### (0) Obligation:

JBC Problem based on JBC Program:
Manifest-Version: 1.0 Created-By: 1.6.0_16 (Sun Microsystems Inc.) Main-Class: Double3
`/** * A recursive loop. * * All calls terminate. * * Julia + BinTerm prove that all calls terminate. * * @author <A HREF="mailto:fausto.spoto@univr.it">Fausto Spoto</A> */public class Double3 {    private static void test(int n) {	while (--n > 0) test(n);    }    public static void main(String[] args) {	test(10);    }}`

### (1) JBC2FIG (SOUND transformation)

Constructed FIGraph.

### (2) Obligation:

FIGraph based on JBC Program:
Double3.main([Ljava/lang/String;)V: Graph of 10 nodes with 0 SCCs.

Double3.test(I)V: Graph of 15 nodes with 1 SCC.

### (3) FIGtoITRSProof (SOUND transformation)

Transformed FIGraph SCCs to IDPs. Logs:

Log for SCC 0:

Generated 13 rules for P and 2 rules for R.

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

Filtered ground terms:

150_0_test_Inc(x1, x2) → 150_0_test_Inc(x2)
158_0_test_Return(x1) → 158_0_test_Return
Cond_150_0_test_Inc(x1, x2, x3) → Cond_150_0_test_Inc(x1, x3)

Filtered duplicate args:

167_1_test_InvokeMethod(x1, x2, x3) → 167_1_test_InvokeMethod(x1, x3)

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

Finished conversion. Obtained 2 rules for P and 0 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

R is empty.

The integer pair graph contains the following rules and edges:
(0): 150_0_TEST_INC(x0[0]) → COND_150_0_TEST_INC(0 < x0[0] + -1, x0[0])
(1): COND_150_0_TEST_INC(TRUE, x0[1]) → 167_1_TEST_INVOKEMETHOD(150_0_test_Inc(x0[1] + -1), x0[1] + -1)
(2): COND_150_0_TEST_INC(TRUE, x0[2]) → 150_0_TEST_INC(x0[2] + -1)
(3): 167_1_TEST_INVOKEMETHOD(158_0_test_Return, x0[3]) → 150_0_TEST_INC(x0[3])

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

(0) -> (2), if ((0 < x0[0] + -1* TRUE)∧(x0[0]* x0[2]))

(1) -> (3), if ((150_0_test_Inc(x0[1] + -1) →* 158_0_test_Return)∧(x0[1] + -1* x0[3]))

(2) -> (0), if ((x0[2] + -1* x0[0]))

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

The set Q is empty.

### (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 150_0_TEST_INC(x0) → COND_150_0_TEST_INC(<(0, +(x0, -1)), x0) the following chains were created:
• We consider the chain 150_0_TEST_INC(x0[0]) → COND_150_0_TEST_INC(<(0, +(x0[0], -1)), x0[0]), COND_150_0_TEST_INC(TRUE, x0[1]) → 167_1_TEST_INVOKEMETHOD(150_0_test_Inc(+(x0[1], -1)), +(x0[1], -1)) which results in the following constraint:

(1)    (<(0, +(x0[0], -1))=TRUEx0[0]=x0[1]150_0_TEST_INC(x0[0])≥NonInfC∧150_0_TEST_INC(x0[0])≥COND_150_0_TEST_INC(<(0, +(x0[0], -1)), x0[0])∧(UIncreasing(COND_150_0_TEST_INC(<(0, +(x0[0], -1)), x0[0])), ≥))

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

(2)    (<(0, +(x0[0], -1))=TRUE150_0_TEST_INC(x0[0])≥NonInfC∧150_0_TEST_INC(x0[0])≥COND_150_0_TEST_INC(<(0, +(x0[0], -1)), x0[0])∧(UIncreasing(COND_150_0_TEST_INC(<(0, +(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_150_0_TEST_INC(<(0, +(x0[0], -1)), x0[0])), ≥)∧[(-1)Bound*bni_12] + [(2)bni_12]x0[0] ≥ 0∧[(-1)bso_13] ≥ 0)

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

(4)    (x0[0] + [-2] ≥ 0 ⇒ (UIncreasing(COND_150_0_TEST_INC(<(0, +(x0[0], -1)), x0[0])), ≥)∧[(-1)Bound*bni_12] + [(2)bni_12]x0[0] ≥ 0∧[(-1)bso_13] ≥ 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_150_0_TEST_INC(<(0, +(x0[0], -1)), x0[0])), ≥)∧[(-1)Bound*bni_12] + [(2)bni_12]x0[0] ≥ 0∧[(-1)bso_13] ≥ 0)

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

(6)    (x0[0] ≥ 0 ⇒ (UIncreasing(COND_150_0_TEST_INC(<(0, +(x0[0], -1)), x0[0])), ≥)∧[(-1)Bound*bni_12 + (4)bni_12] + [(2)bni_12]x0[0] ≥ 0∧[(-1)bso_13] ≥ 0)

• We consider the chain 150_0_TEST_INC(x0[0]) → COND_150_0_TEST_INC(<(0, +(x0[0], -1)), x0[0]), COND_150_0_TEST_INC(TRUE, x0[2]) → 150_0_TEST_INC(+(x0[2], -1)) which results in the following constraint:

(7)    (<(0, +(x0[0], -1))=TRUEx0[0]=x0[2]150_0_TEST_INC(x0[0])≥NonInfC∧150_0_TEST_INC(x0[0])≥COND_150_0_TEST_INC(<(0, +(x0[0], -1)), x0[0])∧(UIncreasing(COND_150_0_TEST_INC(<(0, +(x0[0], -1)), x0[0])), ≥))

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

(8)    (<(0, +(x0[0], -1))=TRUE150_0_TEST_INC(x0[0])≥NonInfC∧150_0_TEST_INC(x0[0])≥COND_150_0_TEST_INC(<(0, +(x0[0], -1)), x0[0])∧(UIncreasing(COND_150_0_TEST_INC(<(0, +(x0[0], -1)), x0[0])), ≥))

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

(9)    (x0[0] + [-2] ≥ 0 ⇒ (UIncreasing(COND_150_0_TEST_INC(<(0, +(x0[0], -1)), x0[0])), ≥)∧[(-1)Bound*bni_12] + [(2)bni_12]x0[0] ≥ 0∧[(-1)bso_13] ≥ 0)

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

(10)    (x0[0] + [-2] ≥ 0 ⇒ (UIncreasing(COND_150_0_TEST_INC(<(0, +(x0[0], -1)), x0[0])), ≥)∧[(-1)Bound*bni_12] + [(2)bni_12]x0[0] ≥ 0∧[(-1)bso_13] ≥ 0)

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

(11)    (x0[0] + [-2] ≥ 0 ⇒ (UIncreasing(COND_150_0_TEST_INC(<(0, +(x0[0], -1)), x0[0])), ≥)∧[(-1)Bound*bni_12] + [(2)bni_12]x0[0] ≥ 0∧[(-1)bso_13] ≥ 0)

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

(12)    (x0[0] ≥ 0 ⇒ (UIncreasing(COND_150_0_TEST_INC(<(0, +(x0[0], -1)), x0[0])), ≥)∧[(-1)Bound*bni_12 + (4)bni_12] + [(2)bni_12]x0[0] ≥ 0∧[(-1)bso_13] ≥ 0)

For Pair COND_150_0_TEST_INC(TRUE, x0) → 167_1_TEST_INVOKEMETHOD(150_0_test_Inc(+(x0, -1)), +(x0, -1)) the following chains were created:
• We consider the chain COND_150_0_TEST_INC(TRUE, x0[1]) → 167_1_TEST_INVOKEMETHOD(150_0_test_Inc(+(x0[1], -1)), +(x0[1], -1)) which results in the following constraint:

(13)    (COND_150_0_TEST_INC(TRUE, x0[1])≥NonInfC∧COND_150_0_TEST_INC(TRUE, x0[1])≥167_1_TEST_INVOKEMETHOD(150_0_test_Inc(+(x0[1], -1)), +(x0[1], -1))∧(UIncreasing(167_1_TEST_INVOKEMETHOD(150_0_test_Inc(+(x0[1], -1)), +(x0[1], -1))), ≥))

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

(14)    ((UIncreasing(167_1_TEST_INVOKEMETHOD(150_0_test_Inc(+(x0[1], -1)), +(x0[1], -1))), ≥)∧[(-1)bso_15] ≥ 0)

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

(15)    ((UIncreasing(167_1_TEST_INVOKEMETHOD(150_0_test_Inc(+(x0[1], -1)), +(x0[1], -1))), ≥)∧[(-1)bso_15] ≥ 0)

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

(16)    ((UIncreasing(167_1_TEST_INVOKEMETHOD(150_0_test_Inc(+(x0[1], -1)), +(x0[1], -1))), ≥)∧[(-1)bso_15] ≥ 0)

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

(17)    ((UIncreasing(167_1_TEST_INVOKEMETHOD(150_0_test_Inc(+(x0[1], -1)), +(x0[1], -1))), ≥)∧0 = 0∧[(-1)bso_15] ≥ 0)

For Pair COND_150_0_TEST_INC(TRUE, x0) → 150_0_TEST_INC(+(x0, -1)) the following chains were created:
• We consider the chain COND_150_0_TEST_INC(TRUE, x0[2]) → 150_0_TEST_INC(+(x0[2], -1)) which results in the following constraint:

(18)    (COND_150_0_TEST_INC(TRUE, x0[2])≥NonInfC∧COND_150_0_TEST_INC(TRUE, x0[2])≥150_0_TEST_INC(+(x0[2], -1))∧(UIncreasing(150_0_TEST_INC(+(x0[2], -1))), ≥))

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

(19)    ((UIncreasing(150_0_TEST_INC(+(x0[2], -1))), ≥)∧[2 + (-1)bso_17] ≥ 0)

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

(20)    ((UIncreasing(150_0_TEST_INC(+(x0[2], -1))), ≥)∧[2 + (-1)bso_17] ≥ 0)

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

(21)    ((UIncreasing(150_0_TEST_INC(+(x0[2], -1))), ≥)∧[2 + (-1)bso_17] ≥ 0)

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

(22)    ((UIncreasing(150_0_TEST_INC(+(x0[2], -1))), ≥)∧0 = 0∧[2 + (-1)bso_17] ≥ 0)

For Pair 167_1_TEST_INVOKEMETHOD(158_0_test_Return, x0) → 150_0_TEST_INC(x0) the following chains were created:
• We consider the chain 167_1_TEST_INVOKEMETHOD(158_0_test_Return, x0[3]) → 150_0_TEST_INC(x0[3]), 150_0_TEST_INC(x0[0]) → COND_150_0_TEST_INC(<(0, +(x0[0], -1)), x0[0]) which results in the following constraint:

(23)    (x0[3]=x0[0]167_1_TEST_INVOKEMETHOD(158_0_test_Return, x0[3])≥NonInfC∧167_1_TEST_INVOKEMETHOD(158_0_test_Return, x0[3])≥150_0_TEST_INC(x0[3])∧(UIncreasing(150_0_TEST_INC(x0[3])), ≥))

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

(24)    (167_1_TEST_INVOKEMETHOD(158_0_test_Return, x0[3])≥NonInfC∧167_1_TEST_INVOKEMETHOD(158_0_test_Return, x0[3])≥150_0_TEST_INC(x0[3])∧(UIncreasing(150_0_TEST_INC(x0[3])), ≥))

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

(25)    ((UIncreasing(150_0_TEST_INC(x0[3])), ≥)∧[2 + (-1)bso_19] ≥ 0)

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

(26)    ((UIncreasing(150_0_TEST_INC(x0[3])), ≥)∧[2 + (-1)bso_19] ≥ 0)

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

(27)    ((UIncreasing(150_0_TEST_INC(x0[3])), ≥)∧[2 + (-1)bso_19] ≥ 0)

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

(28)    ((UIncreasing(150_0_TEST_INC(x0[3])), ≥)∧0 = 0∧[2 + (-1)bso_19] ≥ 0)

To summarize, we get the following constraints P for the following pairs.
• 150_0_TEST_INC(x0) → COND_150_0_TEST_INC(<(0, +(x0, -1)), x0)
• (x0[0] ≥ 0 ⇒ (UIncreasing(COND_150_0_TEST_INC(<(0, +(x0[0], -1)), x0[0])), ≥)∧[(-1)Bound*bni_12 + (4)bni_12] + [(2)bni_12]x0[0] ≥ 0∧[(-1)bso_13] ≥ 0)
• (x0[0] ≥ 0 ⇒ (UIncreasing(COND_150_0_TEST_INC(<(0, +(x0[0], -1)), x0[0])), ≥)∧[(-1)Bound*bni_12 + (4)bni_12] + [(2)bni_12]x0[0] ≥ 0∧[(-1)bso_13] ≥ 0)

• COND_150_0_TEST_INC(TRUE, x0) → 167_1_TEST_INVOKEMETHOD(150_0_test_Inc(+(x0, -1)), +(x0, -1))
• ((UIncreasing(167_1_TEST_INVOKEMETHOD(150_0_test_Inc(+(x0[1], -1)), +(x0[1], -1))), ≥)∧0 = 0∧[(-1)bso_15] ≥ 0)

• COND_150_0_TEST_INC(TRUE, x0) → 150_0_TEST_INC(+(x0, -1))
• ((UIncreasing(150_0_TEST_INC(+(x0[2], -1))), ≥)∧0 = 0∧[2 + (-1)bso_17] ≥ 0)

• 167_1_TEST_INVOKEMETHOD(158_0_test_Return, x0) → 150_0_TEST_INC(x0)
• ((UIncreasing(150_0_TEST_INC(x0[3])), ≥)∧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(150_0_TEST_INC(x1)) = [2]x1
POL(COND_150_0_TEST_INC(x1, x2)) = [2]x2
POL(<(x1, x2)) = [-1]
POL(0) = 0
POL(+(x1, x2)) = x1 + x2
POL(-1) = [-1]
POL(167_1_TEST_INVOKEMETHOD(x1, x2)) = [2] + [2]x2
POL(150_0_test_Inc(x1)) = x1
POL(158_0_test_Return) = [-1]

The following pairs are in P>:

COND_150_0_TEST_INC(TRUE, x0[2]) → 150_0_TEST_INC(+(x0[2], -1))
167_1_TEST_INVOKEMETHOD(158_0_test_Return, x0[3]) → 150_0_TEST_INC(x0[3])

The following pairs are in Pbound:

150_0_TEST_INC(x0[0]) → COND_150_0_TEST_INC(<(0, +(x0[0], -1)), x0[0])

The following pairs are in P:

150_0_TEST_INC(x0[0]) → COND_150_0_TEST_INC(<(0, +(x0[0], -1)), x0[0])
COND_150_0_TEST_INC(TRUE, x0[1]) → 167_1_TEST_INVOKEMETHOD(150_0_test_Inc(+(x0[1], -1)), +(x0[1], -1))

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

R is empty.

The integer pair graph contains the following rules and edges:
(0): 150_0_TEST_INC(x0[0]) → COND_150_0_TEST_INC(0 < x0[0] + -1, x0[0])
(1): COND_150_0_TEST_INC(TRUE, x0[1]) → 167_1_TEST_INVOKEMETHOD(150_0_test_Inc(x0[1] + -1), x0[1] + -1)

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

The set Q is empty.

### (8) IDependencyGraphProof (EQUIVALENT transformation)

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

### (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

R is empty.

The integer pair graph contains the following rules and edges:
(1): COND_150_0_TEST_INC(TRUE, x0[1]) → 167_1_TEST_INVOKEMETHOD(150_0_test_Inc(x0[1] + -1), x0[1] + -1)
(2): COND_150_0_TEST_INC(TRUE, x0[2]) → 150_0_TEST_INC(x0[2] + -1)
(3): 167_1_TEST_INVOKEMETHOD(158_0_test_Return, x0[3]) → 150_0_TEST_INC(x0[3])

(1) -> (3), if ((150_0_test_Inc(x0[1] + -1) →* 158_0_test_Return)∧(x0[1] + -1* x0[3]))

The set Q is empty.

### (11) IDependencyGraphProof (EQUIVALENT transformation)

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