(0) Obligation:

JBC Problem based on JBC Program:
Manifest-Version: 1.0 Created-By: 1.6.0_16 (Sun Microsystems Inc.) Main-Class: GCD2
public class GCD2 {
public static int mod(int a, int b) {
if (a == b) {
return 0;
}
while(a>b) {
a -= b;
}
return a;
}

public static int gcd(int a, int b) {
int tmp;
while(b != 0 && a >= 0 && b >= 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) JBCToGraph (SOUND transformation)

Constructed TerminationGraph.

(2) Obligation:

Termination Graph based on JBC Program:
GCD2.main([Ljava/lang/String;)V: Graph of 218 nodes with 1 SCC.


(3) TerminationGraphToSCCProof (SOUND transformation)

Splitted TerminationGraph to 1 SCCs.

(4) Obligation:

SCC of termination graph based on JBC Program.
SCC contains nodes from the following methods: GCD2.main([Ljava/lang/String;)V
SCC calls the following helper methods:
Performed SCC analyses: UsedFieldsAnalysis

(5) SCCToIDPv1Proof (SOUND transformation)

Transformed FIGraph SCCs to IDPs. Log:

Generated 43 rules for P and 0 rules for R.


P rules:
1001_0_gcd_EQ(EOS(STATIC_1001), i196, i218, i218) → 1003_0_gcd_EQ(EOS(STATIC_1003), i196, i218, i218)
1003_0_gcd_EQ(EOS(STATIC_1003), i196, i218, i218) → 1006_0_gcd_Load(EOS(STATIC_1006), i196, i218) | !(=(i218, 0))
1006_0_gcd_Load(EOS(STATIC_1006), i196, i218) → 1010_0_gcd_LT(EOS(STATIC_1010), i196, i218, i196)
1010_0_gcd_LT(EOS(STATIC_1010), i196, i218, i196) → 1013_0_gcd_Load(EOS(STATIC_1013), i196, i218) | >=(i196, 0)
1013_0_gcd_Load(EOS(STATIC_1013), i196, i218) → 1017_0_gcd_LT(EOS(STATIC_1017), i196, i218, i218)
1017_0_gcd_LT(EOS(STATIC_1017), i196, i222, i222) → 1022_0_gcd_LT(EOS(STATIC_1022), i196, i222, i222)
1022_0_gcd_LT(EOS(STATIC_1022), i196, i222, i222) → 1029_0_gcd_Load(EOS(STATIC_1029), i196, i222) | >=(i222, 0)
1029_0_gcd_Load(EOS(STATIC_1029), i196, i222) → 1033_0_gcd_Store(EOS(STATIC_1033), i196, i222, i222)
1033_0_gcd_Store(EOS(STATIC_1033), i196, i222, i222) → 1037_0_gcd_Load(EOS(STATIC_1037), i196, i222, i222)
1037_0_gcd_Load(EOS(STATIC_1037), i196, i222, i222) → 1040_0_gcd_Load(EOS(STATIC_1040), i222, i222, i196)
1040_0_gcd_Load(EOS(STATIC_1040), i222, i222, i196) → 1044_0_gcd_InvokeMethod(EOS(STATIC_1044), i222, i196, i222)
1044_0_gcd_InvokeMethod(EOS(STATIC_1044), i222, i196, i222) → 1046_0_mod_Load(EOS(STATIC_1046), i222, i196, i222, i196, i222)
1046_0_mod_Load(EOS(STATIC_1046), i222, i196, i222, i196, i222) → 1048_0_mod_Load(EOS(STATIC_1048), i222, i196, i222, i196, i222, i196)
1048_0_mod_Load(EOS(STATIC_1048), i222, i196, i222, i196, i222, i196) → 1050_0_mod_NE(EOS(STATIC_1050), i222, i196, i222, i196, i222, i196, i222)
1050_0_mod_NE(EOS(STATIC_1050), i222, i196, i222, i196, i222, i196, i222) → 1052_0_mod_NE(EOS(STATIC_1052), i222, i196, i222, i196, i222, i196, i222)
1050_0_mod_NE(EOS(STATIC_1050), i222, i222, i222, i222, i222, i222, i222) → 1053_0_mod_NE(EOS(STATIC_1053), i222, i222, i222, i222, i222, i222, i222)
1052_0_mod_NE(EOS(STATIC_1052), i222, i196, i222, i196, i222, i196, i222) → 1055_0_mod_Load(EOS(STATIC_1055), i222, i196, i222, i196, i222) | !(=(i196, i222))
1055_0_mod_Load(EOS(STATIC_1055), i222, i196, i222, i196, i222) → 1104_0_mod_Load(EOS(STATIC_1104), i222, i196, i222, i196, i222)
1104_0_mod_Load(EOS(STATIC_1104), i222, i196, i222, i229, i222) → 1111_0_mod_Load(EOS(STATIC_1111), i222, i196, i222, i229, i222, i229)
1111_0_mod_Load(EOS(STATIC_1111), i222, i196, i222, i229, i222, i229) → 1451_0_mod_LE(EOS(STATIC_1451), i222, i196, i222, i229, i222, i229, i222)
1451_0_mod_LE(EOS(STATIC_1451), i222, i196, i222, i229, i222, i229, i222) → 1452_0_mod_LE(EOS(STATIC_1452), i222, i196, i222, i229, i222, i229, i222)
1451_0_mod_LE(EOS(STATIC_1451), i222, i196, i222, i229, i222, i229, i222) → 1453_0_mod_LE(EOS(STATIC_1453), i222, i196, i222, i229, i222, i229, i222)
1452_0_mod_LE(EOS(STATIC_1452), i222, i196, i222, i229, i222, i229, i222) → 1455_0_mod_Load(EOS(STATIC_1455), i222, i196, i222, i229) | <=(i229, i222)
1455_0_mod_Load(EOS(STATIC_1455), i222, i196, i222, i229) → 1458_0_mod_Return(EOS(STATIC_1458), i222, i196, i222, i229)
1458_0_mod_Return(EOS(STATIC_1458), i222, i196, i222, i229) → 1462_0_gcd_Store(EOS(STATIC_1462), i222, i229)
1462_0_gcd_Store(EOS(STATIC_1462), i222, i229) → 1465_0_gcd_Load(EOS(STATIC_1465), i229, i222)
1465_0_gcd_Load(EOS(STATIC_1465), i229, i222) → 1468_0_gcd_Store(EOS(STATIC_1468), i229, i222)
1468_0_gcd_Store(EOS(STATIC_1468), i229, i222) → 1471_0_gcd_JMP(EOS(STATIC_1471), i222, i229)
1471_0_gcd_JMP(EOS(STATIC_1471), i222, i229) → 1477_0_gcd_Load(EOS(STATIC_1477), i222, i229)
1477_0_gcd_Load(EOS(STATIC_1477), i222, i229) → 963_0_gcd_Load(EOS(STATIC_963), i222, i229)
963_0_gcd_Load(EOS(STATIC_963), i196, i197) → 1001_0_gcd_EQ(EOS(STATIC_1001), i196, i197, i197)
1453_0_mod_LE(EOS(STATIC_1453), i222, i196, i222, i229, i222, i229, i222) → 1457_0_mod_Load(EOS(STATIC_1457), i222, i196, i222, i229, i222) | >(i229, i222)
1457_0_mod_Load(EOS(STATIC_1457), i222, i196, i222, i229, i222) → 1460_0_mod_Load(EOS(STATIC_1460), i222, i196, i222, i222, i229)
1460_0_mod_Load(EOS(STATIC_1460), i222, i196, i222, i222, i229) → 1464_0_mod_IntArithmetic(EOS(STATIC_1464), i222, i196, i222, i222, i229, i222)
1464_0_mod_IntArithmetic(EOS(STATIC_1464), i222, i196, i222, i222, i229, i222) → 1467_0_mod_Store(EOS(STATIC_1467), i222, i196, i222, i222, -(i229, i222)) | >(i222, 0)
1467_0_mod_Store(EOS(STATIC_1467), i222, i196, i222, i222, i452) → 1469_0_mod_JMP(EOS(STATIC_1469), i222, i196, i222, i452, i222)
1469_0_mod_JMP(EOS(STATIC_1469), i222, i196, i222, i452, i222) → 1474_0_mod_Load(EOS(STATIC_1474), i222, i196, i222, i452, i222)
1474_0_mod_Load(EOS(STATIC_1474), i222, i196, i222, i452, i222) → 1104_0_mod_Load(EOS(STATIC_1104), i222, i196, i222, i452, i222)
1053_0_mod_NE(EOS(STATIC_1053), i222, i222, i222, i222, i222, i222, i222) → 1057_0_mod_ConstantStackPush(EOS(STATIC_1057), i222, i222, i222, i222, i222)
1057_0_mod_ConstantStackPush(EOS(STATIC_1057), i222, i222, i222, i222, i222) → 1060_0_mod_Return(EOS(STATIC_1060), i222, i222, i222, i222, i222, 0)
1060_0_mod_Return(EOS(STATIC_1060), i222, i222, i222, i222, i222, matching1) → 1064_0_gcd_Store(EOS(STATIC_1064), i222, 0) | =(matching1, 0)
1064_0_gcd_Store(EOS(STATIC_1064), i222, matching1) → 1084_0_gcd_Store(EOS(STATIC_1084), i222, 0) | =(matching1, 0)
1084_0_gcd_Store(EOS(STATIC_1084), i222, i196) → 1462_0_gcd_Store(EOS(STATIC_1462), i222, i196)
R rules:

Combined rules. Obtained 4 conditional rules for P and 0 conditional rules for R.


P rules:
1001_0_gcd_EQ(EOS(STATIC_1001), x0, x1, x1) → 1451_0_mod_LE(EOS(STATIC_1451), x1, x0, x1, x0, x1, x0, x1) | &&(&&(>(x1, 0), >(+(x0, 1), 0)), !(=(x0, x1)))
1451_0_mod_LE(EOS(STATIC_1451), x0, x1, x0, x2, x0, x2, x0) → 1001_0_gcd_EQ(EOS(STATIC_1001), x0, x2, x2) | <=(x2, x0)
1451_0_mod_LE(EOS(STATIC_1451), x0, x1, x0, x2, x0, x2, x0) → 1451_0_mod_LE(EOS(STATIC_1451), x0, x1, x0, -(x2, x0), x0, -(x2, x0), x0) | &&(>(x2, x0), >(x0, 0))
1001_0_gcd_EQ(EOS(STATIC_1001), x0, x0, x0) → 1001_0_gcd_EQ(EOS(STATIC_1001), x0, 0, 0) | >(x0, 0)
R rules:

Filtered ground terms:



1001_0_gcd_EQ(x1, x2, x3, x4) → 1001_0_gcd_EQ(x2, x3, x4)
Cond_1001_0_gcd_EQ1(x1, x2, x3, x4, x5) → Cond_1001_0_gcd_EQ1(x1, x3, x4, x5)
1451_0_mod_LE(x1, x2, x3, x4, x5, x6, x7, x8) → 1451_0_mod_LE(x2, x3, x4, x5, x6, x7, x8)
Cond_1451_0_mod_LE1(x1, x2, x3, x4, x5, x6, x7, x8, x9) → Cond_1451_0_mod_LE1(x1, x3, x4, x5, x6, x7, x8, x9)
Cond_1451_0_mod_LE(x1, x2, x3, x4, x5, x6, x7, x8, x9) → Cond_1451_0_mod_LE(x1, x3, x4, x5, x6, x7, x8, x9)
Cond_1001_0_gcd_EQ(x1, x2, x3, x4, x5) → Cond_1001_0_gcd_EQ(x1, x3, x4, x5)

Filtered duplicate args:



1001_0_gcd_EQ(x1, x2, x3) → 1001_0_gcd_EQ(x1, x3)
Cond_1001_0_gcd_EQ(x1, x2, x3, x4) → Cond_1001_0_gcd_EQ(x1, x2, x4)
1451_0_mod_LE(x1, x2, x3, x4, x5, x6, x7) → 1451_0_mod_LE(x2, x4, x6, x7)
Cond_1451_0_mod_LE(x1, x2, x3, x4, x5, x6, x7, x8) → Cond_1451_0_mod_LE(x1, x3, x7, x8)
Cond_1451_0_mod_LE1(x1, x2, x3, x4, x5, x6, x7, x8) → Cond_1451_0_mod_LE1(x1, x3, x7, x8)
Cond_1001_0_gcd_EQ1(x1, x2, x3, x4) → Cond_1001_0_gcd_EQ1(x1, x4)

Filtered unneeded arguments:



1451_0_mod_LE(x1, x2, x3, x4) → 1451_0_mod_LE(x2, x3, x4)
Cond_1451_0_mod_LE(x1, x2, x3, x4) → Cond_1451_0_mod_LE(x1, x3, x4)
Cond_1451_0_mod_LE1(x1, x2, x3, x4) → Cond_1451_0_mod_LE1(x1, x3, x4)

Combined rules. Obtained 4 conditional rules for P and 0 conditional rules for R.


P rules:
1001_0_gcd_EQ(x0, x1) → 1451_0_mod_LE(x0, x0, x1) | &&(&&(>(x1, 0), >(x0, -1)), !(=(x0, x1)))
1451_0_mod_LE(x2, x2, x0) → 1001_0_gcd_EQ(x0, x2) | <=(x2, x0)
1451_0_mod_LE(x2, x2, x0) → 1451_0_mod_LE(-(x2, x0), -(x2, x0), x0) | &&(>(x2, x0), >(x0, 0))
1001_0_gcd_EQ(x0, x0) → 1001_0_gcd_EQ(x0, 0) | >(x0, 0)
R rules:

Finished conversion. Obtained 8 rules for P and 0 rules for R. System has predefined symbols.


P rules:
1001_0_GCD_EQ(x0, x1) → COND_1001_0_GCD_EQ(&&(&&(>(x1, 0), >(x0, -1)), !(=(x0, x1))), x0, x1)
COND_1001_0_GCD_EQ(TRUE, x0, x1) → 1451_0_MOD_LE(x0, x0, x1)
1451_0_MOD_LE(x2, x2, x0) → COND_1451_0_MOD_LE(<=(x2, x0), x2, x2, x0)
COND_1451_0_MOD_LE(TRUE, x2, x2, x0) → 1001_0_GCD_EQ(x0, x2)
1451_0_MOD_LE(x2, x2, x0) → COND_1451_0_MOD_LE1(&&(>(x2, x0), >(x0, 0)), x2, x2, x0)
COND_1451_0_MOD_LE1(TRUE, x2, x2, x0) → 1451_0_MOD_LE(-(x2, x0), -(x2, x0), x0)
1001_0_GCD_EQ(x0, x0) → COND_1001_0_GCD_EQ1(>(x0, 0), x0, x0)
COND_1001_0_GCD_EQ1(TRUE, x0, x0) → 1001_0_GCD_EQ(x0, 0)
R rules:

(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): 1001_0_GCD_EQ(x0[0], x1[0]) → COND_1001_0_GCD_EQ(x1[0] > 0 && x0[0] > -1 && !(x0[0] = x1[0]), x0[0], x1[0])
(1): COND_1001_0_GCD_EQ(TRUE, x0[1], x1[1]) → 1451_0_MOD_LE(x0[1], x0[1], x1[1])
(2): 1451_0_MOD_LE(x2[2], x2[2], x0[2]) → COND_1451_0_MOD_LE(x2[2] <= x0[2], x2[2], x2[2], x0[2])
(3): COND_1451_0_MOD_LE(TRUE, x2[3], x2[3], x0[3]) → 1001_0_GCD_EQ(x0[3], x2[3])
(4): 1451_0_MOD_LE(x2[4], x2[4], x0[4]) → COND_1451_0_MOD_LE1(x2[4] > x0[4] && x0[4] > 0, x2[4], x2[4], x0[4])
(5): COND_1451_0_MOD_LE1(TRUE, x2[5], x2[5], x0[5]) → 1451_0_MOD_LE(x2[5] - x0[5], x2[5] - x0[5], x0[5])
(6): 1001_0_GCD_EQ(x0[6], x0[6]) → COND_1001_0_GCD_EQ1(x0[6] > 0, x0[6], x0[6])
(7): COND_1001_0_GCD_EQ1(TRUE, x0[7], x0[7]) → 1001_0_GCD_EQ(x0[7], 0)

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


(1) -> (2), if (x0[1]* x2[2]x1[1]* x0[2])


(1) -> (4), if (x0[1]* x2[4]x1[1]* x0[4])


(2) -> (3), if (x2[2] <= x0[2]x2[2]* x2[3]x0[2]* x0[3])


(3) -> (0), if (x0[3]* x0[0]x2[3]* x1[0])


(3) -> (6), if (x0[3]* x0[6]x2[3]* x0[6])


(4) -> (5), if (x2[4] > x0[4] && x0[4] > 0x2[4]* x2[5]x0[4]* x0[5])


(5) -> (2), if (x2[5] - x0[5]* x2[2]x0[5]* x0[2])


(5) -> (4), if (x2[5] - x0[5]* x2[4]x0[5]* x0[4])


(6) -> (7), if (x0[6] > 0x0[6]* x0[7])


(7) -> (0), if (x0[7]* x0[0]0* x1[0])


(7) -> (6), if (x0[7]* x0[6]0* x0[6])



The set Q is empty.

(7) IDPNonInfProof (SOUND transformation)

Used the following options for this NonInfProof:
IDPGPoloSolver: Range: [(-1,2)] IsNat: false Interpretation Shape Heuristic: aprove.DPFramework.IDPProblem.Processors.nonInf.poly.IdpDefaultShapeHeuristic@bb6f5c4 Constraint Generator: NonInfConstraintGenerator: PathGenerator: MetricPathGenerator: Max Left Steps: 1 Max Right Steps: 1

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 1001_0_GCD_EQ(x0, x1) → COND_1001_0_GCD_EQ(&&(&&(>(x1, 0), >(x0, -1)), !(=(x0, x1))), x0, x1) the following chains were created:
  • We consider the chain 1001_0_GCD_EQ(x0[0], x1[0]) → COND_1001_0_GCD_EQ(&&(&&(>(x1[0], 0), >(x0[0], -1)), !(=(x0[0], x1[0]))), x0[0], x1[0]), COND_1001_0_GCD_EQ(TRUE, x0[1], x1[1]) → 1451_0_MOD_LE(x0[1], x0[1], x1[1]) which results in the following constraint:

    (1)    (&&(&&(>(x1[0], 0), >(x0[0], -1)), !(=(x0[0], x1[0])))=TRUEx0[0]=x0[1]x1[0]=x1[1]1001_0_GCD_EQ(x0[0], x1[0])≥NonInfC∧1001_0_GCD_EQ(x0[0], x1[0])≥COND_1001_0_GCD_EQ(&&(&&(>(x1[0], 0), >(x0[0], -1)), !(=(x0[0], x1[0]))), x0[0], x1[0])∧(UIncreasing(COND_1001_0_GCD_EQ(&&(&&(>(x1[0], 0), >(x0[0], -1)), !(=(x0[0], x1[0]))), x0[0], x1[0])), ≥))



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

    (2)    (>(x1[0], 0)=TRUE>(x0[0], -1)=TRUE<(x0[0], x1[0])=TRUE1001_0_GCD_EQ(x0[0], x1[0])≥NonInfC∧1001_0_GCD_EQ(x0[0], x1[0])≥COND_1001_0_GCD_EQ(&&(&&(>(x1[0], 0), >(x0[0], -1)), !(=(x0[0], x1[0]))), x0[0], x1[0])∧(UIncreasing(COND_1001_0_GCD_EQ(&&(&&(>(x1[0], 0), >(x0[0], -1)), !(=(x0[0], x1[0]))), x0[0], x1[0])), ≥))


    (3)    (>(x1[0], 0)=TRUE>(x0[0], -1)=TRUE>(x0[0], x1[0])=TRUE1001_0_GCD_EQ(x0[0], x1[0])≥NonInfC∧1001_0_GCD_EQ(x0[0], x1[0])≥COND_1001_0_GCD_EQ(&&(&&(>(x1[0], 0), >(x0[0], -1)), !(=(x0[0], x1[0]))), x0[0], x1[0])∧(UIncreasing(COND_1001_0_GCD_EQ(&&(&&(>(x1[0], 0), >(x0[0], -1)), !(=(x0[0], x1[0]))), x0[0], x1[0])), ≥))



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

    (4)    (x1[0] + [-1] ≥ 0∧x0[0] ≥ 0∧x1[0] + [-1] + [-1]x0[0] ≥ 0 ⇒ (UIncreasing(COND_1001_0_GCD_EQ(&&(&&(>(x1[0], 0), >(x0[0], -1)), !(=(x0[0], x1[0]))), x0[0], x1[0])), ≥)∧[(-1)bni_33 + (-1)Bound*bni_33] + [bni_33]x1[0] ≥ 0∧[(-1)bso_34] ≥ 0)



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

    (5)    (x1[0] + [-1] ≥ 0∧x0[0] ≥ 0∧x0[0] + [-1] + [-1]x1[0] ≥ 0 ⇒ (UIncreasing(COND_1001_0_GCD_EQ(&&(&&(>(x1[0], 0), >(x0[0], -1)), !(=(x0[0], x1[0]))), x0[0], x1[0])), ≥)∧[(-1)bni_33 + (-1)Bound*bni_33] + [bni_33]x1[0] ≥ 0∧[(-1)bso_34] ≥ 0)



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

    (6)    (x1[0] + [-1] ≥ 0∧x0[0] ≥ 0∧x1[0] + [-1] + [-1]x0[0] ≥ 0 ⇒ (UIncreasing(COND_1001_0_GCD_EQ(&&(&&(>(x1[0], 0), >(x0[0], -1)), !(=(x0[0], x1[0]))), x0[0], x1[0])), ≥)∧[(-1)bni_33 + (-1)Bound*bni_33] + [bni_33]x1[0] ≥ 0∧[(-1)bso_34] ≥ 0)



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

    (7)    (x1[0] + [-1] ≥ 0∧x0[0] ≥ 0∧x0[0] + [-1] + [-1]x1[0] ≥ 0 ⇒ (UIncreasing(COND_1001_0_GCD_EQ(&&(&&(>(x1[0], 0), >(x0[0], -1)), !(=(x0[0], x1[0]))), x0[0], x1[0])), ≥)∧[(-1)bni_33 + (-1)Bound*bni_33] + [bni_33]x1[0] ≥ 0∧[(-1)bso_34] ≥ 0)



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

    (8)    (x1[0] + [-1] ≥ 0∧x0[0] ≥ 0∧x1[0] + [-1] + [-1]x0[0] ≥ 0 ⇒ (UIncreasing(COND_1001_0_GCD_EQ(&&(&&(>(x1[0], 0), >(x0[0], -1)), !(=(x0[0], x1[0]))), x0[0], x1[0])), ≥)∧[(-1)bni_33 + (-1)Bound*bni_33] + [bni_33]x1[0] ≥ 0∧[(-1)bso_34] ≥ 0)



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

    (9)    (x1[0] + [-1] ≥ 0∧x0[0] ≥ 0∧x0[0] + [-1] + [-1]x1[0] ≥ 0 ⇒ (UIncreasing(COND_1001_0_GCD_EQ(&&(&&(>(x1[0], 0), >(x0[0], -1)), !(=(x0[0], x1[0]))), x0[0], x1[0])), ≥)∧[(-1)bni_33 + (-1)Bound*bni_33] + [bni_33]x1[0] ≥ 0∧[(-1)bso_34] ≥ 0)



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

    (10)    (x1[0] ≥ 0∧x0[0] ≥ 0∧x1[0] + [-1]x0[0] ≥ 0 ⇒ (UIncreasing(COND_1001_0_GCD_EQ(&&(&&(>(x1[0], 0), >(x0[0], -1)), !(=(x0[0], x1[0]))), x0[0], x1[0])), ≥)∧[(-1)Bound*bni_33] + [bni_33]x1[0] ≥ 0∧[(-1)bso_34] ≥ 0)



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

    (11)    (x1[0] ≥ 0∧x0[0] ≥ 0∧x0[0] + [-2] + [-1]x1[0] ≥ 0 ⇒ (UIncreasing(COND_1001_0_GCD_EQ(&&(&&(>(x1[0], 0), >(x0[0], -1)), !(=(x0[0], x1[0]))), x0[0], x1[0])), ≥)∧[(-1)Bound*bni_33] + [bni_33]x1[0] ≥ 0∧[(-1)bso_34] ≥ 0)



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

    (12)    (x0[0] + x1[0] ≥ 0∧x0[0] ≥ 0∧x1[0] ≥ 0 ⇒ (UIncreasing(COND_1001_0_GCD_EQ(&&(&&(>(x1[0], 0), >(x0[0], -1)), !(=(x0[0], x1[0]))), x0[0], x1[0])), ≥)∧[(-1)Bound*bni_33] + [bni_33]x0[0] + [bni_33]x1[0] ≥ 0∧[(-1)bso_34] ≥ 0)



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

    (13)    (x1[0] ≥ 0∧[2] + x1[0] + x0[0] ≥ 0∧x0[0] ≥ 0 ⇒ (UIncreasing(COND_1001_0_GCD_EQ(&&(&&(>(x1[0], 0), >(x0[0], -1)), !(=(x0[0], x1[0]))), x0[0], x1[0])), ≥)∧[(-1)Bound*bni_33] + [bni_33]x1[0] ≥ 0∧[(-1)bso_34] ≥ 0)







For Pair COND_1001_0_GCD_EQ(TRUE, x0, x1) → 1451_0_MOD_LE(x0, x0, x1) the following chains were created:
  • We consider the chain COND_1001_0_GCD_EQ(TRUE, x0[1], x1[1]) → 1451_0_MOD_LE(x0[1], x0[1], x1[1]), 1451_0_MOD_LE(x2[2], x2[2], x0[2]) → COND_1451_0_MOD_LE(<=(x2[2], x0[2]), x2[2], x2[2], x0[2]) which results in the following constraint:

    (14)    (x0[1]=x2[2]x1[1]=x0[2]COND_1001_0_GCD_EQ(TRUE, x0[1], x1[1])≥NonInfC∧COND_1001_0_GCD_EQ(TRUE, x0[1], x1[1])≥1451_0_MOD_LE(x0[1], x0[1], x1[1])∧(UIncreasing(1451_0_MOD_LE(x0[1], x0[1], x1[1])), ≥))



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

    (15)    (COND_1001_0_GCD_EQ(TRUE, x0[1], x1[1])≥NonInfC∧COND_1001_0_GCD_EQ(TRUE, x0[1], x1[1])≥1451_0_MOD_LE(x0[1], x0[1], x1[1])∧(UIncreasing(1451_0_MOD_LE(x0[1], x0[1], x1[1])), ≥))



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

    (16)    ((UIncreasing(1451_0_MOD_LE(x0[1], x0[1], x1[1])), ≥)∧[bni_35] = 0∧[(-1)bso_36] ≥ 0)



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

    (17)    ((UIncreasing(1451_0_MOD_LE(x0[1], x0[1], x1[1])), ≥)∧[bni_35] = 0∧[(-1)bso_36] ≥ 0)



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

    (18)    ((UIncreasing(1451_0_MOD_LE(x0[1], x0[1], x1[1])), ≥)∧[bni_35] = 0∧[(-1)bso_36] ≥ 0)



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

    (19)    ((UIncreasing(1451_0_MOD_LE(x0[1], x0[1], x1[1])), ≥)∧[bni_35] = 0∧0 = 0∧0 = 0∧[(-1)bso_36] ≥ 0)



  • We consider the chain COND_1001_0_GCD_EQ(TRUE, x0[1], x1[1]) → 1451_0_MOD_LE(x0[1], x0[1], x1[1]), 1451_0_MOD_LE(x2[4], x2[4], x0[4]) → COND_1451_0_MOD_LE1(&&(>(x2[4], x0[4]), >(x0[4], 0)), x2[4], x2[4], x0[4]) which results in the following constraint:

    (20)    (x0[1]=x2[4]x1[1]=x0[4]COND_1001_0_GCD_EQ(TRUE, x0[1], x1[1])≥NonInfC∧COND_1001_0_GCD_EQ(TRUE, x0[1], x1[1])≥1451_0_MOD_LE(x0[1], x0[1], x1[1])∧(UIncreasing(1451_0_MOD_LE(x0[1], x0[1], x1[1])), ≥))



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

    (21)    (COND_1001_0_GCD_EQ(TRUE, x0[1], x1[1])≥NonInfC∧COND_1001_0_GCD_EQ(TRUE, x0[1], x1[1])≥1451_0_MOD_LE(x0[1], x0[1], x1[1])∧(UIncreasing(1451_0_MOD_LE(x0[1], x0[1], x1[1])), ≥))



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

    (22)    ((UIncreasing(1451_0_MOD_LE(x0[1], x0[1], x1[1])), ≥)∧[bni_35] = 0∧[(-1)bso_36] ≥ 0)



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

    (23)    ((UIncreasing(1451_0_MOD_LE(x0[1], x0[1], x1[1])), ≥)∧[bni_35] = 0∧[(-1)bso_36] ≥ 0)



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

    (24)    ((UIncreasing(1451_0_MOD_LE(x0[1], x0[1], x1[1])), ≥)∧[bni_35] = 0∧[(-1)bso_36] ≥ 0)



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

    (25)    ((UIncreasing(1451_0_MOD_LE(x0[1], x0[1], x1[1])), ≥)∧[bni_35] = 0∧0 = 0∧0 = 0∧[(-1)bso_36] ≥ 0)







For Pair 1451_0_MOD_LE(x2, x2, x0) → COND_1451_0_MOD_LE(<=(x2, x0), x2, x2, x0) the following chains were created:
  • We consider the chain 1451_0_MOD_LE(x2[2], x2[2], x0[2]) → COND_1451_0_MOD_LE(<=(x2[2], x0[2]), x2[2], x2[2], x0[2]), COND_1451_0_MOD_LE(TRUE, x2[3], x2[3], x0[3]) → 1001_0_GCD_EQ(x0[3], x2[3]) which results in the following constraint:

    (26)    (<=(x2[2], x0[2])=TRUEx2[2]=x2[3]x0[2]=x0[3]1451_0_MOD_LE(x2[2], x2[2], x0[2])≥NonInfC∧1451_0_MOD_LE(x2[2], x2[2], x0[2])≥COND_1451_0_MOD_LE(<=(x2[2], x0[2]), x2[2], x2[2], x0[2])∧(UIncreasing(COND_1451_0_MOD_LE(<=(x2[2], x0[2]), x2[2], x2[2], x0[2])), ≥))



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

    (27)    (<=(x2[2], x0[2])=TRUE1451_0_MOD_LE(x2[2], x2[2], x0[2])≥NonInfC∧1451_0_MOD_LE(x2[2], x2[2], x0[2])≥COND_1451_0_MOD_LE(<=(x2[2], x0[2]), x2[2], x2[2], x0[2])∧(UIncreasing(COND_1451_0_MOD_LE(<=(x2[2], x0[2]), x2[2], x2[2], x0[2])), ≥))



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

    (28)    (x0[2] + [-1]x2[2] ≥ 0 ⇒ (UIncreasing(COND_1451_0_MOD_LE(<=(x2[2], x0[2]), x2[2], x2[2], x0[2])), ≥)∧[(-1)bni_37 + (-1)Bound*bni_37] + [bni_37]x0[2] ≥ 0∧[(-1)bso_38] + x0[2] + [-1]x2[2] ≥ 0)



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

    (29)    (x0[2] + [-1]x2[2] ≥ 0 ⇒ (UIncreasing(COND_1451_0_MOD_LE(<=(x2[2], x0[2]), x2[2], x2[2], x0[2])), ≥)∧[(-1)bni_37 + (-1)Bound*bni_37] + [bni_37]x0[2] ≥ 0∧[(-1)bso_38] + x0[2] + [-1]x2[2] ≥ 0)



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

    (30)    (x0[2] + [-1]x2[2] ≥ 0 ⇒ (UIncreasing(COND_1451_0_MOD_LE(<=(x2[2], x0[2]), x2[2], x2[2], x0[2])), ≥)∧[(-1)bni_37 + (-1)Bound*bni_37] + [bni_37]x0[2] ≥ 0∧[(-1)bso_38] + x0[2] + [-1]x2[2] ≥ 0)



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

    (31)    (x0[2] ≥ 0 ⇒ (UIncreasing(COND_1451_0_MOD_LE(<=(x2[2], x0[2]), x2[2], x2[2], x0[2])), ≥)∧[(-1)bni_37 + (-1)Bound*bni_37] + [bni_37]x2[2] + [bni_37]x0[2] ≥ 0∧[(-1)bso_38] + x0[2] ≥ 0)



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

    (32)    (x0[2] ≥ 0∧x2[2] ≥ 0 ⇒ (UIncreasing(COND_1451_0_MOD_LE(<=(x2[2], x0[2]), x2[2], x2[2], x0[2])), ≥)∧[(-1)bni_37 + (-1)Bound*bni_37] + [bni_37]x2[2] + [bni_37]x0[2] ≥ 0∧[(-1)bso_38] + x0[2] ≥ 0)


    (33)    (x0[2] ≥ 0∧x2[2] ≥ 0 ⇒ (UIncreasing(COND_1451_0_MOD_LE(<=(x2[2], x0[2]), x2[2], x2[2], x0[2])), ≥)∧[(-1)bni_37 + (-1)Bound*bni_37] + [(-1)bni_37]x2[2] + [bni_37]x0[2] ≥ 0∧[(-1)bso_38] + x0[2] ≥ 0)







For Pair COND_1451_0_MOD_LE(TRUE, x2, x2, x0) → 1001_0_GCD_EQ(x0, x2) the following chains were created:
  • We consider the chain COND_1451_0_MOD_LE(TRUE, x2[3], x2[3], x0[3]) → 1001_0_GCD_EQ(x0[3], x2[3]), 1001_0_GCD_EQ(x0[0], x1[0]) → COND_1001_0_GCD_EQ(&&(&&(>(x1[0], 0), >(x0[0], -1)), !(=(x0[0], x1[0]))), x0[0], x1[0]) which results in the following constraint:

    (34)    (x0[3]=x0[0]x2[3]=x1[0]COND_1451_0_MOD_LE(TRUE, x2[3], x2[3], x0[3])≥NonInfC∧COND_1451_0_MOD_LE(TRUE, x2[3], x2[3], x0[3])≥1001_0_GCD_EQ(x0[3], x2[3])∧(UIncreasing(1001_0_GCD_EQ(x0[3], x2[3])), ≥))



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

    (35)    (COND_1451_0_MOD_LE(TRUE, x2[3], x2[3], x0[3])≥NonInfC∧COND_1451_0_MOD_LE(TRUE, x2[3], x2[3], x0[3])≥1001_0_GCD_EQ(x0[3], x2[3])∧(UIncreasing(1001_0_GCD_EQ(x0[3], x2[3])), ≥))



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

    (36)    ((UIncreasing(1001_0_GCD_EQ(x0[3], x2[3])), ≥)∧[bni_39] = 0∧[(-1)bso_40] ≥ 0)



    We simplified constraint (36) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:

    (37)    ((UIncreasing(1001_0_GCD_EQ(x0[3], x2[3])), ≥)∧[bni_39] = 0∧[(-1)bso_40] ≥ 0)



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

    (38)    ((UIncreasing(1001_0_GCD_EQ(x0[3], x2[3])), ≥)∧[bni_39] = 0∧[(-1)bso_40] ≥ 0)



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

    (39)    ((UIncreasing(1001_0_GCD_EQ(x0[3], x2[3])), ≥)∧[bni_39] = 0∧0 = 0∧0 = 0∧[(-1)bso_40] ≥ 0)



  • We consider the chain COND_1451_0_MOD_LE(TRUE, x2[3], x2[3], x0[3]) → 1001_0_GCD_EQ(x0[3], x2[3]), 1001_0_GCD_EQ(x0[6], x0[6]) → COND_1001_0_GCD_EQ1(>(x0[6], 0), x0[6], x0[6]) which results in the following constraint:

    (40)    (x0[3]=x0[6]x2[3]=x0[6]COND_1451_0_MOD_LE(TRUE, x2[3], x2[3], x0[3])≥NonInfC∧COND_1451_0_MOD_LE(TRUE, x2[3], x2[3], x0[3])≥1001_0_GCD_EQ(x0[3], x2[3])∧(UIncreasing(1001_0_GCD_EQ(x0[3], x2[3])), ≥))



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

    (41)    (COND_1451_0_MOD_LE(TRUE, x2[3], x2[3], x2[3])≥NonInfC∧COND_1451_0_MOD_LE(TRUE, x2[3], x2[3], x2[3])≥1001_0_GCD_EQ(x2[3], x2[3])∧(UIncreasing(1001_0_GCD_EQ(x0[3], x2[3])), ≥))



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

    (42)    ((UIncreasing(1001_0_GCD_EQ(x0[3], x2[3])), ≥)∧[bni_39] = 0∧[(-1)bso_40] ≥ 0)



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

    (43)    ((UIncreasing(1001_0_GCD_EQ(x0[3], x2[3])), ≥)∧[bni_39] = 0∧[(-1)bso_40] ≥ 0)



    We simplified constraint (43) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:

    (44)    ((UIncreasing(1001_0_GCD_EQ(x0[3], x2[3])), ≥)∧[bni_39] = 0∧[(-1)bso_40] ≥ 0)



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

    (45)    ((UIncreasing(1001_0_GCD_EQ(x0[3], x2[3])), ≥)∧[bni_39] = 0∧0 = 0∧[(-1)bso_40] ≥ 0)







For Pair 1451_0_MOD_LE(x2, x2, x0) → COND_1451_0_MOD_LE1(&&(>(x2, x0), >(x0, 0)), x2, x2, x0) the following chains were created:
  • We consider the chain 1451_0_MOD_LE(x2[4], x2[4], x0[4]) → COND_1451_0_MOD_LE1(&&(>(x2[4], x0[4]), >(x0[4], 0)), x2[4], x2[4], x0[4]), COND_1451_0_MOD_LE1(TRUE, x2[5], x2[5], x0[5]) → 1451_0_MOD_LE(-(x2[5], x0[5]), -(x2[5], x0[5]), x0[5]) which results in the following constraint:

    (46)    (&&(>(x2[4], x0[4]), >(x0[4], 0))=TRUEx2[4]=x2[5]x0[4]=x0[5]1451_0_MOD_LE(x2[4], x2[4], x0[4])≥NonInfC∧1451_0_MOD_LE(x2[4], x2[4], x0[4])≥COND_1451_0_MOD_LE1(&&(>(x2[4], x0[4]), >(x0[4], 0)), x2[4], x2[4], x0[4])∧(UIncreasing(COND_1451_0_MOD_LE1(&&(>(x2[4], x0[4]), >(x0[4], 0)), x2[4], x2[4], x0[4])), ≥))



    We simplified constraint (46) using rules (IV), (IDP_BOOLEAN) which results in the following new constraint:

    (47)    (>(x2[4], x0[4])=TRUE>(x0[4], 0)=TRUE1451_0_MOD_LE(x2[4], x2[4], x0[4])≥NonInfC∧1451_0_MOD_LE(x2[4], x2[4], x0[4])≥COND_1451_0_MOD_LE1(&&(>(x2[4], x0[4]), >(x0[4], 0)), x2[4], x2[4], x0[4])∧(UIncreasing(COND_1451_0_MOD_LE1(&&(>(x2[4], x0[4]), >(x0[4], 0)), x2[4], x2[4], x0[4])), ≥))



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

    (48)    (x2[4] + [-1] + [-1]x0[4] ≥ 0∧x0[4] + [-1] ≥ 0 ⇒ (UIncreasing(COND_1451_0_MOD_LE1(&&(>(x2[4], x0[4]), >(x0[4], 0)), x2[4], x2[4], x0[4])), ≥)∧[(-1)bni_41 + (-1)Bound*bni_41] + [bni_41]x0[4] ≥ 0∧[(-1)bso_42] ≥ 0)



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

    (49)    (x2[4] + [-1] + [-1]x0[4] ≥ 0∧x0[4] + [-1] ≥ 0 ⇒ (UIncreasing(COND_1451_0_MOD_LE1(&&(>(x2[4], x0[4]), >(x0[4], 0)), x2[4], x2[4], x0[4])), ≥)∧[(-1)bni_41 + (-1)Bound*bni_41] + [bni_41]x0[4] ≥ 0∧[(-1)bso_42] ≥ 0)



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

    (50)    (x2[4] + [-1] + [-1]x0[4] ≥ 0∧x0[4] + [-1] ≥ 0 ⇒ (UIncreasing(COND_1451_0_MOD_LE1(&&(>(x2[4], x0[4]), >(x0[4], 0)), x2[4], x2[4], x0[4])), ≥)∧[(-1)bni_41 + (-1)Bound*bni_41] + [bni_41]x0[4] ≥ 0∧[(-1)bso_42] ≥ 0)



    We simplified constraint (50) using rule (IDP_SMT_SPLIT) which results in the following new constraint:

    (51)    (x2[4] ≥ 0∧x0[4] + [-1] ≥ 0 ⇒ (UIncreasing(COND_1451_0_MOD_LE1(&&(>(x2[4], x0[4]), >(x0[4], 0)), x2[4], x2[4], x0[4])), ≥)∧[(-1)bni_41 + (-1)Bound*bni_41] + [bni_41]x0[4] ≥ 0∧[(-1)bso_42] ≥ 0)



    We simplified constraint (51) using rule (IDP_SMT_SPLIT) which results in the following new constraint:

    (52)    (x2[4] ≥ 0∧x0[4] ≥ 0 ⇒ (UIncreasing(COND_1451_0_MOD_LE1(&&(>(x2[4], x0[4]), >(x0[4], 0)), x2[4], x2[4], x0[4])), ≥)∧[(-1)Bound*bni_41] + [bni_41]x0[4] ≥ 0∧[(-1)bso_42] ≥ 0)







For Pair COND_1451_0_MOD_LE1(TRUE, x2, x2, x0) → 1451_0_MOD_LE(-(x2, x0), -(x2, x0), x0) the following chains were created:
  • We consider the chain 1451_0_MOD_LE(x2[4], x2[4], x0[4]) → COND_1451_0_MOD_LE1(&&(>(x2[4], x0[4]), >(x0[4], 0)), x2[4], x2[4], x0[4]), COND_1451_0_MOD_LE1(TRUE, x2[5], x2[5], x0[5]) → 1451_0_MOD_LE(-(x2[5], x0[5]), -(x2[5], x0[5]), x0[5]), 1451_0_MOD_LE(x2[2], x2[2], x0[2]) → COND_1451_0_MOD_LE(<=(x2[2], x0[2]), x2[2], x2[2], x0[2]) which results in the following constraint:

    (53)    (&&(>(x2[4], x0[4]), >(x0[4], 0))=TRUEx2[4]=x2[5]x0[4]=x0[5]-(x2[5], x0[5])=x2[2]x0[5]=x0[2]COND_1451_0_MOD_LE1(TRUE, x2[5], x2[5], x0[5])≥NonInfC∧COND_1451_0_MOD_LE1(TRUE, x2[5], x2[5], x0[5])≥1451_0_MOD_LE(-(x2[5], x0[5]), -(x2[5], x0[5]), x0[5])∧(UIncreasing(1451_0_MOD_LE(-(x2[5], x0[5]), -(x2[5], x0[5]), x0[5])), ≥))



    We simplified constraint (53) using rules (III), (IV), (IDP_BOOLEAN) which results in the following new constraint:

    (54)    (>(x2[4], x0[4])=TRUE>(x0[4], 0)=TRUECOND_1451_0_MOD_LE1(TRUE, x2[4], x2[4], x0[4])≥NonInfC∧COND_1451_0_MOD_LE1(TRUE, x2[4], x2[4], x0[4])≥1451_0_MOD_LE(-(x2[4], x0[4]), -(x2[4], x0[4]), x0[4])∧(UIncreasing(1451_0_MOD_LE(-(x2[5], x0[5]), -(x2[5], x0[5]), x0[5])), ≥))



    We simplified constraint (54) using rule (POLY_CONSTRAINTS) which results in the following new constraint:

    (55)    (x2[4] + [-1] + [-1]x0[4] ≥ 0∧x0[4] + [-1] ≥ 0 ⇒ (UIncreasing(1451_0_MOD_LE(-(x2[5], x0[5]), -(x2[5], x0[5]), x0[5])), ≥)∧[(-1)bni_43 + (-1)Bound*bni_43] + [bni_43]x0[4] ≥ 0∧[(-1)bso_44] ≥ 0)



    We simplified constraint (55) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:

    (56)    (x2[4] + [-1] + [-1]x0[4] ≥ 0∧x0[4] + [-1] ≥ 0 ⇒ (UIncreasing(1451_0_MOD_LE(-(x2[5], x0[5]), -(x2[5], x0[5]), x0[5])), ≥)∧[(-1)bni_43 + (-1)Bound*bni_43] + [bni_43]x0[4] ≥ 0∧[(-1)bso_44] ≥ 0)



    We simplified constraint (56) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:

    (57)    (x2[4] + [-1] + [-1]x0[4] ≥ 0∧x0[4] + [-1] ≥ 0 ⇒ (UIncreasing(1451_0_MOD_LE(-(x2[5], x0[5]), -(x2[5], x0[5]), x0[5])), ≥)∧[(-1)bni_43 + (-1)Bound*bni_43] + [bni_43]x0[4] ≥ 0∧[(-1)bso_44] ≥ 0)



    We simplified constraint (57) using rule (IDP_SMT_SPLIT) which results in the following new constraint:

    (58)    (x2[4] ≥ 0∧x0[4] + [-1] ≥ 0 ⇒ (UIncreasing(1451_0_MOD_LE(-(x2[5], x0[5]), -(x2[5], x0[5]), x0[5])), ≥)∧[(-1)bni_43 + (-1)Bound*bni_43] + [bni_43]x0[4] ≥ 0∧[(-1)bso_44] ≥ 0)



    We simplified constraint (58) using rule (IDP_SMT_SPLIT) which results in the following new constraint:

    (59)    (x2[4] ≥ 0∧x0[4] ≥ 0 ⇒ (UIncreasing(1451_0_MOD_LE(-(x2[5], x0[5]), -(x2[5], x0[5]), x0[5])), ≥)∧[(-1)Bound*bni_43] + [bni_43]x0[4] ≥ 0∧[(-1)bso_44] ≥ 0)



  • We consider the chain 1451_0_MOD_LE(x2[4], x2[4], x0[4]) → COND_1451_0_MOD_LE1(&&(>(x2[4], x0[4]), >(x0[4], 0)), x2[4], x2[4], x0[4]), COND_1451_0_MOD_LE1(TRUE, x2[5], x2[5], x0[5]) → 1451_0_MOD_LE(-(x2[5], x0[5]), -(x2[5], x0[5]), x0[5]), 1451_0_MOD_LE(x2[4], x2[4], x0[4]) → COND_1451_0_MOD_LE1(&&(>(x2[4], x0[4]), >(x0[4], 0)), x2[4], x2[4], x0[4]) which results in the following constraint:

    (60)    (&&(>(x2[4], x0[4]), >(x0[4], 0))=TRUEx2[4]=x2[5]x0[4]=x0[5]-(x2[5], x0[5])=x2[4]1x0[5]=x0[4]1COND_1451_0_MOD_LE1(TRUE, x2[5], x2[5], x0[5])≥NonInfC∧COND_1451_0_MOD_LE1(TRUE, x2[5], x2[5], x0[5])≥1451_0_MOD_LE(-(x2[5], x0[5]), -(x2[5], x0[5]), x0[5])∧(UIncreasing(1451_0_MOD_LE(-(x2[5], x0[5]), -(x2[5], x0[5]), x0[5])), ≥))



    We simplified constraint (60) using rules (III), (IV), (IDP_BOOLEAN) which results in the following new constraint:

    (61)    (>(x2[4], x0[4])=TRUE>(x0[4], 0)=TRUECOND_1451_0_MOD_LE1(TRUE, x2[4], x2[4], x0[4])≥NonInfC∧COND_1451_0_MOD_LE1(TRUE, x2[4], x2[4], x0[4])≥1451_0_MOD_LE(-(x2[4], x0[4]), -(x2[4], x0[4]), x0[4])∧(UIncreasing(1451_0_MOD_LE(-(x2[5], x0[5]), -(x2[5], x0[5]), x0[5])), ≥))



    We simplified constraint (61) using rule (POLY_CONSTRAINTS) which results in the following new constraint:

    (62)    (x2[4] + [-1] + [-1]x0[4] ≥ 0∧x0[4] + [-1] ≥ 0 ⇒ (UIncreasing(1451_0_MOD_LE(-(x2[5], x0[5]), -(x2[5], x0[5]), x0[5])), ≥)∧[(-1)bni_43 + (-1)Bound*bni_43] + [bni_43]x0[4] ≥ 0∧[(-1)bso_44] ≥ 0)



    We simplified constraint (62) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:

    (63)    (x2[4] + [-1] + [-1]x0[4] ≥ 0∧x0[4] + [-1] ≥ 0 ⇒ (UIncreasing(1451_0_MOD_LE(-(x2[5], x0[5]), -(x2[5], x0[5]), x0[5])), ≥)∧[(-1)bni_43 + (-1)Bound*bni_43] + [bni_43]x0[4] ≥ 0∧[(-1)bso_44] ≥ 0)



    We simplified constraint (63) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:

    (64)    (x2[4] + [-1] + [-1]x0[4] ≥ 0∧x0[4] + [-1] ≥ 0 ⇒ (UIncreasing(1451_0_MOD_LE(-(x2[5], x0[5]), -(x2[5], x0[5]), x0[5])), ≥)∧[(-1)bni_43 + (-1)Bound*bni_43] + [bni_43]x0[4] ≥ 0∧[(-1)bso_44] ≥ 0)



    We simplified constraint (64) using rule (IDP_SMT_SPLIT) which results in the following new constraint:

    (65)    (x2[4] ≥ 0∧x0[4] + [-1] ≥ 0 ⇒ (UIncreasing(1451_0_MOD_LE(-(x2[5], x0[5]), -(x2[5], x0[5]), x0[5])), ≥)∧[(-1)bni_43 + (-1)Bound*bni_43] + [bni_43]x0[4] ≥ 0∧[(-1)bso_44] ≥ 0)



    We simplified constraint (65) using rule (IDP_SMT_SPLIT) which results in the following new constraint:

    (66)    (x2[4] ≥ 0∧x0[4] ≥ 0 ⇒ (UIncreasing(1451_0_MOD_LE(-(x2[5], x0[5]), -(x2[5], x0[5]), x0[5])), ≥)∧[(-1)Bound*bni_43] + [bni_43]x0[4] ≥ 0∧[(-1)bso_44] ≥ 0)







For Pair 1001_0_GCD_EQ(x0, x0) → COND_1001_0_GCD_EQ1(>(x0, 0), x0, x0) the following chains were created:
  • We consider the chain 1001_0_GCD_EQ(x0[6], x0[6]) → COND_1001_0_GCD_EQ1(>(x0[6], 0), x0[6], x0[6]), COND_1001_0_GCD_EQ1(TRUE, x0[7], x0[7]) → 1001_0_GCD_EQ(x0[7], 0) which results in the following constraint:

    (67)    (>(x0[6], 0)=TRUEx0[6]=x0[7]1001_0_GCD_EQ(x0[6], x0[6])≥NonInfC∧1001_0_GCD_EQ(x0[6], x0[6])≥COND_1001_0_GCD_EQ1(>(x0[6], 0), x0[6], x0[6])∧(UIncreasing(COND_1001_0_GCD_EQ1(>(x0[6], 0), x0[6], x0[6])), ≥))



    We simplified constraint (67) using rule (IV) which results in the following new constraint:

    (68)    (>(x0[6], 0)=TRUE1001_0_GCD_EQ(x0[6], x0[6])≥NonInfC∧1001_0_GCD_EQ(x0[6], x0[6])≥COND_1001_0_GCD_EQ1(>(x0[6], 0), x0[6], x0[6])∧(UIncreasing(COND_1001_0_GCD_EQ1(>(x0[6], 0), x0[6], x0[6])), ≥))



    We simplified constraint (68) using rule (POLY_CONSTRAINTS) which results in the following new constraint:

    (69)    (x0[6] + [-1] ≥ 0 ⇒ (UIncreasing(COND_1001_0_GCD_EQ1(>(x0[6], 0), x0[6], x0[6])), ≥)∧[(-1)bni_45 + (-1)Bound*bni_45] + [bni_45]x0[6] ≥ 0∧[-1 + (-1)bso_46] + x0[6] ≥ 0)



    We simplified constraint (69) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:

    (70)    (x0[6] + [-1] ≥ 0 ⇒ (UIncreasing(COND_1001_0_GCD_EQ1(>(x0[6], 0), x0[6], x0[6])), ≥)∧[(-1)bni_45 + (-1)Bound*bni_45] + [bni_45]x0[6] ≥ 0∧[-1 + (-1)bso_46] + x0[6] ≥ 0)



    We simplified constraint (70) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:

    (71)    (x0[6] + [-1] ≥ 0 ⇒ (UIncreasing(COND_1001_0_GCD_EQ1(>(x0[6], 0), x0[6], x0[6])), ≥)∧[(-1)bni_45 + (-1)Bound*bni_45] + [bni_45]x0[6] ≥ 0∧[-1 + (-1)bso_46] + x0[6] ≥ 0)



    We simplified constraint (71) using rule (IDP_SMT_SPLIT) which results in the following new constraint:

    (72)    (x0[6] ≥ 0 ⇒ (UIncreasing(COND_1001_0_GCD_EQ1(>(x0[6], 0), x0[6], x0[6])), ≥)∧[(-1)Bound*bni_45] + [bni_45]x0[6] ≥ 0∧[(-1)bso_46] + x0[6] ≥ 0)







For Pair COND_1001_0_GCD_EQ1(TRUE, x0, x0) → 1001_0_GCD_EQ(x0, 0) the following chains were created:
  • We consider the chain 1001_0_GCD_EQ(x0[6], x0[6]) → COND_1001_0_GCD_EQ1(>(x0[6], 0), x0[6], x0[6]), COND_1001_0_GCD_EQ1(TRUE, x0[7], x0[7]) → 1001_0_GCD_EQ(x0[7], 0), 1001_0_GCD_EQ(x0[0], x1[0]) → COND_1001_0_GCD_EQ(&&(&&(>(x1[0], 0), >(x0[0], -1)), !(=(x0[0], x1[0]))), x0[0], x1[0]) which results in the following constraint:

    (73)    (>(x0[6], 0)=TRUEx0[6]=x0[7]x0[7]=x0[0]0=x1[0]COND_1001_0_GCD_EQ1(TRUE, x0[7], x0[7])≥NonInfC∧COND_1001_0_GCD_EQ1(TRUE, x0[7], x0[7])≥1001_0_GCD_EQ(x0[7], 0)∧(UIncreasing(1001_0_GCD_EQ(x0[7], 0)), ≥))



    We simplified constraint (73) using rules (III), (IV) which results in the following new constraint:

    (74)    (>(x0[6], 0)=TRUECOND_1001_0_GCD_EQ1(TRUE, x0[6], x0[6])≥NonInfC∧COND_1001_0_GCD_EQ1(TRUE, x0[6], x0[6])≥1001_0_GCD_EQ(x0[6], 0)∧(UIncreasing(1001_0_GCD_EQ(x0[7], 0)), ≥))



    We simplified constraint (74) using rule (POLY_CONSTRAINTS) which results in the following new constraint:

    (75)    (x0[6] + [-1] ≥ 0 ⇒ (UIncreasing(1001_0_GCD_EQ(x0[7], 0)), ≥)∧[(-1)Bound*bni_47] ≥ 0∧[1 + (-1)bso_48] ≥ 0)



    We simplified constraint (75) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:

    (76)    (x0[6] + [-1] ≥ 0 ⇒ (UIncreasing(1001_0_GCD_EQ(x0[7], 0)), ≥)∧[(-1)Bound*bni_47] ≥ 0∧[1 + (-1)bso_48] ≥ 0)



    We simplified constraint (76) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:

    (77)    (x0[6] + [-1] ≥ 0 ⇒ (UIncreasing(1001_0_GCD_EQ(x0[7], 0)), ≥)∧[(-1)Bound*bni_47] ≥ 0∧[1 + (-1)bso_48] ≥ 0)



    We simplified constraint (77) using rule (IDP_SMT_SPLIT) which results in the following new constraint:

    (78)    (x0[6] ≥ 0 ⇒ (UIncreasing(1001_0_GCD_EQ(x0[7], 0)), ≥)∧[(-1)Bound*bni_47] ≥ 0∧[1 + (-1)bso_48] ≥ 0)



  • We consider the chain 1001_0_GCD_EQ(x0[6], x0[6]) → COND_1001_0_GCD_EQ1(>(x0[6], 0), x0[6], x0[6]), COND_1001_0_GCD_EQ1(TRUE, x0[7], x0[7]) → 1001_0_GCD_EQ(x0[7], 0), 1001_0_GCD_EQ(x0[6], x0[6]) → COND_1001_0_GCD_EQ1(>(x0[6], 0), x0[6], x0[6]) which results in the following constraint:

    (79)    (>(x0[6], 0)=TRUEx0[6]=x0[7]x0[7]=x0[6]10=x0[6]1COND_1001_0_GCD_EQ1(TRUE, x0[7], x0[7])≥NonInfC∧COND_1001_0_GCD_EQ1(TRUE, x0[7], x0[7])≥1001_0_GCD_EQ(x0[7], 0)∧(UIncreasing(1001_0_GCD_EQ(x0[7], 0)), ≥))



    We solved constraint (79) using rules (I), (II), (III), (IDP_CONSTANT_FOLD).




To summarize, we get the following constraints P for the following pairs.
  • 1001_0_GCD_EQ(x0, x1) → COND_1001_0_GCD_EQ(&&(&&(>(x1, 0), >(x0, -1)), !(=(x0, x1))), x0, x1)
    • (x0[0] + x1[0] ≥ 0∧x0[0] ≥ 0∧x1[0] ≥ 0 ⇒ (UIncreasing(COND_1001_0_GCD_EQ(&&(&&(>(x1[0], 0), >(x0[0], -1)), !(=(x0[0], x1[0]))), x0[0], x1[0])), ≥)∧[(-1)Bound*bni_33] + [bni_33]x0[0] + [bni_33]x1[0] ≥ 0∧[(-1)bso_34] ≥ 0)
    • (x1[0] ≥ 0∧[2] + x1[0] + x0[0] ≥ 0∧x0[0] ≥ 0 ⇒ (UIncreasing(COND_1001_0_GCD_EQ(&&(&&(>(x1[0], 0), >(x0[0], -1)), !(=(x0[0], x1[0]))), x0[0], x1[0])), ≥)∧[(-1)Bound*bni_33] + [bni_33]x1[0] ≥ 0∧[(-1)bso_34] ≥ 0)

  • COND_1001_0_GCD_EQ(TRUE, x0, x1) → 1451_0_MOD_LE(x0, x0, x1)
    • ((UIncreasing(1451_0_MOD_LE(x0[1], x0[1], x1[1])), ≥)∧[bni_35] = 0∧0 = 0∧0 = 0∧[(-1)bso_36] ≥ 0)
    • ((UIncreasing(1451_0_MOD_LE(x0[1], x0[1], x1[1])), ≥)∧[bni_35] = 0∧0 = 0∧0 = 0∧[(-1)bso_36] ≥ 0)

  • 1451_0_MOD_LE(x2, x2, x0) → COND_1451_0_MOD_LE(<=(x2, x0), x2, x2, x0)
    • (x0[2] ≥ 0∧x2[2] ≥ 0 ⇒ (UIncreasing(COND_1451_0_MOD_LE(<=(x2[2], x0[2]), x2[2], x2[2], x0[2])), ≥)∧[(-1)bni_37 + (-1)Bound*bni_37] + [bni_37]x2[2] + [bni_37]x0[2] ≥ 0∧[(-1)bso_38] + x0[2] ≥ 0)
    • (x0[2] ≥ 0∧x2[2] ≥ 0 ⇒ (UIncreasing(COND_1451_0_MOD_LE(<=(x2[2], x0[2]), x2[2], x2[2], x0[2])), ≥)∧[(-1)bni_37 + (-1)Bound*bni_37] + [(-1)bni_37]x2[2] + [bni_37]x0[2] ≥ 0∧[(-1)bso_38] + x0[2] ≥ 0)

  • COND_1451_0_MOD_LE(TRUE, x2, x2, x0) → 1001_0_GCD_EQ(x0, x2)
    • ((UIncreasing(1001_0_GCD_EQ(x0[3], x2[3])), ≥)∧[bni_39] = 0∧0 = 0∧0 = 0∧[(-1)bso_40] ≥ 0)
    • ((UIncreasing(1001_0_GCD_EQ(x0[3], x2[3])), ≥)∧[bni_39] = 0∧0 = 0∧[(-1)bso_40] ≥ 0)

  • 1451_0_MOD_LE(x2, x2, x0) → COND_1451_0_MOD_LE1(&&(>(x2, x0), >(x0, 0)), x2, x2, x0)
    • (x2[4] ≥ 0∧x0[4] ≥ 0 ⇒ (UIncreasing(COND_1451_0_MOD_LE1(&&(>(x2[4], x0[4]), >(x0[4], 0)), x2[4], x2[4], x0[4])), ≥)∧[(-1)Bound*bni_41] + [bni_41]x0[4] ≥ 0∧[(-1)bso_42] ≥ 0)

  • COND_1451_0_MOD_LE1(TRUE, x2, x2, x0) → 1451_0_MOD_LE(-(x2, x0), -(x2, x0), x0)
    • (x2[4] ≥ 0∧x0[4] ≥ 0 ⇒ (UIncreasing(1451_0_MOD_LE(-(x2[5], x0[5]), -(x2[5], x0[5]), x0[5])), ≥)∧[(-1)Bound*bni_43] + [bni_43]x0[4] ≥ 0∧[(-1)bso_44] ≥ 0)
    • (x2[4] ≥ 0∧x0[4] ≥ 0 ⇒ (UIncreasing(1451_0_MOD_LE(-(x2[5], x0[5]), -(x2[5], x0[5]), x0[5])), ≥)∧[(-1)Bound*bni_43] + [bni_43]x0[4] ≥ 0∧[(-1)bso_44] ≥ 0)

  • 1001_0_GCD_EQ(x0, x0) → COND_1001_0_GCD_EQ1(>(x0, 0), x0, x0)
    • (x0[6] ≥ 0 ⇒ (UIncreasing(COND_1001_0_GCD_EQ1(>(x0[6], 0), x0[6], x0[6])), ≥)∧[(-1)Bound*bni_45] + [bni_45]x0[6] ≥ 0∧[(-1)bso_46] + x0[6] ≥ 0)

  • COND_1001_0_GCD_EQ1(TRUE, x0, x0) → 1001_0_GCD_EQ(x0, 0)
    • (x0[6] ≥ 0 ⇒ (UIncreasing(1001_0_GCD_EQ(x0[7], 0)), ≥)∧[(-1)Bound*bni_47] ≥ 0∧[1 + (-1)bso_48] ≥ 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(1001_0_GCD_EQ(x1, x2)) = [-1] + x2   
POL(COND_1001_0_GCD_EQ(x1, x2, x3)) = [-1] + x3 + [-1]x1   
POL(&&(x1, x2)) = 0   
POL(>(x1, x2)) = [-1]   
POL(0) = 0   
POL(-1) = [-1]   
POL(!(x1)) = [-1]   
POL(=(x1, x2)) = [-1]   
POL(1451_0_MOD_LE(x1, x2, x3)) = [-1] + x3 + [-1]x2 + x1   
POL(COND_1451_0_MOD_LE(x1, x2, x3, x4)) = [-1] + [-1]x3 + [2]x2   
POL(<=(x1, x2)) = 0   
POL(COND_1451_0_MOD_LE1(x1, x2, x3, x4)) = [-1] + x4 + [-1]x3 + x2 + [-1]x1   
POL(-(x1, x2)) = x1 + [-1]x2   
POL(COND_1001_0_GCD_EQ1(x1, x2, x3)) = [-1]x3 + x2   

The following pairs are in P>:

COND_1001_0_GCD_EQ1(TRUE, x0[7], x0[7]) → 1001_0_GCD_EQ(x0[7], 0)

The following pairs are in Pbound:

1001_0_GCD_EQ(x0[0], x1[0]) → COND_1001_0_GCD_EQ(&&(&&(>(x1[0], 0), >(x0[0], -1)), !(=(x0[0], x1[0]))), x0[0], x1[0])
1451_0_MOD_LE(x2[4], x2[4], x0[4]) → COND_1451_0_MOD_LE1(&&(>(x2[4], x0[4]), >(x0[4], 0)), x2[4], x2[4], x0[4])
COND_1451_0_MOD_LE1(TRUE, x2[5], x2[5], x0[5]) → 1451_0_MOD_LE(-(x2[5], x0[5]), -(x2[5], x0[5]), x0[5])
1001_0_GCD_EQ(x0[6], x0[6]) → COND_1001_0_GCD_EQ1(>(x0[6], 0), x0[6], x0[6])
COND_1001_0_GCD_EQ1(TRUE, x0[7], x0[7]) → 1001_0_GCD_EQ(x0[7], 0)

The following pairs are in P:

1001_0_GCD_EQ(x0[0], x1[0]) → COND_1001_0_GCD_EQ(&&(&&(>(x1[0], 0), >(x0[0], -1)), !(=(x0[0], x1[0]))), x0[0], x1[0])
COND_1001_0_GCD_EQ(TRUE, x0[1], x1[1]) → 1451_0_MOD_LE(x0[1], x0[1], x1[1])
1451_0_MOD_LE(x2[2], x2[2], x0[2]) → COND_1451_0_MOD_LE(<=(x2[2], x0[2]), x2[2], x2[2], x0[2])
COND_1451_0_MOD_LE(TRUE, x2[3], x2[3], x0[3]) → 1001_0_GCD_EQ(x0[3], x2[3])
1451_0_MOD_LE(x2[4], x2[4], x0[4]) → COND_1451_0_MOD_LE1(&&(>(x2[4], x0[4]), >(x0[4], 0)), x2[4], x2[4], x0[4])
COND_1451_0_MOD_LE1(TRUE, x2[5], x2[5], x0[5]) → 1451_0_MOD_LE(-(x2[5], x0[5]), -(x2[5], x0[5]), x0[5])
1001_0_GCD_EQ(x0[6], x0[6]) → COND_1001_0_GCD_EQ1(>(x0[6], 0), x0[6], x0[6])

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

(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:
(0): 1001_0_GCD_EQ(x0[0], x1[0]) → COND_1001_0_GCD_EQ(x1[0] > 0 && x0[0] > -1 && !(x0[0] = x1[0]), x0[0], x1[0])
(1): COND_1001_0_GCD_EQ(TRUE, x0[1], x1[1]) → 1451_0_MOD_LE(x0[1], x0[1], x1[1])
(2): 1451_0_MOD_LE(x2[2], x2[2], x0[2]) → COND_1451_0_MOD_LE(x2[2] <= x0[2], x2[2], x2[2], x0[2])
(3): COND_1451_0_MOD_LE(TRUE, x2[3], x2[3], x0[3]) → 1001_0_GCD_EQ(x0[3], x2[3])
(4): 1451_0_MOD_LE(x2[4], x2[4], x0[4]) → COND_1451_0_MOD_LE1(x2[4] > x0[4] && x0[4] > 0, x2[4], x2[4], x0[4])
(5): COND_1451_0_MOD_LE1(TRUE, x2[5], x2[5], x0[5]) → 1451_0_MOD_LE(x2[5] - x0[5], x2[5] - x0[5], x0[5])
(6): 1001_0_GCD_EQ(x0[6], x0[6]) → COND_1001_0_GCD_EQ1(x0[6] > 0, x0[6], x0[6])

(3) -> (0), if (x0[3]* x0[0]x2[3]* x1[0])


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


(1) -> (2), if (x0[1]* x2[2]x1[1]* x0[2])


(5) -> (2), if (x2[5] - x0[5]* x2[2]x0[5]* x0[2])


(2) -> (3), if (x2[2] <= x0[2]x2[2]* x2[3]x0[2]* x0[3])


(1) -> (4), if (x0[1]* x2[4]x1[1]* x0[4])


(5) -> (4), if (x2[5] - x0[5]* x2[4]x0[5]* x0[4])


(4) -> (5), if (x2[4] > x0[4] && x0[4] > 0x2[4]* x2[5]x0[4]* x0[5])


(3) -> (6), if (x0[3]* x0[6]x2[3]* x0[6])



The set Q is empty.

(9) IDependencyGraphProof (EQUIVALENT transformation)

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

(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:
(5): COND_1451_0_MOD_LE1(TRUE, x2[5], x2[5], x0[5]) → 1451_0_MOD_LE(x2[5] - x0[5], x2[5] - x0[5], x0[5])
(4): 1451_0_MOD_LE(x2[4], x2[4], x0[4]) → COND_1451_0_MOD_LE1(x2[4] > x0[4] && x0[4] > 0, x2[4], x2[4], x0[4])
(3): COND_1451_0_MOD_LE(TRUE, x2[3], x2[3], x0[3]) → 1001_0_GCD_EQ(x0[3], x2[3])
(2): 1451_0_MOD_LE(x2[2], x2[2], x0[2]) → COND_1451_0_MOD_LE(x2[2] <= x0[2], x2[2], x2[2], x0[2])
(1): COND_1001_0_GCD_EQ(TRUE, x0[1], x1[1]) → 1451_0_MOD_LE(x0[1], x0[1], x1[1])
(0): 1001_0_GCD_EQ(x0[0], x1[0]) → COND_1001_0_GCD_EQ(x1[0] > 0 && x0[0] > -1 && !(x0[0] = x1[0]), x0[0], x1[0])

(3) -> (0), if (x0[3]* x0[0]x2[3]* x1[0])


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


(1) -> (2), if (x0[1]* x2[2]x1[1]* x0[2])


(5) -> (2), if (x2[5] - x0[5]* x2[2]x0[5]* x0[2])


(2) -> (3), if (x2[2] <= x0[2]x2[2]* x2[3]x0[2]* x0[3])


(1) -> (4), if (x0[1]* x2[4]x1[1]* x0[4])


(5) -> (4), if (x2[5] - x0[5]* x2[4]x0[5]* x0[4])


(4) -> (5), if (x2[4] > x0[4] && x0[4] > 0x2[4]* x2[5]x0[4]* x0[5])



The set Q is empty.

(11) IDPNonInfProof (SOUND transformation)

Used the following options for this NonInfProof:
IDPGPoloSolver: Range: [(-1,2)] IsNat: false Interpretation Shape Heuristic: aprove.DPFramework.IDPProblem.Processors.nonInf.poly.IdpDefaultShapeHeuristic@bb6f5c4 Constraint Generator: NonInfConstraintGenerator: PathGenerator: MetricPathGenerator: Max Left Steps: 1 Max Right Steps: 1

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_1451_0_MOD_LE1(TRUE, x2[5], x2[5], x0[5]) → 1451_0_MOD_LE(-(x2[5], x0[5]), -(x2[5], x0[5]), x0[5]) the following chains were created:
  • We consider the chain 1451_0_MOD_LE(x2[4], x2[4], x0[4]) → COND_1451_0_MOD_LE1(&&(>(x2[4], x0[4]), >(x0[4], 0)), x2[4], x2[4], x0[4]), COND_1451_0_MOD_LE1(TRUE, x2[5], x2[5], x0[5]) → 1451_0_MOD_LE(-(x2[5], x0[5]), -(x2[5], x0[5]), x0[5]), 1451_0_MOD_LE(x2[2], x2[2], x0[2]) → COND_1451_0_MOD_LE(<=(x2[2], x0[2]), x2[2], x2[2], x0[2]) which results in the following constraint:

    (1)    (&&(>(x2[4], x0[4]), >(x0[4], 0))=TRUEx2[4]=x2[5]x0[4]=x0[5]-(x2[5], x0[5])=x2[2]x0[5]=x0[2]COND_1451_0_MOD_LE1(TRUE, x2[5], x2[5], x0[5])≥NonInfC∧COND_1451_0_MOD_LE1(TRUE, x2[5], x2[5], x0[5])≥1451_0_MOD_LE(-(x2[5], x0[5]), -(x2[5], x0[5]), x0[5])∧(UIncreasing(1451_0_MOD_LE(-(x2[5], x0[5]), -(x2[5], x0[5]), x0[5])), ≥))



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

    (2)    (>(x2[4], x0[4])=TRUE>(x0[4], 0)=TRUECOND_1451_0_MOD_LE1(TRUE, x2[4], x2[4], x0[4])≥NonInfC∧COND_1451_0_MOD_LE1(TRUE, x2[4], x2[4], x0[4])≥1451_0_MOD_LE(-(x2[4], x0[4]), -(x2[4], x0[4]), x0[4])∧(UIncreasing(1451_0_MOD_LE(-(x2[5], x0[5]), -(x2[5], x0[5]), x0[5])), ≥))



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

    (3)    (x2[4] + [-1] + [-1]x0[4] ≥ 0∧x0[4] + [-1] ≥ 0 ⇒ (UIncreasing(1451_0_MOD_LE(-(x2[5], x0[5]), -(x2[5], x0[5]), x0[5])), ≥)∧[(-1)Bound*bni_29] + [bni_29]x2[4] ≥ 0∧[1 + (-1)bso_30] ≥ 0)



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

    (4)    (x2[4] + [-1] + [-1]x0[4] ≥ 0∧x0[4] + [-1] ≥ 0 ⇒ (UIncreasing(1451_0_MOD_LE(-(x2[5], x0[5]), -(x2[5], x0[5]), x0[5])), ≥)∧[(-1)Bound*bni_29] + [bni_29]x2[4] ≥ 0∧[1 + (-1)bso_30] ≥ 0)



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

    (5)    (x2[4] + [-1] + [-1]x0[4] ≥ 0∧x0[4] + [-1] ≥ 0 ⇒ (UIncreasing(1451_0_MOD_LE(-(x2[5], x0[5]), -(x2[5], x0[5]), x0[5])), ≥)∧[(-1)Bound*bni_29] + [bni_29]x2[4] ≥ 0∧[1 + (-1)bso_30] ≥ 0)



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

    (6)    (x2[4] ≥ 0∧x0[4] + [-1] ≥ 0 ⇒ (UIncreasing(1451_0_MOD_LE(-(x2[5], x0[5]), -(x2[5], x0[5]), x0[5])), ≥)∧[(-1)Bound*bni_29 + bni_29] + [bni_29]x0[4] + [bni_29]x2[4] ≥ 0∧[1 + (-1)bso_30] ≥ 0)



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

    (7)    (x2[4] ≥ 0∧x0[4] ≥ 0 ⇒ (UIncreasing(1451_0_MOD_LE(-(x2[5], x0[5]), -(x2[5], x0[5]), x0[5])), ≥)∧[(-1)Bound*bni_29 + (2)bni_29] + [bni_29]x0[4] + [bni_29]x2[4] ≥ 0∧[1 + (-1)bso_30] ≥ 0)



  • We consider the chain 1451_0_MOD_LE(x2[4], x2[4], x0[4]) → COND_1451_0_MOD_LE1(&&(>(x2[4], x0[4]), >(x0[4], 0)), x2[4], x2[4], x0[4]), COND_1451_0_MOD_LE1(TRUE, x2[5], x2[5], x0[5]) → 1451_0_MOD_LE(-(x2[5], x0[5]), -(x2[5], x0[5]), x0[5]), 1451_0_MOD_LE(x2[4], x2[4], x0[4]) → COND_1451_0_MOD_LE1(&&(>(x2[4], x0[4]), >(x0[4], 0)), x2[4], x2[4], x0[4]) which results in the following constraint:

    (8)    (&&(>(x2[4], x0[4]), >(x0[4], 0))=TRUEx2[4]=x2[5]x0[4]=x0[5]-(x2[5], x0[5])=x2[4]1x0[5]=x0[4]1COND_1451_0_MOD_LE1(TRUE, x2[5], x2[5], x0[5])≥NonInfC∧COND_1451_0_MOD_LE1(TRUE, x2[5], x2[5], x0[5])≥1451_0_MOD_LE(-(x2[5], x0[5]), -(x2[5], x0[5]), x0[5])∧(UIncreasing(1451_0_MOD_LE(-(x2[5], x0[5]), -(x2[5], x0[5]), x0[5])), ≥))



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

    (9)    (>(x2[4], x0[4])=TRUE>(x0[4], 0)=TRUECOND_1451_0_MOD_LE1(TRUE, x2[4], x2[4], x0[4])≥NonInfC∧COND_1451_0_MOD_LE1(TRUE, x2[4], x2[4], x0[4])≥1451_0_MOD_LE(-(x2[4], x0[4]), -(x2[4], x0[4]), x0[4])∧(UIncreasing(1451_0_MOD_LE(-(x2[5], x0[5]), -(x2[5], x0[5]), x0[5])), ≥))



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

    (10)    (x2[4] + [-1] + [-1]x0[4] ≥ 0∧x0[4] + [-1] ≥ 0 ⇒ (UIncreasing(1451_0_MOD_LE(-(x2[5], x0[5]), -(x2[5], x0[5]), x0[5])), ≥)∧[(-1)Bound*bni_29] + [bni_29]x2[4] ≥ 0∧[1 + (-1)bso_30] ≥ 0)



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

    (11)    (x2[4] + [-1] + [-1]x0[4] ≥ 0∧x0[4] + [-1] ≥ 0 ⇒ (UIncreasing(1451_0_MOD_LE(-(x2[5], x0[5]), -(x2[5], x0[5]), x0[5])), ≥)∧[(-1)Bound*bni_29] + [bni_29]x2[4] ≥ 0∧[1 + (-1)bso_30] ≥ 0)



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

    (12)    (x2[4] + [-1] + [-1]x0[4] ≥ 0∧x0[4] + [-1] ≥ 0 ⇒ (UIncreasing(1451_0_MOD_LE(-(x2[5], x0[5]), -(x2[5], x0[5]), x0[5])), ≥)∧[(-1)Bound*bni_29] + [bni_29]x2[4] ≥ 0∧[1 + (-1)bso_30] ≥ 0)



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

    (13)    (x2[4] ≥ 0∧x0[4] + [-1] ≥ 0 ⇒ (UIncreasing(1451_0_MOD_LE(-(x2[5], x0[5]), -(x2[5], x0[5]), x0[5])), ≥)∧[(-1)Bound*bni_29 + bni_29] + [bni_29]x0[4] + [bni_29]x2[4] ≥ 0∧[1 + (-1)bso_30] ≥ 0)



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

    (14)    (x2[4] ≥ 0∧x0[4] ≥ 0 ⇒ (UIncreasing(1451_0_MOD_LE(-(x2[5], x0[5]), -(x2[5], x0[5]), x0[5])), ≥)∧[(-1)Bound*bni_29 + (2)bni_29] + [bni_29]x0[4] + [bni_29]x2[4] ≥ 0∧[1 + (-1)bso_30] ≥ 0)







For Pair 1451_0_MOD_LE(x2[4], x2[4], x0[4]) → COND_1451_0_MOD_LE1(&&(>(x2[4], x0[4]), >(x0[4], 0)), x2[4], x2[4], x0[4]) the following chains were created:
  • We consider the chain 1451_0_MOD_LE(x2[4], x2[4], x0[4]) → COND_1451_0_MOD_LE1(&&(>(x2[4], x0[4]), >(x0[4], 0)), x2[4], x2[4], x0[4]), COND_1451_0_MOD_LE1(TRUE, x2[5], x2[5], x0[5]) → 1451_0_MOD_LE(-(x2[5], x0[5]), -(x2[5], x0[5]), x0[5]) which results in the following constraint:

    (15)    (&&(>(x2[4], x0[4]), >(x0[4], 0))=TRUEx2[4]=x2[5]x0[4]=x0[5]1451_0_MOD_LE(x2[4], x2[4], x0[4])≥NonInfC∧1451_0_MOD_LE(x2[4], x2[4], x0[4])≥COND_1451_0_MOD_LE1(&&(>(x2[4], x0[4]), >(x0[4], 0)), x2[4], x2[4], x0[4])∧(UIncreasing(COND_1451_0_MOD_LE1(&&(>(x2[4], x0[4]), >(x0[4], 0)), x2[4], x2[4], x0[4])), ≥))



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

    (16)    (>(x2[4], x0[4])=TRUE>(x0[4], 0)=TRUE1451_0_MOD_LE(x2[4], x2[4], x0[4])≥NonInfC∧1451_0_MOD_LE(x2[4], x2[4], x0[4])≥COND_1451_0_MOD_LE1(&&(>(x2[4], x0[4]), >(x0[4], 0)), x2[4], x2[4], x0[4])∧(UIncreasing(COND_1451_0_MOD_LE1(&&(>(x2[4], x0[4]), >(x0[4], 0)), x2[4], x2[4], x0[4])), ≥))



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

    (17)    (x2[4] + [-1] + [-1]x0[4] ≥ 0∧x0[4] + [-1] ≥ 0 ⇒ (UIncreasing(COND_1451_0_MOD_LE1(&&(>(x2[4], x0[4]), >(x0[4], 0)), x2[4], x2[4], x0[4])), ≥)∧[(-1)bni_31 + (-1)Bound*bni_31] + [bni_31]x0[4] + [bni_31]x2[4] ≥ 0∧[-1 + (-1)bso_32] + x0[4] ≥ 0)



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

    (18)    (x2[4] + [-1] + [-1]x0[4] ≥ 0∧x0[4] + [-1] ≥ 0 ⇒ (UIncreasing(COND_1451_0_MOD_LE1(&&(>(x2[4], x0[4]), >(x0[4], 0)), x2[4], x2[4], x0[4])), ≥)∧[(-1)bni_31 + (-1)Bound*bni_31] + [bni_31]x0[4] + [bni_31]x2[4] ≥ 0∧[-1 + (-1)bso_32] + x0[4] ≥ 0)



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

    (19)    (x2[4] + [-1] + [-1]x0[4] ≥ 0∧x0[4] + [-1] ≥ 0 ⇒ (UIncreasing(COND_1451_0_MOD_LE1(&&(>(x2[4], x0[4]), >(x0[4], 0)), x2[4], x2[4], x0[4])), ≥)∧[(-1)bni_31 + (-1)Bound*bni_31] + [bni_31]x0[4] + [bni_31]x2[4] ≥ 0∧[-1 + (-1)bso_32] + x0[4] ≥ 0)



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

    (20)    (x2[4] ≥ 0∧x0[4] + [-1] ≥ 0 ⇒ (UIncreasing(COND_1451_0_MOD_LE1(&&(>(x2[4], x0[4]), >(x0[4], 0)), x2[4], x2[4], x0[4])), ≥)∧[(-1)Bound*bni_31] + [(2)bni_31]x0[4] + [bni_31]x2[4] ≥ 0∧[-1 + (-1)bso_32] + x0[4] ≥ 0)



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

    (21)    (x2[4] ≥ 0∧x0[4] ≥ 0 ⇒ (UIncreasing(COND_1451_0_MOD_LE1(&&(>(x2[4], x0[4]), >(x0[4], 0)), x2[4], x2[4], x0[4])), ≥)∧[(-1)Bound*bni_31 + (2)bni_31] + [(2)bni_31]x0[4] + [bni_31]x2[4] ≥ 0∧[(-1)bso_32] + x0[4] ≥ 0)







For Pair COND_1451_0_MOD_LE(TRUE, x2[3], x2[3], x0[3]) → 1001_0_GCD_EQ(x0[3], x2[3]) the following chains were created:
  • We consider the chain COND_1451_0_MOD_LE(TRUE, x2[3], x2[3], x0[3]) → 1001_0_GCD_EQ(x0[3], x2[3]), 1001_0_GCD_EQ(x0[0], x1[0]) → COND_1001_0_GCD_EQ(&&(&&(>(x1[0], 0), >(x0[0], -1)), !(=(x0[0], x1[0]))), x0[0], x1[0]) which results in the following constraint:

    (22)    (x0[3]=x0[0]x2[3]=x1[0]COND_1451_0_MOD_LE(TRUE, x2[3], x2[3], x0[3])≥NonInfC∧COND_1451_0_MOD_LE(TRUE, x2[3], x2[3], x0[3])≥1001_0_GCD_EQ(x0[3], x2[3])∧(UIncreasing(1001_0_GCD_EQ(x0[3], x2[3])), ≥))



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

    (23)    (COND_1451_0_MOD_LE(TRUE, x2[3], x2[3], x0[3])≥NonInfC∧COND_1451_0_MOD_LE(TRUE, x2[3], x2[3], x0[3])≥1001_0_GCD_EQ(x0[3], x2[3])∧(UIncreasing(1001_0_GCD_EQ(x0[3], x2[3])), ≥))



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

    (24)    ((UIncreasing(1001_0_GCD_EQ(x0[3], x2[3])), ≥)∧[bni_33] = 0∧[(-1)bso_34] ≥ 0)



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

    (25)    ((UIncreasing(1001_0_GCD_EQ(x0[3], x2[3])), ≥)∧[bni_33] = 0∧[(-1)bso_34] ≥ 0)



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

    (26)    ((UIncreasing(1001_0_GCD_EQ(x0[3], x2[3])), ≥)∧[bni_33] = 0∧[(-1)bso_34] ≥ 0)



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

    (27)    ((UIncreasing(1001_0_GCD_EQ(x0[3], x2[3])), ≥)∧[bni_33] = 0∧0 = 0∧0 = 0∧[(-1)bso_34] ≥ 0)







For Pair 1451_0_MOD_LE(x2[2], x2[2], x0[2]) → COND_1451_0_MOD_LE(<=(x2[2], x0[2]), x2[2], x2[2], x0[2]) the following chains were created:
  • We consider the chain 1451_0_MOD_LE(x2[2], x2[2], x0[2]) → COND_1451_0_MOD_LE(<=(x2[2], x0[2]), x2[2], x2[2], x0[2]), COND_1451_0_MOD_LE(TRUE, x2[3], x2[3], x0[3]) → 1001_0_GCD_EQ(x0[3], x2[3]) which results in the following constraint:

    (28)    (<=(x2[2], x0[2])=TRUEx2[2]=x2[3]x0[2]=x0[3]1451_0_MOD_LE(x2[2], x2[2], x0[2])≥NonInfC∧1451_0_MOD_LE(x2[2], x2[2], x0[2])≥COND_1451_0_MOD_LE(<=(x2[2], x0[2]), x2[2], x2[2], x0[2])∧(UIncreasing(COND_1451_0_MOD_LE(<=(x2[2], x0[2]), x2[2], x2[2], x0[2])), ≥))



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

    (29)    (<=(x2[2], x0[2])=TRUE1451_0_MOD_LE(x2[2], x2[2], x0[2])≥NonInfC∧1451_0_MOD_LE(x2[2], x2[2], x0[2])≥COND_1451_0_MOD_LE(<=(x2[2], x0[2]), x2[2], x2[2], x0[2])∧(UIncreasing(COND_1451_0_MOD_LE(<=(x2[2], x0[2]), x2[2], x2[2], x0[2])), ≥))



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

    (30)    (x0[2] + [-1]x2[2] ≥ 0 ⇒ (UIncreasing(COND_1451_0_MOD_LE(<=(x2[2], x0[2]), x2[2], x2[2], x0[2])), ≥)∧[(-1)bni_35 + (-1)Bound*bni_35] + [bni_35]x0[2] + [bni_35]x2[2] ≥ 0∧[(-1)bso_36] ≥ 0)



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

    (31)    (x0[2] + [-1]x2[2] ≥ 0 ⇒ (UIncreasing(COND_1451_0_MOD_LE(<=(x2[2], x0[2]), x2[2], x2[2], x0[2])), ≥)∧[(-1)bni_35 + (-1)Bound*bni_35] + [bni_35]x0[2] + [bni_35]x2[2] ≥ 0∧[(-1)bso_36] ≥ 0)



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

    (32)    (x0[2] + [-1]x2[2] ≥ 0 ⇒ (UIncreasing(COND_1451_0_MOD_LE(<=(x2[2], x0[2]), x2[2], x2[2], x0[2])), ≥)∧[(-1)bni_35 + (-1)Bound*bni_35] + [bni_35]x0[2] + [bni_35]x2[2] ≥ 0∧[(-1)bso_36] ≥ 0)



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

    (33)    (x0[2] ≥ 0 ⇒ (UIncreasing(COND_1451_0_MOD_LE(<=(x2[2], x0[2]), x2[2], x2[2], x0[2])), ≥)∧[(-1)bni_35 + (-1)Bound*bni_35] + [(2)bni_35]x2[2] + [bni_35]x0[2] ≥ 0∧[(-1)bso_36] ≥ 0)



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

    (34)    (x0[2] ≥ 0∧x2[2] ≥ 0 ⇒ (UIncreasing(COND_1451_0_MOD_LE(<=(x2[2], x0[2]), x2[2], x2[2], x0[2])), ≥)∧[(-1)bni_35 + (-1)Bound*bni_35] + [(-2)bni_35]x2[2] + [bni_35]x0[2] ≥ 0∧[(-1)bso_36] ≥ 0)


    (35)    (x0[2] ≥ 0∧x2[2] ≥ 0 ⇒ (UIncreasing(COND_1451_0_MOD_LE(<=(x2[2], x0[2]), x2[2], x2[2], x0[2])), ≥)∧[(-1)bni_35 + (-1)Bound*bni_35] + [(2)bni_35]x2[2] + [bni_35]x0[2] ≥ 0∧[(-1)bso_36] ≥ 0)







For Pair COND_1001_0_GCD_EQ(TRUE, x0[1], x1[1]) → 1451_0_MOD_LE(x0[1], x0[1], x1[1]) the following chains were created:
  • We consider the chain COND_1001_0_GCD_EQ(TRUE, x0[1], x1[1]) → 1451_0_MOD_LE(x0[1], x0[1], x1[1]), 1451_0_MOD_LE(x2[2], x2[2], x0[2]) → COND_1451_0_MOD_LE(<=(x2[2], x0[2]), x2[2], x2[2], x0[2]) which results in the following constraint:

    (36)    (x0[1]=x2[2]x1[1]=x0[2]COND_1001_0_GCD_EQ(TRUE, x0[1], x1[1])≥NonInfC∧COND_1001_0_GCD_EQ(TRUE, x0[1], x1[1])≥1451_0_MOD_LE(x0[1], x0[1], x1[1])∧(UIncreasing(1451_0_MOD_LE(x0[1], x0[1], x1[1])), ≥))



    We simplified constraint (36) using rule (IV) which results in the following new constraint:

    (37)    (COND_1001_0_GCD_EQ(TRUE, x0[1], x1[1])≥NonInfC∧COND_1001_0_GCD_EQ(TRUE, x0[1], x1[1])≥1451_0_MOD_LE(x0[1], x0[1], x1[1])∧(UIncreasing(1451_0_MOD_LE(x0[1], x0[1], x1[1])), ≥))



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

    (38)    ((UIncreasing(1451_0_MOD_LE(x0[1], x0[1], x1[1])), ≥)∧[bni_37] = 0∧[(-1)bso_38] ≥ 0)



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

    (39)    ((UIncreasing(1451_0_MOD_LE(x0[1], x0[1], x1[1])), ≥)∧[bni_37] = 0∧[(-1)bso_38] ≥ 0)



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

    (40)    ((UIncreasing(1451_0_MOD_LE(x0[1], x0[1], x1[1])), ≥)∧[bni_37] = 0∧[(-1)bso_38] ≥ 0)



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

    (41)    ((UIncreasing(1451_0_MOD_LE(x0[1], x0[1], x1[1])), ≥)∧[bni_37] = 0∧0 = 0∧0 = 0∧[(-1)bso_38] ≥ 0)



  • We consider the chain COND_1001_0_GCD_EQ(TRUE, x0[1], x1[1]) → 1451_0_MOD_LE(x0[1], x0[1], x1[1]), 1451_0_MOD_LE(x2[4], x2[4], x0[4]) → COND_1451_0_MOD_LE1(&&(>(x2[4], x0[4]), >(x0[4], 0)), x2[4], x2[4], x0[4]) which results in the following constraint:

    (42)    (x0[1]=x2[4]x1[1]=x0[4]COND_1001_0_GCD_EQ(TRUE, x0[1], x1[1])≥NonInfC∧COND_1001_0_GCD_EQ(TRUE, x0[1], x1[1])≥1451_0_MOD_LE(x0[1], x0[1], x1[1])∧(UIncreasing(1451_0_MOD_LE(x0[1], x0[1], x1[1])), ≥))



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

    (43)    (COND_1001_0_GCD_EQ(TRUE, x0[1], x1[1])≥NonInfC∧COND_1001_0_GCD_EQ(TRUE, x0[1], x1[1])≥1451_0_MOD_LE(x0[1], x0[1], x1[1])∧(UIncreasing(1451_0_MOD_LE(x0[1], x0[1], x1[1])), ≥))



    We simplified constraint (43) using rule (POLY_CONSTRAINTS) which results in the following new constraint:

    (44)    ((UIncreasing(1451_0_MOD_LE(x0[1], x0[1], x1[1])), ≥)∧[bni_37] = 0∧[(-1)bso_38] ≥ 0)



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

    (45)    ((UIncreasing(1451_0_MOD_LE(x0[1], x0[1], x1[1])), ≥)∧[bni_37] = 0∧[(-1)bso_38] ≥ 0)



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

    (46)    ((UIncreasing(1451_0_MOD_LE(x0[1], x0[1], x1[1])), ≥)∧[bni_37] = 0∧[(-1)bso_38] ≥ 0)



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

    (47)    ((UIncreasing(1451_0_MOD_LE(x0[1], x0[1], x1[1])), ≥)∧[bni_37] = 0∧0 = 0∧0 = 0∧[(-1)bso_38] ≥ 0)







For Pair 1001_0_GCD_EQ(x0[0], x1[0]) → COND_1001_0_GCD_EQ(&&(&&(>(x1[0], 0), >(x0[0], -1)), !(=(x0[0], x1[0]))), x0[0], x1[0]) the following chains were created:
  • We consider the chain 1001_0_GCD_EQ(x0[0], x1[0]) → COND_1001_0_GCD_EQ(&&(&&(>(x1[0], 0), >(x0[0], -1)), !(=(x0[0], x1[0]))), x0[0], x1[0]), COND_1001_0_GCD_EQ(TRUE, x0[1], x1[1]) → 1451_0_MOD_LE(x0[1], x0[1], x1[1]) which results in the following constraint:

    (48)    (&&(&&(>(x1[0], 0), >(x0[0], -1)), !(=(x0[0], x1[0])))=TRUEx0[0]=x0[1]x1[0]=x1[1]1001_0_GCD_EQ(x0[0], x1[0])≥NonInfC∧1001_0_GCD_EQ(x0[0], x1[0])≥COND_1001_0_GCD_EQ(&&(&&(>(x1[0], 0), >(x0[0], -1)), !(=(x0[0], x1[0]))), x0[0], x1[0])∧(UIncreasing(COND_1001_0_GCD_EQ(&&(&&(>(x1[0], 0), >(x0[0], -1)), !(=(x0[0], x1[0]))), x0[0], x1[0])), ≥))



    We simplified constraint (48) using rules (IV), (IDP_BOOLEAN) which results in the following new constraints:

    (49)    (>(x1[0], 0)=TRUE>(x0[0], -1)=TRUE<(x0[0], x1[0])=TRUE1001_0_GCD_EQ(x0[0], x1[0])≥NonInfC∧1001_0_GCD_EQ(x0[0], x1[0])≥COND_1001_0_GCD_EQ(&&(&&(>(x1[0], 0), >(x0[0], -1)), !(=(x0[0], x1[0]))), x0[0], x1[0])∧(UIncreasing(COND_1001_0_GCD_EQ(&&(&&(>(x1[0], 0), >(x0[0], -1)), !(=(x0[0], x1[0]))), x0[0], x1[0])), ≥))


    (50)    (>(x1[0], 0)=TRUE>(x0[0], -1)=TRUE>(x0[0], x1[0])=TRUE1001_0_GCD_EQ(x0[0], x1[0])≥NonInfC∧1001_0_GCD_EQ(x0[0], x1[0])≥COND_1001_0_GCD_EQ(&&(&&(>(x1[0], 0), >(x0[0], -1)), !(=(x0[0], x1[0]))), x0[0], x1[0])∧(UIncreasing(COND_1001_0_GCD_EQ(&&(&&(>(x1[0], 0), >(x0[0], -1)), !(=(x0[0], x1[0]))), x0[0], x1[0])), ≥))



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

    (51)    (x1[0] + [-1] ≥ 0∧x0[0] ≥ 0∧x1[0] + [-1] + [-1]x0[0] ≥ 0 ⇒ (UIncreasing(COND_1001_0_GCD_EQ(&&(&&(>(x1[0], 0), >(x0[0], -1)), !(=(x0[0], x1[0]))), x0[0], x1[0])), ≥)∧[(-1)bni_39 + (-1)Bound*bni_39] + [bni_39]x1[0] + [bni_39]x0[0] ≥ 0∧[(-1)bso_40] ≥ 0)



    We simplified constraint (50) using rule (POLY_CONSTRAINTS) which results in the following new constraint:

    (52)    (x1[0] + [-1] ≥ 0∧x0[0] ≥ 0∧x0[0] + [-1] + [-1]x1[0] ≥ 0 ⇒ (UIncreasing(COND_1001_0_GCD_EQ(&&(&&(>(x1[0], 0), >(x0[0], -1)), !(=(x0[0], x1[0]))), x0[0], x1[0])), ≥)∧[(-1)bni_39 + (-1)Bound*bni_39] + [bni_39]x1[0] + [bni_39]x0[0] ≥ 0∧[(-1)bso_40] ≥ 0)



    We simplified constraint (51) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:

    (53)    (x1[0] + [-1] ≥ 0∧x0[0] ≥ 0∧x1[0] + [-1] + [-1]x0[0] ≥ 0 ⇒ (UIncreasing(COND_1001_0_GCD_EQ(&&(&&(>(x1[0], 0), >(x0[0], -1)), !(=(x0[0], x1[0]))), x0[0], x1[0])), ≥)∧[(-1)bni_39 + (-1)Bound*bni_39] + [bni_39]x1[0] + [bni_39]x0[0] ≥ 0∧[(-1)bso_40] ≥ 0)



    We simplified constraint (52) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:

    (54)    (x1[0] + [-1] ≥ 0∧x0[0] ≥ 0∧x0[0] + [-1] + [-1]x1[0] ≥ 0 ⇒ (UIncreasing(COND_1001_0_GCD_EQ(&&(&&(>(x1[0], 0), >(x0[0], -1)), !(=(x0[0], x1[0]))), x0[0], x1[0])), ≥)∧[(-1)bni_39 + (-1)Bound*bni_39] + [bni_39]x1[0] + [bni_39]x0[0] ≥ 0∧[(-1)bso_40] ≥ 0)



    We simplified constraint (53) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:

    (55)    (x1[0] + [-1] ≥ 0∧x0[0] ≥ 0∧x1[0] + [-1] + [-1]x0[0] ≥ 0 ⇒ (UIncreasing(COND_1001_0_GCD_EQ(&&(&&(>(x1[0], 0), >(x0[0], -1)), !(=(x0[0], x1[0]))), x0[0], x1[0])), ≥)∧[(-1)bni_39 + (-1)Bound*bni_39] + [bni_39]x1[0] + [bni_39]x0[0] ≥ 0∧[(-1)bso_40] ≥ 0)



    We simplified constraint (54) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:

    (56)    (x1[0] + [-1] ≥ 0∧x0[0] ≥ 0∧x0[0] + [-1] + [-1]x1[0] ≥ 0 ⇒ (UIncreasing(COND_1001_0_GCD_EQ(&&(&&(>(x1[0], 0), >(x0[0], -1)), !(=(x0[0], x1[0]))), x0[0], x1[0])), ≥)∧[(-1)bni_39 + (-1)Bound*bni_39] + [bni_39]x1[0] + [bni_39]x0[0] ≥ 0∧[(-1)bso_40] ≥ 0)



    We simplified constraint (55) using rule (IDP_SMT_SPLIT) which results in the following new constraint:

    (57)    (x1[0] ≥ 0∧x0[0] ≥ 0∧x1[0] + [-1]x0[0] ≥ 0 ⇒ (UIncreasing(COND_1001_0_GCD_EQ(&&(&&(>(x1[0], 0), >(x0[0], -1)), !(=(x0[0], x1[0]))), x0[0], x1[0])), ≥)∧[(-1)Bound*bni_39] + [bni_39]x1[0] + [bni_39]x0[0] ≥ 0∧[(-1)bso_40] ≥ 0)



    We simplified constraint (56) using rule (IDP_SMT_SPLIT) which results in the following new constraint:

    (58)    (x1[0] ≥ 0∧x0[0] ≥ 0∧x0[0] + [-2] + [-1]x1[0] ≥ 0 ⇒ (UIncreasing(COND_1001_0_GCD_EQ(&&(&&(>(x1[0], 0), >(x0[0], -1)), !(=(x0[0], x1[0]))), x0[0], x1[0])), ≥)∧[(-1)Bound*bni_39] + [bni_39]x1[0] + [bni_39]x0[0] ≥ 0∧[(-1)bso_40] ≥ 0)



    We simplified constraint (57) using rule (IDP_SMT_SPLIT) which results in the following new constraint:

    (59)    (x0[0] + x1[0] ≥ 0∧x0[0] ≥ 0∧x1[0] ≥ 0 ⇒ (UIncreasing(COND_1001_0_GCD_EQ(&&(&&(>(x1[0], 0), >(x0[0], -1)), !(=(x0[0], x1[0]))), x0[0], x1[0])), ≥)∧[(-1)Bound*bni_39] + [(2)bni_39]x0[0] + [bni_39]x1[0] ≥ 0∧[(-1)bso_40] ≥ 0)



    We simplified constraint (58) using rule (IDP_SMT_SPLIT) which results in the following new constraint:

    (60)    (x1[0] ≥ 0∧[2] + x1[0] + x0[0] ≥ 0∧x0[0] ≥ 0 ⇒ (UIncreasing(COND_1001_0_GCD_EQ(&&(&&(>(x1[0], 0), >(x0[0], -1)), !(=(x0[0], x1[0]))), x0[0], x1[0])), ≥)∧[(-1)Bound*bni_39 + (2)bni_39] + [(2)bni_39]x1[0] + [bni_39]x0[0] ≥ 0∧[(-1)bso_40] ≥ 0)







To summarize, we get the following constraints P for the following pairs.
  • COND_1451_0_MOD_LE1(TRUE, x2[5], x2[5], x0[5]) → 1451_0_MOD_LE(-(x2[5], x0[5]), -(x2[5], x0[5]), x0[5])
    • (x2[4] ≥ 0∧x0[4] ≥ 0 ⇒ (UIncreasing(1451_0_MOD_LE(-(x2[5], x0[5]), -(x2[5], x0[5]), x0[5])), ≥)∧[(-1)Bound*bni_29 + (2)bni_29] + [bni_29]x0[4] + [bni_29]x2[4] ≥ 0∧[1 + (-1)bso_30] ≥ 0)
    • (x2[4] ≥ 0∧x0[4] ≥ 0 ⇒ (UIncreasing(1451_0_MOD_LE(-(x2[5], x0[5]), -(x2[5], x0[5]), x0[5])), ≥)∧[(-1)Bound*bni_29 + (2)bni_29] + [bni_29]x0[4] + [bni_29]x2[4] ≥ 0∧[1 + (-1)bso_30] ≥ 0)

  • 1451_0_MOD_LE(x2[4], x2[4], x0[4]) → COND_1451_0_MOD_LE1(&&(>(x2[4], x0[4]), >(x0[4], 0)), x2[4], x2[4], x0[4])
    • (x2[4] ≥ 0∧x0[4] ≥ 0 ⇒ (UIncreasing(COND_1451_0_MOD_LE1(&&(>(x2[4], x0[4]), >(x0[4], 0)), x2[4], x2[4], x0[4])), ≥)∧[(-1)Bound*bni_31 + (2)bni_31] + [(2)bni_31]x0[4] + [bni_31]x2[4] ≥ 0∧[(-1)bso_32] + x0[4] ≥ 0)

  • COND_1451_0_MOD_LE(TRUE, x2[3], x2[3], x0[3]) → 1001_0_GCD_EQ(x0[3], x2[3])
    • ((UIncreasing(1001_0_GCD_EQ(x0[3], x2[3])), ≥)∧[bni_33] = 0∧0 = 0∧0 = 0∧[(-1)bso_34] ≥ 0)

  • 1451_0_MOD_LE(x2[2], x2[2], x0[2]) → COND_1451_0_MOD_LE(<=(x2[2], x0[2]), x2[2], x2[2], x0[2])
    • (x0[2] ≥ 0∧x2[2] ≥ 0 ⇒ (UIncreasing(COND_1451_0_MOD_LE(<=(x2[2], x0[2]), x2[2], x2[2], x0[2])), ≥)∧[(-1)bni_35 + (-1)Bound*bni_35] + [(-2)bni_35]x2[2] + [bni_35]x0[2] ≥ 0∧[(-1)bso_36] ≥ 0)
    • (x0[2] ≥ 0∧x2[2] ≥ 0 ⇒ (UIncreasing(COND_1451_0_MOD_LE(<=(x2[2], x0[2]), x2[2], x2[2], x0[2])), ≥)∧[(-1)bni_35 + (-1)Bound*bni_35] + [(2)bni_35]x2[2] + [bni_35]x0[2] ≥ 0∧[(-1)bso_36] ≥ 0)

  • COND_1001_0_GCD_EQ(TRUE, x0[1], x1[1]) → 1451_0_MOD_LE(x0[1], x0[1], x1[1])
    • ((UIncreasing(1451_0_MOD_LE(x0[1], x0[1], x1[1])), ≥)∧[bni_37] = 0∧0 = 0∧0 = 0∧[(-1)bso_38] ≥ 0)
    • ((UIncreasing(1451_0_MOD_LE(x0[1], x0[1], x1[1])), ≥)∧[bni_37] = 0∧0 = 0∧0 = 0∧[(-1)bso_38] ≥ 0)

  • 1001_0_GCD_EQ(x0[0], x1[0]) → COND_1001_0_GCD_EQ(&&(&&(>(x1[0], 0), >(x0[0], -1)), !(=(x0[0], x1[0]))), x0[0], x1[0])
    • (x0[0] + x1[0] ≥ 0∧x0[0] ≥ 0∧x1[0] ≥ 0 ⇒ (UIncreasing(COND_1001_0_GCD_EQ(&&(&&(>(x1[0], 0), >(x0[0], -1)), !(=(x0[0], x1[0]))), x0[0], x1[0])), ≥)∧[(-1)Bound*bni_39] + [(2)bni_39]x0[0] + [bni_39]x1[0] ≥ 0∧[(-1)bso_40] ≥ 0)
    • (x1[0] ≥ 0∧[2] + x1[0] + x0[0] ≥ 0∧x0[0] ≥ 0 ⇒ (UIncreasing(COND_1001_0_GCD_EQ(&&(&&(>(x1[0], 0), >(x0[0], -1)), !(=(x0[0], x1[0]))), x0[0], x1[0])), ≥)∧[(-1)Bound*bni_39 + (2)bni_39] + [(2)bni_39]x1[0] + [bni_39]x0[0] ≥ 0∧[(-1)bso_40] ≥ 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_1451_0_MOD_LE1(x1, x2, x3, x4)) = [2]x3 + [-1]x2   
POL(1451_0_MOD_LE(x1, x2, x3)) = [-1] + x3 + [-1]x2 + [2]x1   
POL(-(x1, x2)) = x1 + [-1]x2   
POL(&&(x1, x2)) = [-1]   
POL(>(x1, x2)) = [-1]   
POL(0) = 0   
POL(COND_1451_0_MOD_LE(x1, x2, x3, x4)) = [-1] + x4 + x3   
POL(1001_0_GCD_EQ(x1, x2)) = [-1] + x2 + x1   
POL(<=(x1, x2)) = [-1]   
POL(COND_1001_0_GCD_EQ(x1, x2, x3)) = [-1] + x3 + x2   
POL(-1) = [-1]   
POL(!(x1)) = [-1]   
POL(=(x1, x2)) = [-1]   

The following pairs are in P>:

COND_1451_0_MOD_LE1(TRUE, x2[5], x2[5], x0[5]) → 1451_0_MOD_LE(-(x2[5], x0[5]), -(x2[5], x0[5]), x0[5])

The following pairs are in Pbound:

COND_1451_0_MOD_LE1(TRUE, x2[5], x2[5], x0[5]) → 1451_0_MOD_LE(-(x2[5], x0[5]), -(x2[5], x0[5]), x0[5])
1451_0_MOD_LE(x2[4], x2[4], x0[4]) → COND_1451_0_MOD_LE1(&&(>(x2[4], x0[4]), >(x0[4], 0)), x2[4], x2[4], x0[4])
1001_0_GCD_EQ(x0[0], x1[0]) → COND_1001_0_GCD_EQ(&&(&&(>(x1[0], 0), >(x0[0], -1)), !(=(x0[0], x1[0]))), x0[0], x1[0])

The following pairs are in P:

1451_0_MOD_LE(x2[4], x2[4], x0[4]) → COND_1451_0_MOD_LE1(&&(>(x2[4], x0[4]), >(x0[4], 0)), x2[4], x2[4], x0[4])
COND_1451_0_MOD_LE(TRUE, x2[3], x2[3], x0[3]) → 1001_0_GCD_EQ(x0[3], x2[3])
1451_0_MOD_LE(x2[2], x2[2], x0[2]) → COND_1451_0_MOD_LE(<=(x2[2], x0[2]), x2[2], x2[2], x0[2])
COND_1001_0_GCD_EQ(TRUE, x0[1], x1[1]) → 1451_0_MOD_LE(x0[1], x0[1], x1[1])
1001_0_GCD_EQ(x0[0], x1[0]) → COND_1001_0_GCD_EQ(&&(&&(>(x1[0], 0), >(x0[0], -1)), !(=(x0[0], x1[0]))), x0[0], x1[0])

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

FALSE1&&(FALSE, FALSE)1

(12) 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:
(4): 1451_0_MOD_LE(x2[4], x2[4], x0[4]) → COND_1451_0_MOD_LE1(x2[4] > x0[4] && x0[4] > 0, x2[4], x2[4], x0[4])
(3): COND_1451_0_MOD_LE(TRUE, x2[3], x2[3], x0[3]) → 1001_0_GCD_EQ(x0[3], x2[3])
(2): 1451_0_MOD_LE(x2[2], x2[2], x0[2]) → COND_1451_0_MOD_LE(x2[2] <= x0[2], x2[2], x2[2], x0[2])
(1): COND_1001_0_GCD_EQ(TRUE, x0[1], x1[1]) → 1451_0_MOD_LE(x0[1], x0[1], x1[1])
(0): 1001_0_GCD_EQ(x0[0], x1[0]) → COND_1001_0_GCD_EQ(x1[0] > 0 && x0[0] > -1 && !(x0[0] = x1[0]), x0[0], x1[0])

(3) -> (0), if (x0[3]* x0[0]x2[3]* x1[0])


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


(1) -> (2), if (x0[1]* x2[2]x1[1]* x0[2])


(2) -> (3), if (x2[2] <= x0[2]x2[2]* x2[3]x0[2]* x0[3])


(1) -> (4), if (x0[1]* x2[4]x1[1]* x0[4])



The set Q is empty.

(13) IDependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 1 SCC 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, Boolean


R is empty.

The integer pair graph contains the following rules and edges:
(2): 1451_0_MOD_LE(x2[2], x2[2], x0[2]) → COND_1451_0_MOD_LE(x2[2] <= x0[2], x2[2], x2[2], x0[2])
(1): COND_1001_0_GCD_EQ(TRUE, x0[1], x1[1]) → 1451_0_MOD_LE(x0[1], x0[1], x1[1])
(0): 1001_0_GCD_EQ(x0[0], x1[0]) → COND_1001_0_GCD_EQ(x1[0] > 0 && x0[0] > -1 && !(x0[0] = x1[0]), x0[0], x1[0])
(3): COND_1451_0_MOD_LE(TRUE, x2[3], x2[3], x0[3]) → 1001_0_GCD_EQ(x0[3], x2[3])

(3) -> (0), if (x0[3]* x0[0]x2[3]* x1[0])


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


(1) -> (2), if (x0[1]* x2[2]x1[1]* x0[2])


(2) -> (3), if (x2[2] <= x0[2]x2[2]* x2[3]x0[2]* x0[3])



The set Q is empty.

(15) IDPNonInfProof (SOUND transformation)

Used the following options for this NonInfProof:
IDPGPoloSolver: Range: [(-1,2)] IsNat: false Interpretation Shape Heuristic: aprove.DPFramework.IDPProblem.Processors.nonInf.poly.IdpDefaultShapeHeuristic@7a62173f Constraint Generator: NonInfConstraintGenerator: PathGenerator: MetricPathGenerator: Max Left Steps: 3 Max Right Steps: 2

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 1451_0_MOD_LE(x2[2], x2[2], x0[2]) → COND_1451_0_MOD_LE(<=(x2[2], x0[2]), x2[2], x2[2], x0[2]) the following chains were created:
  • We consider the chain COND_1451_0_MOD_LE(TRUE, x2[3], x2[3], x0[3]) → 1001_0_GCD_EQ(x0[3], x2[3]), 1001_0_GCD_EQ(x0[0], x1[0]) → COND_1001_0_GCD_EQ(&&(&&(>(x1[0], 0), >(x0[0], -1)), !(=(x0[0], x1[0]))), x0[0], x1[0]), COND_1001_0_GCD_EQ(TRUE, x0[1], x1[1]) → 1451_0_MOD_LE(x0[1], x0[1], x1[1]), 1451_0_MOD_LE(x2[2], x2[2], x0[2]) → COND_1451_0_MOD_LE(<=(x2[2], x0[2]), x2[2], x2[2], x0[2]), COND_1451_0_MOD_LE(TRUE, x2[3], x2[3], x0[3]) → 1001_0_GCD_EQ(x0[3], x2[3]), 1001_0_GCD_EQ(x0[0], x1[0]) → COND_1001_0_GCD_EQ(&&(&&(>(x1[0], 0), >(x0[0], -1)), !(=(x0[0], x1[0]))), x0[0], x1[0]) which results in the following constraint:

    (1)    (x0[3]=x0[0]x2[3]=x1[0]&&(&&(>(x1[0], 0), >(x0[0], -1)), !(=(x0[0], x1[0])))=TRUEx0[0]=x0[1]x1[0]=x1[1]x0[1]=x2[2]x1[1]=x0[2]<=(x2[2], x0[2])=TRUEx2[2]=x2[3]1x0[2]=x0[3]1x0[3]1=x0[0]1x2[3]1=x1[0]11451_0_MOD_LE(x2[2], x2[2], x0[2])≥NonInfC∧1451_0_MOD_LE(x2[2], x2[2], x0[2])≥COND_1451_0_MOD_LE(<=(x2[2], x0[2]), x2[2], x2[2], x0[2])∧(UIncreasing(COND_1451_0_MOD_LE(<=(x2[2], x0[2]), x2[2], x2[2], x0[2])), ≥))



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

    (2)    (<=(x0[0], x1[0])=TRUE>(x1[0], 0)=TRUE>(x0[0], -1)=TRUE<(x0[0], x1[0])=TRUE1451_0_MOD_LE(x0[0], x0[0], x1[0])≥NonInfC∧1451_0_MOD_LE(x0[0], x0[0], x1[0])≥COND_1451_0_MOD_LE(<=(x0[0], x1[0]), x0[0], x0[0], x1[0])∧(UIncreasing(COND_1451_0_MOD_LE(<=(x2[2], x0[2]), x2[2], x2[2], x0[2])), ≥))



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

    (3)    (x1[0] + [-1]x0[0] ≥ 0∧x1[0] + [-1] ≥ 0∧x0[0] ≥ 0∧x1[0] + [-1] + [-1]x0[0] ≥ 0 ⇒ (UIncreasing(COND_1451_0_MOD_LE(<=(x2[2], x0[2]), x2[2], x2[2], x0[2])), ≥)∧[(-1)bni_24 + (-1)Bound*bni_24] + [(-1)bni_24]x1[0] + [(2)bni_24]x0[0] ≥ 0∧[(-1)bso_25] ≥ 0)



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

    (4)    (x1[0] + [-1]x0[0] ≥ 0∧x1[0] + [-1] ≥ 0∧x0[0] ≥ 0∧x1[0] + [-1] + [-1]x0[0] ≥ 0 ⇒ (UIncreasing(COND_1451_0_MOD_LE(<=(x2[2], x0[2]), x2[2], x2[2], x0[2])), ≥)∧[(-1)bni_24 + (-1)Bound*bni_24] + [(-1)bni_24]x1[0] + [(2)bni_24]x0[0] ≥ 0∧[(-1)bso_25] ≥ 0)



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

    (6)    (x1[0] + [-1]x0[0] ≥ 0∧x1[0] + [-1] ≥ 0∧x0[0] ≥ 0∧x0[0] + [-1] + [-1]x1[0] ≥ 0 ⇒ (UIncreasing(COND_1451_0_MOD_LE(<=(x2[2], x0[2]), x2[2], x2[2], x0[2])), ≥)∧[(-1)bni_24 + (-1)Bound*bni_24] + [(-1)bni_24]x1[0] + [(2)bni_24]x0[0] ≥ 0∧[(-1)bso_25] ≥ 0)



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

    (7)    (x1[0] + [-1]x0[0] ≥ 0∧x1[0] + [-1] ≥ 0∧x0[0] ≥ 0∧x1[0] + [-1] + [-1]x0[0] ≥ 0 ⇒ (UIncreasing(COND_1451_0_MOD_LE(<=(x2[2], x0[2]), x2[2], x2[2], x0[2])), ≥)∧[(-1)bni_24 + (-1)Bound*bni_24] + [(-1)bni_24]x1[0] + [(2)bni_24]x0[0] ≥ 0∧[(-1)bso_25] ≥ 0)



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

    (8)    (x1[0] + [-1]x0[0] ≥ 0∧x1[0] + [-1] ≥ 0∧x0[0] ≥ 0∧x0[0] + [-1] + [-1]x1[0] ≥ 0 ⇒ (UIncreasing(COND_1451_0_MOD_LE(<=(x2[2], x0[2]), x2[2], x2[2], x0[2])), ≥)∧[(-1)bni_24 + (-1)Bound*bni_24] + [(-1)bni_24]x1[0] + [(2)bni_24]x0[0] ≥ 0∧[(-1)bso_25] ≥ 0)



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

    (9)    (x1[0] ≥ 0∧x0[0] + [-1] + x1[0] ≥ 0∧x0[0] ≥ 0∧[-1] + x1[0] ≥ 0 ⇒ (UIncreasing(COND_1451_0_MOD_LE(<=(x2[2], x0[2]), x2[2], x2[2], x0[2])), ≥)∧[(-1)bni_24 + (-1)Bound*bni_24] + [bni_24]x0[0] + [(-1)bni_24]x1[0] ≥ 0∧[(-1)bso_25] ≥ 0)



    We solved constraint (8) using rule (IDP_SMT_SPLIT).We simplified constraint (9) using rule (IDP_SMT_SPLIT) which results in the following new constraint:

    (10)    ([1] + x1[0] ≥ 0∧x0[0] + x1[0] ≥ 0∧x0[0] ≥ 0∧x1[0] ≥ 0 ⇒ (UIncreasing(COND_1451_0_MOD_LE(<=(x2[2], x0[2]), x2[2], x2[2], x0[2])), ≥)∧[(-2)bni_24 + (-1)Bound*bni_24] + [bni_24]x0[0] + [(-1)bni_24]x1[0] ≥ 0∧[(-1)bso_25] ≥ 0)







For Pair COND_1001_0_GCD_EQ(TRUE, x0[1], x1[1]) → 1451_0_MOD_LE(x0[1], x0[1], x1[1]) the following chains were created:
  • We consider the chain 1451_0_MOD_LE(x2[2], x2[2], x0[2]) → COND_1451_0_MOD_LE(<=(x2[2], x0[2]), x2[2], x2[2], x0[2]), COND_1451_0_MOD_LE(TRUE, x2[3], x2[3], x0[3]) → 1001_0_GCD_EQ(x0[3], x2[3]), 1001_0_GCD_EQ(x0[0], x1[0]) → COND_1001_0_GCD_EQ(&&(&&(>(x1[0], 0), >(x0[0], -1)), !(=(x0[0], x1[0]))), x0[0], x1[0]), COND_1001_0_GCD_EQ(TRUE, x0[1], x1[1]) → 1451_0_MOD_LE(x0[1], x0[1], x1[1]), 1451_0_MOD_LE(x2[2], x2[2], x0[2]) → COND_1451_0_MOD_LE(<=(x2[2], x0[2]), x2[2], x2[2], x0[2]), COND_1451_0_MOD_LE(TRUE, x2[3], x2[3], x0[3]) → 1001_0_GCD_EQ(x0[3], x2[3]) which results in the following constraint:

    (11)    (<=(x2[2], x0[2])=TRUEx2[2]=x2[3]x0[2]=x0[3]x0[3]=x0[0]x2[3]=x1[0]&&(&&(>(x1[0], 0), >(x0[0], -1)), !(=(x0[0], x1[0])))=TRUEx0[0]=x0[1]x1[0]=x1[1]x0[1]=x2[2]1x1[1]=x0[2]1<=(x2[2]1, x0[2]1)=TRUEx2[2]1=x2[3]1x0[2]1=x0[3]1COND_1001_0_GCD_EQ(TRUE, x0[1], x1[1])≥NonInfC∧COND_1001_0_GCD_EQ(TRUE, x0[1], x1[1])≥1451_0_MOD_LE(x0[1], x0[1], x1[1])∧(UIncreasing(1451_0_MOD_LE(x0[1], x0[1], x1[1])), ≥))



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

    (12)    (<=(x2[2], x0[2])=TRUE<=(x0[2], x2[2])=TRUE>(x2[2], 0)=TRUE>(x0[2], -1)=TRUE<(x0[2], x2[2])=TRUECOND_1001_0_GCD_EQ(TRUE, x0[2], x2[2])≥NonInfC∧COND_1001_0_GCD_EQ(TRUE, x0[2], x2[2])≥1451_0_MOD_LE(x0[2], x0[2], x2[2])∧(UIncreasing(1451_0_MOD_LE(x0[1], x0[1], x1[1])), ≥))



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

    (13)    (x0[2] + [-1]x2[2] ≥ 0∧x2[2] + [-1]x0[2] ≥ 0∧x2[2] + [-1] ≥ 0∧x0[2] ≥ 0∧x2[2] + [-1] + [-1]x0[2] ≥ 0 ⇒ (UIncreasing(1451_0_MOD_LE(x0[1], x0[1], x1[1])), ≥)∧[(-1)bni_26 + (-1)Bound*bni_26] + [(-1)bni_26]x0[2] ≥ 0∧[(-1)bso_27] + x2[2] + [-3]x0[2] ≥ 0)



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

    (14)    (x0[2] + [-1]x2[2] ≥ 0∧x2[2] + [-1]x0[2] ≥ 0∧x2[2] + [-1] ≥ 0∧x0[2] ≥ 0∧x2[2] + [-1] + [-1]x0[2] ≥ 0 ⇒ (UIncreasing(1451_0_MOD_LE(x0[1], x0[1], x1[1])), ≥)∧[(-1)bni_26 + (-1)Bound*bni_26] + [(-1)bni_26]x0[2] ≥ 0∧[(-1)bso_27] + x2[2] + [-3]x0[2] ≥ 0)



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

    (16)    (x0[2] + [-1]x2[2] ≥ 0∧x2[2] + [-1]x0[2] ≥ 0∧x2[2] + [-1] ≥ 0∧x0[2] ≥ 0∧x0[2] + [-1] + [-1]x2[2] ≥ 0 ⇒ (UIncreasing(1451_0_MOD_LE(x0[1], x0[1], x1[1])), ≥)∧[(-1)bni_26 + (-1)Bound*bni_26] + [(-1)bni_26]x0[2] ≥ 0∧[(-1)bso_27] + x2[2] + [-3]x0[2] ≥ 0)



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

    (17)    (x0[2] + [-1]x2[2] ≥ 0∧x2[2] + [-1]x0[2] ≥ 0∧x2[2] + [-1] ≥ 0∧x0[2] ≥ 0∧x2[2] + [-1] + [-1]x0[2] ≥ 0 ⇒ (UIncreasing(1451_0_MOD_LE(x0[1], x0[1], x1[1])), ≥)∧[(-1)bni_26 + (-1)Bound*bni_26] + [(-1)bni_26]x0[2] ≥ 0∧[(-1)bso_27] + x2[2] + [-3]x0[2] ≥ 0)



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

    (18)    (x0[2] + [-1]x2[2] ≥ 0∧x2[2] + [-1]x0[2] ≥ 0∧x2[2] + [-1] ≥ 0∧x0[2] ≥ 0∧x0[2] + [-1] + [-1]x2[2] ≥ 0 ⇒ (UIncreasing(1451_0_MOD_LE(x0[1], x0[1], x1[1])), ≥)∧[(-1)bni_26 + (-1)Bound*bni_26] + [(-1)bni_26]x0[2] ≥ 0∧[(-1)bso_27] + x2[2] + [-3]x0[2] ≥ 0)



    We solved constraint (17) using rule (IDP_SMT_SPLIT).We solved constraint (18) using rule (IDP_SMT_SPLIT).




For Pair 1001_0_GCD_EQ(x0[0], x1[0]) → COND_1001_0_GCD_EQ(&&(&&(>(x1[0], 0), >(x0[0], -1)), !(=(x0[0], x1[0]))), x0[0], x1[0]) the following chains were created:
  • We consider the chain COND_1001_0_GCD_EQ(TRUE, x0[1], x1[1]) → 1451_0_MOD_LE(x0[1], x0[1], x1[1]), 1451_0_MOD_LE(x2[2], x2[2], x0[2]) → COND_1451_0_MOD_LE(<=(x2[2], x0[2]), x2[2], x2[2], x0[2]), COND_1451_0_MOD_LE(TRUE, x2[3], x2[3], x0[3]) → 1001_0_GCD_EQ(x0[3], x2[3]), 1001_0_GCD_EQ(x0[0], x1[0]) → COND_1001_0_GCD_EQ(&&(&&(>(x1[0], 0), >(x0[0], -1)), !(=(x0[0], x1[0]))), x0[0], x1[0]), COND_1001_0_GCD_EQ(TRUE, x0[1], x1[1]) → 1451_0_MOD_LE(x0[1], x0[1], x1[1]), 1451_0_MOD_LE(x2[2], x2[2], x0[2]) → COND_1451_0_MOD_LE(<=(x2[2], x0[2]), x2[2], x2[2], x0[2]) which results in the following constraint:

    (19)    (x0[1]=x2[2]x1[1]=x0[2]<=(x2[2], x0[2])=TRUEx2[2]=x2[3]x0[2]=x0[3]x0[3]=x0[0]x2[3]=x1[0]&&(&&(>(x1[0], 0), >(x0[0], -1)), !(=(x0[0], x1[0])))=TRUEx0[0]=x0[1]1x1[0]=x1[1]1x0[1]1=x2[2]1x1[1]1=x0[2]11001_0_GCD_EQ(x0[0], x1[0])≥NonInfC∧1001_0_GCD_EQ(x0[0], x1[0])≥COND_1001_0_GCD_EQ(&&(&&(>(x1[0], 0), >(x0[0], -1)), !(=(x0[0], x1[0]))), x0[0], x1[0])∧(UIncreasing(COND_1001_0_GCD_EQ(&&(&&(>(x1[0], 0), >(x0[0], -1)), !(=(x0[0], x1[0]))), x0[0], x1[0])), ≥))



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

    (20)    (<=(x2[2], x0[2])=TRUE>(x2[2], 0)=TRUE>(x0[2], -1)=TRUE<(x0[2], x2[2])=TRUE1001_0_GCD_EQ(x0[2], x2[2])≥NonInfC∧1001_0_GCD_EQ(x0[2], x2[2])≥COND_1001_0_GCD_EQ(&&(&&(>(x2[2], 0), >(x0[2], -1)), !(=(x0[2], x2[2]))), x0[2], x2[2])∧(UIncreasing(COND_1001_0_GCD_EQ(&&(&&(>(x1[0], 0), >(x0[0], -1)), !(=(x0[0], x1[0]))), x0[0], x1[0])), ≥))



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

    (21)    (x0[2] + [-1]x2[2] ≥ 0∧x2[2] + [-1] ≥ 0∧x0[2] ≥ 0∧x2[2] + [-1] + [-1]x0[2] ≥ 0 ⇒ (UIncreasing(COND_1001_0_GCD_EQ(&&(&&(>(x1[0], 0), >(x0[0], -1)), !(=(x0[0], x1[0]))), x0[0], x1[0])), ≥)∧[bni_28 + (-1)Bound*bni_28] + [(-1)bni_28]x0[2] ≥ 0∧[1 + (-1)bso_29] ≥ 0)



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

    (22)    (x0[2] + [-1]x2[2] ≥ 0∧x2[2] + [-1] ≥ 0∧x0[2] ≥ 0∧x2[2] + [-1] + [-1]x0[2] ≥ 0 ⇒ (UIncreasing(COND_1001_0_GCD_EQ(&&(&&(>(x1[0], 0), >(x0[0], -1)), !(=(x0[0], x1[0]))), x0[0], x1[0])), ≥)∧[bni_28 + (-1)Bound*bni_28] + [(-1)bni_28]x0[2] ≥ 0∧[1 + (-1)bso_29] ≥ 0)



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

    (24)    (x0[2] + [-1]x2[2] ≥ 0∧x2[2] + [-1] ≥ 0∧x0[2] ≥ 0∧x0[2] + [-1] + [-1]x2[2] ≥ 0 ⇒ (UIncreasing(COND_1001_0_GCD_EQ(&&(&&(>(x1[0], 0), >(x0[0], -1)), !(=(x0[0], x1[0]))), x0[0], x1[0])), ≥)∧[bni_28 + (-1)Bound*bni_28] + [(-1)bni_28]x0[2] ≥ 0∧[1 + (-1)bso_29] ≥ 0)



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

    (25)    (x0[2] + [-1]x2[2] ≥ 0∧x2[2] + [-1] ≥ 0∧x0[2] ≥ 0∧x2[2] + [-1] + [-1]x0[2] ≥ 0 ⇒ (UIncreasing(COND_1001_0_GCD_EQ(&&(&&(>(x1[0], 0), >(x0[0], -1)), !(=(x0[0], x1[0]))), x0[0], x1[0])), ≥)∧[bni_28 + (-1)Bound*bni_28] + [(-1)bni_28]x0[2] ≥ 0∧[1 + (-1)bso_29] ≥ 0)



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

    (26)    (x0[2] + [-1]x2[2] ≥ 0∧x2[2] + [-1] ≥ 0∧x0[2] ≥ 0∧x0[2] + [-1] + [-1]x2[2] ≥ 0 ⇒ (UIncreasing(COND_1001_0_GCD_EQ(&&(&&(>(x1[0], 0), >(x0[0], -1)), !(=(x0[0], x1[0]))), x0[0], x1[0])), ≥)∧[bni_28 + (-1)Bound*bni_28] + [(-1)bni_28]x0[2] ≥ 0∧[1 + (-1)bso_29] ≥ 0)



    We solved constraint (25) using rule (IDP_SMT_SPLIT).We simplified constraint (26) using rule (IDP_SMT_SPLIT) which results in the following new constraint:

    (27)    (x0[2] ≥ 0∧x2[2] + [-1] ≥ 0∧x2[2] + x0[2] ≥ 0∧[-1] + x0[2] ≥ 0 ⇒ (UIncreasing(COND_1001_0_GCD_EQ(&&(&&(>(x1[0], 0), >(x0[0], -1)), !(=(x0[0], x1[0]))), x0[0], x1[0])), ≥)∧[bni_28 + (-1)Bound*bni_28] + [(-1)bni_28]x2[2] + [(-1)bni_28]x0[2] ≥ 0∧[1 + (-1)bso_29] ≥ 0)



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

    (28)    (x0[2] ≥ 0∧x2[2] ≥ 0∧[1] + x2[2] + x0[2] ≥ 0∧[-1] + x0[2] ≥ 0 ⇒ (UIncreasing(COND_1001_0_GCD_EQ(&&(&&(>(x1[0], 0), >(x0[0], -1)), !(=(x0[0], x1[0]))), x0[0], x1[0])), ≥)∧[(-1)Bound*bni_28] + [(-1)bni_28]x2[2] + [(-1)bni_28]x0[2] ≥ 0∧[1 + (-1)bso_29] ≥ 0)



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

    (29)    ([1] + x0[2] ≥ 0∧x2[2] ≥ 0∧[2] + x2[2] + x0[2] ≥ 0∧x0[2] ≥ 0 ⇒ (UIncreasing(COND_1001_0_GCD_EQ(&&(&&(>(x1[0], 0), >(x0[0], -1)), !(=(x0[0], x1[0]))), x0[0], x1[0])), ≥)∧[(-1)bni_28 + (-1)Bound*bni_28] + [(-1)bni_28]x2[2] + [(-1)bni_28]x0[2] ≥ 0∧[1 + (-1)bso_29] ≥ 0)







For Pair COND_1451_0_MOD_LE(TRUE, x2[3], x2[3], x0[3]) → 1001_0_GCD_EQ(x0[3], x2[3]) the following chains were created:
  • We consider the chain 1001_0_GCD_EQ(x0[0], x1[0]) → COND_1001_0_GCD_EQ(&&(&&(>(x1[0], 0), >(x0[0], -1)), !(=(x0[0], x1[0]))), x0[0], x1[0]), COND_1001_0_GCD_EQ(TRUE, x0[1], x1[1]) → 1451_0_MOD_LE(x0[1], x0[1], x1[1]), 1451_0_MOD_LE(x2[2], x2[2], x0[2]) → COND_1451_0_MOD_LE(<=(x2[2], x0[2]), x2[2], x2[2], x0[2]), COND_1451_0_MOD_LE(TRUE, x2[3], x2[3], x0[3]) → 1001_0_GCD_EQ(x0[3], x2[3]), 1001_0_GCD_EQ(x0[0], x1[0]) → COND_1001_0_GCD_EQ(&&(&&(>(x1[0], 0), >(x0[0], -1)), !(=(x0[0], x1[0]))), x0[0], x1[0]), COND_1001_0_GCD_EQ(TRUE, x0[1], x1[1]) → 1451_0_MOD_LE(x0[1], x0[1], x1[1]) which results in the following constraint:

    (30)    (&&(&&(>(x1[0], 0), >(x0[0], -1)), !(=(x0[0], x1[0])))=TRUEx0[0]=x0[1]x1[0]=x1[1]x0[1]=x2[2]x1[1]=x0[2]<=(x2[2], x0[2])=TRUEx2[2]=x2[3]x0[2]=x0[3]x0[3]=x0[0]1x2[3]=x1[0]1&&(&&(>(x1[0]1, 0), >(x0[0]1, -1)), !(=(x0[0]1, x1[0]1)))=TRUEx0[0]1=x0[1]1x1[0]1=x1[1]1COND_1451_0_MOD_LE(TRUE, x2[3], x2[3], x0[3])≥NonInfC∧COND_1451_0_MOD_LE(TRUE, x2[3], x2[3], x0[3])≥1001_0_GCD_EQ(x0[3], x2[3])∧(UIncreasing(1001_0_GCD_EQ(x0[3], x2[3])), ≥))



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

    (31)    (<=(x0[0], x1[0])=TRUE>(x1[0], 0)=TRUE>(x0[0], -1)=TRUE>(x0[0], 0)=TRUE>(x1[0], -1)=TRUE<(x0[0], x1[0])=TRUE<(x1[0], x0[0])=TRUECOND_1451_0_MOD_LE(TRUE, x0[0], x0[0], x1[0])≥NonInfC∧COND_1451_0_MOD_LE(TRUE, x0[0], x0[0], x1[0])≥1001_0_GCD_EQ(x1[0], x0[0])∧(UIncreasing(1001_0_GCD_EQ(x0[3], x2[3])), ≥))



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

    (32)    (x1[0] + [-1]x0[0] ≥ 0∧x1[0] + [-1] ≥ 0∧x0[0] ≥ 0∧x0[0] + [-1] ≥ 0∧x1[0] ≥ 0∧x1[0] + [-1] + [-1]x0[0] ≥ 0∧x0[0] + [-1] + [-1]x1[0] ≥ 0 ⇒ (UIncreasing(1001_0_GCD_EQ(x0[3], x2[3])), ≥)∧[(-1)bni_30 + (-1)Bound*bni_30] + [(-1)bni_30]x1[0] + [(2)bni_30]x0[0] ≥ 0∧[-2 + (-1)bso_31] + [2]x0[0] ≥ 0)



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

    (33)    (x1[0] + [-1]x0[0] ≥ 0∧x1[0] + [-1] ≥ 0∧x0[0] ≥ 0∧x0[0] + [-1] ≥ 0∧x1[0] ≥ 0∧x1[0] + [-1] + [-1]x0[0] ≥ 0∧x0[0] + [-1] + [-1]x1[0] ≥ 0 ⇒ (UIncreasing(1001_0_GCD_EQ(x0[3], x2[3])), ≥)∧[(-1)bni_30 + (-1)Bound*bni_30] + [(-1)bni_30]x1[0] + [(2)bni_30]x0[0] ≥ 0∧[-2 + (-1)bso_31] + [2]x0[0] ≥ 0)



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

    (35)    (x1[0] + [-1]x0[0] ≥ 0∧x1[0] + [-1] ≥ 0∧x0[0] ≥ 0∧x0[0] + [-1] ≥ 0∧x1[0] ≥ 0∧x1[0] + [-1] + [-1]x0[0] ≥ 0∧x1[0] + [-1] + [-1]x0[0] ≥ 0 ⇒ (UIncreasing(1001_0_GCD_EQ(x0[3], x2[3])), ≥)∧[(-1)bni_30 + (-1)Bound*bni_30] + [(-1)bni_30]x1[0] + [(2)bni_30]x0[0] ≥ 0∧[-2 + (-1)bso_31] + [2]x0[0] ≥ 0)



    We simplified constraint (36) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:

    (37)    (x1[0] + [-1]x0[0] ≥ 0∧x1[0] + [-1] ≥ 0∧x0[0] ≥ 0∧x0[0] + [-1] ≥ 0∧x1[0] ≥ 0∧x0[0] + [-1] + [-1]x1[0] ≥ 0∧x0[0] + [-1] + [-1]x1[0] ≥ 0 ⇒ (UIncreasing(1001_0_GCD_EQ(x0[3], x2[3])), ≥)∧[(-1)bni_30 + (-1)Bound*bni_30] + [(-1)bni_30]x1[0] + [(2)bni_30]x0[0] ≥ 0∧[-2 + (-1)bso_31] + [2]x0[0] ≥ 0)



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

    (39)    (x1[0] + [-1]x0[0] ≥ 0∧x1[0] + [-1] ≥ 0∧x0[0] ≥ 0∧x0[0] + [-1] ≥ 0∧x1[0] ≥ 0∧x0[0] + [-1] + [-1]x1[0] ≥ 0∧x1[0] + [-1] + [-1]x0[0] ≥ 0 ⇒ (UIncreasing(1001_0_GCD_EQ(x0[3], x2[3])), ≥)∧[(-1)bni_30 + (-1)Bound*bni_30] + [(-1)bni_30]x1[0] + [(2)bni_30]x0[0] ≥ 0∧[-2 + (-1)bso_31] + [2]x0[0] ≥ 0)



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

    (40)    (x1[0] + [-1]x0[0] ≥ 0∧x1[0] + [-1] ≥ 0∧x0[0] ≥ 0∧x0[0] + [-1] ≥ 0∧x1[0] ≥ 0∧x1[0] + [-1] + [-1]x0[0] ≥ 0∧x0[0] + [-1] + [-1]x1[0] ≥ 0 ⇒ (UIncreasing(1001_0_GCD_EQ(x0[3], x2[3])), ≥)∧[(-1)bni_30 + (-1)Bound*bni_30] + [(-1)bni_30]x1[0] + [(2)bni_30]x0[0] ≥ 0∧[-2 + (-1)bso_31] + [2]x0[0] ≥ 0)



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

    (41)    (x1[0] + [-1]x0[0] ≥ 0∧x1[0] + [-1] ≥ 0∧x0[0] ≥ 0∧x0[0] + [-1] ≥ 0∧x1[0] ≥ 0∧x1[0] + [-1] + [-1]x0[0] ≥ 0∧x1[0] + [-1] + [-1]x0[0] ≥ 0 ⇒ (UIncreasing(1001_0_GCD_EQ(x0[3], x2[3])), ≥)∧[(-1)bni_30 + (-1)Bound*bni_30] + [(-1)bni_30]x1[0] + [(2)bni_30]x0[0] ≥ 0∧[-2 + (-1)bso_31] + [2]x0[0] ≥ 0)



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

    (42)    (x1[0] + [-1]x0[0] ≥ 0∧x1[0] + [-1] ≥ 0∧x0[0] ≥ 0∧x0[0] + [-1] ≥ 0∧x1[0] ≥ 0∧x0[0] + [-1] + [-1]x1[0] ≥ 0∧x0[0] + [-1] + [-1]x1[0] ≥ 0 ⇒ (UIncreasing(1001_0_GCD_EQ(x0[3], x2[3])), ≥)∧[(-1)bni_30 + (-1)Bound*bni_30] + [(-1)bni_30]x1[0] + [(2)bni_30]x0[0] ≥ 0∧[-2 + (-1)bso_31] + [2]x0[0] ≥ 0)



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

    (43)    (x1[0] + [-1]x0[0] ≥ 0∧x1[0] + [-1] ≥ 0∧x0[0] ≥ 0∧x0[0] + [-1] ≥ 0∧x1[0] ≥ 0∧x0[0] + [-1] + [-1]x1[0] ≥ 0∧x1[0] + [-1] + [-1]x0[0] ≥ 0 ⇒ (UIncreasing(1001_0_GCD_EQ(x0[3], x2[3])), ≥)∧[(-1)bni_30 + (-1)Bound*bni_30] + [(-1)bni_30]x1[0] + [(2)bni_30]x0[0] ≥ 0∧[-2 + (-1)bso_31] + [2]x0[0] ≥ 0)



    We solved constraint (40) using rule (IDP_SMT_SPLIT).We simplified constraint (41) using rule (IDP_SMT_SPLIT) which results in the following new constraint:

    (44)    (x1[0] ≥ 0∧x0[0] + [-1] + x1[0] ≥ 0∧x0[0] ≥ 0∧x0[0] + [-1] ≥ 0∧x0[0] + x1[0] ≥ 0∧[-1] + x1[0] ≥ 0∧[-1] + x1[0] ≥ 0 ⇒ (UIncreasing(1001_0_GCD_EQ(x0[3], x2[3])), ≥)∧[(-1)bni_30 + (-1)Bound*bni_30] + [bni_30]x0[0] + [(-1)bni_30]x1[0] ≥ 0∧[-2 + (-1)bso_31] + [2]x0[0] ≥ 0)



    We solved constraint (42) using rule (IDP_SMT_SPLIT).We solved constraint (43) using rule (IDP_SMT_SPLIT).We simplified constraint (44) using rule (IDP_SMT_SPLIT) which results in the following new constraint:

    (45)    (x1[0] ≥ 0∧x0[0] + x1[0] ≥ 0∧[1] + x0[0] ≥ 0∧x0[0] ≥ 0∧[1] + x0[0] + x1[0] ≥ 0∧[-1] + x1[0] ≥ 0∧[-1] + x1[0] ≥ 0 ⇒ (UIncreasing(1001_0_GCD_EQ(x0[3], x2[3])), ≥)∧[(-1)Bound*bni_30] + [bni_30]x0[0] + [(-1)bni_30]x1[0] ≥ 0∧[(-1)bso_31] + [2]x0[0] ≥ 0)



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

    (46)    ([1] + x1[0] ≥ 0∧[1] + x0[0] + x1[0] ≥ 0∧[1] + x0[0] ≥ 0∧x0[0] ≥ 0∧[2] + x0[0] + x1[0] ≥ 0∧x1[0] ≥ 0∧x1[0] ≥ 0 ⇒ (UIncreasing(1001_0_GCD_EQ(x0[3], x2[3])), ≥)∧[(-1)bni_30 + (-1)Bound*bni_30] + [bni_30]x0[0] + [(-1)bni_30]x1[0] ≥ 0∧[(-1)bso_31] + [2]x0[0] ≥ 0)







To summarize, we get the following constraints P for the following pairs.
  • 1451_0_MOD_LE(x2[2], x2[2], x0[2]) → COND_1451_0_MOD_LE(<=(x2[2], x0[2]), x2[2], x2[2], x0[2])
    • ([1] + x1[0] ≥ 0∧x0[0] + x1[0] ≥ 0∧x0[0] ≥ 0∧x1[0] ≥ 0 ⇒ (UIncreasing(COND_1451_0_MOD_LE(<=(x2[2], x0[2]), x2[2], x2[2], x0[2])), ≥)∧[(-2)bni_24 + (-1)Bound*bni_24] + [bni_24]x0[0] + [(-1)bni_24]x1[0] ≥ 0∧[(-1)bso_25] ≥ 0)

  • COND_1001_0_GCD_EQ(TRUE, x0[1], x1[1]) → 1451_0_MOD_LE(x0[1], x0[1], x1[1])

  • 1001_0_GCD_EQ(x0[0], x1[0]) → COND_1001_0_GCD_EQ(&&(&&(>(x1[0], 0), >(x0[0], -1)), !(=(x0[0], x1[0]))), x0[0], x1[0])
    • ([1] + x0[2] ≥ 0∧x2[2] ≥ 0∧[2] + x2[2] + x0[2] ≥ 0∧x0[2] ≥ 0 ⇒ (UIncreasing(COND_1001_0_GCD_EQ(&&(&&(>(x1[0], 0), >(x0[0], -1)), !(=(x0[0], x1[0]))), x0[0], x1[0])), ≥)∧[(-1)bni_28 + (-1)Bound*bni_28] + [(-1)bni_28]x2[2] + [(-1)bni_28]x0[2] ≥ 0∧[1 + (-1)bso_29] ≥ 0)

  • COND_1451_0_MOD_LE(TRUE, x2[3], x2[3], x0[3]) → 1001_0_GCD_EQ(x0[3], x2[3])
    • ([1] + x1[0] ≥ 0∧[1] + x0[0] + x1[0] ≥ 0∧[1] + x0[0] ≥ 0∧x0[0] ≥ 0∧[2] + x0[0] + x1[0] ≥ 0∧x1[0] ≥ 0∧x1[0] ≥ 0 ⇒ (UIncreasing(1001_0_GCD_EQ(x0[3], x2[3])), ≥)∧[(-1)bni_30 + (-1)Bound*bni_30] + [bni_30]x0[0] + [(-1)bni_30]x1[0] ≥ 0∧[(-1)bso_31] + [2]x0[0] ≥ 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(1451_0_MOD_LE(x1, x2, x3)) = [-1] + [-1]x3 + [2]x1   
POL(COND_1451_0_MOD_LE(x1, x2, x3, x4)) = [-1] + [-1]x4 + x3 + x2   
POL(<=(x1, x2)) = [-1]   
POL(COND_1001_0_GCD_EQ(x1, x2, x3)) = [-1] + [-1]x2 + x1   
POL(1001_0_GCD_EQ(x1, x2)) = [1] + [-1]x1   
POL(&&(x1, x2)) = [1]   
POL(>(x1, x2)) = [-1]   
POL(0) = 0   
POL(-1) = [-1]   
POL(!(x1)) = [-1]   
POL(=(x1, x2)) = [-1]   

The following pairs are in P>:

COND_1001_0_GCD_EQ(TRUE, x0[1], x1[1]) → 1451_0_MOD_LE(x0[1], x0[1], x1[1])
1001_0_GCD_EQ(x0[0], x1[0]) → COND_1001_0_GCD_EQ(&&(&&(>(x1[0], 0), >(x0[0], -1)), !(=(x0[0], x1[0]))), x0[0], x1[0])

The following pairs are in Pbound:

COND_1001_0_GCD_EQ(TRUE, x0[1], x1[1]) → 1451_0_MOD_LE(x0[1], x0[1], x1[1])

The following pairs are in P:

1451_0_MOD_LE(x2[2], x2[2], x0[2]) → COND_1451_0_MOD_LE(<=(x2[2], x0[2]), x2[2], x2[2], x0[2])
COND_1451_0_MOD_LE(TRUE, x2[3], x2[3], x0[3]) → 1001_0_GCD_EQ(x0[3], x2[3])

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

(16) Obligation:

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


The following domains are used:

Integer, Boolean


R is empty.

The integer pair graph contains the following rules and edges:
(2): 1451_0_MOD_LE(x2[2], x2[2], x0[2]) → COND_1451_0_MOD_LE(x2[2] <= x0[2], x2[2], x2[2], x0[2])
(0): 1001_0_GCD_EQ(x0[0], x1[0]) → COND_1001_0_GCD_EQ(x1[0] > 0 && x0[0] > -1 && !(x0[0] = x1[0]), x0[0], x1[0])
(3): COND_1451_0_MOD_LE(TRUE, x2[3], x2[3], x0[3]) → 1001_0_GCD_EQ(x0[3], x2[3])

(3) -> (0), if (x0[3]* x0[0]x2[3]* x1[0])


(2) -> (3), if (x2[2] <= x0[2]x2[2]* x2[3]x0[2]* x0[3])



The set Q is empty.

(17) IDependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 3 less nodes.

(18) TRUE