x1[0]), x1[0]))), ≥))

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

(2)    (>(x0[0], 0)=TRUE>(x1[0], x0[0])=TRUE>(x1[0], 0)=TRUE1129_2_MAIN_INVOKEMETHOD(1129_1_gcd_InvokeMethod(1129_0_mod_LT(x0[0], x1[0]), x1[0]))≥NonInfC∧1129_2_MAIN_INVOKEMETHOD(1129_1_gcd_InvokeMethod(1129_0_mod_LT(x0[0], x1[0]), x1[0]))≥COND_1129_2_MAIN_INVOKEMETHOD(&&(&&(>(x1[0], x0[0]), >(x1[0], 0)), >(x0[0], 0)), 1129_1_gcd_InvokeMethod(1129_0_mod_LT(x0[0], x1[0]), x1[0]))∧(UIncreasing(COND_1129_2_MAIN_INVOKEMETHOD(&&(&&(>(x1[0], x0[0]), >(x1[0], 0)), >(x0[0], 0)), 1129_1_gcd_InvokeMethod(1129_0_mod_LT(x0[0], x1[0]), x1[0]))), ≥))

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

(3)    (x0[0] + [-1] ≥ 0∧x1[0] + [-1] + [-1]x0[0] ≥ 0∧x1[0] + [-1] ≥ 0 ⇒ (UIncreasing(COND_1129_2_MAIN_INVOKEMETHOD(&&(&&(>(x1[0], x0[0]), >(x1[0], 0)), >(x0[0], 0)), 1129_1_gcd_InvokeMethod(1129_0_mod_LT(x0[0], x1[0]), x1[0]))), ≥)∧[(-1)Bound*bni_20] + [bni_20]x1[0] + [bni_20]x0[0] ≥ 0∧[(-1)bso_21] ≥ 0)

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

(4)    (x0[0] + [-1] ≥ 0∧x1[0] + [-1] + [-1]x0[0] ≥ 0∧x1[0] + [-1] ≥ 0 ⇒ (UIncreasing(COND_1129_2_MAIN_INVOKEMETHOD(&&(&&(>(x1[0], x0[0]), >(x1[0], 0)), >(x0[0], 0)), 1129_1_gcd_InvokeMethod(1129_0_mod_LT(x0[0], x1[0]), x1[0]))), ≥)∧[(-1)Bound*bni_20] + [bni_20]x1[0] + [bni_20]x0[0] ≥ 0∧[(-1)bso_21] ≥ 0)

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

(5)    (x0[0] + [-1] ≥ 0∧x1[0] + [-1] + [-1]x0[0] ≥ 0∧x1[0] + [-1] ≥ 0 ⇒ (UIncreasing(COND_1129_2_MAIN_INVOKEMETHOD(&&(&&(>(x1[0], x0[0]), >(x1[0], 0)), >(x0[0], 0)), 1129_1_gcd_InvokeMethod(1129_0_mod_LT(x0[0], x1[0]), x1[0]))), ≥)∧[(-1)Bound*bni_20] + [bni_20]x1[0] + [bni_20]x0[0] ≥ 0∧[(-1)bso_21] ≥ 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∧x1[0] + [-1] ≥ 0 ⇒ (UIncreasing(COND_1129_2_MAIN_INVOKEMETHOD(&&(&&(>(x1[0], x0[0]), >(x1[0], 0)), >(x0[0], 0)), 1129_1_gcd_InvokeMethod(1129_0_mod_LT(x0[0], x1[0]), x1[0]))), ≥)∧[(-1)Bound*bni_20 + bni_20] + [bni_20]x1[0] + [bni_20]x0[0] ≥ 0∧[(-1)bso_21] ≥ 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] + x1[0] ≥ 0 ⇒ (UIncreasing(COND_1129_2_MAIN_INVOKEMETHOD(&&(&&(>(x1[0], x0[0]), >(x1[0], 0)), >(x0[0], 0)), 1129_1_gcd_InvokeMethod(1129_0_mod_LT(x0[0], x1[0]), x1[0]))), ≥)∧[(-1)Bound*bni_20 + (3)bni_20] + [(2)bni_20]x0[0] + [bni_20]x1[0] ≥ 0∧[(-1)bso_21] ≥ 0)

For Pair COND_1129_2_MAIN_INVOKEMETHOD(TRUE, 1129_1_gcd_InvokeMethod(1129_0_mod_LT(x0, x1), x1)) → 1129_2_MAIN_INVOKEMETHOD(1129_1_gcd_InvokeMethod(1129_0_mod_LT(x1, x0), x0)) the following chains were created:
• We consider the chain 1129_2_MAIN_INVOKEMETHOD(1129_1_gcd_InvokeMethod(1129_0_mod_LT(x0[0], x1[0]), x1[0])) → COND_1129_2_MAIN_INVOKEMETHOD(&&(&&(>(x1[0], x0[0]), >(x1[0], 0)), >(x0[0], 0)), 1129_1_gcd_InvokeMethod(1129_0_mod_LT(x0[0], x1[0]), x1[0])), COND_1129_2_MAIN_INVOKEMETHOD(TRUE, 1129_1_gcd_InvokeMethod(1129_0_mod_LT(x0[1], x1[1]), x1[1])) → 1129_2_MAIN_INVOKEMETHOD(1129_1_gcd_InvokeMethod(1129_0_mod_LT(x1[1], x0[1]), x0[1])), 1129_2_MAIN_INVOKEMETHOD(1129_1_gcd_InvokeMethod(1129_0_mod_LT(x0[0], x1[0]), x1[0])) → COND_1129_2_MAIN_INVOKEMETHOD(&&(&&(>(x1[0], x0[0]), >(x1[0], 0)), >(x0[0], 0)), 1129_1_gcd_InvokeMethod(1129_0_mod_LT(x0[0], x1[0]), x1[0])) which results in the following constraint:

(8)    (&&(&&(>(x1[0], x0[0]), >(x1[0], 0)), >(x0[0], 0))=TRUE1129_1_gcd_InvokeMethod(1129_0_mod_LT(x0[0], x1[0]), x1[0])=1129_1_gcd_InvokeMethod(1129_0_mod_LT(x0[1], x1[1]), x1[1])∧1129_1_gcd_InvokeMethod(1129_0_mod_LT(x1[1], x0[1]), x0[1])=1129_1_gcd_InvokeMethod(1129_0_mod_LT(x0[0]1, x1[0]1), x1[0]1) ⇒ COND_1129_2_MAIN_INVOKEMETHOD(TRUE, 1129_1_gcd_InvokeMethod(1129_0_mod_LT(x0[1], x1[1]), x1[1]))≥NonInfC∧COND_1129_2_MAIN_INVOKEMETHOD(TRUE, 1129_1_gcd_InvokeMethod(1129_0_mod_LT(x0[1], x1[1]), x1[1]))≥1129_2_MAIN_INVOKEMETHOD(1129_1_gcd_InvokeMethod(1129_0_mod_LT(x1[1], x0[1]), x0[1]))∧(UIncreasing(1129_2_MAIN_INVOKEMETHOD(1129_1_gcd_InvokeMethod(1129_0_mod_LT(x1[1], x0[1]), x0[1]))), ≥))

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

(9)    (>(x0[0], 0)=TRUE>(x1[0], x0[0])=TRUE>(x1[0], 0)=TRUECOND_1129_2_MAIN_INVOKEMETHOD(TRUE, 1129_1_gcd_InvokeMethod(1129_0_mod_LT(x0[0], x1[0]), x1[0]))≥NonInfC∧COND_1129_2_MAIN_INVOKEMETHOD(TRUE, 1129_1_gcd_InvokeMethod(1129_0_mod_LT(x0[0], x1[0]), x1[0]))≥1129_2_MAIN_INVOKEMETHOD(1129_1_gcd_InvokeMethod(1129_0_mod_LT(x1[0], x0[0]), x0[0]))∧(UIncreasing(1129_2_MAIN_INVOKEMETHOD(1129_1_gcd_InvokeMethod(1129_0_mod_LT(x1[1], x0[1]), x0[1]))), ≥))

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

(10)    (x0[0] + [-1] ≥ 0∧x1[0] + [-1] + [-1]x0[0] ≥ 0∧x1[0] + [-1] ≥ 0 ⇒ (UIncreasing(1129_2_MAIN_INVOKEMETHOD(1129_1_gcd_InvokeMethod(1129_0_mod_LT(x1[1], x0[1]), x0[1]))), ≥)∧[(-1)Bound*bni_22] + [bni_22]x1[0] + [bni_22]x0[0] ≥ 0∧[(-1)bso_23] ≥ 0)

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

(11)    (x0[0] + [-1] ≥ 0∧x1[0] + [-1] + [-1]x0[0] ≥ 0∧x1[0] + [-1] ≥ 0 ⇒ (UIncreasing(1129_2_MAIN_INVOKEMETHOD(1129_1_gcd_InvokeMethod(1129_0_mod_LT(x1[1], x0[1]), x0[1]))), ≥)∧[(-1)Bound*bni_22] + [bni_22]x1[0] + [bni_22]x0[0] ≥ 0∧[(-1)bso_23] ≥ 0)

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

(12)    (x0[0] + [-1] ≥ 0∧x1[0] + [-1] + [-1]x0[0] ≥ 0∧x1[0] + [-1] ≥ 0 ⇒ (UIncreasing(1129_2_MAIN_INVOKEMETHOD(1129_1_gcd_InvokeMethod(1129_0_mod_LT(x1[1], x0[1]), x0[1]))), ≥)∧[(-1)Bound*bni_22] + [bni_22]x1[0] + [bni_22]x0[0] ≥ 0∧[(-1)bso_23] ≥ 0)

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

(13)    (x0[0] ≥ 0∧x1[0] + [-2] + [-1]x0[0] ≥ 0∧x1[0] + [-1] ≥ 0 ⇒ (UIncreasing(1129_2_MAIN_INVOKEMETHOD(1129_1_gcd_InvokeMethod(1129_0_mod_LT(x1[1], x0[1]), x0[1]))), ≥)∧[(-1)Bound*bni_22 + bni_22] + [bni_22]x1[0] + [bni_22]x0[0] ≥ 0∧[(-1)bso_23] ≥ 0)

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

(14)    (x0[0] ≥ 0∧x1[0] ≥ 0∧[1] + x0[0] + x1[0] ≥ 0 ⇒ (UIncreasing(1129_2_MAIN_INVOKEMETHOD(1129_1_gcd_InvokeMethod(1129_0_mod_LT(x1[1], x0[1]), x0[1]))), ≥)∧[(-1)Bound*bni_22 + (3)bni_22] + [(2)bni_22]x0[0] + [bni_22]x1[0] ≥ 0∧[(-1)bso_23] ≥ 0)

• We consider the chain 1129_2_MAIN_INVOKEMETHOD(1129_1_gcd_InvokeMethod(1129_0_mod_LT(x0[0], x1[0]), x1[0])) → COND_1129_2_MAIN_INVOKEMETHOD(&&(&&(>(x1[0], x0[0]), >(x1[0], 0)), >(x0[0], 0)), 1129_1_gcd_InvokeMethod(1129_0_mod_LT(x0[0], x1[0]), x1[0])), COND_1129_2_MAIN_INVOKEMETHOD(TRUE, 1129_1_gcd_InvokeMethod(1129_0_mod_LT(x0[1], x1[1]), x1[1])) → 1129_2_MAIN_INVOKEMETHOD(1129_1_gcd_InvokeMethod(1129_0_mod_LT(x1[1], x0[1]), x0[1])), 1129_2_MAIN_INVOKEMETHOD(1129_1_gcd_InvokeMethod(1129_0_mod_LT(x0[2], x1[2]), x1[2])) → COND_1129_2_MAIN_INVOKEMETHOD1(&&(>(x1[2], 0), <=(x1[2], x0[2])), 1129_1_gcd_InvokeMethod(1129_0_mod_LT(x0[2], x1[2]), x1[2])) which results in the following constraint:

(15)    (&&(&&(>(x1[0], x0[0]), >(x1[0], 0)), >(x0[0], 0))=TRUE1129_1_gcd_InvokeMethod(1129_0_mod_LT(x0[0], x1[0]), x1[0])=1129_1_gcd_InvokeMethod(1129_0_mod_LT(x0[1], x1[1]), x1[1])∧1129_1_gcd_InvokeMethod(1129_0_mod_LT(x1[1], x0[1]), x0[1])=1129_1_gcd_InvokeMethod(1129_0_mod_LT(x0[2], x1[2]), x1[2]) ⇒ COND_1129_2_MAIN_INVOKEMETHOD(TRUE, 1129_1_gcd_InvokeMethod(1129_0_mod_LT(x0[1], x1[1]), x1[1]))≥NonInfC∧COND_1129_2_MAIN_INVOKEMETHOD(TRUE, 1129_1_gcd_InvokeMethod(1129_0_mod_LT(x0[1], x1[1]), x1[1]))≥1129_2_MAIN_INVOKEMETHOD(1129_1_gcd_InvokeMethod(1129_0_mod_LT(x1[1], x0[1]), x0[1]))∧(UIncreasing(1129_2_MAIN_INVOKEMETHOD(1129_1_gcd_InvokeMethod(1129_0_mod_LT(x1[1], x0[1]), x0[1]))), ≥))

We simplified constraint (15) using rules (I), (II), (III), (IV), (IDP_BOOLEAN) which results in the following new constraint:

(16)    (>(x0[0], 0)=TRUE>(x1[0], x0[0])=TRUE>(x1[0], 0)=TRUECOND_1129_2_MAIN_INVOKEMETHOD(TRUE, 1129_1_gcd_InvokeMethod(1129_0_mod_LT(x0[0], x1[0]), x1[0]))≥NonInfC∧COND_1129_2_MAIN_INVOKEMETHOD(TRUE, 1129_1_gcd_InvokeMethod(1129_0_mod_LT(x0[0], x1[0]), x1[0]))≥1129_2_MAIN_INVOKEMETHOD(1129_1_gcd_InvokeMethod(1129_0_mod_LT(x1[0], x0[0]), x0[0]))∧(UIncreasing(1129_2_MAIN_INVOKEMETHOD(1129_1_gcd_InvokeMethod(1129_0_mod_LT(x1[1], x0[1]), x0[1]))), ≥))

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

(17)    (x0[0] + [-1] ≥ 0∧x1[0] + [-1] + [-1]x0[0] ≥ 0∧x1[0] + [-1] ≥ 0 ⇒ (UIncreasing(1129_2_MAIN_INVOKEMETHOD(1129_1_gcd_InvokeMethod(1129_0_mod_LT(x1[1], x0[1]), x0[1]))), ≥)∧[(-1)Bound*bni_22] + [bni_22]x1[0] + [bni_22]x0[0] ≥ 0∧[(-1)bso_23] ≥ 0)

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

(18)    (x0[0] + [-1] ≥ 0∧x1[0] + [-1] + [-1]x0[0] ≥ 0∧x1[0] + [-1] ≥ 0 ⇒ (UIncreasing(1129_2_MAIN_INVOKEMETHOD(1129_1_gcd_InvokeMethod(1129_0_mod_LT(x1[1], x0[1]), x0[1]))), ≥)∧[(-1)Bound*bni_22] + [bni_22]x1[0] + [bni_22]x0[0] ≥ 0∧[(-1)bso_23] ≥ 0)

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

(19)    (x0[0] + [-1] ≥ 0∧x1[0] + [-1] + [-1]x0[0] ≥ 0∧x1[0] + [-1] ≥ 0 ⇒ (UIncreasing(1129_2_MAIN_INVOKEMETHOD(1129_1_gcd_InvokeMethod(1129_0_mod_LT(x1[1], x0[1]), x0[1]))), ≥)∧[(-1)Bound*bni_22] + [bni_22]x1[0] + [bni_22]x0[0] ≥ 0∧[(-1)bso_23] ≥ 0)

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

(20)    (x0[0] ≥ 0∧x1[0] + [-2] + [-1]x0[0] ≥ 0∧x1[0] + [-1] ≥ 0 ⇒ (UIncreasing(1129_2_MAIN_INVOKEMETHOD(1129_1_gcd_InvokeMethod(1129_0_mod_LT(x1[1], x0[1]), x0[1]))), ≥)∧[(-1)Bound*bni_22 + bni_22] + [bni_22]x1[0] + [bni_22]x0[0] ≥ 0∧[(-1)bso_23] ≥ 0)

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

(21)    (x0[0] ≥ 0∧x1[0] ≥ 0∧[1] + x0[0] + x1[0] ≥ 0 ⇒ (UIncreasing(1129_2_MAIN_INVOKEMETHOD(1129_1_gcd_InvokeMethod(1129_0_mod_LT(x1[1], x0[1]), x0[1]))), ≥)∧[(-1)Bound*bni_22 + (3)bni_22] + [(2)bni_22]x0[0] + [bni_22]x1[0] ≥ 0∧[(-1)bso_23] ≥ 0)

For Pair 1129_2_MAIN_INVOKEMETHOD(1129_1_gcd_InvokeMethod(1129_0_mod_LT(x0, x1), x1)) → COND_1129_2_MAIN_INVOKEMETHOD1(&&(>(x1, 0), <=(x1, x0)), 1129_1_gcd_InvokeMethod(1129_0_mod_LT(x0, x1), x1)) the following chains were created:
• We consider the chain 1129_2_MAIN_INVOKEMETHOD(1129_1_gcd_InvokeMethod(1129_0_mod_LT(x0[2], x1[2]), x1[2])) → COND_1129_2_MAIN_INVOKEMETHOD1(&&(>(x1[2], 0), <=(x1[2], x0[2])), 1129_1_gcd_InvokeMethod(1129_0_mod_LT(x0[2], x1[2]), x1[2])), COND_1129_2_MAIN_INVOKEMETHOD1(TRUE, 1129_1_gcd_InvokeMethod(1129_0_mod_LT(x0[3], x1[3]), x1[3])) → 1129_2_MAIN_INVOKEMETHOD(1129_1_gcd_InvokeMethod(1129_0_mod_LT(-(x0[3], x1[3]), x1[3]), x1[3])) which results in the following constraint:

(22)    (&&(>(x1[2], 0), <=(x1[2], x0[2]))=TRUE1129_1_gcd_InvokeMethod(1129_0_mod_LT(x0[2], x1[2]), x1[2])=1129_1_gcd_InvokeMethod(1129_0_mod_LT(x0[3], x1[3]), x1[3]) ⇒ 1129_2_MAIN_INVOKEMETHOD(1129_1_gcd_InvokeMethod(1129_0_mod_LT(x0[2], x1[2]), x1[2]))≥NonInfC∧1129_2_MAIN_INVOKEMETHOD(1129_1_gcd_InvokeMethod(1129_0_mod_LT(x0[2], x1[2]), x1[2]))≥COND_1129_2_MAIN_INVOKEMETHOD1(&&(>(x1[2], 0), <=(x1[2], x0[2])), 1129_1_gcd_InvokeMethod(1129_0_mod_LT(x0[2], x1[2]), x1[2]))∧(UIncreasing(COND_1129_2_MAIN_INVOKEMETHOD1(&&(>(x1[2], 0), <=(x1[2], x0[2])), 1129_1_gcd_InvokeMethod(1129_0_mod_LT(x0[2], x1[2]), x1[2]))), ≥))

We simplified constraint (22) using rules (I), (II), (IV), (IDP_BOOLEAN) which results in the following new constraint:

(23)    (>(x1[2], 0)=TRUE<=(x1[2], x0[2])=TRUE1129_2_MAIN_INVOKEMETHOD(1129_1_gcd_InvokeMethod(1129_0_mod_LT(x0[2], x1[2]), x1[2]))≥NonInfC∧1129_2_MAIN_INVOKEMETHOD(1129_1_gcd_InvokeMethod(1129_0_mod_LT(x0[2], x1[2]), x1[2]))≥COND_1129_2_MAIN_INVOKEMETHOD1(&&(>(x1[2], 0), <=(x1[2], x0[2])), 1129_1_gcd_InvokeMethod(1129_0_mod_LT(x0[2], x1[2]), x1[2]))∧(UIncreasing(COND_1129_2_MAIN_INVOKEMETHOD1(&&(>(x1[2], 0), <=(x1[2], x0[2])), 1129_1_gcd_InvokeMethod(1129_0_mod_LT(x0[2], x1[2]), x1[2]))), ≥))

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

(24)    (x1[2] + [-1] ≥ 0∧x0[2] + [-1]x1[2] ≥ 0 ⇒ (UIncreasing(COND_1129_2_MAIN_INVOKEMETHOD1(&&(>(x1[2], 0), <=(x1[2], x0[2])), 1129_1_gcd_InvokeMethod(1129_0_mod_LT(x0[2], x1[2]), x1[2]))), ≥)∧[(-1)Bound*bni_24] + [bni_24]x1[2] + [bni_24]x0[2] ≥ 0∧[(-1)bso_25] ≥ 0)

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

(25)    (x1[2] + [-1] ≥ 0∧x0[2] + [-1]x1[2] ≥ 0 ⇒ (UIncreasing(COND_1129_2_MAIN_INVOKEMETHOD1(&&(>(x1[2], 0), <=(x1[2], x0[2])), 1129_1_gcd_InvokeMethod(1129_0_mod_LT(x0[2], x1[2]), x1[2]))), ≥)∧[(-1)Bound*bni_24] + [bni_24]x1[2] + [bni_24]x0[2] ≥ 0∧[(-1)bso_25] ≥ 0)

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

(26)    (x1[2] + [-1] ≥ 0∧x0[2] + [-1]x1[2] ≥ 0 ⇒ (UIncreasing(COND_1129_2_MAIN_INVOKEMETHOD1(&&(>(x1[2], 0), <=(x1[2], x0[2])), 1129_1_gcd_InvokeMethod(1129_0_mod_LT(x0[2], x1[2]), x1[2]))), ≥)∧[(-1)Bound*bni_24] + [bni_24]x1[2] + [bni_24]x0[2] ≥ 0∧[(-1)bso_25] ≥ 0)

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

(27)    (x1[2] ≥ 0∧x0[2] + [-1] + [-1]x1[2] ≥ 0 ⇒ (UIncreasing(COND_1129_2_MAIN_INVOKEMETHOD1(&&(>(x1[2], 0), <=(x1[2], x0[2])), 1129_1_gcd_InvokeMethod(1129_0_mod_LT(x0[2], x1[2]), x1[2]))), ≥)∧[(-1)Bound*bni_24 + bni_24] + [bni_24]x1[2] + [bni_24]x0[2] ≥ 0∧[(-1)bso_25] ≥ 0)

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

(28)    (x1[2] ≥ 0∧x0[2] ≥ 0 ⇒ (UIncreasing(COND_1129_2_MAIN_INVOKEMETHOD1(&&(>(x1[2], 0), <=(x1[2], x0[2])), 1129_1_gcd_InvokeMethod(1129_0_mod_LT(x0[2], x1[2]), x1[2]))), ≥)∧[(-1)Bound*bni_24 + (2)bni_24] + [(2)bni_24]x1[2] + [bni_24]x0[2] ≥ 0∧[(-1)bso_25] ≥ 0)

For Pair COND_1129_2_MAIN_INVOKEMETHOD1(TRUE, 1129_1_gcd_InvokeMethod(1129_0_mod_LT(x0, x1), x1)) → 1129_2_MAIN_INVOKEMETHOD(1129_1_gcd_InvokeMethod(1129_0_mod_LT(-(x0, x1), x1), x1)) the following chains were created:
• We consider the chain 1129_2_MAIN_INVOKEMETHOD(1129_1_gcd_InvokeMethod(1129_0_mod_LT(x0[2], x1[2]), x1[2])) → COND_1129_2_MAIN_INVOKEMETHOD1(&&(>(x1[2], 0), <=(x1[2], x0[2])), 1129_1_gcd_InvokeMethod(1129_0_mod_LT(x0[2], x1[2]), x1[2])), COND_1129_2_MAIN_INVOKEMETHOD1(TRUE, 1129_1_gcd_InvokeMethod(1129_0_mod_LT(x0[3], x1[3]), x1[3])) → 1129_2_MAIN_INVOKEMETHOD(1129_1_gcd_InvokeMethod(1129_0_mod_LT(-(x0[3], x1[3]), x1[3]), x1[3])), 1129_2_MAIN_INVOKEMETHOD(1129_1_gcd_InvokeMethod(1129_0_mod_LT(x0[0], x1[0]), x1[0])) → COND_1129_2_MAIN_INVOKEMETHOD(&&(&&(>(x1[0], x0[0]), >(x1[0], 0)), >(x0[0], 0)), 1129_1_gcd_InvokeMethod(1129_0_mod_LT(x0[0], x1[0]), x1[0])) which results in the following constraint:

(29)    (&&(>(x1[2], 0), <=(x1[2], x0[2]))=TRUE1129_1_gcd_InvokeMethod(1129_0_mod_LT(x0[2], x1[2]), x1[2])=1129_1_gcd_InvokeMethod(1129_0_mod_LT(x0[3], x1[3]), x1[3])∧1129_1_gcd_InvokeMethod(1129_0_mod_LT(-(x0[3], x1[3]), x1[3]), x1[3])=1129_1_gcd_InvokeMethod(1129_0_mod_LT(x0[0], x1[0]), x1[0]) ⇒ COND_1129_2_MAIN_INVOKEMETHOD1(TRUE, 1129_1_gcd_InvokeMethod(1129_0_mod_LT(x0[3], x1[3]), x1[3]))≥NonInfC∧COND_1129_2_MAIN_INVOKEMETHOD1(TRUE, 1129_1_gcd_InvokeMethod(1129_0_mod_LT(x0[3], x1[3]), x1[3]))≥1129_2_MAIN_INVOKEMETHOD(1129_1_gcd_InvokeMethod(1129_0_mod_LT(-(x0[3], x1[3]), x1[3]), x1[3]))∧(UIncreasing(1129_2_MAIN_INVOKEMETHOD(1129_1_gcd_InvokeMethod(1129_0_mod_LT(-(x0[3], x1[3]), x1[3]), x1[3]))), ≥))

We simplified constraint (29) using rules (I), (II), (III), (IV), (IDP_BOOLEAN) which results in the following new constraint:

(30)    (>(x1[2], 0)=TRUE<=(x1[2], x0[2])=TRUECOND_1129_2_MAIN_INVOKEMETHOD1(TRUE, 1129_1_gcd_InvokeMethod(1129_0_mod_LT(x0[2], x1[2]), x1[2]))≥NonInfC∧COND_1129_2_MAIN_INVOKEMETHOD1(TRUE, 1129_1_gcd_InvokeMethod(1129_0_mod_LT(x0[2], x1[2]), x1[2]))≥1129_2_MAIN_INVOKEMETHOD(1129_1_gcd_InvokeMethod(1129_0_mod_LT(-(x0[2], x1[2]), x1[2]), x1[2]))∧(UIncreasing(1129_2_MAIN_INVOKEMETHOD(1129_1_gcd_InvokeMethod(1129_0_mod_LT(-(x0[3], x1[3]), x1[3]), x1[3]))), ≥))

We simplified constraint (30) using rule (POLY_CONSTRAINTS) which results in the following new constraint:

(31)    (x1[2] + [-1] ≥ 0∧x0[2] + [-1]x1[2] ≥ 0 ⇒ (UIncreasing(1129_2_MAIN_INVOKEMETHOD(1129_1_gcd_InvokeMethod(1129_0_mod_LT(-(x0[3], x1[3]), x1[3]), x1[3]))), ≥)∧[(-1)Bound*bni_26] + [bni_26]x1[2] + [bni_26]x0[2] ≥ 0∧[(-1)bso_27] + x1[2] ≥ 0)

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

(32)    (x1[2] + [-1] ≥ 0∧x0[2] + [-1]x1[2] ≥ 0 ⇒ (UIncreasing(1129_2_MAIN_INVOKEMETHOD(1129_1_gcd_InvokeMethod(1129_0_mod_LT(-(x0[3], x1[3]), x1[3]), x1[3]))), ≥)∧[(-1)Bound*bni_26] + [bni_26]x1[2] + [bni_26]x0[2] ≥ 0∧[(-1)bso_27] + x1[2] ≥ 0)

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

(33)    (x1[2] + [-1] ≥ 0∧x0[2] + [-1]x1[2] ≥ 0 ⇒ (UIncreasing(1129_2_MAIN_INVOKEMETHOD(1129_1_gcd_InvokeMethod(1129_0_mod_LT(-(x0[3], x1[3]), x1[3]), x1[3]))), ≥)∧[(-1)Bound*bni_26] + [bni_26]x1[2] + [bni_26]x0[2] ≥ 0∧[(-1)bso_27] + x1[2] ≥ 0)

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

(34)    (x1[2] ≥ 0∧x0[2] + [-1] + [-1]x1[2] ≥ 0 ⇒ (UIncreasing(1129_2_MAIN_INVOKEMETHOD(1129_1_gcd_InvokeMethod(1129_0_mod_LT(-(x0[3], x1[3]), x1[3]), x1[3]))), ≥)∧[(-1)Bound*bni_26 + bni_26] + [bni_26]x1[2] + [bni_26]x0[2] ≥ 0∧[1 + (-1)bso_27] + x1[2] ≥ 0)

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

(35)    (x1[2] ≥ 0∧x0[2] ≥ 0 ⇒ (UIncreasing(1129_2_MAIN_INVOKEMETHOD(1129_1_gcd_InvokeMethod(1129_0_mod_LT(-(x0[3], x1[3]), x1[3]), x1[3]))), ≥)∧[(-1)Bound*bni_26 + (2)bni_26] + [(2)bni_26]x1[2] + [bni_26]x0[2] ≥ 0∧[1 + (-1)bso_27] + x1[2] ≥ 0)

• We consider the chain 1129_2_MAIN_INVOKEMETHOD(1129_1_gcd_InvokeMethod(1129_0_mod_LT(x0[2], x1[2]), x1[2])) → COND_1129_2_MAIN_INVOKEMETHOD1(&&(>(x1[2], 0), <=(x1[2], x0[2])), 1129_1_gcd_InvokeMethod(1129_0_mod_LT(x0[2], x1[2]), x1[2])), COND_1129_2_MAIN_INVOKEMETHOD1(TRUE, 1129_1_gcd_InvokeMethod(1129_0_mod_LT(x0[3], x1[3]), x1[3])) → 1129_2_MAIN_INVOKEMETHOD(1129_1_gcd_InvokeMethod(1129_0_mod_LT(-(x0[3], x1[3]), x1[3]), x1[3])), 1129_2_MAIN_INVOKEMETHOD(1129_1_gcd_InvokeMethod(1129_0_mod_LT(x0[2], x1[2]), x1[2])) → COND_1129_2_MAIN_INVOKEMETHOD1(&&(>(x1[2], 0), <=(x1[2], x0[2])), 1129_1_gcd_InvokeMethod(1129_0_mod_LT(x0[2], x1[2]), x1[2])) which results in the following constraint:

(36)    (&&(>(x1[2], 0), <=(x1[2], x0[2]))=TRUE1129_1_gcd_InvokeMethod(1129_0_mod_LT(x0[2], x1[2]), x1[2])=1129_1_gcd_InvokeMethod(1129_0_mod_LT(x0[3], x1[3]), x1[3])∧1129_1_gcd_InvokeMethod(1129_0_mod_LT(-(x0[3], x1[3]), x1[3]), x1[3])=1129_1_gcd_InvokeMethod(1129_0_mod_LT(x0[2]1, x1[2]1), x1[2]1) ⇒ COND_1129_2_MAIN_INVOKEMETHOD1(TRUE, 1129_1_gcd_InvokeMethod(1129_0_mod_LT(x0[3], x1[3]), x1[3]))≥NonInfC∧COND_1129_2_MAIN_INVOKEMETHOD1(TRUE, 1129_1_gcd_InvokeMethod(1129_0_mod_LT(x0[3], x1[3]), x1[3]))≥1129_2_MAIN_INVOKEMETHOD(1129_1_gcd_InvokeMethod(1129_0_mod_LT(-(x0[3], x1[3]), x1[3]), x1[3]))∧(UIncreasing(1129_2_MAIN_INVOKEMETHOD(1129_1_gcd_InvokeMethod(1129_0_mod_LT(-(x0[3], x1[3]), x1[3]), x1[3]))), ≥))

We simplified constraint (36) using rules (I), (II), (III), (IV), (IDP_BOOLEAN) which results in the following new constraint:

(37)    (>(x1[2], 0)=TRUE<=(x1[2], x0[2])=TRUECOND_1129_2_MAIN_INVOKEMETHOD1(TRUE, 1129_1_gcd_InvokeMethod(1129_0_mod_LT(x0[2], x1[2]), x1[2]))≥NonInfC∧COND_1129_2_MAIN_INVOKEMETHOD1(TRUE, 1129_1_gcd_InvokeMethod(1129_0_mod_LT(x0[2], x1[2]), x1[2]))≥1129_2_MAIN_INVOKEMETHOD(1129_1_gcd_InvokeMethod(1129_0_mod_LT(-(x0[2], x1[2]), x1[2]), x1[2]))∧(UIncreasing(1129_2_MAIN_INVOKEMETHOD(1129_1_gcd_InvokeMethod(1129_0_mod_LT(-(x0[3], x1[3]), x1[3]), x1[3]))), ≥))

We simplified constraint (37) using rule (POLY_CONSTRAINTS) which results in the following new constraint:

(38)    (x1[2] + [-1] ≥ 0∧x0[2] + [-1]x1[2] ≥ 0 ⇒ (UIncreasing(1129_2_MAIN_INVOKEMETHOD(1129_1_gcd_InvokeMethod(1129_0_mod_LT(-(x0[3], x1[3]), x1[3]), x1[3]))), ≥)∧[(-1)Bound*bni_26] + [bni_26]x1[2] + [bni_26]x0[2] ≥ 0∧[(-1)bso_27] + x1[2] ≥ 0)

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

(39)    (x1[2] + [-1] ≥ 0∧x0[2] + [-1]x1[2] ≥ 0 ⇒ (UIncreasing(1129_2_MAIN_INVOKEMETHOD(1129_1_gcd_InvokeMethod(1129_0_mod_LT(-(x0[3], x1[3]), x1[3]), x1[3]))), ≥)∧[(-1)Bound*bni_26] + [bni_26]x1[2] + [bni_26]x0[2] ≥ 0∧[(-1)bso_27] + x1[2] ≥ 0)

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

(40)    (x1[2] + [-1] ≥ 0∧x0[2] + [-1]x1[2] ≥ 0 ⇒ (UIncreasing(1129_2_MAIN_INVOKEMETHOD(1129_1_gcd_InvokeMethod(1129_0_mod_LT(-(x0[3], x1[3]), x1[3]), x1[3]))), ≥)∧[(-1)Bound*bni_26] + [bni_26]x1[2] + [bni_26]x0[2] ≥ 0∧[(-1)bso_27] + x1[2] ≥ 0)

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

(41)    (x1[2] ≥ 0∧x0[2] + [-1] + [-1]x1[2] ≥ 0 ⇒ (UIncreasing(1129_2_MAIN_INVOKEMETHOD(1129_1_gcd_InvokeMethod(1129_0_mod_LT(-(x0[3], x1[3]), x1[3]), x1[3]))), ≥)∧[(-1)Bound*bni_26 + bni_26] + [bni_26]x1[2] + [bni_26]x0[2] ≥ 0∧[1 + (-1)bso_27] + x1[2] ≥ 0)

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

(42)    (x1[2] ≥ 0∧x0[2] ≥ 0 ⇒ (UIncreasing(1129_2_MAIN_INVOKEMETHOD(1129_1_gcd_InvokeMethod(1129_0_mod_LT(-(x0[3], x1[3]), x1[3]), x1[3]))), ≥)∧[(-1)Bound*bni_26 + (2)bni_26] + [(2)bni_26]x1[2] + [bni_26]x0[2] ≥ 0∧[1 + (-1)bso_27] + x1[2] ≥ 0)

To summarize, we get the following constraints P for the following pairs.
• 1129_2_MAIN_INVOKEMETHOD(1129_1_gcd_InvokeMethod(1129_0_mod_LT(x0, x1), x1)) → COND_1129_2_MAIN_INVOKEMETHOD(&&(&&(>(x1, x0), >(x1, 0)), >(x0, 0)), 1129_1_gcd_InvokeMethod(1129_0_mod_LT(x0, x1), x1))
• (x0[0] ≥ 0∧x1[0] ≥ 0∧[1] + x0[0] + x1[0] ≥ 0 ⇒ (UIncreasing(COND_1129_2_MAIN_INVOKEMETHOD(&&(&&(>(x1[0], x0[0]), >(x1[0], 0)), >(x0[0], 0)), 1129_1_gcd_InvokeMethod(1129_0_mod_LT(x0[0], x1[0]), x1[0]))), ≥)∧[(-1)Bound*bni_20 + (3)bni_20] + [(2)bni_20]x0[0] + [bni_20]x1[0] ≥ 0∧[(-1)bso_21] ≥ 0)

• COND_1129_2_MAIN_INVOKEMETHOD(TRUE, 1129_1_gcd_InvokeMethod(1129_0_mod_LT(x0, x1), x1)) → 1129_2_MAIN_INVOKEMETHOD(1129_1_gcd_InvokeMethod(1129_0_mod_LT(x1, x0), x0))
• (x0[0] ≥ 0∧x1[0] ≥ 0∧[1] + x0[0] + x1[0] ≥ 0 ⇒ (UIncreasing(1129_2_MAIN_INVOKEMETHOD(1129_1_gcd_InvokeMethod(1129_0_mod_LT(x1[1], x0[1]), x0[1]))), ≥)∧[(-1)Bound*bni_22 + (3)bni_22] + [(2)bni_22]x0[0] + [bni_22]x1[0] ≥ 0∧[(-1)bso_23] ≥ 0)
• (x0[0] ≥ 0∧x1[0] ≥ 0∧[1] + x0[0] + x1[0] ≥ 0 ⇒ (UIncreasing(1129_2_MAIN_INVOKEMETHOD(1129_1_gcd_InvokeMethod(1129_0_mod_LT(x1[1], x0[1]), x0[1]))), ≥)∧[(-1)Bound*bni_22 + (3)bni_22] + [(2)bni_22]x0[0] + [bni_22]x1[0] ≥ 0∧[(-1)bso_23] ≥ 0)

• 1129_2_MAIN_INVOKEMETHOD(1129_1_gcd_InvokeMethod(1129_0_mod_LT(x0, x1), x1)) → COND_1129_2_MAIN_INVOKEMETHOD1(&&(>(x1, 0), <=(x1, x0)), 1129_1_gcd_InvokeMethod(1129_0_mod_LT(x0, x1), x1))
• (x1[2] ≥ 0∧x0[2] ≥ 0 ⇒ (UIncreasing(COND_1129_2_MAIN_INVOKEMETHOD1(&&(>(x1[2], 0), <=(x1[2], x0[2])), 1129_1_gcd_InvokeMethod(1129_0_mod_LT(x0[2], x1[2]), x1[2]))), ≥)∧[(-1)Bound*bni_24 + (2)bni_24] + [(2)bni_24]x1[2] + [bni_24]x0[2] ≥ 0∧[(-1)bso_25] ≥ 0)

• COND_1129_2_MAIN_INVOKEMETHOD1(TRUE, 1129_1_gcd_InvokeMethod(1129_0_mod_LT(x0, x1), x1)) → 1129_2_MAIN_INVOKEMETHOD(1129_1_gcd_InvokeMethod(1129_0_mod_LT(-(x0, x1), x1), x1))
• (x1[2] ≥ 0∧x0[2] ≥ 0 ⇒ (UIncreasing(1129_2_MAIN_INVOKEMETHOD(1129_1_gcd_InvokeMethod(1129_0_mod_LT(-(x0[3], x1[3]), x1[3]), x1[3]))), ≥)∧[(-1)Bound*bni_26 + (2)bni_26] + [(2)bni_26]x1[2] + [bni_26]x0[2] ≥ 0∧[1 + (-1)bso_27] + x1[2] ≥ 0)
• (x1[2] ≥ 0∧x0[2] ≥ 0 ⇒ (UIncreasing(1129_2_MAIN_INVOKEMETHOD(1129_1_gcd_InvokeMethod(1129_0_mod_LT(-(x0[3], x1[3]), x1[3]), x1[3]))), ≥)∧[(-1)Bound*bni_26 + (2)bni_26] + [(2)bni_26]x1[2] + [bni_26]x0[2] ≥ 0∧[1 + (-1)bso_27] + x1[2] ≥ 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(1129_2_MAIN_INVOKEMETHOD(x1)) = [-1] + [-1]x1
POL(1129_1_gcd_InvokeMethod(x1, x2)) = x1
POL(1129_0_mod_LT(x1, x2)) = [-1] + [-1]x2 + [-1]x1
POL(COND_1129_2_MAIN_INVOKEMETHOD(x1, x2)) = [-1] + [-1]x2
POL(&&(x1, x2)) = [-1]
POL(>(x1, x2)) = [-1]
POL(0) = 0
POL(COND_1129_2_MAIN_INVOKEMETHOD1(x1, x2)) = [-1] + [-1]x2
POL(<=(x1, x2)) = [-1]
POL(-(x1, x2)) = x1 + [-1]x2

The following pairs are in P>:

COND_1129_2_MAIN_INVOKEMETHOD1(TRUE, 1129_1_gcd_InvokeMethod(1129_0_mod_LT(x0[3], x1[3]), x1[3])) → 1129_2_MAIN_INVOKEMETHOD(1129_1_gcd_InvokeMethod(1129_0_mod_LT(-(x0[3], x1[3]), x1[3]), x1[3]))

The following pairs are in Pbound:

1129_2_MAIN_INVOKEMETHOD(1129_1_gcd_InvokeMethod(1129_0_mod_LT(x0[0], x1[0]), x1[0])) → COND_1129_2_MAIN_INVOKEMETHOD(&&(&&(>(x1[0], x0[0]), >(x1[0], 0)), >(x0[0], 0)), 1129_1_gcd_InvokeMethod(1129_0_mod_LT(x0[0], x1[0]), x1[0]))
COND_1129_2_MAIN_INVOKEMETHOD(TRUE, 1129_1_gcd_InvokeMethod(1129_0_mod_LT(x0[1], x1[1]), x1[1])) → 1129_2_MAIN_INVOKEMETHOD(1129_1_gcd_InvokeMethod(1129_0_mod_LT(x1[1], x0[1]), x0[1]))
1129_2_MAIN_INVOKEMETHOD(1129_1_gcd_InvokeMethod(1129_0_mod_LT(x0[2], x1[2]), x1[2])) → COND_1129_2_MAIN_INVOKEMETHOD1(&&(>(x1[2], 0), <=(x1[2], x0[2])), 1129_1_gcd_InvokeMethod(1129_0_mod_LT(x0[2], x1[2]), x1[2]))
COND_1129_2_MAIN_INVOKEMETHOD1(TRUE, 1129_1_gcd_InvokeMethod(1129_0_mod_LT(x0[3], x1[3]), x1[3])) → 1129_2_MAIN_INVOKEMETHOD(1129_1_gcd_InvokeMethod(1129_0_mod_LT(-(x0[3], x1[3]), x1[3]), x1[3]))

The following pairs are in P:

1129_2_MAIN_INVOKEMETHOD(1129_1_gcd_InvokeMethod(1129_0_mod_LT(x0[0], x1[0]), x1[0])) → COND_1129_2_MAIN_INVOKEMETHOD(&&(&&(>(x1[0], x0[0]), >(x1[0], 0)), >(x0[0], 0)), 1129_1_gcd_InvokeMethod(1129_0_mod_LT(x0[0], x1[0]), x1[0]))
COND_1129_2_MAIN_INVOKEMETHOD(TRUE, 1129_1_gcd_InvokeMethod(1129_0_mod_LT(x0[1], x1[1]), x1[1])) → 1129_2_MAIN_INVOKEMETHOD(1129_1_gcd_InvokeMethod(1129_0_mod_LT(x1[1], x0[1]), x0[1]))
1129_2_MAIN_INVOKEMETHOD(1129_1_gcd_InvokeMethod(1129_0_mod_LT(x0[2], x1[2]), x1[2])) → COND_1129_2_MAIN_INVOKEMETHOD1(&&(>(x1[2], 0), <=(x1[2], x0[2])), 1129_1_gcd_InvokeMethod(1129_0_mod_LT(x0[2], x1[2]), x1[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

### (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:

Boolean, Integer

R is empty.

The integer pair graph contains the following rules and edges:
(0): 1129_2_MAIN_INVOKEMETHOD(1129_1_gcd_InvokeMethod(1129_0_mod_LT(x0[0], x1[0]), x1[0])) → COND_1129_2_MAIN_INVOKEMETHOD(x1[0] > x0[0] && x1[0] > 0 && x0[0] > 0, 1129_1_gcd_InvokeMethod(1129_0_mod_LT(x0[0], x1[0]), x1[0]))
(1): COND_1129_2_MAIN_INVOKEMETHOD(TRUE, 1129_1_gcd_InvokeMethod(1129_0_mod_LT(x0[1], x1[1]), x1[1])) → 1129_2_MAIN_INVOKEMETHOD(1129_1_gcd_InvokeMethod(1129_0_mod_LT(x1[1], x0[1]), x0[1]))
(2): 1129_2_MAIN_INVOKEMETHOD(1129_1_gcd_InvokeMethod(1129_0_mod_LT(x0[2], x1[2]), x1[2])) → COND_1129_2_MAIN_INVOKEMETHOD1(x1[2] > 0 && x1[2] <= x0[2], 1129_1_gcd_InvokeMethod(1129_0_mod_LT(x0[2], x1[2]), x1[2]))

(1) -> (0), if ((1129_1_gcd_InvokeMethod(1129_0_mod_LT(x1[1], x0[1]), x0[1]) →* 1129_1_gcd_InvokeMethod(1129_0_mod_LT(x0[0], x1[0]), x1[0])))

(0) -> (1), if ((x1[0] > x0[0] && x1[0] > 0 && x0[0] > 0* TRUE)∧(1129_1_gcd_InvokeMethod(1129_0_mod_LT(x0[0], x1[0]), x1[0]) →* 1129_1_gcd_InvokeMethod(1129_0_mod_LT(x0[1], x1[1]), x1[1])))

(1) -> (2), if ((1129_1_gcd_InvokeMethod(1129_0_mod_LT(x1[1], x0[1]), x0[1]) →* 1129_1_gcd_InvokeMethod(1129_0_mod_LT(x0[2], x1[2]), x1[2])))

The set Q is empty.

### (7) IDependencyGraphProof (EQUIVALENT transformation)

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

### (8) Obligation:

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

The following domains are used:

Boolean, Integer

R is empty.

The integer pair graph contains the following rules and edges:
(1): COND_1129_2_MAIN_INVOKEMETHOD(TRUE, 1129_1_gcd_InvokeMethod(1129_0_mod_LT(x0[1], x1[1]), x1[1])) → 1129_2_MAIN_INVOKEMETHOD(1129_1_gcd_InvokeMethod(1129_0_mod_LT(x1[1], x0[1]), x0[1]))
(0): 1129_2_MAIN_INVOKEMETHOD(1129_1_gcd_InvokeMethod(1129_0_mod_LT(x0[0], x1[0]), x1[0])) → COND_1129_2_MAIN_INVOKEMETHOD(x1[0] > x0[0] && x1[0] > 0 && x0[0] > 0, 1129_1_gcd_InvokeMethod(1129_0_mod_LT(x0[0], x1[0]), x1[0]))

(1) -> (0), if ((1129_1_gcd_InvokeMethod(1129_0_mod_LT(x1[1], x0[1]), x0[1]) →* 1129_1_gcd_InvokeMethod(1129_0_mod_LT(x0[0], x1[0]), x1[0])))

(0) -> (1), if ((x1[0] > x0[0] && x1[0] > 0 && x0[0] > 0* TRUE)∧(1129_1_gcd_InvokeMethod(1129_0_mod_LT(x0[0], x1[0]), x1[0]) →* 1129_1_gcd_InvokeMethod(1129_0_mod_LT(x0[1], x1[1]), x1[1])))

The set Q is empty.

### (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 COND_1129_2_MAIN_INVOKEMETHOD(TRUE, 1129_1_gcd_InvokeMethod(1129_0_mod_LT(x0[1], x1[1]), x1[1])) → 1129_2_MAIN_INVOKEMETHOD(1129_1_gcd_InvokeMethod(1129_0_mod_LT(x1[1], x0[1]), x0[1])) the following chains were created:
• We consider the chain 1129_2_MAIN_INVOKEMETHOD(1129_1_gcd_InvokeMethod(1129_0_mod_LT(x0[0], x1[0]), x1[0])) → COND_1129_2_MAIN_INVOKEMETHOD(&&(&&(>(x1[0], x0[0]), >(x1[0], 0)), >(x0[0], 0)), 1129_1_gcd_InvokeMethod(1129_0_mod_LT(x0[0], x1[0]), x1[0])), COND_1129_2_MAIN_INVOKEMETHOD(TRUE, 1129_1_gcd_InvokeMethod(1129_0_mod_LT(x0[1], x1[1]), x1[1])) → 1129_2_MAIN_INVOKEMETHOD(1129_1_gcd_InvokeMethod(1129_0_mod_LT(x1[1], x0[1]), x0[1])), 1129_2_MAIN_INVOKEMETHOD(1129_1_gcd_InvokeMethod(1129_0_mod_LT(x0[0], x1[0]), x1[0])) → COND_1129_2_MAIN_INVOKEMETHOD(&&(&&(>(x1[0], x0[0]), >(x1[0], 0)), >(x0[0], 0)), 1129_1_gcd_InvokeMethod(1129_0_mod_LT(x0[0], x1[0]), x1[0])) which results in the following constraint:

(1)    (&&(&&(>(x1[0], x0[0]), >(x1[0], 0)), >(x0[0], 0))=TRUE1129_1_gcd_InvokeMethod(1129_0_mod_LT(x0[0], x1[0]), x1[0])=1129_1_gcd_InvokeMethod(1129_0_mod_LT(x0[1], x1[1]), x1[1])∧1129_1_gcd_InvokeMethod(1129_0_mod_LT(x1[1], x0[1]), x0[1])=1129_1_gcd_InvokeMethod(1129_0_mod_LT(x0[0]1, x1[0]1), x1[0]1) ⇒ COND_1129_2_MAIN_INVOKEMETHOD(TRUE, 1129_1_gcd_InvokeMethod(1129_0_mod_LT(x0[1], x1[1]), x1[1]))≥NonInfC∧COND_1129_2_MAIN_INVOKEMETHOD(TRUE, 1129_1_gcd_InvokeMethod(1129_0_mod_LT(x0[1], x1[1]), x1[1]))≥1129_2_MAIN_INVOKEMETHOD(1129_1_gcd_InvokeMethod(1129_0_mod_LT(x1[1], x0[1]), x0[1]))∧(UIncreasing(1129_2_MAIN_INVOKEMETHOD(1129_1_gcd_InvokeMethod(1129_0_mod_LT(x1[1], x0[1]), x0[1]))), ≥))

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

(2)    (>(x0[0], 0)=TRUE>(x1[0], x0[0])=TRUE>(x1[0], 0)=TRUECOND_1129_2_MAIN_INVOKEMETHOD(TRUE, 1129_1_gcd_InvokeMethod(1129_0_mod_LT(x0[0], x1[0]), x1[0]))≥NonInfC∧COND_1129_2_MAIN_INVOKEMETHOD(TRUE, 1129_1_gcd_InvokeMethod(1129_0_mod_LT(x0[0], x1[0]), x1[0]))≥1129_2_MAIN_INVOKEMETHOD(1129_1_gcd_InvokeMethod(1129_0_mod_LT(x1[0], x0[0]), x0[0]))∧(UIncreasing(1129_2_MAIN_INVOKEMETHOD(1129_1_gcd_InvokeMethod(1129_0_mod_LT(x1[1], x0[1]), x0[1]))), ≥))

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

(3)    (x0[0] + [-1] ≥ 0∧x1[0] + [-1] + [-1]x0[0] ≥ 0∧x1[0] + [-1] ≥ 0 ⇒ (UIncreasing(1129_2_MAIN_INVOKEMETHOD(1129_1_gcd_InvokeMethod(1129_0_mod_LT(x1[1], x0[1]), x0[1]))), ≥)∧[(-2)bni_16 + (-1)Bound*bni_16] + [(2)bni_16]x1[0] + [bni_16]x0[0] ≥ 0∧[-1 + (-1)bso_17] + x1[0] + [-1]x0[0] ≥ 0)

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

(4)    (x0[0] + [-1] ≥ 0∧x1[0] + [-1] + [-1]x0[0] ≥ 0∧x1[0] + [-1] ≥ 0 ⇒ (UIncreasing(1129_2_MAIN_INVOKEMETHOD(1129_1_gcd_InvokeMethod(1129_0_mod_LT(x1[1], x0[1]), x0[1]))), ≥)∧[(-2)bni_16 + (-1)Bound*bni_16] + [(2)bni_16]x1[0] + [bni_16]x0[0] ≥ 0∧[-1 + (-1)bso_17] + x1[0] + [-1]x0[0] ≥ 0)

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

(5)    (x0[0] + [-1] ≥ 0∧x1[0] + [-1] + [-1]x0[0] ≥ 0∧x1[0] + [-1] ≥ 0 ⇒ (UIncreasing(1129_2_MAIN_INVOKEMETHOD(1129_1_gcd_InvokeMethod(1129_0_mod_LT(x1[1], x0[1]), x0[1]))), ≥)∧[(-2)bni_16 + (-1)Bound*bni_16] + [(2)bni_16]x1[0] + [bni_16]x0[0] ≥ 0∧[-1 + (-1)bso_17] + x1[0] + [-1]x0[0] ≥ 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∧x1[0] + [-1] ≥ 0 ⇒ (UIncreasing(1129_2_MAIN_INVOKEMETHOD(1129_1_gcd_InvokeMethod(1129_0_mod_LT(x1[1], x0[1]), x0[1]))), ≥)∧[(-1)bni_16 + (-1)Bound*bni_16] + [(2)bni_16]x1[0] + [bni_16]x0[0] ≥ 0∧[-2 + (-1)bso_17] + x1[0] + [-1]x0[0] ≥ 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] + x1[0] ≥ 0 ⇒ (UIncreasing(1129_2_MAIN_INVOKEMETHOD(1129_1_gcd_InvokeMethod(1129_0_mod_LT(x1[1], x0[1]), x0[1]))), ≥)∧[(3)bni_16 + (-1)Bound*bni_16] + [(3)bni_16]x0[0] + [(2)bni_16]x1[0] ≥ 0∧[(-1)bso_17] + x1[0] ≥ 0)

For Pair 1129_2_MAIN_INVOKEMETHOD(1129_1_gcd_InvokeMethod(1129_0_mod_LT(x0[0], x1[0]), x1[0])) → COND_1129_2_MAIN_INVOKEMETHOD(&&(&&(>(x1[0], x0[0]), >(x1[0], 0)), >(x0[0], 0)), 1129_1_gcd_InvokeMethod(1129_0_mod_LT(x0[0], x1[0]), x1[0])) the following chains were created:
• We consider the chain 1129_2_MAIN_INVOKEMETHOD(1129_1_gcd_InvokeMethod(1129_0_mod_LT(x0[0], x1[0]), x1[0])) → COND_1129_2_MAIN_INVOKEMETHOD(&&(&&(>(x1[0], x0[0]), >(x1[0], 0)), >(x0[0], 0)), 1129_1_gcd_InvokeMethod(1129_0_mod_LT(x0[0], x1[0]), x1[0])), COND_1129_2_MAIN_INVOKEMETHOD(TRUE, 1129_1_gcd_InvokeMethod(1129_0_mod_LT(x0[1], x1[1]), x1[1])) → 1129_2_MAIN_INVOKEMETHOD(1129_1_gcd_InvokeMethod(1129_0_mod_LT(x1[1], x0[1]), x0[1])) which results in the following constraint:

(8)    (&&(&&(>(x1[0], x0[0]), >(x1[0], 0)), >(x0[0], 0))=TRUE1129_1_gcd_InvokeMethod(1129_0_mod_LT(x0[0], x1[0]), x1[0])=1129_1_gcd_InvokeMethod(1129_0_mod_LT(x0[1], x1[1]), x1[1]) ⇒ 1129_2_MAIN_INVOKEMETHOD(1129_1_gcd_InvokeMethod(1129_0_mod_LT(x0[0], x1[0]), x1[0]))≥NonInfC∧1129_2_MAIN_INVOKEMETHOD(1129_1_gcd_InvokeMethod(1129_0_mod_LT(x0[0], x1[0]), x1[0]))≥COND_1129_2_MAIN_INVOKEMETHOD(&&(&&(>(x1[0], x0[0]), >(x1[0], 0)), >(x0[0], 0)), 1129_1_gcd_InvokeMethod(1129_0_mod_LT(x0[0], x1[0]), x1[0]))∧(UIncreasing(COND_1129_2_MAIN_INVOKEMETHOD(&&(&&(>(x1[0], x0[0]), >(x1[0], 0)), >(x0[0], 0)), 1129_1_gcd_InvokeMethod(1129_0_mod_LT(x0[0], x1[0]), x1[0]))), ≥))

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

(9)    (>(x0[0], 0)=TRUE>(x1[0], x0[0])=TRUE>(x1[0], 0)=TRUE1129_2_MAIN_INVOKEMETHOD(1129_1_gcd_InvokeMethod(1129_0_mod_LT(x0[0], x1[0]), x1[0]))≥NonInfC∧1129_2_MAIN_INVOKEMETHOD(1129_1_gcd_InvokeMethod(1129_0_mod_LT(x0[0], x1[0]), x1[0]))≥COND_1129_2_MAIN_INVOKEMETHOD(&&(&&(>(x1[0], x0[0]), >(x1[0], 0)), >(x0[0], 0)), 1129_1_gcd_InvokeMethod(1129_0_mod_LT(x0[0], x1[0]), x1[0]))∧(UIncreasing(COND_1129_2_MAIN_INVOKEMETHOD(&&(&&(>(x1[0], x0[0]), >(x1[0], 0)), >(x0[0], 0)), 1129_1_gcd_InvokeMethod(1129_0_mod_LT(x0[0], x1[0]), x1[0]))), ≥))

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

(10)    (x0[0] + [-1] ≥ 0∧x1[0] + [-1] + [-1]x0[0] ≥ 0∧x1[0] + [-1] ≥ 0 ⇒ (UIncreasing(COND_1129_2_MAIN_INVOKEMETHOD(&&(&&(>(x1[0], x0[0]), >(x1[0], 0)), >(x0[0], 0)), 1129_1_gcd_InvokeMethod(1129_0_mod_LT(x0[0], x1[0]), x1[0]))), ≥)∧[(-1)bni_18 + (-1)Bound*bni_18] + [(2)bni_18]x1[0] + [bni_18]x0[0] ≥ 0∧[1 + (-1)bso_19] ≥ 0)

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

(11)    (x0[0] + [-1] ≥ 0∧x1[0] + [-1] + [-1]x0[0] ≥ 0∧x1[0] + [-1] ≥ 0 ⇒ (UIncreasing(COND_1129_2_MAIN_INVOKEMETHOD(&&(&&(>(x1[0], x0[0]), >(x1[0], 0)), >(x0[0], 0)), 1129_1_gcd_InvokeMethod(1129_0_mod_LT(x0[0], x1[0]), x1[0]))), ≥)∧[(-1)bni_18 + (-1)Bound*bni_18] + [(2)bni_18]x1[0] + [bni_18]x0[0] ≥ 0∧[1 + (-1)bso_19] ≥ 0)

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

(12)    (x0[0] + [-1] ≥ 0∧x1[0] + [-1] + [-1]x0[0] ≥ 0∧x1[0] + [-1] ≥ 0 ⇒ (UIncreasing(COND_1129_2_MAIN_INVOKEMETHOD(&&(&&(>(x1[0], x0[0]), >(x1[0], 0)), >(x0[0], 0)), 1129_1_gcd_InvokeMethod(1129_0_mod_LT(x0[0], x1[0]), x1[0]))), ≥)∧[(-1)bni_18 + (-1)Bound*bni_18] + [(2)bni_18]x1[0] + [bni_18]x0[0] ≥ 0∧[1 + (-1)bso_19] ≥ 0)

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

(13)    (x0[0] ≥ 0∧x1[0] + [-2] + [-1]x0[0] ≥ 0∧x1[0] + [-1] ≥ 0 ⇒ (UIncreasing(COND_1129_2_MAIN_INVOKEMETHOD(&&(&&(>(x1[0], x0[0]), >(x1[0], 0)), >(x0[0], 0)), 1129_1_gcd_InvokeMethod(1129_0_mod_LT(x0[0], x1[0]), x1[0]))), ≥)∧[(-1)Bound*bni_18] + [(2)bni_18]x1[0] + [bni_18]x0[0] ≥ 0∧[1 + (-1)bso_19] ≥ 0)

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

(14)    (x0[0] ≥ 0∧x1[0] ≥ 0∧[1] + x0[0] + x1[0] ≥ 0 ⇒ (UIncreasing(COND_1129_2_MAIN_INVOKEMETHOD(&&(&&(>(x1[0], x0[0]), >(x1[0], 0)), >(x0[0], 0)), 1129_1_gcd_InvokeMethod(1129_0_mod_LT(x0[0], x1[0]), x1[0]))), ≥)∧[(4)bni_18 + (-1)Bound*bni_18] + [(3)bni_18]x0[0] + [(2)bni_18]x1[0] ≥ 0∧[1 + (-1)bso_19] ≥ 0)

To summarize, we get the following constraints P for the following pairs.
• COND_1129_2_MAIN_INVOKEMETHOD(TRUE, 1129_1_gcd_InvokeMethod(1129_0_mod_LT(x0[1], x1[1]), x1[1])) → 1129_2_MAIN_INVOKEMETHOD(1129_1_gcd_InvokeMethod(1129_0_mod_LT(x1[1], x0[1]), x0[1]))
• (x0[0] ≥ 0∧x1[0] ≥ 0∧[1] + x0[0] + x1[0] ≥ 0 ⇒ (UIncreasing(1129_2_MAIN_INVOKEMETHOD(1129_1_gcd_InvokeMethod(1129_0_mod_LT(x1[1], x0[1]), x0[1]))), ≥)∧[(3)bni_16 + (-1)Bound*bni_16] + [(3)bni_16]x0[0] + [(2)bni_16]x1[0] ≥ 0∧[(-1)bso_17] + x1[0] ≥ 0)

• 1129_2_MAIN_INVOKEMETHOD(1129_1_gcd_InvokeMethod(1129_0_mod_LT(x0[0], x1[0]), x1[0])) → COND_1129_2_MAIN_INVOKEMETHOD(&&(&&(>(x1[0], x0[0]), >(x1[0], 0)), >(x0[0], 0)), 1129_1_gcd_InvokeMethod(1129_0_mod_LT(x0[0], x1[0]), x1[0]))
• (x0[0] ≥ 0∧x1[0] ≥ 0∧[1] + x0[0] + x1[0] ≥ 0 ⇒ (UIncreasing(COND_1129_2_MAIN_INVOKEMETHOD(&&(&&(>(x1[0], x0[0]), >(x1[0], 0)), >(x0[0], 0)), 1129_1_gcd_InvokeMethod(1129_0_mod_LT(x0[0], x1[0]), x1[0]))), ≥)∧[(4)bni_18 + (-1)Bound*bni_18] + [(3)bni_18]x0[0] + [(2)bni_18]x1[0] ≥ 0∧[1 + (-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) = [1]
POL(FALSE) = [1]
POL(COND_1129_2_MAIN_INVOKEMETHOD(x1, x2)) = [-1] + [-1]x2 + [-1]x1
POL(1129_1_gcd_InvokeMethod(x1, x2)) = [1] + [-1]x2 + x1
POL(1129_0_mod_LT(x1, x2)) = [-1] + [-1]x2 + [-1]x1
POL(1129_2_MAIN_INVOKEMETHOD(x1)) = [-1] + [-1]x1
POL(&&(x1, x2)) = [1]
POL(>(x1, x2)) = [-1]
POL(0) = 0

The following pairs are in P>:

1129_2_MAIN_INVOKEMETHOD(1129_1_gcd_InvokeMethod(1129_0_mod_LT(x0[0], x1[0]), x1[0])) → COND_1129_2_MAIN_INVOKEMETHOD(&&(&&(>(x1[0], x0[0]), >(x1[0], 0)), >(x0[0], 0)), 1129_1_gcd_InvokeMethod(1129_0_mod_LT(x0[0], x1[0]), x1[0]))

The following pairs are in Pbound:

COND_1129_2_MAIN_INVOKEMETHOD(TRUE, 1129_1_gcd_InvokeMethod(1129_0_mod_LT(x0[1], x1[1]), x1[1])) → 1129_2_MAIN_INVOKEMETHOD(1129_1_gcd_InvokeMethod(1129_0_mod_LT(x1[1], x0[1]), x0[1]))
1129_2_MAIN_INVOKEMETHOD(1129_1_gcd_InvokeMethod(1129_0_mod_LT(x0[0], x1[0]), x1[0])) → COND_1129_2_MAIN_INVOKEMETHOD(&&(&&(>(x1[0], x0[0]), >(x1[0], 0)), >(x0[0], 0)), 1129_1_gcd_InvokeMethod(1129_0_mod_LT(x0[0], x1[0]), x1[0]))

The following pairs are in P:

COND_1129_2_MAIN_INVOKEMETHOD(TRUE, 1129_1_gcd_InvokeMethod(1129_0_mod_LT(x0[1], x1[1]), x1[1])) → 1129_2_MAIN_INVOKEMETHOD(1129_1_gcd_InvokeMethod(1129_0_mod_LT(x1[1], x0[1]), x0[1]))

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

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

### (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:
none

R is empty.

The integer pair graph contains the following rules and edges:
(1): COND_1129_2_MAIN_INVOKEMETHOD(TRUE, 1129_1_gcd_InvokeMethod(1129_0_mod_LT(x0[1], x1[1]), x1[1])) → 1129_2_MAIN_INVOKEMETHOD(1129_1_gcd_InvokeMethod(1129_0_mod_LT(x1[1], x0[1]), x0[1]))

The set Q is empty.

### (11) IDependencyGraphProof (EQUIVALENT transformation)

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