(0) Obligation:

JBC Problem based on JBC Program:
Manifest-Version: 1.0 Created-By: 1.6.0_16 (Sun Microsystems Inc.) Main-Class: MinusMin
`public class MinusMin{  public static int min (int x, int y) {    if (x < y) return x;    else return y;  }  public static void main(String[] args) {    Random.args = args;    int x = Random.random();    int y = Random.random();    int res = 0;    while (min(x-1,y) == y) {      y++;      res++;    }  }}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:
MinusMin.main([Ljava/lang/String;)V: Graph of 197 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: MinusMin.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 25 rules for P and 0 rules for R.

P rules:
675_0_main_ConstantStackPush(EOS(STATIC_675), i18, i106, i18) → 677_0_main_IntArithmetic(EOS(STATIC_677), i18, i106, i18, 1)
677_0_main_IntArithmetic(EOS(STATIC_677), i18, i106, i18, matching1) → 679_0_main_Load(EOS(STATIC_679), i18, i106, -(i18, 1)) | &&(>=(i18, 0), =(matching1, 1))
679_0_main_Load(EOS(STATIC_679), i18, i106, i111) → 681_0_main_InvokeMethod(EOS(STATIC_681), i18, i106, i111, i106)
681_0_main_InvokeMethod(EOS(STATIC_681), i18, i106, i111, i106) → 682_0_min_Load(EOS(STATIC_682), i18, i106, i111, i106, i111, i106)
682_0_min_Load(EOS(STATIC_682), i18, i106, i111, i106, i111, i106) → 685_0_min_Load(EOS(STATIC_685), i18, i106, i111, i106, i111, i106, i111)
685_0_min_Load(EOS(STATIC_685), i18, i106, i111, i106, i111, i106, i111) → 687_0_min_GE(EOS(STATIC_687), i18, i106, i111, i106, i111, i106, i111, i106)
687_0_min_GE(EOS(STATIC_687), i18, i106, i111, i106, i111, i106, i111, i106) → 689_0_min_GE(EOS(STATIC_689), i18, i106, i111, i106, i111, i106, i111, i106)
687_0_min_GE(EOS(STATIC_687), i18, i106, i111, i106, i111, i106, i111, i106) → 690_0_min_GE(EOS(STATIC_690), i18, i106, i111, i106, i111, i106, i111, i106)
689_0_min_GE(EOS(STATIC_689), i18, i106, i111, i106, i111, i106, i111, i106) → 692_0_min_Load(EOS(STATIC_692), i18, i106, i111, i106, i106) | >=(i111, i106)
692_0_min_Load(EOS(STATIC_692), i18, i106, i111, i106, i106) → 697_0_min_Return(EOS(STATIC_697), i18, i106, i111, i106, i106)
697_0_min_Return(EOS(STATIC_697), i18, i106, i111, i106, i106) → 701_0_main_Load(EOS(STATIC_701), i18, i106, i106)
701_0_main_Load(EOS(STATIC_701), i18, i106, i106) → 706_0_main_NE(EOS(STATIC_706), i18, i106, i106, i106)
706_0_main_NE(EOS(STATIC_706), i18, i106, i106, i106) → 713_0_main_Inc(EOS(STATIC_713), i18, i106)
713_0_main_Inc(EOS(STATIC_713), i18, i106) → 726_0_main_Inc(EOS(STATIC_726), i18, i106)
726_0_main_Inc(EOS(STATIC_726), i18, i106) → 729_0_main_Inc(EOS(STATIC_729), i18, +(i106, 1))
729_0_main_Inc(EOS(STATIC_729), i18, i114) → 731_0_main_JMP(EOS(STATIC_731), i18, i114)
731_0_main_JMP(EOS(STATIC_731), i18, i114) → 735_0_main_Load(EOS(STATIC_735), i18, i114)
735_0_main_Load(EOS(STATIC_735), i18, i114) → 672_0_main_Load(EOS(STATIC_672), i18, i114)
672_0_main_Load(EOS(STATIC_672), i18, i106) → 675_0_main_ConstantStackPush(EOS(STATIC_675), i18, i106, i18)
690_0_min_GE(EOS(STATIC_690), i18, i106, i111, i106, i111, i106, i111, i106) → 695_0_min_Load(EOS(STATIC_695), i18, i106, i111, i106, i111, i106) | <(i111, i106)
695_0_min_Load(EOS(STATIC_695), i18, i106, i111, i106, i111, i106) → 699_0_min_Return(EOS(STATIC_699), i18, i106, i111, i106, i106, i111)
699_0_min_Return(EOS(STATIC_699), i18, i106, i111, i106, i106, i111) → 704_0_main_Load(EOS(STATIC_704), i18, i106, i111)
704_0_main_Load(EOS(STATIC_704), i18, i106, i111) → 711_0_main_NE(EOS(STATIC_711), i18, i106, i111, i106)
711_0_main_NE(EOS(STATIC_711), i18, i106, i106, i106) → 717_0_main_NE(EOS(STATIC_717), i18, i106, i106, i106)
717_0_main_NE(EOS(STATIC_717), i18, i106, i106, i106) → 726_0_main_Inc(EOS(STATIC_726), i18, i106)
R rules:

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

P rules:
675_0_main_ConstantStackPush(EOS(STATIC_675), x0, x1, x0) → 675_0_main_ConstantStackPush(EOS(STATIC_675), x0, +(x1, 1), x0) | &&(<=(x1, -(x0, 1)), >(+(x0, 1), 0))
675_0_main_ConstantStackPush(EOS(STATIC_675), x0, -(x0, 1), x0) → 675_0_main_ConstantStackPush(EOS(STATIC_675), x0, +(-(x0, 1), 1), x0) | &&(>(+(x0, 1), 0), <(-(x0, 1), -(x0, 1)))
R rules:

Filtered ground terms:

675_0_main_ConstantStackPush(x1, x2, x3, x4) → 675_0_main_ConstantStackPush(x2, x3, x4)
EOS(x1) → EOS
Cond_675_0_main_ConstantStackPush1(x1, x2, x3, x4, x5) → Cond_675_0_main_ConstantStackPush1(x1, x3, x4, x5)
Cond_675_0_main_ConstantStackPush(x1, x2, x3, x4, x5) → Cond_675_0_main_ConstantStackPush(x1, x3, x4, x5)

Filtered duplicate args:

675_0_main_ConstantStackPush(x1, x2, x3) → 675_0_main_ConstantStackPush(x2, x3)
Cond_675_0_main_ConstantStackPush(x1, x2, x3, x4) → Cond_675_0_main_ConstantStackPush(x1, x3, x4)
Cond_675_0_main_ConstantStackPush1(x1, x2, x3, x4) → Cond_675_0_main_ConstantStackPush1(x1, x3)

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

P rules:
675_0_main_ConstantStackPush(x1, x0) → 675_0_main_ConstantStackPush(+(x1, 1), x0) | &&(<=(x1, -(x0, 1)), >(x0, -1))
675_0_main_ConstantStackPush(x0_-1, x0) → 675_0_main_ConstantStackPush(+(-(x0, 1), 1), x0) | &&(&&(>(x0, -1), <(-(x0, 1), -(x0, 1))), =(x0_-1, -(x0, 1)))
R rules:

Finished conversion. Obtained 4 rules for P and 0 rules for R. System has predefined symbols.

P rules:
675_0_MAIN_CONSTANTSTACKPUSH(x1, x0) → COND_675_0_MAIN_CONSTANTSTACKPUSH(&&(<=(x1, -(x0, 1)), >(x0, -1)), x1, x0)
COND_675_0_MAIN_CONSTANTSTACKPUSH(TRUE, x1, x0) → 675_0_MAIN_CONSTANTSTACKPUSH(+(x1, 1), x0)
675_0_MAIN_CONSTANTSTACKPUSH(x0_-1, x0) → COND_675_0_MAIN_CONSTANTSTACKPUSH1(&&(&&(>(x0, -1), <(-(x0, 1), -(x0, 1))), =(x0_-1, -(x0, 1))), x0_-1, x0)
COND_675_0_MAIN_CONSTANTSTACKPUSH1(TRUE, x0_-1, x0) → 675_0_MAIN_CONSTANTSTACKPUSH(+(-(x0, 1), 1), x0)
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): 675_0_MAIN_CONSTANTSTACKPUSH(x1[0], x0[0]) → COND_675_0_MAIN_CONSTANTSTACKPUSH(x1[0] <= x0[0] - 1 && x0[0] > -1, x1[0], x0[0])
(1): COND_675_0_MAIN_CONSTANTSTACKPUSH(TRUE, x1[1], x0[1]) → 675_0_MAIN_CONSTANTSTACKPUSH(x1[1] + 1, x0[1])
(2): 675_0_MAIN_CONSTANTSTACKPUSH(x0_-1[2], x0[2]) → COND_675_0_MAIN_CONSTANTSTACKPUSH1(x0[2] > -1 && x0[2] - 1 < x0[2] - 1 && x0_-1[2] = x0[2] - 1, x0_-1[2], x0[2])
(3): COND_675_0_MAIN_CONSTANTSTACKPUSH1(TRUE, x0_-1[3], x0[3]) → 675_0_MAIN_CONSTANTSTACKPUSH(x0[3] - 1 + 1, x0[3])

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

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

(1) -> (2), if (x1[1] + 1* x0_-1[2]x0[1]* x0[2])

(2) -> (3), if (x0[2] > -1 && x0[2] - 1 < x0[2] - 1 && x0_-1[2] = x0[2] - 1x0_-1[2]* x0_-1[3]x0[2]* x0[3])

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

(3) -> (2), if (x0[3] - 1 + 1* x0_-1[2]x0[3]* x0[2])

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.IdpCand1ShapeHeuristic@1d87a604 Constraint Generator: NonInfConstraintGenerator: PathGenerator: MetricPathGenerator: Max Left Steps: 0 Max Right Steps: 0

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 675_0_MAIN_CONSTANTSTACKPUSH(x1, x0) → COND_675_0_MAIN_CONSTANTSTACKPUSH(&&(<=(x1, -(x0, 1)), >(x0, -1)), x1, x0) the following chains were created:
• We consider the chain 675_0_MAIN_CONSTANTSTACKPUSH(x1[0], x0[0]) → COND_675_0_MAIN_CONSTANTSTACKPUSH(&&(<=(x1[0], -(x0[0], 1)), >(x0[0], -1)), x1[0], x0[0]), COND_675_0_MAIN_CONSTANTSTACKPUSH(TRUE, x1[1], x0[1]) → 675_0_MAIN_CONSTANTSTACKPUSH(+(x1[1], 1), x0[1]) which results in the following constraint:

(1)    (&&(<=(x1[0], -(x0[0], 1)), >(x0[0], -1))=TRUEx1[0]=x1[1]x0[0]=x0[1]675_0_MAIN_CONSTANTSTACKPUSH(x1[0], x0[0])≥NonInfC∧675_0_MAIN_CONSTANTSTACKPUSH(x1[0], x0[0])≥COND_675_0_MAIN_CONSTANTSTACKPUSH(&&(<=(x1[0], -(x0[0], 1)), >(x0[0], -1)), x1[0], x0[0])∧(UIncreasing(COND_675_0_MAIN_CONSTANTSTACKPUSH(&&(<=(x1[0], -(x0[0], 1)), >(x0[0], -1)), x1[0], x0[0])), ≥))

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

(2)    (<=(x1[0], -(x0[0], 1))=TRUE>(x0[0], -1)=TRUE675_0_MAIN_CONSTANTSTACKPUSH(x1[0], x0[0])≥NonInfC∧675_0_MAIN_CONSTANTSTACKPUSH(x1[0], x0[0])≥COND_675_0_MAIN_CONSTANTSTACKPUSH(&&(<=(x1[0], -(x0[0], 1)), >(x0[0], -1)), x1[0], x0[0])∧(UIncreasing(COND_675_0_MAIN_CONSTANTSTACKPUSH(&&(<=(x1[0], -(x0[0], 1)), >(x0[0], -1)), x1[0], x0[0])), ≥))

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

(3)    (x0[0] + [-1] + [-1]x1[0] ≥ 0∧x0[0] ≥ 0 ⇒ (UIncreasing(COND_675_0_MAIN_CONSTANTSTACKPUSH(&&(<=(x1[0], -(x0[0], 1)), >(x0[0], -1)), x1[0], x0[0])), ≥)∧[(-1)bni_13 + (-1)Bound*bni_13] + [bni_13]x0[0] ≥ 0∧[(-1)bso_14] ≥ 0)

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

(4)    (x0[0] + [-1] + [-1]x1[0] ≥ 0∧x0[0] ≥ 0 ⇒ (UIncreasing(COND_675_0_MAIN_CONSTANTSTACKPUSH(&&(<=(x1[0], -(x0[0], 1)), >(x0[0], -1)), x1[0], x0[0])), ≥)∧[(-1)bni_13 + (-1)Bound*bni_13] + [bni_13]x0[0] ≥ 0∧[(-1)bso_14] ≥ 0)

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

(5)    (x0[0] + [-1] + [-1]x1[0] ≥ 0∧x0[0] ≥ 0 ⇒ (UIncreasing(COND_675_0_MAIN_CONSTANTSTACKPUSH(&&(<=(x1[0], -(x0[0], 1)), >(x0[0], -1)), x1[0], x0[0])), ≥)∧[(-1)bni_13 + (-1)Bound*bni_13] + [bni_13]x0[0] ≥ 0∧[(-1)bso_14] ≥ 0)

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

(6)    (x0[0] + [-1] + [-1]x1[0] ≥ 0∧x0[0] ≥ 0∧x1[0] ≥ 0 ⇒ (UIncreasing(COND_675_0_MAIN_CONSTANTSTACKPUSH(&&(<=(x1[0], -(x0[0], 1)), >(x0[0], -1)), x1[0], x0[0])), ≥)∧[(-1)bni_13 + (-1)Bound*bni_13] + [bni_13]x0[0] ≥ 0∧[(-1)bso_14] ≥ 0)

(7)    (x0[0] + [-1] + x1[0] ≥ 0∧x0[0] ≥ 0∧x1[0] ≥ 0 ⇒ (UIncreasing(COND_675_0_MAIN_CONSTANTSTACKPUSH(&&(<=(x1[0], -(x0[0], 1)), >(x0[0], -1)), x1[0], x0[0])), ≥)∧[(-1)bni_13 + (-1)Bound*bni_13] + [bni_13]x0[0] ≥ 0∧[(-1)bso_14] ≥ 0)

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

(8)    (x0[0] ≥ 0∧[1] + x1[0] + x0[0] ≥ 0∧x1[0] ≥ 0 ⇒ (UIncreasing(COND_675_0_MAIN_CONSTANTSTACKPUSH(&&(<=(x1[0], -(x0[0], 1)), >(x0[0], -1)), x1[0], x0[0])), ≥)∧[(-1)Bound*bni_13] + [bni_13]x1[0] + [bni_13]x0[0] ≥ 0∧[(-1)bso_14] ≥ 0)

For Pair COND_675_0_MAIN_CONSTANTSTACKPUSH(TRUE, x1, x0) → 675_0_MAIN_CONSTANTSTACKPUSH(+(x1, 1), x0) the following chains were created:
• We consider the chain COND_675_0_MAIN_CONSTANTSTACKPUSH(TRUE, x1[1], x0[1]) → 675_0_MAIN_CONSTANTSTACKPUSH(+(x1[1], 1), x0[1]) which results in the following constraint:

(9)    (COND_675_0_MAIN_CONSTANTSTACKPUSH(TRUE, x1[1], x0[1])≥NonInfC∧COND_675_0_MAIN_CONSTANTSTACKPUSH(TRUE, x1[1], x0[1])≥675_0_MAIN_CONSTANTSTACKPUSH(+(x1[1], 1), x0[1])∧(UIncreasing(675_0_MAIN_CONSTANTSTACKPUSH(+(x1[1], 1), x0[1])), ≥))

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

(10)    ((UIncreasing(675_0_MAIN_CONSTANTSTACKPUSH(+(x1[1], 1), x0[1])), ≥)∧[bni_15] = 0∧[(-1)bso_16] ≥ 0)

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

(11)    ((UIncreasing(675_0_MAIN_CONSTANTSTACKPUSH(+(x1[1], 1), x0[1])), ≥)∧[bni_15] = 0∧[(-1)bso_16] ≥ 0)

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

(12)    ((UIncreasing(675_0_MAIN_CONSTANTSTACKPUSH(+(x1[1], 1), x0[1])), ≥)∧[bni_15] = 0∧[(-1)bso_16] ≥ 0)

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

(13)    ((UIncreasing(675_0_MAIN_CONSTANTSTACKPUSH(+(x1[1], 1), x0[1])), ≥)∧[bni_15] = 0∧0 = 0∧0 = 0∧[(-1)bso_16] ≥ 0)

For Pair 675_0_MAIN_CONSTANTSTACKPUSH(x0_-1, x0) → COND_675_0_MAIN_CONSTANTSTACKPUSH1(&&(&&(>(x0, -1), <(-(x0, 1), -(x0, 1))), =(x0_-1, -(x0, 1))), x0_-1, x0) the following chains were created:
• We consider the chain 675_0_MAIN_CONSTANTSTACKPUSH(x0_-1[2], x0[2]) → COND_675_0_MAIN_CONSTANTSTACKPUSH1(&&(&&(>(x0[2], -1), <(-(x0[2], 1), -(x0[2], 1))), =(x0_-1[2], -(x0[2], 1))), x0_-1[2], x0[2]), COND_675_0_MAIN_CONSTANTSTACKPUSH1(TRUE, x0_-1[3], x0[3]) → 675_0_MAIN_CONSTANTSTACKPUSH(+(-(x0[3], 1), 1), x0[3]) which results in the following constraint:

(14)    (&&(&&(>(x0[2], -1), <(-(x0[2], 1), -(x0[2], 1))), =(x0_-1[2], -(x0[2], 1)))=TRUEx0_-1[2]=x0_-1[3]x0[2]=x0[3]675_0_MAIN_CONSTANTSTACKPUSH(x0_-1[2], x0[2])≥NonInfC∧675_0_MAIN_CONSTANTSTACKPUSH(x0_-1[2], x0[2])≥COND_675_0_MAIN_CONSTANTSTACKPUSH1(&&(&&(>(x0[2], -1), <(-(x0[2], 1), -(x0[2], 1))), =(x0_-1[2], -(x0[2], 1))), x0_-1[2], x0[2])∧(UIncreasing(COND_675_0_MAIN_CONSTANTSTACKPUSH1(&&(&&(>(x0[2], -1), <(-(x0[2], 1), -(x0[2], 1))), =(x0_-1[2], -(x0[2], 1))), x0_-1[2], x0[2])), ≥))

We simplified constraint (14) using rules (IV), (IDP_BOOLEAN) which results in the following new constraint:

(15)    (>(x0[2], -1)=TRUE<(-(x0[2], 1), -(x0[2], 1))=TRUE>=(x0_-1[2], -(x0[2], 1))=TRUE<=(x0_-1[2], -(x0[2], 1))=TRUE675_0_MAIN_CONSTANTSTACKPUSH(x0_-1[2], x0[2])≥NonInfC∧675_0_MAIN_CONSTANTSTACKPUSH(x0_-1[2], x0[2])≥COND_675_0_MAIN_CONSTANTSTACKPUSH1(&&(&&(>(x0[2], -1), <(-(x0[2], 1), -(x0[2], 1))), =(x0_-1[2], -(x0[2], 1))), x0_-1[2], x0[2])∧(UIncreasing(COND_675_0_MAIN_CONSTANTSTACKPUSH1(&&(&&(>(x0[2], -1), <(-(x0[2], 1), -(x0[2], 1))), =(x0_-1[2], -(x0[2], 1))), x0_-1[2], x0[2])), ≥))

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

(16)    (x0[2] ≥ 0∧[-1] ≥ 0∧x0-1[2] + [1] + [-1]x0[2] ≥ 0∧x0[2] + [-1] + [-1]x0-1[2] ≥ 0 ⇒ (UIncreasing(COND_675_0_MAIN_CONSTANTSTACKPUSH1(&&(&&(>(x0[2], -1), <(-(x0[2], 1), -(x0[2], 1))), =(x0_-1[2], -(x0[2], 1))), x0_-1[2], x0[2])), ≥)∧[(-1)bni_17 + (-1)Bound*bni_17] + [bni_17]x0[2] ≥ 0∧[-2 + (-1)bso_18] ≥ 0)

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

(17)    (x0[2] ≥ 0∧[-1] ≥ 0∧x0-1[2] + [1] + [-1]x0[2] ≥ 0∧x0[2] + [-1] + [-1]x0-1[2] ≥ 0 ⇒ (UIncreasing(COND_675_0_MAIN_CONSTANTSTACKPUSH1(&&(&&(>(x0[2], -1), <(-(x0[2], 1), -(x0[2], 1))), =(x0_-1[2], -(x0[2], 1))), x0_-1[2], x0[2])), ≥)∧[(-1)bni_17 + (-1)Bound*bni_17] + [bni_17]x0[2] ≥ 0∧[-2 + (-1)bso_18] ≥ 0)

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

(18)    (x0[2] ≥ 0∧[-1] ≥ 0∧x0-1[2] + [1] + [-1]x0[2] ≥ 0∧x0[2] + [-1] + [-1]x0-1[2] ≥ 0 ⇒ (UIncreasing(COND_675_0_MAIN_CONSTANTSTACKPUSH1(&&(&&(>(x0[2], -1), <(-(x0[2], 1), -(x0[2], 1))), =(x0_-1[2], -(x0[2], 1))), x0_-1[2], x0[2])), ≥)∧[(-1)bni_17 + (-1)Bound*bni_17] + [bni_17]x0[2] ≥ 0∧[-2 + (-1)bso_18] ≥ 0)

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

For Pair COND_675_0_MAIN_CONSTANTSTACKPUSH1(TRUE, x0_-1, x0) → 675_0_MAIN_CONSTANTSTACKPUSH(+(-(x0, 1), 1), x0) the following chains were created:
• We consider the chain COND_675_0_MAIN_CONSTANTSTACKPUSH1(TRUE, x0_-1[3], x0[3]) → 675_0_MAIN_CONSTANTSTACKPUSH(+(-(x0[3], 1), 1), x0[3]) which results in the following constraint:

(19)    (COND_675_0_MAIN_CONSTANTSTACKPUSH1(TRUE, x0_-1[3], x0[3])≥NonInfC∧COND_675_0_MAIN_CONSTANTSTACKPUSH1(TRUE, x0_-1[3], x0[3])≥675_0_MAIN_CONSTANTSTACKPUSH(+(-(x0[3], 1), 1), x0[3])∧(UIncreasing(675_0_MAIN_CONSTANTSTACKPUSH(+(-(x0[3], 1), 1), x0[3])), ≥))

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

(20)    ((UIncreasing(675_0_MAIN_CONSTANTSTACKPUSH(+(-(x0[3], 1), 1), x0[3])), ≥)∧[bni_19] = 0∧[2 + (-1)bso_20] ≥ 0)

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

(21)    ((UIncreasing(675_0_MAIN_CONSTANTSTACKPUSH(+(-(x0[3], 1), 1), x0[3])), ≥)∧[bni_19] = 0∧[2 + (-1)bso_20] ≥ 0)

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

(22)    ((UIncreasing(675_0_MAIN_CONSTANTSTACKPUSH(+(-(x0[3], 1), 1), x0[3])), ≥)∧[bni_19] = 0∧[2 + (-1)bso_20] ≥ 0)

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

(23)    ((UIncreasing(675_0_MAIN_CONSTANTSTACKPUSH(+(-(x0[3], 1), 1), x0[3])), ≥)∧[bni_19] = 0∧0 = 0∧0 = 0∧[2 + (-1)bso_20] ≥ 0)

To summarize, we get the following constraints P for the following pairs.
• 675_0_MAIN_CONSTANTSTACKPUSH(x1, x0) → COND_675_0_MAIN_CONSTANTSTACKPUSH(&&(<=(x1, -(x0, 1)), >(x0, -1)), x1, x0)
• (x0[0] + [-1] + x1[0] ≥ 0∧x0[0] ≥ 0∧x1[0] ≥ 0 ⇒ (UIncreasing(COND_675_0_MAIN_CONSTANTSTACKPUSH(&&(<=(x1[0], -(x0[0], 1)), >(x0[0], -1)), x1[0], x0[0])), ≥)∧[(-1)bni_13 + (-1)Bound*bni_13] + [bni_13]x0[0] ≥ 0∧[(-1)bso_14] ≥ 0)
• (x0[0] ≥ 0∧[1] + x1[0] + x0[0] ≥ 0∧x1[0] ≥ 0 ⇒ (UIncreasing(COND_675_0_MAIN_CONSTANTSTACKPUSH(&&(<=(x1[0], -(x0[0], 1)), >(x0[0], -1)), x1[0], x0[0])), ≥)∧[(-1)Bound*bni_13] + [bni_13]x1[0] + [bni_13]x0[0] ≥ 0∧[(-1)bso_14] ≥ 0)

• COND_675_0_MAIN_CONSTANTSTACKPUSH(TRUE, x1, x0) → 675_0_MAIN_CONSTANTSTACKPUSH(+(x1, 1), x0)
• ((UIncreasing(675_0_MAIN_CONSTANTSTACKPUSH(+(x1[1], 1), x0[1])), ≥)∧[bni_15] = 0∧0 = 0∧0 = 0∧[(-1)bso_16] ≥ 0)

• 675_0_MAIN_CONSTANTSTACKPUSH(x0_-1, x0) → COND_675_0_MAIN_CONSTANTSTACKPUSH1(&&(&&(>(x0, -1), <(-(x0, 1), -(x0, 1))), =(x0_-1, -(x0, 1))), x0_-1, x0)

• COND_675_0_MAIN_CONSTANTSTACKPUSH1(TRUE, x0_-1, x0) → 675_0_MAIN_CONSTANTSTACKPUSH(+(-(x0, 1), 1), x0)
• ((UIncreasing(675_0_MAIN_CONSTANTSTACKPUSH(+(-(x0[3], 1), 1), x0[3])), ≥)∧[bni_19] = 0∧0 = 0∧0 = 0∧[2 + (-1)bso_20] ≥ 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(675_0_MAIN_CONSTANTSTACKPUSH(x1, x2)) = [-1] + x2
POL(COND_675_0_MAIN_CONSTANTSTACKPUSH(x1, x2, x3)) = [-1] + x3
POL(&&(x1, x2)) = [-1]
POL(<=(x1, x2)) = [-1]
POL(-(x1, x2)) = x1 + [-1]x2
POL(1) = [1]
POL(>(x1, x2)) = [-1]
POL(-1) = [-1]
POL(+(x1, x2)) = x1 + x2
POL(COND_675_0_MAIN_CONSTANTSTACKPUSH1(x1, x2, x3)) = [1] + x3
POL(<(x1, x2)) = [-1]
POL(=(x1, x2)) = [-1]

The following pairs are in P>:

675_0_MAIN_CONSTANTSTACKPUSH(x0_-1[2], x0[2]) → COND_675_0_MAIN_CONSTANTSTACKPUSH1(&&(&&(>(x0[2], -1), <(-(x0[2], 1), -(x0[2], 1))), =(x0_-1[2], -(x0[2], 1))), x0_-1[2], x0[2])
COND_675_0_MAIN_CONSTANTSTACKPUSH1(TRUE, x0_-1[3], x0[3]) → 675_0_MAIN_CONSTANTSTACKPUSH(+(-(x0[3], 1), 1), x0[3])

The following pairs are in Pbound:

675_0_MAIN_CONSTANTSTACKPUSH(x1[0], x0[0]) → COND_675_0_MAIN_CONSTANTSTACKPUSH(&&(<=(x1[0], -(x0[0], 1)), >(x0[0], -1)), x1[0], x0[0])
675_0_MAIN_CONSTANTSTACKPUSH(x0_-1[2], x0[2]) → COND_675_0_MAIN_CONSTANTSTACKPUSH1(&&(&&(>(x0[2], -1), <(-(x0[2], 1), -(x0[2], 1))), =(x0_-1[2], -(x0[2], 1))), x0_-1[2], x0[2])

The following pairs are in P:

675_0_MAIN_CONSTANTSTACKPUSH(x1[0], x0[0]) → COND_675_0_MAIN_CONSTANTSTACKPUSH(&&(<=(x1[0], -(x0[0], 1)), >(x0[0], -1)), x1[0], x0[0])
COND_675_0_MAIN_CONSTANTSTACKPUSH(TRUE, x1[1], x0[1]) → 675_0_MAIN_CONSTANTSTACKPUSH(+(x1[1], 1), x0[1])

There are no usable rules.

(9) Obligation:

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

The following domains are used:

Boolean, Integer

R is empty.

The integer pair graph contains the following rules and edges:
(0): 675_0_MAIN_CONSTANTSTACKPUSH(x1[0], x0[0]) → COND_675_0_MAIN_CONSTANTSTACKPUSH(x1[0] <= x0[0] - 1 && x0[0] > -1, x1[0], x0[0])
(1): COND_675_0_MAIN_CONSTANTSTACKPUSH(TRUE, x1[1], x0[1]) → 675_0_MAIN_CONSTANTSTACKPUSH(x1[1] + 1, x0[1])

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

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

The set Q is empty.

(10) 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.IdpCand1ShapeHeuristic@1d87a604 Constraint Generator: NonInfConstraintGenerator: PathGenerator: MetricPathGenerator: Max Left Steps: 0 Max Right Steps: 0

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 675_0_MAIN_CONSTANTSTACKPUSH(x1[0], x0[0]) → COND_675_0_MAIN_CONSTANTSTACKPUSH(&&(<=(x1[0], -(x0[0], 1)), >(x0[0], -1)), x1[0], x0[0]) the following chains were created:
• We consider the chain 675_0_MAIN_CONSTANTSTACKPUSH(x1[0], x0[0]) → COND_675_0_MAIN_CONSTANTSTACKPUSH(&&(<=(x1[0], -(x0[0], 1)), >(x0[0], -1)), x1[0], x0[0]), COND_675_0_MAIN_CONSTANTSTACKPUSH(TRUE, x1[1], x0[1]) → 675_0_MAIN_CONSTANTSTACKPUSH(+(x1[1], 1), x0[1]) which results in the following constraint:

(1)    (&&(<=(x1[0], -(x0[0], 1)), >(x0[0], -1))=TRUEx1[0]=x1[1]x0[0]=x0[1]675_0_MAIN_CONSTANTSTACKPUSH(x1[0], x0[0])≥NonInfC∧675_0_MAIN_CONSTANTSTACKPUSH(x1[0], x0[0])≥COND_675_0_MAIN_CONSTANTSTACKPUSH(&&(<=(x1[0], -(x0[0], 1)), >(x0[0], -1)), x1[0], x0[0])∧(UIncreasing(COND_675_0_MAIN_CONSTANTSTACKPUSH(&&(<=(x1[0], -(x0[0], 1)), >(x0[0], -1)), x1[0], x0[0])), ≥))

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

(2)    (<=(x1[0], -(x0[0], 1))=TRUE>(x0[0], -1)=TRUE675_0_MAIN_CONSTANTSTACKPUSH(x1[0], x0[0])≥NonInfC∧675_0_MAIN_CONSTANTSTACKPUSH(x1[0], x0[0])≥COND_675_0_MAIN_CONSTANTSTACKPUSH(&&(<=(x1[0], -(x0[0], 1)), >(x0[0], -1)), x1[0], x0[0])∧(UIncreasing(COND_675_0_MAIN_CONSTANTSTACKPUSH(&&(<=(x1[0], -(x0[0], 1)), >(x0[0], -1)), x1[0], x0[0])), ≥))

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

(3)    (x0[0] + [-1] + [-1]x1[0] ≥ 0∧x0[0] ≥ 0 ⇒ (UIncreasing(COND_675_0_MAIN_CONSTANTSTACKPUSH(&&(<=(x1[0], -(x0[0], 1)), >(x0[0], -1)), x1[0], x0[0])), ≥)∧[bni_8 + (-1)Bound*bni_8] + [bni_8]x0[0] + [(-1)bni_8]x1[0] ≥ 0∧[(-1)bso_9] ≥ 0)

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

(4)    (x0[0] + [-1] + [-1]x1[0] ≥ 0∧x0[0] ≥ 0 ⇒ (UIncreasing(COND_675_0_MAIN_CONSTANTSTACKPUSH(&&(<=(x1[0], -(x0[0], 1)), >(x0[0], -1)), x1[0], x0[0])), ≥)∧[bni_8 + (-1)Bound*bni_8] + [bni_8]x0[0] + [(-1)bni_8]x1[0] ≥ 0∧[(-1)bso_9] ≥ 0)

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

(5)    (x0[0] + [-1] + [-1]x1[0] ≥ 0∧x0[0] ≥ 0 ⇒ (UIncreasing(COND_675_0_MAIN_CONSTANTSTACKPUSH(&&(<=(x1[0], -(x0[0], 1)), >(x0[0], -1)), x1[0], x0[0])), ≥)∧[bni_8 + (-1)Bound*bni_8] + [bni_8]x0[0] + [(-1)bni_8]x1[0] ≥ 0∧[(-1)bso_9] ≥ 0)

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

(6)    (x0[0] + [-1] + [-1]x1[0] ≥ 0∧x0[0] ≥ 0∧x1[0] ≥ 0 ⇒ (UIncreasing(COND_675_0_MAIN_CONSTANTSTACKPUSH(&&(<=(x1[0], -(x0[0], 1)), >(x0[0], -1)), x1[0], x0[0])), ≥)∧[bni_8 + (-1)Bound*bni_8] + [bni_8]x0[0] + [(-1)bni_8]x1[0] ≥ 0∧[(-1)bso_9] ≥ 0)

(7)    (x0[0] + [-1] + x1[0] ≥ 0∧x0[0] ≥ 0∧x1[0] ≥ 0 ⇒ (UIncreasing(COND_675_0_MAIN_CONSTANTSTACKPUSH(&&(<=(x1[0], -(x0[0], 1)), >(x0[0], -1)), x1[0], x0[0])), ≥)∧[bni_8 + (-1)Bound*bni_8] + [bni_8]x0[0] + [bni_8]x1[0] ≥ 0∧[(-1)bso_9] ≥ 0)

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

(8)    (x0[0] ≥ 0∧[1] + x1[0] + x0[0] ≥ 0∧x1[0] ≥ 0 ⇒ (UIncreasing(COND_675_0_MAIN_CONSTANTSTACKPUSH(&&(<=(x1[0], -(x0[0], 1)), >(x0[0], -1)), x1[0], x0[0])), ≥)∧[(2)bni_8 + (-1)Bound*bni_8] + [bni_8]x0[0] ≥ 0∧[(-1)bso_9] ≥ 0)

For Pair COND_675_0_MAIN_CONSTANTSTACKPUSH(TRUE, x1[1], x0[1]) → 675_0_MAIN_CONSTANTSTACKPUSH(+(x1[1], 1), x0[1]) the following chains were created:
• We consider the chain COND_675_0_MAIN_CONSTANTSTACKPUSH(TRUE, x1[1], x0[1]) → 675_0_MAIN_CONSTANTSTACKPUSH(+(x1[1], 1), x0[1]) which results in the following constraint:

(9)    (COND_675_0_MAIN_CONSTANTSTACKPUSH(TRUE, x1[1], x0[1])≥NonInfC∧COND_675_0_MAIN_CONSTANTSTACKPUSH(TRUE, x1[1], x0[1])≥675_0_MAIN_CONSTANTSTACKPUSH(+(x1[1], 1), x0[1])∧(UIncreasing(675_0_MAIN_CONSTANTSTACKPUSH(+(x1[1], 1), x0[1])), ≥))

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

(10)    ((UIncreasing(675_0_MAIN_CONSTANTSTACKPUSH(+(x1[1], 1), x0[1])), ≥)∧[bni_10] = 0∧[1 + (-1)bso_11] ≥ 0)

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

(11)    ((UIncreasing(675_0_MAIN_CONSTANTSTACKPUSH(+(x1[1], 1), x0[1])), ≥)∧[bni_10] = 0∧[1 + (-1)bso_11] ≥ 0)

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

(12)    ((UIncreasing(675_0_MAIN_CONSTANTSTACKPUSH(+(x1[1], 1), x0[1])), ≥)∧[bni_10] = 0∧[1 + (-1)bso_11] ≥ 0)

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

(13)    ((UIncreasing(675_0_MAIN_CONSTANTSTACKPUSH(+(x1[1], 1), x0[1])), ≥)∧[bni_10] = 0∧0 = 0∧0 = 0∧[1 + (-1)bso_11] ≥ 0)

To summarize, we get the following constraints P for the following pairs.
• 675_0_MAIN_CONSTANTSTACKPUSH(x1[0], x0[0]) → COND_675_0_MAIN_CONSTANTSTACKPUSH(&&(<=(x1[0], -(x0[0], 1)), >(x0[0], -1)), x1[0], x0[0])
• (x0[0] + [-1] + x1[0] ≥ 0∧x0[0] ≥ 0∧x1[0] ≥ 0 ⇒ (UIncreasing(COND_675_0_MAIN_CONSTANTSTACKPUSH(&&(<=(x1[0], -(x0[0], 1)), >(x0[0], -1)), x1[0], x0[0])), ≥)∧[bni_8 + (-1)Bound*bni_8] + [bni_8]x0[0] + [bni_8]x1[0] ≥ 0∧[(-1)bso_9] ≥ 0)
• (x0[0] ≥ 0∧[1] + x1[0] + x0[0] ≥ 0∧x1[0] ≥ 0 ⇒ (UIncreasing(COND_675_0_MAIN_CONSTANTSTACKPUSH(&&(<=(x1[0], -(x0[0], 1)), >(x0[0], -1)), x1[0], x0[0])), ≥)∧[(2)bni_8 + (-1)Bound*bni_8] + [bni_8]x0[0] ≥ 0∧[(-1)bso_9] ≥ 0)

• COND_675_0_MAIN_CONSTANTSTACKPUSH(TRUE, x1[1], x0[1]) → 675_0_MAIN_CONSTANTSTACKPUSH(+(x1[1], 1), x0[1])
• ((UIncreasing(675_0_MAIN_CONSTANTSTACKPUSH(+(x1[1], 1), x0[1])), ≥)∧[bni_10] = 0∧0 = 0∧0 = 0∧[1 + (-1)bso_11] ≥ 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(675_0_MAIN_CONSTANTSTACKPUSH(x1, x2)) = [1] + x2 + [-1]x1
POL(COND_675_0_MAIN_CONSTANTSTACKPUSH(x1, x2, x3)) = [1] + x3 + [-1]x2
POL(&&(x1, x2)) = [-1]
POL(<=(x1, x2)) = [-1]
POL(-(x1, x2)) = x1 + [-1]x2
POL(1) = [1]
POL(>(x1, x2)) = [-1]
POL(-1) = [-1]
POL(+(x1, x2)) = x1 + x2

The following pairs are in P>:

COND_675_0_MAIN_CONSTANTSTACKPUSH(TRUE, x1[1], x0[1]) → 675_0_MAIN_CONSTANTSTACKPUSH(+(x1[1], 1), x0[1])

The following pairs are in Pbound:

675_0_MAIN_CONSTANTSTACKPUSH(x1[0], x0[0]) → COND_675_0_MAIN_CONSTANTSTACKPUSH(&&(<=(x1[0], -(x0[0], 1)), >(x0[0], -1)), x1[0], x0[0])

The following pairs are in P:

675_0_MAIN_CONSTANTSTACKPUSH(x1[0], x0[0]) → COND_675_0_MAIN_CONSTANTSTACKPUSH(&&(<=(x1[0], -(x0[0], 1)), >(x0[0], -1)), x1[0], x0[0])

There are no usable rules.

(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:
(0): 675_0_MAIN_CONSTANTSTACKPUSH(x1[0], x0[0]) → COND_675_0_MAIN_CONSTANTSTACKPUSH(x1[0] <= x0[0] - 1 && x0[0] > -1, x1[0], x0[0])

The set Q is empty.

(13) IDependencyGraphProof (EQUIVALENT transformation)

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

(15) Obligation:

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

The following domains are used:

Integer

R is empty.

The integer pair graph contains the following rules and edges:
(1): COND_675_0_MAIN_CONSTANTSTACKPUSH(TRUE, x1[1], x0[1]) → 675_0_MAIN_CONSTANTSTACKPUSH(x1[1] + 1, x0[1])

The set Q is empty.

(16) IDependencyGraphProof (EQUIVALENT transformation)

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

(18) Obligation:

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

The following domains are used:

Integer

R is empty.

The integer pair graph contains the following rules and edges:
(1): COND_675_0_MAIN_CONSTANTSTACKPUSH(TRUE, x1[1], x0[1]) → 675_0_MAIN_CONSTANTSTACKPUSH(x1[1] + 1, x0[1])
(3): COND_675_0_MAIN_CONSTANTSTACKPUSH1(TRUE, x0_-1[3], x0[3]) → 675_0_MAIN_CONSTANTSTACKPUSH(x0[3] - 1 + 1, x0[3])

The set Q is empty.

(19) IDependencyGraphProof (EQUIVALENT transformation)

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