(0) Obligation:
JBC Problem based on JBC Program:
Manifest-Version: 1.0
Created-By: 1.6.0_22 (Sun Microsystems Inc.)
Main-Class: Nest/Nest
package Nest;
public class Nest{
public static int nest(int x){
if (x == 0) return 0;
else return nest(nest(x-1));
}
public static void main(final String[] args) {
final int x = args[0].length();
final int y = nest(x);
}
}
(1) JBC2FIG (SOUND transformation)
Constructed FIGraph.
(2) Obligation:
FIGraph based on JBC Program:
Nest.Nest.main([Ljava/lang/String;)V: Graph of 69 nodes with 0 SCCs.
Nest.Nest.nest(I)I: Graph of 23 nodes with 0 SCCs.
(3) FIGtoITRSProof (SOUND transformation)
Transformed FIGraph SCCs to IDPs. Logs:
Log for SCC 0: Generated 17 rules for P and 6 rules for R.
Combined rules. Obtained 3 rules for P and 2 rules for R.
Filtered ground terms:
242_1_nest_InvokeMethod(x1, x2) → 242_1_nest_InvokeMethod(x1)
89_0_nest_NE(x1, x2, x3) → 89_0_nest_NE(x2, x3)
320_0_nest_Return(x1, x2) → 320_0_nest_Return
135_0_nest_Return(x1, x2, x3) → 135_0_nest_Return
Cond_89_0_nest_NE(x1, x2, x3, x4) → Cond_89_0_nest_NE(x1, x3, x4)
Filtered duplicate args:
89_0_nest_NE(x1, x2) → 89_0_nest_NE(x2)
Cond_89_0_nest_NE(x1, x2, x3) → Cond_89_0_nest_NE(x1, x3)
Combined rules. Obtained 3 rules for P and 2 rules for R.
Finished conversion. Obtained 3 rules for P and 2 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
The ITRS R consists of the following rules:
89_0_nest_NE(
0) →
135_0_nest_Return242_1_nest_InvokeMethod(
135_0_nest_Return) →
320_0_nest_ReturnThe integer pair graph contains the following rules and edges:
(0):
89_0_NEST_NE(
x0[0]) →
COND_89_0_NEST_NE(
x0[0] > 0,
x0[0])
(1):
COND_89_0_NEST_NE(
TRUE,
x0[1]) →
200_1_NEST_INVOKEMETHOD(
89_0_nest_NE(
x0[1] - 1),
x0[1] - 1)
(2):
COND_89_0_NEST_NE(
TRUE,
x0[2]) →
89_0_NEST_NE(
x0[2] - 1)
(3):
200_1_NEST_INVOKEMETHOD(
135_0_nest_Return,
0) →
89_0_NEST_NE(
0)
(4):
200_1_NEST_INVOKEMETHOD(
320_0_nest_Return,
x1[4]) →
89_0_NEST_NE(
0)
(0) -> (1), if ((x0[0] > 0 →* TRUE)∧(x0[0] →* x0[1]))
(0) -> (2), if ((x0[0] > 0 →* TRUE)∧(x0[0] →* x0[2]))
(1) -> (3), if ((89_0_nest_NE(x0[1] - 1) →* 135_0_nest_Return)∧(x0[1] - 1 →* 0))
(1) -> (4), if ((89_0_nest_NE(x0[1] - 1) →* 320_0_nest_Return)∧(x0[1] - 1 →* x1[4]))
(2) -> (0), if ((x0[2] - 1 →* x0[0]))
(3) -> (0), if ((0 →* x0[0]))
(4) -> (0), if ((0 →* x0[0]))
The set Q consists of the following terms:
89_0_nest_NE(
0)
242_1_nest_InvokeMethod(
135_0_nest_Return)
(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
89_0_NEST_NE(
x0) →
COND_89_0_NEST_NE(
>(
x0,
0),
x0) the following chains were created:
- We consider the chain 89_0_NEST_NE(x0[0]) → COND_89_0_NEST_NE(>(x0[0], 0), x0[0]), COND_89_0_NEST_NE(TRUE, x0[1]) → 200_1_NEST_INVOKEMETHOD(89_0_nest_NE(-(x0[1], 1)), -(x0[1], 1)) which results in the following constraint:
(1) (>(x0[0], 0)=TRUE∧x0[0]=x0[1] ⇒ 89_0_NEST_NE(x0[0])≥NonInfC∧89_0_NEST_NE(x0[0])≥COND_89_0_NEST_NE(>(x0[0], 0), x0[0])∧(UIncreasing(COND_89_0_NEST_NE(>(x0[0], 0), x0[0])), ≥))
We simplified constraint (1) using rule (IV) which results in the following new constraint:
(2) (>(x0[0], 0)=TRUE ⇒ 89_0_NEST_NE(x0[0])≥NonInfC∧89_0_NEST_NE(x0[0])≥COND_89_0_NEST_NE(>(x0[0], 0), x0[0])∧(UIncreasing(COND_89_0_NEST_NE(>(x0[0], 0), x0[0])), ≥))
We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(3) (x0[0] + [-1] ≥ 0 ⇒ (UIncreasing(COND_89_0_NEST_NE(>(x0[0], 0), x0[0])), ≥)∧[(-1)bni_16 + (-1)Bound*bni_16] ≥ 0∧[(-1)bso_17] ≥ 0)
We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(4) (x0[0] + [-1] ≥ 0 ⇒ (UIncreasing(COND_89_0_NEST_NE(>(x0[0], 0), x0[0])), ≥)∧[(-1)bni_16 + (-1)Bound*bni_16] ≥ 0∧[(-1)bso_17] ≥ 0)
We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(5) (x0[0] + [-1] ≥ 0 ⇒ (UIncreasing(COND_89_0_NEST_NE(>(x0[0], 0), x0[0])), ≥)∧[(-1)bni_16 + (-1)Bound*bni_16] ≥ 0∧[(-1)bso_17] ≥ 0)
We simplified constraint (5) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(6) (x0[0] ≥ 0 ⇒ (UIncreasing(COND_89_0_NEST_NE(>(x0[0], 0), x0[0])), ≥)∧[(-1)bni_16 + (-1)Bound*bni_16] ≥ 0∧[(-1)bso_17] ≥ 0)
- We consider the chain 89_0_NEST_NE(x0[0]) → COND_89_0_NEST_NE(>(x0[0], 0), x0[0]), COND_89_0_NEST_NE(TRUE, x0[2]) → 89_0_NEST_NE(-(x0[2], 1)) which results in the following constraint:
(7) (>(x0[0], 0)=TRUE∧x0[0]=x0[2] ⇒ 89_0_NEST_NE(x0[0])≥NonInfC∧89_0_NEST_NE(x0[0])≥COND_89_0_NEST_NE(>(x0[0], 0), x0[0])∧(UIncreasing(COND_89_0_NEST_NE(>(x0[0], 0), x0[0])), ≥))
We simplified constraint (7) using rule (IV) which results in the following new constraint:
(8) (>(x0[0], 0)=TRUE ⇒ 89_0_NEST_NE(x0[0])≥NonInfC∧89_0_NEST_NE(x0[0])≥COND_89_0_NEST_NE(>(x0[0], 0), x0[0])∧(UIncreasing(COND_89_0_NEST_NE(>(x0[0], 0), x0[0])), ≥))
We simplified constraint (8) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(9) (x0[0] + [-1] ≥ 0 ⇒ (UIncreasing(COND_89_0_NEST_NE(>(x0[0], 0), x0[0])), ≥)∧[(-1)bni_16 + (-1)Bound*bni_16] ≥ 0∧[(-1)bso_17] ≥ 0)
We simplified constraint (9) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(10) (x0[0] + [-1] ≥ 0 ⇒ (UIncreasing(COND_89_0_NEST_NE(>(x0[0], 0), x0[0])), ≥)∧[(-1)bni_16 + (-1)Bound*bni_16] ≥ 0∧[(-1)bso_17] ≥ 0)
We simplified constraint (10) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(11) (x0[0] + [-1] ≥ 0 ⇒ (UIncreasing(COND_89_0_NEST_NE(>(x0[0], 0), x0[0])), ≥)∧[(-1)bni_16 + (-1)Bound*bni_16] ≥ 0∧[(-1)bso_17] ≥ 0)
We simplified constraint (11) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(12) (x0[0] ≥ 0 ⇒ (UIncreasing(COND_89_0_NEST_NE(>(x0[0], 0), x0[0])), ≥)∧[(-1)bni_16 + (-1)Bound*bni_16] ≥ 0∧[(-1)bso_17] ≥ 0)
For Pair
COND_89_0_NEST_NE(
TRUE,
x0) →
200_1_NEST_INVOKEMETHOD(
89_0_nest_NE(
-(
x0,
1)),
-(
x0,
1)) the following chains were created:
- We consider the chain COND_89_0_NEST_NE(TRUE, x0[1]) → 200_1_NEST_INVOKEMETHOD(89_0_nest_NE(-(x0[1], 1)), -(x0[1], 1)) which results in the following constraint:
(13) (COND_89_0_NEST_NE(TRUE, x0[1])≥NonInfC∧COND_89_0_NEST_NE(TRUE, x0[1])≥200_1_NEST_INVOKEMETHOD(89_0_nest_NE(-(x0[1], 1)), -(x0[1], 1))∧(UIncreasing(200_1_NEST_INVOKEMETHOD(89_0_nest_NE(-(x0[1], 1)), -(x0[1], 1))), ≥))
We simplified constraint (13) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(14) ((UIncreasing(200_1_NEST_INVOKEMETHOD(89_0_nest_NE(-(x0[1], 1)), -(x0[1], 1))), ≥)∧[(-1)bso_19] ≥ 0)
We simplified constraint (14) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(15) ((UIncreasing(200_1_NEST_INVOKEMETHOD(89_0_nest_NE(-(x0[1], 1)), -(x0[1], 1))), ≥)∧[(-1)bso_19] ≥ 0)
We simplified constraint (15) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(16) ((UIncreasing(200_1_NEST_INVOKEMETHOD(89_0_nest_NE(-(x0[1], 1)), -(x0[1], 1))), ≥)∧[(-1)bso_19] ≥ 0)
We simplified constraint (16) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint:
(17) ((UIncreasing(200_1_NEST_INVOKEMETHOD(89_0_nest_NE(-(x0[1], 1)), -(x0[1], 1))), ≥)∧0 = 0∧[(-1)bso_19] ≥ 0)
For Pair
COND_89_0_NEST_NE(
TRUE,
x0) →
89_0_NEST_NE(
-(
x0,
1)) the following chains were created:
- We consider the chain COND_89_0_NEST_NE(TRUE, x0[2]) → 89_0_NEST_NE(-(x0[2], 1)) which results in the following constraint:
(18) (COND_89_0_NEST_NE(TRUE, x0[2])≥NonInfC∧COND_89_0_NEST_NE(TRUE, x0[2])≥89_0_NEST_NE(-(x0[2], 1))∧(UIncreasing(89_0_NEST_NE(-(x0[2], 1))), ≥))
We simplified constraint (18) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(19) ((UIncreasing(89_0_NEST_NE(-(x0[2], 1))), ≥)∧[(-1)bso_21] ≥ 0)
We simplified constraint (19) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(20) ((UIncreasing(89_0_NEST_NE(-(x0[2], 1))), ≥)∧[(-1)bso_21] ≥ 0)
We simplified constraint (20) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(21) ((UIncreasing(89_0_NEST_NE(-(x0[2], 1))), ≥)∧[(-1)bso_21] ≥ 0)
We simplified constraint (21) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint:
(22) ((UIncreasing(89_0_NEST_NE(-(x0[2], 1))), ≥)∧0 = 0∧[(-1)bso_21] ≥ 0)
For Pair
200_1_NEST_INVOKEMETHOD(
135_0_nest_Return,
0) →
89_0_NEST_NE(
0) the following chains were created:
- We consider the chain 200_1_NEST_INVOKEMETHOD(135_0_nest_Return, 0) → 89_0_NEST_NE(0), 89_0_NEST_NE(x0[0]) → COND_89_0_NEST_NE(>(x0[0], 0), x0[0]) which results in the following constraint:
(23) (0=x0[0] ⇒ 200_1_NEST_INVOKEMETHOD(135_0_nest_Return, 0)≥NonInfC∧200_1_NEST_INVOKEMETHOD(135_0_nest_Return, 0)≥89_0_NEST_NE(0)∧(UIncreasing(89_0_NEST_NE(0)), ≥))
We simplified constraint (23) using rule (IV) which results in the following new constraint:
(24) (200_1_NEST_INVOKEMETHOD(135_0_nest_Return, 0)≥NonInfC∧200_1_NEST_INVOKEMETHOD(135_0_nest_Return, 0)≥89_0_NEST_NE(0)∧(UIncreasing(89_0_NEST_NE(0)), ≥))
We simplified constraint (24) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(25) ((UIncreasing(89_0_NEST_NE(0)), ≥)∧[(-1)bso_23] ≥ 0)
We simplified constraint (25) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(26) ((UIncreasing(89_0_NEST_NE(0)), ≥)∧[(-1)bso_23] ≥ 0)
We simplified constraint (26) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(27) ((UIncreasing(89_0_NEST_NE(0)), ≥)∧[(-1)bso_23] ≥ 0)
For Pair
200_1_NEST_INVOKEMETHOD(
320_0_nest_Return,
x1) →
89_0_NEST_NE(
0) the following chains were created:
- We consider the chain 200_1_NEST_INVOKEMETHOD(320_0_nest_Return, x1[4]) → 89_0_NEST_NE(0) which results in the following constraint:
(28) (200_1_NEST_INVOKEMETHOD(320_0_nest_Return, x1[4])≥NonInfC∧200_1_NEST_INVOKEMETHOD(320_0_nest_Return, x1[4])≥89_0_NEST_NE(0)∧(UIncreasing(89_0_NEST_NE(0)), ≥))
We simplified constraint (28) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(29) ((UIncreasing(89_0_NEST_NE(0)), ≥)∧[3 + (-1)bso_25] ≥ 0)
We simplified constraint (29) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(30) ((UIncreasing(89_0_NEST_NE(0)), ≥)∧[3 + (-1)bso_25] ≥ 0)
We simplified constraint (30) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(31) ((UIncreasing(89_0_NEST_NE(0)), ≥)∧[3 + (-1)bso_25] ≥ 0)
We simplified constraint (31) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint:
(32) ((UIncreasing(89_0_NEST_NE(0)), ≥)∧0 = 0∧[3 + (-1)bso_25] ≥ 0)
To summarize, we get the following constraints P
≥ for the following pairs.
- 89_0_NEST_NE(x0) → COND_89_0_NEST_NE(>(x0, 0), x0)
- (x0[0] ≥ 0 ⇒ (UIncreasing(COND_89_0_NEST_NE(>(x0[0], 0), x0[0])), ≥)∧[(-1)bni_16 + (-1)Bound*bni_16] ≥ 0∧[(-1)bso_17] ≥ 0)
- (x0[0] ≥ 0 ⇒ (UIncreasing(COND_89_0_NEST_NE(>(x0[0], 0), x0[0])), ≥)∧[(-1)bni_16 + (-1)Bound*bni_16] ≥ 0∧[(-1)bso_17] ≥ 0)
- COND_89_0_NEST_NE(TRUE, x0) → 200_1_NEST_INVOKEMETHOD(89_0_nest_NE(-(x0, 1)), -(x0, 1))
- ((UIncreasing(200_1_NEST_INVOKEMETHOD(89_0_nest_NE(-(x0[1], 1)), -(x0[1], 1))), ≥)∧0 = 0∧[(-1)bso_19] ≥ 0)
- COND_89_0_NEST_NE(TRUE, x0) → 89_0_NEST_NE(-(x0, 1))
- ((UIncreasing(89_0_NEST_NE(-(x0[2], 1))), ≥)∧0 = 0∧[(-1)bso_21] ≥ 0)
- 200_1_NEST_INVOKEMETHOD(135_0_nest_Return, 0) → 89_0_NEST_NE(0)
- ((UIncreasing(89_0_NEST_NE(0)), ≥)∧[(-1)bso_23] ≥ 0)
- 200_1_NEST_INVOKEMETHOD(320_0_nest_Return, x1) → 89_0_NEST_NE(0)
- ((UIncreasing(89_0_NEST_NE(0)), ≥)∧0 = 0∧[3 + (-1)bso_25] ≥ 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(89_0_nest_NE(x1)) = [2]
POL(0) = 0
POL(135_0_nest_Return) = [2]
POL(242_1_nest_InvokeMethod(x1)) = [-1]
POL(320_0_nest_Return) = [-1]
POL(89_0_NEST_NE(x1)) = [-1]
POL(COND_89_0_NEST_NE(x1, x2)) = [-1]
POL(>(x1, x2)) = [-1]
POL(200_1_NEST_INVOKEMETHOD(x1, x2)) = [1] + [-1]x1
POL(-(x1, x2)) = x1 + [-1]x2
POL(1) = [1]
The following pairs are in P
>:
200_1_NEST_INVOKEMETHOD(320_0_nest_Return, x1[4]) → 89_0_NEST_NE(0)
The following pairs are in P
bound:
89_0_NEST_NE(x0[0]) → COND_89_0_NEST_NE(>(x0[0], 0), x0[0])
The following pairs are in P
≥:
89_0_NEST_NE(x0[0]) → COND_89_0_NEST_NE(>(x0[0], 0), x0[0])
COND_89_0_NEST_NE(TRUE, x0[1]) → 200_1_NEST_INVOKEMETHOD(89_0_nest_NE(-(x0[1], 1)), -(x0[1], 1))
COND_89_0_NEST_NE(TRUE, x0[2]) → 89_0_NEST_NE(-(x0[2], 1))
200_1_NEST_INVOKEMETHOD(135_0_nest_Return, 0) → 89_0_NEST_NE(0)
At least the following rules have been oriented under context sensitive arithmetic replacement:
89_0_nest_NE(0)1 ↔ 135_0_nest_Return1
(6) Complex Obligation (AND)
(7) Obligation:
IDP problem:
The following function symbols are pre-defined:
!= | ~ | Neq: (Integer, Integer) -> Boolean |
* | ~ | Mul: (Integer, Integer) -> Integer |
>= | ~ | Ge: (Integer, Integer) -> Boolean |
-1 | ~ | UnaryMinus: (Integer) -> Integer |
| | ~ | Bwor: (Integer, Integer) -> Integer |
/ | ~ | Div: (Integer, Integer) -> Integer |
= | ~ | Eq: (Integer, Integer) -> Boolean |
| ~ | Bwxor: (Integer, Integer) -> Integer |
|| | ~ | Lor: (Boolean, Boolean) -> Boolean |
! | ~ | Lnot: (Boolean) -> Boolean |
< | ~ | Lt: (Integer, Integer) -> Boolean |
- | ~ | Sub: (Integer, Integer) -> Integer |
<= | ~ | Le: (Integer, Integer) -> Boolean |
> | ~ | Gt: (Integer, Integer) -> Boolean |
~ | ~ | Bwnot: (Integer) -> Integer |
% | ~ | Mod: (Integer, Integer) -> Integer |
& | ~ | Bwand: (Integer, Integer) -> Integer |
+ | ~ | Add: (Integer, Integer) -> Integer |
&& | ~ | Land: (Boolean, Boolean) -> Boolean |
The following domains are used:
Integer
The ITRS R consists of the following rules:
89_0_nest_NE(
0) →
135_0_nest_Return242_1_nest_InvokeMethod(
135_0_nest_Return) →
320_0_nest_ReturnThe integer pair graph contains the following rules and edges:
(0):
89_0_NEST_NE(
x0[0]) →
COND_89_0_NEST_NE(
x0[0] > 0,
x0[0])
(1):
COND_89_0_NEST_NE(
TRUE,
x0[1]) →
200_1_NEST_INVOKEMETHOD(
89_0_nest_NE(
x0[1] - 1),
x0[1] - 1)
(2):
COND_89_0_NEST_NE(
TRUE,
x0[2]) →
89_0_NEST_NE(
x0[2] - 1)
(3):
200_1_NEST_INVOKEMETHOD(
135_0_nest_Return,
0) →
89_0_NEST_NE(
0)
(2) -> (0), if ((x0[2] - 1 →* x0[0]))
(3) -> (0), if ((0 →* x0[0]))
(0) -> (1), if ((x0[0] > 0 →* TRUE)∧(x0[0] →* x0[1]))
(0) -> (2), if ((x0[0] > 0 →* TRUE)∧(x0[0] →* x0[2]))
(1) -> (3), if ((89_0_nest_NE(x0[1] - 1) →* 135_0_nest_Return)∧(x0[1] - 1 →* 0))
The set Q consists of the following terms:
89_0_nest_NE(
0)
242_1_nest_InvokeMethod(
135_0_nest_Return)
(8) 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
89_0_NEST_NE(
x0[0]) →
COND_89_0_NEST_NE(
>(
x0[0],
0),
x0[0]) the following chains were created:
- We consider the chain 89_0_NEST_NE(x0[0]) → COND_89_0_NEST_NE(>(x0[0], 0), x0[0]), COND_89_0_NEST_NE(TRUE, x0[1]) → 200_1_NEST_INVOKEMETHOD(89_0_nest_NE(-(x0[1], 1)), -(x0[1], 1)) which results in the following constraint:
(1) (>(x0[0], 0)=TRUE∧x0[0]=x0[1] ⇒ 89_0_NEST_NE(x0[0])≥NonInfC∧89_0_NEST_NE(x0[0])≥COND_89_0_NEST_NE(>(x0[0], 0), x0[0])∧(UIncreasing(COND_89_0_NEST_NE(>(x0[0], 0), x0[0])), ≥))
We simplified constraint (1) using rule (IV) which results in the following new constraint:
(2) (>(x0[0], 0)=TRUE ⇒ 89_0_NEST_NE(x0[0])≥NonInfC∧89_0_NEST_NE(x0[0])≥COND_89_0_NEST_NE(>(x0[0], 0), x0[0])∧(UIncreasing(COND_89_0_NEST_NE(>(x0[0], 0), x0[0])), ≥))
We simplified constraint (2) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(3) (x0[0] + [-1] ≥ 0 ⇒ (UIncreasing(COND_89_0_NEST_NE(>(x0[0], 0), x0[0])), ≥)∧[(-1)bni_11 + (-1)Bound*bni_11] + [bni_11]x0[0] ≥ 0∧[(-1)bso_12] ≥ 0)
We simplified constraint (3) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(4) (x0[0] + [-1] ≥ 0 ⇒ (UIncreasing(COND_89_0_NEST_NE(>(x0[0], 0), x0[0])), ≥)∧[(-1)bni_11 + (-1)Bound*bni_11] + [bni_11]x0[0] ≥ 0∧[(-1)bso_12] ≥ 0)
We simplified constraint (4) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(5) (x0[0] + [-1] ≥ 0 ⇒ (UIncreasing(COND_89_0_NEST_NE(>(x0[0], 0), x0[0])), ≥)∧[(-1)bni_11 + (-1)Bound*bni_11] + [bni_11]x0[0] ≥ 0∧[(-1)bso_12] ≥ 0)
We simplified constraint (5) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(6) (x0[0] ≥ 0 ⇒ (UIncreasing(COND_89_0_NEST_NE(>(x0[0], 0), x0[0])), ≥)∧[(-1)Bound*bni_11] + [bni_11]x0[0] ≥ 0∧[(-1)bso_12] ≥ 0)
- We consider the chain 89_0_NEST_NE(x0[0]) → COND_89_0_NEST_NE(>(x0[0], 0), x0[0]), COND_89_0_NEST_NE(TRUE, x0[2]) → 89_0_NEST_NE(-(x0[2], 1)) which results in the following constraint:
(7) (>(x0[0], 0)=TRUE∧x0[0]=x0[2] ⇒ 89_0_NEST_NE(x0[0])≥NonInfC∧89_0_NEST_NE(x0[0])≥COND_89_0_NEST_NE(>(x0[0], 0), x0[0])∧(UIncreasing(COND_89_0_NEST_NE(>(x0[0], 0), x0[0])), ≥))
We simplified constraint (7) using rule (IV) which results in the following new constraint:
(8) (>(x0[0], 0)=TRUE ⇒ 89_0_NEST_NE(x0[0])≥NonInfC∧89_0_NEST_NE(x0[0])≥COND_89_0_NEST_NE(>(x0[0], 0), x0[0])∧(UIncreasing(COND_89_0_NEST_NE(>(x0[0], 0), x0[0])), ≥))
We simplified constraint (8) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(9) (x0[0] + [-1] ≥ 0 ⇒ (UIncreasing(COND_89_0_NEST_NE(>(x0[0], 0), x0[0])), ≥)∧[(-1)bni_11 + (-1)Bound*bni_11] + [bni_11]x0[0] ≥ 0∧[(-1)bso_12] ≥ 0)
We simplified constraint (9) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(10) (x0[0] + [-1] ≥ 0 ⇒ (UIncreasing(COND_89_0_NEST_NE(>(x0[0], 0), x0[0])), ≥)∧[(-1)bni_11 + (-1)Bound*bni_11] + [bni_11]x0[0] ≥ 0∧[(-1)bso_12] ≥ 0)
We simplified constraint (10) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(11) (x0[0] + [-1] ≥ 0 ⇒ (UIncreasing(COND_89_0_NEST_NE(>(x0[0], 0), x0[0])), ≥)∧[(-1)bni_11 + (-1)Bound*bni_11] + [bni_11]x0[0] ≥ 0∧[(-1)bso_12] ≥ 0)
We simplified constraint (11) using rule (IDP_SMT_SPLIT) which results in the following new constraint:
(12) (x0[0] ≥ 0 ⇒ (UIncreasing(COND_89_0_NEST_NE(>(x0[0], 0), x0[0])), ≥)∧[(-1)Bound*bni_11] + [bni_11]x0[0] ≥ 0∧[(-1)bso_12] ≥ 0)
For Pair
COND_89_0_NEST_NE(
TRUE,
x0[1]) →
200_1_NEST_INVOKEMETHOD(
89_0_nest_NE(
-(
x0[1],
1)),
-(
x0[1],
1)) the following chains were created:
- We consider the chain COND_89_0_NEST_NE(TRUE, x0[1]) → 200_1_NEST_INVOKEMETHOD(89_0_nest_NE(-(x0[1], 1)), -(x0[1], 1)) which results in the following constraint:
(13) (COND_89_0_NEST_NE(TRUE, x0[1])≥NonInfC∧COND_89_0_NEST_NE(TRUE, x0[1])≥200_1_NEST_INVOKEMETHOD(89_0_nest_NE(-(x0[1], 1)), -(x0[1], 1))∧(UIncreasing(200_1_NEST_INVOKEMETHOD(89_0_nest_NE(-(x0[1], 1)), -(x0[1], 1))), ≥))
We simplified constraint (13) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(14) ((UIncreasing(200_1_NEST_INVOKEMETHOD(89_0_nest_NE(-(x0[1], 1)), -(x0[1], 1))), ≥)∧[(-1)bso_14] ≥ 0)
We simplified constraint (14) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(15) ((UIncreasing(200_1_NEST_INVOKEMETHOD(89_0_nest_NE(-(x0[1], 1)), -(x0[1], 1))), ≥)∧[(-1)bso_14] ≥ 0)
We simplified constraint (15) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(16) ((UIncreasing(200_1_NEST_INVOKEMETHOD(89_0_nest_NE(-(x0[1], 1)), -(x0[1], 1))), ≥)∧[(-1)bso_14] ≥ 0)
We simplified constraint (16) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint:
(17) ((UIncreasing(200_1_NEST_INVOKEMETHOD(89_0_nest_NE(-(x0[1], 1)), -(x0[1], 1))), ≥)∧0 = 0∧[(-1)bso_14] ≥ 0)
For Pair
COND_89_0_NEST_NE(
TRUE,
x0[2]) →
89_0_NEST_NE(
-(
x0[2],
1)) the following chains were created:
- We consider the chain COND_89_0_NEST_NE(TRUE, x0[2]) → 89_0_NEST_NE(-(x0[2], 1)) which results in the following constraint:
(18) (COND_89_0_NEST_NE(TRUE, x0[2])≥NonInfC∧COND_89_0_NEST_NE(TRUE, x0[2])≥89_0_NEST_NE(-(x0[2], 1))∧(UIncreasing(89_0_NEST_NE(-(x0[2], 1))), ≥))
We simplified constraint (18) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(19) ((UIncreasing(89_0_NEST_NE(-(x0[2], 1))), ≥)∧[1 + (-1)bso_16] ≥ 0)
We simplified constraint (19) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(20) ((UIncreasing(89_0_NEST_NE(-(x0[2], 1))), ≥)∧[1 + (-1)bso_16] ≥ 0)
We simplified constraint (20) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(21) ((UIncreasing(89_0_NEST_NE(-(x0[2], 1))), ≥)∧[1 + (-1)bso_16] ≥ 0)
We simplified constraint (21) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint:
(22) ((UIncreasing(89_0_NEST_NE(-(x0[2], 1))), ≥)∧0 = 0∧[1 + (-1)bso_16] ≥ 0)
For Pair
200_1_NEST_INVOKEMETHOD(
135_0_nest_Return,
0) →
89_0_NEST_NE(
0) the following chains were created:
- We consider the chain 200_1_NEST_INVOKEMETHOD(135_0_nest_Return, 0) → 89_0_NEST_NE(0), 89_0_NEST_NE(x0[0]) → COND_89_0_NEST_NE(>(x0[0], 0), x0[0]) which results in the following constraint:
(23) (0=x0[0] ⇒ 200_1_NEST_INVOKEMETHOD(135_0_nest_Return, 0)≥NonInfC∧200_1_NEST_INVOKEMETHOD(135_0_nest_Return, 0)≥89_0_NEST_NE(0)∧(UIncreasing(89_0_NEST_NE(0)), ≥))
We simplified constraint (23) using rule (IV) which results in the following new constraint:
(24) (200_1_NEST_INVOKEMETHOD(135_0_nest_Return, 0)≥NonInfC∧200_1_NEST_INVOKEMETHOD(135_0_nest_Return, 0)≥89_0_NEST_NE(0)∧(UIncreasing(89_0_NEST_NE(0)), ≥))
We simplified constraint (24) using rule (POLY_CONSTRAINTS) which results in the following new constraint:
(25) ((UIncreasing(89_0_NEST_NE(0)), ≥)∧[1 + (-1)bso_18] ≥ 0)
We simplified constraint (25) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:
(26) ((UIncreasing(89_0_NEST_NE(0)), ≥)∧[1 + (-1)bso_18] ≥ 0)
We simplified constraint (26) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:
(27) ((UIncreasing(89_0_NEST_NE(0)), ≥)∧[1 + (-1)bso_18] ≥ 0)
To summarize, we get the following constraints P
≥ for the following pairs.
- 89_0_NEST_NE(x0[0]) → COND_89_0_NEST_NE(>(x0[0], 0), x0[0])
- (x0[0] ≥ 0 ⇒ (UIncreasing(COND_89_0_NEST_NE(>(x0[0], 0), x0[0])), ≥)∧[(-1)Bound*bni_11] + [bni_11]x0[0] ≥ 0∧[(-1)bso_12] ≥ 0)
- (x0[0] ≥ 0 ⇒ (UIncreasing(COND_89_0_NEST_NE(>(x0[0], 0), x0[0])), ≥)∧[(-1)Bound*bni_11] + [bni_11]x0[0] ≥ 0∧[(-1)bso_12] ≥ 0)
- COND_89_0_NEST_NE(TRUE, x0[1]) → 200_1_NEST_INVOKEMETHOD(89_0_nest_NE(-(x0[1], 1)), -(x0[1], 1))
- ((UIncreasing(200_1_NEST_INVOKEMETHOD(89_0_nest_NE(-(x0[1], 1)), -(x0[1], 1))), ≥)∧0 = 0∧[(-1)bso_14] ≥ 0)
- COND_89_0_NEST_NE(TRUE, x0[2]) → 89_0_NEST_NE(-(x0[2], 1))
- ((UIncreasing(89_0_NEST_NE(-(x0[2], 1))), ≥)∧0 = 0∧[1 + (-1)bso_16] ≥ 0)
- 200_1_NEST_INVOKEMETHOD(135_0_nest_Return, 0) → 89_0_NEST_NE(0)
- ((UIncreasing(89_0_NEST_NE(0)), ≥)∧[1 + (-1)bso_18] ≥ 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(89_0_nest_NE(x1)) = [-1] + [-1]x1
POL(0) = 0
POL(135_0_nest_Return) = [-1]
POL(242_1_nest_InvokeMethod(x1)) = [-1]
POL(320_0_nest_Return) = [-1]
POL(89_0_NEST_NE(x1)) = [-1] + x1
POL(COND_89_0_NEST_NE(x1, x2)) = [-1] + x2
POL(>(x1, x2)) = [1]
POL(200_1_NEST_INVOKEMETHOD(x1, x2)) = [-1] + [-1]x1
POL(-(x1, x2)) = x1 + [-1]x2
POL(1) = [1]
The following pairs are in P
>:
COND_89_0_NEST_NE(TRUE, x0[2]) → 89_0_NEST_NE(-(x0[2], 1))
200_1_NEST_INVOKEMETHOD(135_0_nest_Return, 0) → 89_0_NEST_NE(0)
The following pairs are in P
bound:
89_0_NEST_NE(x0[0]) → COND_89_0_NEST_NE(>(x0[0], 0), x0[0])
The following pairs are in P
≥:
89_0_NEST_NE(x0[0]) → COND_89_0_NEST_NE(>(x0[0], 0), x0[0])
COND_89_0_NEST_NE(TRUE, x0[1]) → 200_1_NEST_INVOKEMETHOD(89_0_nest_NE(-(x0[1], 1)), -(x0[1], 1))
At least the following rules have been oriented under context sensitive arithmetic replacement:
89_0_nest_NE(0)1 ↔ 135_0_nest_Return1
(9) Complex Obligation (AND)
(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
The ITRS R consists of the following rules:
89_0_nest_NE(
0) →
135_0_nest_Return242_1_nest_InvokeMethod(
135_0_nest_Return) →
320_0_nest_ReturnThe integer pair graph contains the following rules and edges:
(0):
89_0_NEST_NE(
x0[0]) →
COND_89_0_NEST_NE(
x0[0] > 0,
x0[0])
(1):
COND_89_0_NEST_NE(
TRUE,
x0[1]) →
200_1_NEST_INVOKEMETHOD(
89_0_nest_NE(
x0[1] - 1),
x0[1] - 1)
(0) -> (1), if ((x0[0] > 0 →* TRUE)∧(x0[0] →* x0[1]))
The set Q consists of the following terms:
89_0_nest_NE(
0)
242_1_nest_InvokeMethod(
135_0_nest_Return)
(11) IDependencyGraphProof (EQUIVALENT transformation)
The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 2 less nodes.
(12) TRUE
(13) 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
The ITRS R consists of the following rules:
89_0_nest_NE(
0) →
135_0_nest_Return242_1_nest_InvokeMethod(
135_0_nest_Return) →
320_0_nest_ReturnThe integer pair graph contains the following rules and edges:
(1):
COND_89_0_NEST_NE(
TRUE,
x0[1]) →
200_1_NEST_INVOKEMETHOD(
89_0_nest_NE(
x0[1] - 1),
x0[1] - 1)
(2):
COND_89_0_NEST_NE(
TRUE,
x0[2]) →
89_0_NEST_NE(
x0[2] - 1)
(3):
200_1_NEST_INVOKEMETHOD(
135_0_nest_Return,
0) →
89_0_NEST_NE(
0)
(1) -> (3), if ((89_0_nest_NE(x0[1] - 1) →* 135_0_nest_Return)∧(x0[1] - 1 →* 0))
The set Q consists of the following terms:
89_0_nest_NE(
0)
242_1_nest_InvokeMethod(
135_0_nest_Return)
(14) IDependencyGraphProof (EQUIVALENT transformation)
The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 3 less nodes.
(15) TRUE
(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
The ITRS R consists of the following rules:
89_0_nest_NE(
0) →
135_0_nest_Return242_1_nest_InvokeMethod(
135_0_nest_Return) →
320_0_nest_ReturnThe integer pair graph contains the following rules and edges:
(1):
COND_89_0_NEST_NE(
TRUE,
x0[1]) →
200_1_NEST_INVOKEMETHOD(
89_0_nest_NE(
x0[1] - 1),
x0[1] - 1)
(2):
COND_89_0_NEST_NE(
TRUE,
x0[2]) →
89_0_NEST_NE(
x0[2] - 1)
(3):
200_1_NEST_INVOKEMETHOD(
135_0_nest_Return,
0) →
89_0_NEST_NE(
0)
(4):
200_1_NEST_INVOKEMETHOD(
320_0_nest_Return,
x1[4]) →
89_0_NEST_NE(
0)
(1) -> (3), if ((89_0_nest_NE(x0[1] - 1) →* 135_0_nest_Return)∧(x0[1] - 1 →* 0))
(1) -> (4), if ((89_0_nest_NE(x0[1] - 1) →* 320_0_nest_Return)∧(x0[1] - 1 →* x1[4]))
The set Q consists of the following terms:
89_0_nest_NE(
0)
242_1_nest_InvokeMethod(
135_0_nest_Return)
(17) IDependencyGraphProof (EQUIVALENT transformation)
The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 4 less nodes.
(18) TRUE