### (0) Obligation:

JBC Problem based on JBC Program:
Manifest-Version: 1.0 Created-By: 1.6.0_22 (Sun Microsystems Inc.) Main-Class: TerminatorRec02
`public class TerminatorRec02 {	public static void main(String[] args) {		fact(args.length);	}	public static int fact(int x) {		if (x > 1) {			int y = fact(x - 1);			return y * x;		}		return 1;	}}`

### (1) JBC2FIG (SOUND transformation)

Constructed FIGraph.

### (2) Obligation:

FIGraph based on JBC Program:
TerminatorRec02.main([Ljava/lang/String;)V: Graph of 15 nodes with 0 SCCs.

TerminatorRec02.fact(I)I: Graph of 27 nodes with 0 SCCs.

### (3) FIGtoITRSProof (SOUND transformation)

Transformed FIGraph SCCs to IDPs. Logs:

Log for SCC 0:

Generated 10 rules for P and 16 rules for R.

Combined rules. Obtained 1 rules for P and 2 rules for R.

Filtered ground terms:

26_0_fact_ConstantStackPush(x1, x2, x3) → 26_0_fact_ConstantStackPush(x2, x3)
Cond_26_0_fact_ConstantStackPush(x1, x2, x3, x4) → Cond_26_0_fact_ConstantStackPush(x1, x3, x4)
75_0_fact_Return(x1) → 75_0_fact_Return
Cond_57_1_fact_InvokeMethod1(x1, x2, x3, x4) → Cond_57_1_fact_InvokeMethod1(x1, x3, x4)
Cond_57_1_fact_InvokeMethod(x1, x2, x3, x4) → Cond_57_1_fact_InvokeMethod(x1, x3)
37_0_fact_Return(x1, x2) → 37_0_fact_Return

Filtered duplicate args:

26_0_fact_ConstantStackPush(x1, x2) → 26_0_fact_ConstantStackPush(x2)
Cond_26_0_fact_ConstantStackPush(x1, x2, x3) → Cond_26_0_fact_ConstantStackPush(x1, x3)

Filtered unneeded arguments:

Cond_57_1_fact_InvokeMethod(x1, x2) → Cond_57_1_fact_InvokeMethod(x1)
Cond_57_1_fact_InvokeMethod1(x1, x2, x3) → Cond_57_1_fact_InvokeMethod1(x1)

Combined rules. Obtained 1 rules for P and 2 rules for R.

Finished conversion. Obtained 1 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:
57_1_fact_InvokeMethod(37_0_fact_Return, x1, 1) → Cond_57_1_fact_InvokeMethod(x1 > 1, 37_0_fact_Return, x1, 1)
Cond_57_1_fact_InvokeMethod(TRUE, 37_0_fact_Return, x1, 1) → 75_0_fact_Return
57_1_fact_InvokeMethod(75_0_fact_Return, x0, x1) → Cond_57_1_fact_InvokeMethod1(x0 > 1, 75_0_fact_Return, x0, x1)
Cond_57_1_fact_InvokeMethod1(TRUE, 75_0_fact_Return, x0, x1) → 75_0_fact_Return

The integer pair graph contains the following rules and edges:
(0): 26_0_FACT_CONSTANTSTACKPUSH(x0[0]) → COND_26_0_FACT_CONSTANTSTACKPUSH(x0[0] > 1, x0[0])
(1): COND_26_0_FACT_CONSTANTSTACKPUSH(TRUE, x0[1]) → 26_0_FACT_CONSTANTSTACKPUSH(x0[1] - 1)

(0) -> (1), if ((x0[0] > 1* TRUE)∧(x0[0]* x0[1]))

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

The set Q consists of the following terms:
57_1_fact_InvokeMethod(37_0_fact_Return, x0, 1)
Cond_57_1_fact_InvokeMethod(TRUE, 37_0_fact_Return, x0, 1)
57_1_fact_InvokeMethod(75_0_fact_Return, x0, x1)
Cond_57_1_fact_InvokeMethod1(TRUE, 75_0_fact_Return, x0, x1)

### (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 26_0_FACT_CONSTANTSTACKPUSH(x0) → COND_26_0_FACT_CONSTANTSTACKPUSH(>(x0, 1), x0) the following chains were created:
• We consider the chain 26_0_FACT_CONSTANTSTACKPUSH(x0[0]) → COND_26_0_FACT_CONSTANTSTACKPUSH(>(x0[0], 1), x0[0]), COND_26_0_FACT_CONSTANTSTACKPUSH(TRUE, x0[1]) → 26_0_FACT_CONSTANTSTACKPUSH(-(x0[1], 1)) which results in the following constraint:

(1)    (>(x0[0], 1)=TRUEx0[0]=x0[1]26_0_FACT_CONSTANTSTACKPUSH(x0[0])≥NonInfC∧26_0_FACT_CONSTANTSTACKPUSH(x0[0])≥COND_26_0_FACT_CONSTANTSTACKPUSH(>(x0[0], 1), x0[0])∧(UIncreasing(COND_26_0_FACT_CONSTANTSTACKPUSH(>(x0[0], 1), x0[0])), ≥))

We simplified constraint (1) using rule (IV) which results in the following new constraint:

(2)    (>(x0[0], 1)=TRUE26_0_FACT_CONSTANTSTACKPUSH(x0[0])≥NonInfC∧26_0_FACT_CONSTANTSTACKPUSH(x0[0])≥COND_26_0_FACT_CONSTANTSTACKPUSH(>(x0[0], 1), x0[0])∧(UIncreasing(COND_26_0_FACT_CONSTANTSTACKPUSH(>(x0[0], 1), x0[0])), ≥))

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

(3)    (x0[0] + [-2] ≥ 0 ⇒ (UIncreasing(COND_26_0_FACT_CONSTANTSTACKPUSH(>(x0[0], 1), x0[0])), ≥)∧[(-1)Bound*bni_16] + [(2)bni_16]x0[0] ≥ 0∧[(-1)bso_17] ≥ 0)

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

(4)    (x0[0] + [-2] ≥ 0 ⇒ (UIncreasing(COND_26_0_FACT_CONSTANTSTACKPUSH(>(x0[0], 1), x0[0])), ≥)∧[(-1)Bound*bni_16] + [(2)bni_16]x0[0] ≥ 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] + [-2] ≥ 0 ⇒ (UIncreasing(COND_26_0_FACT_CONSTANTSTACKPUSH(>(x0[0], 1), x0[0])), ≥)∧[(-1)Bound*bni_16] + [(2)bni_16]x0[0] ≥ 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_26_0_FACT_CONSTANTSTACKPUSH(>(x0[0], 1), x0[0])), ≥)∧[(-1)Bound*bni_16 + (4)bni_16] + [(2)bni_16]x0[0] ≥ 0∧[(-1)bso_17] ≥ 0)

For Pair COND_26_0_FACT_CONSTANTSTACKPUSH(TRUE, x0) → 26_0_FACT_CONSTANTSTACKPUSH(-(x0, 1)) the following chains were created:
• We consider the chain COND_26_0_FACT_CONSTANTSTACKPUSH(TRUE, x0[1]) → 26_0_FACT_CONSTANTSTACKPUSH(-(x0[1], 1)) which results in the following constraint:

(7)    (COND_26_0_FACT_CONSTANTSTACKPUSH(TRUE, x0[1])≥NonInfC∧COND_26_0_FACT_CONSTANTSTACKPUSH(TRUE, x0[1])≥26_0_FACT_CONSTANTSTACKPUSH(-(x0[1], 1))∧(UIncreasing(26_0_FACT_CONSTANTSTACKPUSH(-(x0[1], 1))), ≥))

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

(8)    ((UIncreasing(26_0_FACT_CONSTANTSTACKPUSH(-(x0[1], 1))), ≥)∧[2 + (-1)bso_19] ≥ 0)

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

(9)    ((UIncreasing(26_0_FACT_CONSTANTSTACKPUSH(-(x0[1], 1))), ≥)∧[2 + (-1)bso_19] ≥ 0)

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

(10)    ((UIncreasing(26_0_FACT_CONSTANTSTACKPUSH(-(x0[1], 1))), ≥)∧[2 + (-1)bso_19] ≥ 0)

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

(11)    ((UIncreasing(26_0_FACT_CONSTANTSTACKPUSH(-(x0[1], 1))), ≥)∧0 = 0∧[2 + (-1)bso_19] ≥ 0)

To summarize, we get the following constraints P for the following pairs.
• 26_0_FACT_CONSTANTSTACKPUSH(x0) → COND_26_0_FACT_CONSTANTSTACKPUSH(>(x0, 1), x0)
• (x0[0] ≥ 0 ⇒ (UIncreasing(COND_26_0_FACT_CONSTANTSTACKPUSH(>(x0[0], 1), x0[0])), ≥)∧[(-1)Bound*bni_16 + (4)bni_16] + [(2)bni_16]x0[0] ≥ 0∧[(-1)bso_17] ≥ 0)

• COND_26_0_FACT_CONSTANTSTACKPUSH(TRUE, x0) → 26_0_FACT_CONSTANTSTACKPUSH(-(x0, 1))
• ((UIncreasing(26_0_FACT_CONSTANTSTACKPUSH(-(x0[1], 1))), ≥)∧0 = 0∧[2 + (-1)bso_19] ≥ 0)

The constraints for P> respective Pbound are constructed from P where we just replace every occurence of "t ≥ s" in P by "t > s" respective "t ≥ c". Here c stands for the fresh constant used for Pbound.
Using the following integer polynomial ordering the resulting constraints can be solved
Polynomial interpretation over integers[POLO]:

POL(TRUE) = 0
POL(FALSE) = 0
POL(57_1_fact_InvokeMethod(x1, x2, x3)) = [-1] + [-1]x2
POL(37_0_fact_Return) = [-1]
POL(1) = [1]
POL(Cond_57_1_fact_InvokeMethod(x1, x2, x3, x4)) = [-1] + [-1]x3
POL(>(x1, x2)) = [-1]
POL(75_0_fact_Return) = [-1]
POL(Cond_57_1_fact_InvokeMethod1(x1, x2, x3, x4)) = [-1] + [-1]x3
POL(26_0_FACT_CONSTANTSTACKPUSH(x1)) = [2]x1
POL(COND_26_0_FACT_CONSTANTSTACKPUSH(x1, x2)) = [2]x2
POL(-(x1, x2)) = x1 + [-1]x2

The following pairs are in P>:

COND_26_0_FACT_CONSTANTSTACKPUSH(TRUE, x0[1]) → 26_0_FACT_CONSTANTSTACKPUSH(-(x0[1], 1))

The following pairs are in Pbound:

26_0_FACT_CONSTANTSTACKPUSH(x0[0]) → COND_26_0_FACT_CONSTANTSTACKPUSH(>(x0[0], 1), x0[0])

The following pairs are in P:

26_0_FACT_CONSTANTSTACKPUSH(x0[0]) → COND_26_0_FACT_CONSTANTSTACKPUSH(>(x0[0], 1), x0[0])

There are no usable rules.

### (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:
57_1_fact_InvokeMethod(37_0_fact_Return, x1, 1) → Cond_57_1_fact_InvokeMethod(x1 > 1, 37_0_fact_Return, x1, 1)
Cond_57_1_fact_InvokeMethod(TRUE, 37_0_fact_Return, x1, 1) → 75_0_fact_Return
57_1_fact_InvokeMethod(75_0_fact_Return, x0, x1) → Cond_57_1_fact_InvokeMethod1(x0 > 1, 75_0_fact_Return, x0, x1)
Cond_57_1_fact_InvokeMethod1(TRUE, 75_0_fact_Return, x0, x1) → 75_0_fact_Return

The integer pair graph contains the following rules and edges:
(0): 26_0_FACT_CONSTANTSTACKPUSH(x0[0]) → COND_26_0_FACT_CONSTANTSTACKPUSH(x0[0] > 1, x0[0])

The set Q consists of the following terms:
57_1_fact_InvokeMethod(37_0_fact_Return, x0, 1)
Cond_57_1_fact_InvokeMethod(TRUE, 37_0_fact_Return, x0, 1)
57_1_fact_InvokeMethod(75_0_fact_Return, x0, x1)
Cond_57_1_fact_InvokeMethod1(TRUE, 75_0_fact_Return, x0, x1)

### (8) IDependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs 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

The ITRS R consists of the following rules:
57_1_fact_InvokeMethod(37_0_fact_Return, x1, 1) → Cond_57_1_fact_InvokeMethod(x1 > 1, 37_0_fact_Return, x1, 1)
Cond_57_1_fact_InvokeMethod(TRUE, 37_0_fact_Return, x1, 1) → 75_0_fact_Return
57_1_fact_InvokeMethod(75_0_fact_Return, x0, x1) → Cond_57_1_fact_InvokeMethod1(x0 > 1, 75_0_fact_Return, x0, x1)
Cond_57_1_fact_InvokeMethod1(TRUE, 75_0_fact_Return, x0, x1) → 75_0_fact_Return

The integer pair graph contains the following rules and edges:
(1): COND_26_0_FACT_CONSTANTSTACKPUSH(TRUE, x0[1]) → 26_0_FACT_CONSTANTSTACKPUSH(x0[1] - 1)

The set Q consists of the following terms:
57_1_fact_InvokeMethod(37_0_fact_Return, x0, 1)
Cond_57_1_fact_InvokeMethod(TRUE, 37_0_fact_Return, x0, 1)
57_1_fact_InvokeMethod(75_0_fact_Return, x0, x1)
Cond_57_1_fact_InvokeMethod1(TRUE, 75_0_fact_Return, x0, x1)

### (11) IDependencyGraphProof (EQUIVALENT transformation)

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