Termination proof of /tmp/tmp70brwG/AG313.jar

Termination of the given JBC could be proven:

0 JBC

↳1 JBC2FIG (⇒)

↳2 JBCTerminationGraph

↳3 FIGtoITRSProof (⇒)

↳4 IDP

↳5 IDPNonInfProof (⇒)

↳6 IDP

↳7 IDependencyGraphProof (⇔)

↳8 TRUE

(0) Obligation:

JBC Problem based on JBC Program:
Manifest-Version: 1.0 Created-By: 1.6.0_20 (Sun Microsystems Inc.) Main-Class: AG313

public class AG313 {
  public static void main(String[] args) {
    int x, y;
    x = args[0].length();
    y = args[1].length() + 1;
    quot(x,y);

  }


  public static int quot(int x, int y) {
    int i = 0;
    if(x==0) return 0;
    while (x > 0 && y > 0) {
      i += 1;
      x = (x - 1)- (y - 1);

    }
    return i;
  }
}

(1) JBC2FIG (SOUND transformation)

Constructed FIGraph.

(2) Obligation:

FIGraph based on JBC Program:
AG313.main([Ljava/lang/String;)V: Graph of 157 nodes with 1 SCC.

(3) FIGtoITRSProof (SOUND transformation)

Transformed FIGraph SCCs to IDPs. Logs:

Log for SCC 0:

Generated 16 rules for P and 5 rules for R.

Combined rules. Obtained 1 rules for P and 1 rules for R.

Filtered ground terms:

898_0_quot_LE(x1, x2, x3, x4) → 898_0_quot_LE(x2, x3, x4)
911_0_main_Return(x1) → 911_0_main_Return

Filtered duplicate args:

898_0_quot_LE(x1, x2, x3) → 898_0_quot_LE(x2, x3)

Combined rules. Obtained 1 rules for P and 1 rules for R.

Finished conversion. Obtained 1 rules for P and 1 rules for R. System has predefined symbols.

(4) Obligation:

IDP problem:
The following function symbols are pre-defined:

!=	~	Neq: (Integer, Integer) -> Boolean
*	~	Mul: (Integer, Integer) -> Integer
>=	~	Ge: (Integer, Integer) -> Boolean
-1	~	UnaryMinus: (Integer) -> Integer
\|	~	Bwor: (Integer, Integer) -> Integer
/	~	Div: (Integer, Integer) -> Integer
=	~	Eq: (Integer, Integer) -> Boolean
	~	Bwxor: (Integer, Integer) -> Integer
\|\|	~	Lor: (Boolean, Boolean) -> Boolean
!	~	Lnot: (Boolean) -> Boolean
<	~	Lt: (Integer, Integer) -> Boolean
-	~	Sub: (Integer, Integer) -> Integer
<=	~	Le: (Integer, Integer) -> Boolean
>	~	Gt: (Integer, Integer) -> Boolean
~	~	Bwnot: (Integer) -> Integer
%	~	Mod: (Integer, Integer) -> Integer
&	~	Bwand: (Integer, Integer) -> Integer
+	~	Add: (Integer, Integer) -> Integer
&&	~	Land: (Boolean, Boolean) -> Boolean

The following domains are used:

Integer, Boolean

The ITRS R consists of the following rules:
898_1_main_InvokeMethod(898_0_quot_LE(x1, x0), x1) → Cond_898_1_main_InvokeMethod(x0 <= 0, 898_0_quot_LE(x1, x0), x1)
Cond_898_1_main_InvokeMethod(TRUE, 898_0_quot_LE(x1, x0), x1) → 911_0_main_Return

The integer pair graph contains the following rules and edges:
(0): 898_1_MAIN_INVOKEMETHOD(898_0_quot_LE(x1[0], x0[0]), x1[0]) → COND_898_1_MAIN_INVOKEMETHOD(x1[0] > 0 && x0[0] > 0 && 0 <= x1[0] - 1 && 0 <= x0[0] - 1, 898_0_quot_LE(x1[0], x0[0]), x1[0])
(1): COND_898_1_MAIN_INVOKEMETHOD(TRUE, 898_0_quot_LE(x1[1], x0[1]), x1[1]) → 898_1_MAIN_INVOKEMETHOD(898_0_quot_LE(x1[1], x0[1] - 1 - x1[1] - 1), x1[1])

(0) -> (1), if ((x1[0] > 0 && x0[0] > 0 && 0 <= x1[0] - 1 && 0 <= x0[0] - 1 →^* TRUE)∧(898_0_quot_LE(x1[0], x0[0]) →^* 898_0_quot_LE(x1[1], x0[1]))∧(x1[0] →^* x1[1]))

(1) -> (0), if ((898_0_quot_LE(x1[1], x0[1] - 1 - x1[1] - 1) →^* 898_0_quot_LE(x1[0], x0[0]))∧(x1[1] →^* x1[0]))

The set Q consists of the following terms:
898_1_main_InvokeMethod(898_0_quot_LE(x0, x1), x0)
Cond_898_1_main_InvokeMethod(TRUE, 898_0_quot_LE(x0, x1), x0)

(5) IDPNonInfProof (SOUND transformation)

The constraints were generated the following way:
The DP Problem is simplified using the Induction Calculus [NONINF] with the following steps:
Note that final constraints are written in bold face.

For Pair 898_1_MAIN_INVOKEMETHOD(898_0_quot_LE(x1, x0), x1) → COND_898_1_MAIN_INVOKEMETHOD(&&(&&(&&(>(x1, 0), >(x0, 0)), <=(0, -(x1, 1))), <=(0, -(x0, 1))), 898_0_quot_LE(x1, x0), x1) the following chains were created:

We consider the chain 898_1_MAIN_INVOKEMETHOD(898_0_quot_LE(x1[0], x0[0]), x1[0]) → COND_898_1_MAIN_INVOKEMETHOD(&&(&&(&&(>(x1[0], 0), >(x0[0], 0)), <=(0, -(x1[0], 1))), <=(0, -(x0[0], 1))), 898_0_quot_LE(x1[0], x0[0]), x1[0]), COND_898_1_MAIN_INVOKEMETHOD(TRUE, 898_0_quot_LE(x1[1], x0[1]), x1[1]) → 898_1_MAIN_INVOKEMETHOD(898_0_quot_LE(x1[1], -(-(x0[1], 1), -(x1[1], 1))), x1[1]) which results in the following constraint:
(1)    (&&(&&(&&(>(x1[0], 0), >(x0[0], 0)), <=(0, -(x1[0], 1))), <=(0, -(x0[0], 1)))=TRUE∧898_0_quot_LE(x1[0], x0[0])=898_0_quot_LE(x1[1], x0[1])∧x1[0]=x1[1] ⇒ 898_1_MAIN_INVOKEMETHOD(898_0_quot_LE(x1[0], x0[0]), x1[0])≥NonInfC∧898_1_MAIN_INVOKEMETHOD(898_0_quot_LE(x1[0], x0[0]), x1[0])≥COND_898_1_MAIN_INVOKEMETHOD(&&(&&(&&(>(x1[0], 0), >(x0[0], 0)), <=(0, -(x1[0], 1))), <=(0, -(x0[0], 1))), 898_0_quot_LE(x1[0], x0[0]), x1[0])∧(U^Increasing(COND_898_1_MAIN_INVOKEMETHOD(&&(&&(&&(>(x1[0], 0), >(x0[0], 0)), <=(0, -(x1[0], 1))), <=(0, -(x0[0], 1))), 898_0_quot_LE(x1[0], x0[0]), x1[0])), ≥))

We simplified constraint (1) using rules (I), (II), (IV), (IDP_BOOLEAN) which results in the following new constraint:
(2)    (<=(0, -(x0[0], 1))=TRUE∧<=(0, -(x1[0], 1))=TRUE∧>(x1[0], 0)=TRUE∧>(x0[0], 0)=TRUE ⇒ 898_1_MAIN_INVOKEMETHOD(898_0_quot_LE(x1[0], x0[0]), x1[0])≥NonInfC∧898_1_MAIN_INVOKEMETHOD(898_0_quot_LE(x1[0], x0[0]), x1[0])≥COND_898_1_MAIN_INVOKEMETHOD(&&(&&(&&(>(x1[0], 0), >(x0[0], 0)), <=(0, -(x1[0], 1))), <=(0, -(x0[0], 1))), 898_0_quot_LE(x1[0], x0[0]), x1[0])∧(U^Increasing(COND_898_1_MAIN_INVOKEMETHOD(&&(&&(&&(>(x1[0], 0), >(x0[0], 0)), <=(0, -(x1[0], 1))), <=(0, -(x0[0], 1))), 898_0_quot_LE(x1[0], x0[0]), x1[0])), ≥))

We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(3)    (x0[0] + [-1] ≥ 0∧x1[0] + [-1] ≥ 0∧x1[0] + [-1] ≥ 0∧x0[0] + [-1] ≥ 0 ⇒ (U^Increasing(COND_898_1_MAIN_INVOKEMETHOD(&&(&&(&&(>(x1[0], 0), >(x0[0], 0)), <=(0, -(x1[0], 1))), <=(0, -(x0[0], 1))), 898_0_quot_LE(x1[0], x0[0]), x1[0])), ≥)∧[(-1)bni_24 + (-1)Bound*bni_24] + [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)    (x0[0] + [-1] ≥ 0∧x1[0] + [-1] ≥ 0∧x1[0] + [-1] ≥ 0∧x0[0] + [-1] ≥ 0 ⇒ (U^Increasing(COND_898_1_MAIN_INVOKEMETHOD(&&(&&(&&(>(x1[0], 0), >(x0[0], 0)), <=(0, -(x1[0], 1))), <=(0, -(x0[0], 1))), 898_0_quot_LE(x1[0], x0[0]), x1[0])), ≥)∧[(-1)bni_24 + (-1)Bound*bni_24] + [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:
(5)    (x0[0] + [-1] ≥ 0∧x1[0] + [-1] ≥ 0∧x1[0] + [-1] ≥ 0∧x0[0] + [-1] ≥ 0 ⇒ (U^Increasing(COND_898_1_MAIN_INVOKEMETHOD(&&(&&(&&(>(x1[0], 0), >(x0[0], 0)), <=(0, -(x1[0], 1))), <=(0, -(x0[0], 1))), 898_0_quot_LE(x1[0], x0[0]), x1[0])), ≥)∧[(-1)bni_24 + (-1)Bound*bni_24] + [bni_24]x0[0] ≥ 0∧[(-1)bso_25] ≥ 0)

We simplified constraint (5) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(6)    (x0[0] ≥ 0∧x1[0] + [-1] ≥ 0∧x1[0] + [-1] ≥ 0∧x0[0] ≥ 0 ⇒ (U^Increasing(COND_898_1_MAIN_INVOKEMETHOD(&&(&&(&&(>(x1[0], 0), >(x0[0], 0)), <=(0, -(x1[0], 1))), <=(0, -(x0[0], 1))), 898_0_quot_LE(x1[0], x0[0]), x1[0])), ≥)∧[(-1)Bound*bni_24] + [bni_24]x0[0] ≥ 0∧[(-1)bso_25] ≥ 0)

We simplified constraint (6) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(7)    (x0[0] ≥ 0∧x1[0] ≥ 0∧x1[0] ≥ 0∧x0[0] ≥ 0 ⇒ (U^Increasing(COND_898_1_MAIN_INVOKEMETHOD(&&(&&(&&(>(x1[0], 0), >(x0[0], 0)), <=(0, -(x1[0], 1))), <=(0, -(x0[0], 1))), 898_0_quot_LE(x1[0], x0[0]), x1[0])), ≥)∧[(-1)Bound*bni_24] + [bni_24]x0[0] ≥ 0∧[(-1)bso_25] ≥ 0)

For Pair COND_898_1_MAIN_INVOKEMETHOD(TRUE, 898_0_quot_LE(x1, x0), x1) → 898_1_MAIN_INVOKEMETHOD(898_0_quot_LE(x1, -(-(x0, 1), -(x1, 1))), x1) the following chains were created:

We consider the chain 898_1_MAIN_INVOKEMETHOD(898_0_quot_LE(x1[0], x0[0]), x1[0]) → COND_898_1_MAIN_INVOKEMETHOD(&&(&&(&&(>(x1[0], 0), >(x0[0], 0)), <=(0, -(x1[0], 1))), <=(0, -(x0[0], 1))), 898_0_quot_LE(x1[0], x0[0]), x1[0]), COND_898_1_MAIN_INVOKEMETHOD(TRUE, 898_0_quot_LE(x1[1], x0[1]), x1[1]) → 898_1_MAIN_INVOKEMETHOD(898_0_quot_LE(x1[1], -(-(x0[1], 1), -(x1[1], 1))), x1[1]), 898_1_MAIN_INVOKEMETHOD(898_0_quot_LE(x1[0], x0[0]), x1[0]) → COND_898_1_MAIN_INVOKEMETHOD(&&(&&(&&(>(x1[0], 0), >(x0[0], 0)), <=(0, -(x1[0], 1))), <=(0, -(x0[0], 1))), 898_0_quot_LE(x1[0], x0[0]), x1[0]) which results in the following constraint:
(8)    (&&(&&(&&(>(x1[0], 0), >(x0[0], 0)), <=(0, -(x1[0], 1))), <=(0, -(x0[0], 1)))=TRUE∧898_0_quot_LE(x1[0], x0[0])=898_0_quot_LE(x1[1], x0[1])∧x1[0]=x1[1]∧898_0_quot_LE(x1[1], -(-(x0[1], 1), -(x1[1], 1)))=898_0_quot_LE(x1[0]1, x0[0]1)∧x1[1]=x1[0]1 ⇒ COND_898_1_MAIN_INVOKEMETHOD(TRUE, 898_0_quot_LE(x1[1], x0[1]), x1[1])≥NonInfC∧COND_898_1_MAIN_INVOKEMETHOD(TRUE, 898_0_quot_LE(x1[1], x0[1]), x1[1])≥898_1_MAIN_INVOKEMETHOD(898_0_quot_LE(x1[1], -(-(x0[1], 1), -(x1[1], 1))), x1[1])∧(U^Increasing(898_1_MAIN_INVOKEMETHOD(898_0_quot_LE(x1[1], -(-(x0[1], 1), -(x1[1], 1))), x1[1])), ≥))

We simplified constraint (8) using rules (I), (II), (III), (IV), (IDP_BOOLEAN) which results in the following new constraint:
(9)    (<=(0, -(x0[0], 1))=TRUE∧<=(0, -(x1[0], 1))=TRUE∧>(x1[0], 0)=TRUE∧>(x0[0], 0)=TRUE ⇒ COND_898_1_MAIN_INVOKEMETHOD(TRUE, 898_0_quot_LE(x1[0], x0[0]), x1[0])≥NonInfC∧COND_898_1_MAIN_INVOKEMETHOD(TRUE, 898_0_quot_LE(x1[0], x0[0]), x1[0])≥898_1_MAIN_INVOKEMETHOD(898_0_quot_LE(x1[0], -(-(x0[0], 1), -(x1[0], 1))), x1[0])∧(U^Increasing(898_1_MAIN_INVOKEMETHOD(898_0_quot_LE(x1[1], -(-(x0[1], 1), -(x1[1], 1))), x1[1])), ≥))

We simplified constraint (9) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(10)    (x0[0] + [-1] ≥ 0∧x1[0] + [-1] ≥ 0∧x1[0] + [-1] ≥ 0∧x0[0] + [-1] ≥ 0 ⇒ (U^Increasing(898_1_MAIN_INVOKEMETHOD(898_0_quot_LE(x1[1], -(-(x0[1], 1), -(x1[1], 1))), x1[1])), ≥)∧[(-1)bni_26 + (-1)Bound*bni_26] + [bni_26]x0[0] ≥ 0∧[(-1)bso_27] + x1[0] ≥ 0)

We simplified constraint (10) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(11)    (x0[0] + [-1] ≥ 0∧x1[0] + [-1] ≥ 0∧x1[0] + [-1] ≥ 0∧x0[0] + [-1] ≥ 0 ⇒ (U^Increasing(898_1_MAIN_INVOKEMETHOD(898_0_quot_LE(x1[1], -(-(x0[1], 1), -(x1[1], 1))), x1[1])), ≥)∧[(-1)bni_26 + (-1)Bound*bni_26] + [bni_26]x0[0] ≥ 0∧[(-1)bso_27] + x1[0] ≥ 0)

We simplified constraint (11) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(12)    (x0[0] + [-1] ≥ 0∧x1[0] + [-1] ≥ 0∧x1[0] + [-1] ≥ 0∧x0[0] + [-1] ≥ 0 ⇒ (U^Increasing(898_1_MAIN_INVOKEMETHOD(898_0_quot_LE(x1[1], -(-(x0[1], 1), -(x1[1], 1))), x1[1])), ≥)∧[(-1)bni_26 + (-1)Bound*bni_26] + [bni_26]x0[0] ≥ 0∧[(-1)bso_27] + x1[0] ≥ 0)

We simplified constraint (12) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(13)    (x0[0] ≥ 0∧x1[0] + [-1] ≥ 0∧x1[0] + [-1] ≥ 0∧x0[0] ≥ 0 ⇒ (U^Increasing(898_1_MAIN_INVOKEMETHOD(898_0_quot_LE(x1[1], -(-(x0[1], 1), -(x1[1], 1))), x1[1])), ≥)∧[(-1)Bound*bni_26] + [bni_26]x0[0] ≥ 0∧[(-1)bso_27] + x1[0] ≥ 0)

We simplified constraint (13) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(14)    (x0[0] ≥ 0∧x1[0] ≥ 0∧x1[0] ≥ 0∧x0[0] ≥ 0 ⇒ (U^Increasing(898_1_MAIN_INVOKEMETHOD(898_0_quot_LE(x1[1], -(-(x0[1], 1), -(x1[1], 1))), x1[1])), ≥)∧[(-1)Bound*bni_26] + [bni_26]x0[0] ≥ 0∧[1 + (-1)bso_27] + x1[0] ≥ 0)

To summarize, we get the following constraints P_≥ for the following pairs.

898_1_MAIN_INVOKEMETHOD(898_0_quot_LE(x1, x0), x1) → COND_898_1_MAIN_INVOKEMETHOD(&&(&&(&&(>(x1, 0), >(x0, 0)), <=(0, -(x1, 1))), <=(0, -(x0, 1))), 898_0_quot_LE(x1, x0), x1)
- (x0[0] ≥ 0∧x1[0] ≥ 0∧x1[0] ≥ 0∧x0[0] ≥ 0 ⇒ (U^Increasing(COND_898_1_MAIN_INVOKEMETHOD(&&(&&(&&(>(x1[0], 0), >(x0[0], 0)), <=(0, -(x1[0], 1))), <=(0, -(x0[0], 1))), 898_0_quot_LE(x1[0], x0[0]), x1[0])), ≥)∧[(-1)Bound*bni_24] + [bni_24]x0[0] ≥ 0∧[(-1)bso_25] ≥ 0)
COND_898_1_MAIN_INVOKEMETHOD(TRUE, 898_0_quot_LE(x1, x0), x1) → 898_1_MAIN_INVOKEMETHOD(898_0_quot_LE(x1, -(-(x0, 1), -(x1, 1))), x1)
- (x0[0] ≥ 0∧x1[0] ≥ 0∧x1[0] ≥ 0∧x0[0] ≥ 0 ⇒ (U^Increasing(898_1_MAIN_INVOKEMETHOD(898_0_quot_LE(x1[1], -(-(x0[1], 1), -(x1[1], 1))), x1[1])), ≥)∧[(-1)Bound*bni_26] + [bni_26]x0[0] ≥ 0∧[1 + (-1)bso_27] + x1[0] ≥ 0)

The constraints for P_> respective P_bound 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 P_bound.
Using the following integer polynomial ordering the resulting constraints can be solved
Polynomial interpretation over integers[POLO]:

POL(TRUE) = 0
POL(FALSE) = 0
POL(898_1_main_InvokeMethod(x₁, x₂)) = [-1] + [-1]x₂ + [-1]x₁
POL(898_0_quot_LE(x₁, x₂)) = [-1]x₂ + [-1]x₁
POL(Cond_898_1_main_InvokeMethod(x₁, x₂, x₃)) = [-1] + [-1]x₃ + [-1]x₂ + [-1]x₁
POL(<=(x₁, x₂)) = [-1]
POL(0) = 0
POL(911_0_main_Return) = [-1]
POL(898_1_MAIN_INVOKEMETHOD(x₁, x₂)) = [-1] + [-1]x₂ + [-1]x₁
POL(COND_898_1_MAIN_INVOKEMETHOD(x₁, x₂, x₃)) = [-1] + [-1]x₃ + [-1]x₂ + [-1]x₁
POL(&&(x₁, x₂)) = 0
POL(>(x₁, x₂)) = [-1]
POL(-(x₁, x₂)) = x₁ + [-1]x₂
POL(1) = [1]

The following pairs are in P_>:

COND_898_1_MAIN_INVOKEMETHOD(TRUE, 898_0_quot_LE(x1[1], x0[1]), x1[1]) → 898_1_MAIN_INVOKEMETHOD(898_0_quot_LE(x1[1], -(-(x0[1], 1), -(x1[1], 1))), x1[1])

The following pairs are in P_bound:

898_1_MAIN_INVOKEMETHOD(898_0_quot_LE(x1[0], x0[0]), x1[0]) → COND_898_1_MAIN_INVOKEMETHOD(&&(&&(&&(>(x1[0], 0), >(x0[0], 0)), <=(0, -(x1[0], 1))), <=(0, -(x0[0], 1))), 898_0_quot_LE(x1[0], x0[0]), x1[0])
COND_898_1_MAIN_INVOKEMETHOD(TRUE, 898_0_quot_LE(x1[1], x0[1]), x1[1]) → 898_1_MAIN_INVOKEMETHOD(898_0_quot_LE(x1[1], -(-(x0[1], 1), -(x1[1], 1))), x1[1])

The following pairs are in P_≥:

898_1_MAIN_INVOKEMETHOD(898_0_quot_LE(x1[0], x0[0]), x1[0]) → COND_898_1_MAIN_INVOKEMETHOD(&&(&&(&&(>(x1[0], 0), >(x0[0], 0)), <=(0, -(x1[0], 1))), <=(0, -(x0[0], 1))), 898_0_quot_LE(x1[0], x0[0]), x1[0])

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

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

(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:

Integer, Boolean

The ITRS R consists of the following rules:
898_1_main_InvokeMethod(898_0_quot_LE(x1, x0), x1) → Cond_898_1_main_InvokeMethod(x0 <= 0, 898_0_quot_LE(x1, x0), x1)
Cond_898_1_main_InvokeMethod(TRUE, 898_0_quot_LE(x1, x0), x1) → 911_0_main_Return

The integer pair graph contains the following rules and edges:
(0): 898_1_MAIN_INVOKEMETHOD(898_0_quot_LE(x1[0], x0[0]), x1[0]) → COND_898_1_MAIN_INVOKEMETHOD(x1[0] > 0 && x0[0] > 0 && 0 <= x1[0] - 1 && 0 <= x0[0] - 1, 898_0_quot_LE(x1[0], x0[0]), x1[0])

The set Q consists of the following terms:
898_1_main_InvokeMethod(898_0_quot_LE(x0, x1), x0)
Cond_898_1_main_InvokeMethod(TRUE, 898_0_quot_LE(x0, x1), x0)

(7) IDependencyGraphProof (EQUIVALENT transformation)

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

(0) Obligation:

(1) JBC2FIG (SOUND transformation)

(2) Obligation:

(3) FIGtoITRSProof (SOUND transformation)

(4) Obligation:

(5) IDPNonInfProof (SOUND transformation)

(6) Obligation:

(7) IDependencyGraphProof (EQUIVALENT transformation)

(8) TRUE