### (0) Obligation:

JBC Problem based on JBC Program:
Manifest-Version: 1.0 Created-By: 1.6.0_20 (Apple Inc.) Main-Class: HanoiR
`public class HanoiR {    private void solve(int h, int from, int to, int support) {	if (h < 1) return;	else if (h == 1) {	    //System.out.println("from " + from + " to " + to + "\n");	}	else {	    solve(h - 1, from, support, to);	    //System.out.println("from " + from + " to " + to + "\n");	    solve(h - 1, support, to, from);	}    }    public static void main(String[] args) {	Random.args = args;	new HanoiR().solve(Random.random(),1,2,3);    }}public class Random {  static String[] args;  static int index = 0;  public static int random() {      if (index >= args.length)	  return 0;      String string = args[index];      index++;      return string.length();  }}`

### (1) JBC2FIG (SOUND transformation)

Constructed FIGraph.

### (2) Obligation:

FIGraph based on JBC Program:
HanoiR.main([Ljava/lang/String;)V: Graph of 97 nodes with 0 SCCs.

HanoiR.solve(IIII)V: Graph of 45 nodes with 0 SCCs.

### (3) FIGtoITRSProof (SOUND transformation)

Transformed FIGraph SCCs to IDPs. Logs:

Log for SCC 0:

Generated 33 rules for P and 12 rules for R.

Combined rules. Obtained 3 rules for P and 2 rules for R.

Filtered ground terms:

690_0_solve_ConstantStackPush(x1, x2, x3) → 690_0_solve_ConstantStackPush(x2, x3)
Cond_746_1_solve_InvokeMethod1(x1, x2, x3, x4) → Cond_746_1_solve_InvokeMethod1(x1, x3, x4)
886_0_solve_Return(x1) → 886_0_solve_Return
Cond_746_1_solve_InvokeMethod(x1, x2, x3, x4) → Cond_746_1_solve_InvokeMethod(x1, x3)
723_0_solve_Return(x1) → 723_0_solve_Return
Cond_690_0_solve_ConstantStackPush(x1, x2, x3, x4) → Cond_690_0_solve_ConstantStackPush(x1, x3, x4)

Filtered duplicate args:

690_0_solve_ConstantStackPush(x1, x2) → 690_0_solve_ConstantStackPush(x2)
Cond_690_0_solve_ConstantStackPush(x1, x2, x3) → Cond_690_0_solve_ConstantStackPush(x1, x3)

Filtered unneeded arguments:

Cond_746_1_solve_InvokeMethod1(x1, x2, x3) → Cond_746_1_solve_InvokeMethod1(x1, x2)

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:
836_1_solve_InvokeMethod(723_0_solve_Return, 1) → 886_0_solve_Return
836_1_solve_InvokeMethod(886_0_solve_Return, x0) → 886_0_solve_Return

The integer pair graph contains the following rules and edges:
(0): 690_0_SOLVE_CONSTANTSTACKPUSH(x0[0]) → COND_690_0_SOLVE_CONSTANTSTACKPUSH(x0[0] > 1, x0[0])
(1): COND_690_0_SOLVE_CONSTANTSTACKPUSH(TRUE, x0[1]) → 746_1_SOLVE_INVOKEMETHOD(690_0_solve_ConstantStackPush(x0[1] - 1), x0[1], x0[1] - 1)
(2): COND_690_0_SOLVE_CONSTANTSTACKPUSH(TRUE, x0[2]) → 690_0_SOLVE_CONSTANTSTACKPUSH(x0[2] - 1)
(3): 746_1_SOLVE_INVOKEMETHOD(723_0_solve_Return, x0[3], 1) → COND_746_1_SOLVE_INVOKEMETHOD(x0[3] > 0, 723_0_solve_Return, x0[3], 1)
(4): COND_746_1_SOLVE_INVOKEMETHOD(TRUE, 723_0_solve_Return, x0[4], 1) → 690_0_SOLVE_CONSTANTSTACKPUSH(x0[4] - 1)
(5): 746_1_SOLVE_INVOKEMETHOD(886_0_solve_Return, x0[5], x1[5]) → COND_746_1_SOLVE_INVOKEMETHOD1(x0[5] > 0, 886_0_solve_Return, x0[5], x1[5])
(6): COND_746_1_SOLVE_INVOKEMETHOD1(TRUE, 886_0_solve_Return, x0[6], x1[6]) → 690_0_SOLVE_CONSTANTSTACKPUSH(x0[6] - 1)

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

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

(1) -> (3), if ((690_0_solve_ConstantStackPush(x0[1] - 1) →* 723_0_solve_Return)∧(x0[1]* x0[3])∧(x0[1] - 1* 1))

(1) -> (5), if ((690_0_solve_ConstantStackPush(x0[1] - 1) →* 886_0_solve_Return)∧(x0[1]* x0[5])∧(x0[1] - 1* x1[5]))

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

(3) -> (4), if ((x0[3] > 0* TRUE)∧(x0[3]* x0[4]))

(4) -> (0), if ((x0[4] - 1* x0[0]))

(5) -> (6), if ((x0[5] > 0* TRUE)∧(x0[5]* x0[6])∧(x1[5]* x1[6]))

(6) -> (0), if ((x0[6] - 1* x0[0]))

The set Q consists of the following terms:
836_1_solve_InvokeMethod(723_0_solve_Return, 1)
836_1_solve_InvokeMethod(886_0_solve_Return, 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 690_0_SOLVE_CONSTANTSTACKPUSH(x0) → COND_690_0_SOLVE_CONSTANTSTACKPUSH(>(x0, 1), x0) the following chains were created:
• We consider the chain 690_0_SOLVE_CONSTANTSTACKPUSH(x0[0]) → COND_690_0_SOLVE_CONSTANTSTACKPUSH(>(x0[0], 1), x0[0]), COND_690_0_SOLVE_CONSTANTSTACKPUSH(TRUE, x0[1]) → 746_1_SOLVE_INVOKEMETHOD(690_0_solve_ConstantStackPush(-(x0[1], 1)), x0[1], -(x0[1], 1)) which results in the following constraint:

(1)    (>(x0[0], 1)=TRUEx0[0]=x0[1]690_0_SOLVE_CONSTANTSTACKPUSH(x0[0])≥NonInfC∧690_0_SOLVE_CONSTANTSTACKPUSH(x0[0])≥COND_690_0_SOLVE_CONSTANTSTACKPUSH(>(x0[0], 1), x0[0])∧(UIncreasing(COND_690_0_SOLVE_CONSTANTSTACKPUSH(>(x0[0], 1), x0[0])), ≥))

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

(2)    (>(x0[0], 1)=TRUE690_0_SOLVE_CONSTANTSTACKPUSH(x0[0])≥NonInfC∧690_0_SOLVE_CONSTANTSTACKPUSH(x0[0])≥COND_690_0_SOLVE_CONSTANTSTACKPUSH(>(x0[0], 1), x0[0])∧(UIncreasing(COND_690_0_SOLVE_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_690_0_SOLVE_CONSTANTSTACKPUSH(>(x0[0], 1), x0[0])), ≥)∧[bni_17 + (-1)Bound*bni_17] + [bni_17]x0[0] ≥ 0∧[(-1)bso_18] ≥ 0)

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

(4)    (x0[0] + [-2] ≥ 0 ⇒ (UIncreasing(COND_690_0_SOLVE_CONSTANTSTACKPUSH(>(x0[0], 1), x0[0])), ≥)∧[bni_17 + (-1)Bound*bni_17] + [bni_17]x0[0] ≥ 0∧[(-1)bso_18] ≥ 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_690_0_SOLVE_CONSTANTSTACKPUSH(>(x0[0], 1), x0[0])), ≥)∧[bni_17 + (-1)Bound*bni_17] + [bni_17]x0[0] ≥ 0∧[(-1)bso_18] ≥ 0)

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

(6)    (x0[0] ≥ 0 ⇒ (UIncreasing(COND_690_0_SOLVE_CONSTANTSTACKPUSH(>(x0[0], 1), x0[0])), ≥)∧[(3)bni_17 + (-1)Bound*bni_17] + [bni_17]x0[0] ≥ 0∧[(-1)bso_18] ≥ 0)

• We consider the chain 690_0_SOLVE_CONSTANTSTACKPUSH(x0[0]) → COND_690_0_SOLVE_CONSTANTSTACKPUSH(>(x0[0], 1), x0[0]), COND_690_0_SOLVE_CONSTANTSTACKPUSH(TRUE, x0[2]) → 690_0_SOLVE_CONSTANTSTACKPUSH(-(x0[2], 1)) which results in the following constraint:

(7)    (>(x0[0], 1)=TRUEx0[0]=x0[2]690_0_SOLVE_CONSTANTSTACKPUSH(x0[0])≥NonInfC∧690_0_SOLVE_CONSTANTSTACKPUSH(x0[0])≥COND_690_0_SOLVE_CONSTANTSTACKPUSH(>(x0[0], 1), x0[0])∧(UIncreasing(COND_690_0_SOLVE_CONSTANTSTACKPUSH(>(x0[0], 1), x0[0])), ≥))

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

(8)    (>(x0[0], 1)=TRUE690_0_SOLVE_CONSTANTSTACKPUSH(x0[0])≥NonInfC∧690_0_SOLVE_CONSTANTSTACKPUSH(x0[0])≥COND_690_0_SOLVE_CONSTANTSTACKPUSH(>(x0[0], 1), x0[0])∧(UIncreasing(COND_690_0_SOLVE_CONSTANTSTACKPUSH(>(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_690_0_SOLVE_CONSTANTSTACKPUSH(>(x0[0], 1), x0[0])), ≥)∧[bni_17 + (-1)Bound*bni_17] + [bni_17]x0[0] ≥ 0∧[(-1)bso_18] ≥ 0)

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

(10)    (x0[0] + [-2] ≥ 0 ⇒ (UIncreasing(COND_690_0_SOLVE_CONSTANTSTACKPUSH(>(x0[0], 1), x0[0])), ≥)∧[bni_17 + (-1)Bound*bni_17] + [bni_17]x0[0] ≥ 0∧[(-1)bso_18] ≥ 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_690_0_SOLVE_CONSTANTSTACKPUSH(>(x0[0], 1), x0[0])), ≥)∧[bni_17 + (-1)Bound*bni_17] + [bni_17]x0[0] ≥ 0∧[(-1)bso_18] ≥ 0)

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

(12)    (x0[0] ≥ 0 ⇒ (UIncreasing(COND_690_0_SOLVE_CONSTANTSTACKPUSH(>(x0[0], 1), x0[0])), ≥)∧[(3)bni_17 + (-1)Bound*bni_17] + [bni_17]x0[0] ≥ 0∧[(-1)bso_18] ≥ 0)

For Pair COND_690_0_SOLVE_CONSTANTSTACKPUSH(TRUE, x0) → 746_1_SOLVE_INVOKEMETHOD(690_0_solve_ConstantStackPush(-(x0, 1)), x0, -(x0, 1)) the following chains were created:
• We consider the chain COND_690_0_SOLVE_CONSTANTSTACKPUSH(TRUE, x0[1]) → 746_1_SOLVE_INVOKEMETHOD(690_0_solve_ConstantStackPush(-(x0[1], 1)), x0[1], -(x0[1], 1)) which results in the following constraint:

(13)    (COND_690_0_SOLVE_CONSTANTSTACKPUSH(TRUE, x0[1])≥NonInfC∧COND_690_0_SOLVE_CONSTANTSTACKPUSH(TRUE, x0[1])≥746_1_SOLVE_INVOKEMETHOD(690_0_solve_ConstantStackPush(-(x0[1], 1)), x0[1], -(x0[1], 1))∧(UIncreasing(746_1_SOLVE_INVOKEMETHOD(690_0_solve_ConstantStackPush(-(x0[1], 1)), x0[1], -(x0[1], 1))), ≥))

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

(14)    ((UIncreasing(746_1_SOLVE_INVOKEMETHOD(690_0_solve_ConstantStackPush(-(x0[1], 1)), x0[1], -(x0[1], 1))), ≥)∧[(-1)bso_20] ≥ 0)

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

(15)    ((UIncreasing(746_1_SOLVE_INVOKEMETHOD(690_0_solve_ConstantStackPush(-(x0[1], 1)), x0[1], -(x0[1], 1))), ≥)∧[(-1)bso_20] ≥ 0)

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

(16)    ((UIncreasing(746_1_SOLVE_INVOKEMETHOD(690_0_solve_ConstantStackPush(-(x0[1], 1)), x0[1], -(x0[1], 1))), ≥)∧[(-1)bso_20] ≥ 0)

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

(17)    ((UIncreasing(746_1_SOLVE_INVOKEMETHOD(690_0_solve_ConstantStackPush(-(x0[1], 1)), x0[1], -(x0[1], 1))), ≥)∧0 = 0∧[(-1)bso_20] ≥ 0)

For Pair COND_690_0_SOLVE_CONSTANTSTACKPUSH(TRUE, x0) → 690_0_SOLVE_CONSTANTSTACKPUSH(-(x0, 1)) the following chains were created:
• We consider the chain COND_690_0_SOLVE_CONSTANTSTACKPUSH(TRUE, x0[2]) → 690_0_SOLVE_CONSTANTSTACKPUSH(-(x0[2], 1)) which results in the following constraint:

(18)    (COND_690_0_SOLVE_CONSTANTSTACKPUSH(TRUE, x0[2])≥NonInfC∧COND_690_0_SOLVE_CONSTANTSTACKPUSH(TRUE, x0[2])≥690_0_SOLVE_CONSTANTSTACKPUSH(-(x0[2], 1))∧(UIncreasing(690_0_SOLVE_CONSTANTSTACKPUSH(-(x0[2], 1))), ≥))

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

(19)    ((UIncreasing(690_0_SOLVE_CONSTANTSTACKPUSH(-(x0[2], 1))), ≥)∧[1 + (-1)bso_22] ≥ 0)

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

(20)    ((UIncreasing(690_0_SOLVE_CONSTANTSTACKPUSH(-(x0[2], 1))), ≥)∧[1 + (-1)bso_22] ≥ 0)

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

(21)    ((UIncreasing(690_0_SOLVE_CONSTANTSTACKPUSH(-(x0[2], 1))), ≥)∧[1 + (-1)bso_22] ≥ 0)

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

(22)    ((UIncreasing(690_0_SOLVE_CONSTANTSTACKPUSH(-(x0[2], 1))), ≥)∧0 = 0∧[1 + (-1)bso_22] ≥ 0)

For Pair 746_1_SOLVE_INVOKEMETHOD(723_0_solve_Return, x0, 1) → COND_746_1_SOLVE_INVOKEMETHOD(>(x0, 0), 723_0_solve_Return, x0, 1) the following chains were created:
• We consider the chain 746_1_SOLVE_INVOKEMETHOD(723_0_solve_Return, x0[3], 1) → COND_746_1_SOLVE_INVOKEMETHOD(>(x0[3], 0), 723_0_solve_Return, x0[3], 1), COND_746_1_SOLVE_INVOKEMETHOD(TRUE, 723_0_solve_Return, x0[4], 1) → 690_0_SOLVE_CONSTANTSTACKPUSH(-(x0[4], 1)) which results in the following constraint:

(23)    (>(x0[3], 0)=TRUEx0[3]=x0[4]746_1_SOLVE_INVOKEMETHOD(723_0_solve_Return, x0[3], 1)≥NonInfC∧746_1_SOLVE_INVOKEMETHOD(723_0_solve_Return, x0[3], 1)≥COND_746_1_SOLVE_INVOKEMETHOD(>(x0[3], 0), 723_0_solve_Return, x0[3], 1)∧(UIncreasing(COND_746_1_SOLVE_INVOKEMETHOD(>(x0[3], 0), 723_0_solve_Return, x0[3], 1)), ≥))

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

(24)    (>(x0[3], 0)=TRUE746_1_SOLVE_INVOKEMETHOD(723_0_solve_Return, x0[3], 1)≥NonInfC∧746_1_SOLVE_INVOKEMETHOD(723_0_solve_Return, x0[3], 1)≥COND_746_1_SOLVE_INVOKEMETHOD(>(x0[3], 0), 723_0_solve_Return, x0[3], 1)∧(UIncreasing(COND_746_1_SOLVE_INVOKEMETHOD(>(x0[3], 0), 723_0_solve_Return, x0[3], 1)), ≥))

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

(25)    (x0[3] + [-1] ≥ 0 ⇒ (UIncreasing(COND_746_1_SOLVE_INVOKEMETHOD(>(x0[3], 0), 723_0_solve_Return, x0[3], 1)), ≥)∧[bni_23 + (-1)Bound*bni_23] + [bni_23]x0[3] ≥ 0∧[1 + (-1)bso_24] ≥ 0)

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

(26)    (x0[3] + [-1] ≥ 0 ⇒ (UIncreasing(COND_746_1_SOLVE_INVOKEMETHOD(>(x0[3], 0), 723_0_solve_Return, x0[3], 1)), ≥)∧[bni_23 + (-1)Bound*bni_23] + [bni_23]x0[3] ≥ 0∧[1 + (-1)bso_24] ≥ 0)

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

(27)    (x0[3] + [-1] ≥ 0 ⇒ (UIncreasing(COND_746_1_SOLVE_INVOKEMETHOD(>(x0[3], 0), 723_0_solve_Return, x0[3], 1)), ≥)∧[bni_23 + (-1)Bound*bni_23] + [bni_23]x0[3] ≥ 0∧[1 + (-1)bso_24] ≥ 0)

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

(28)    (x0[3] ≥ 0 ⇒ (UIncreasing(COND_746_1_SOLVE_INVOKEMETHOD(>(x0[3], 0), 723_0_solve_Return, x0[3], 1)), ≥)∧[(2)bni_23 + (-1)Bound*bni_23] + [bni_23]x0[3] ≥ 0∧[1 + (-1)bso_24] ≥ 0)

For Pair COND_746_1_SOLVE_INVOKEMETHOD(TRUE, 723_0_solve_Return, x0, 1) → 690_0_SOLVE_CONSTANTSTACKPUSH(-(x0, 1)) the following chains were created:
• We consider the chain COND_746_1_SOLVE_INVOKEMETHOD(TRUE, 723_0_solve_Return, x0[4], 1) → 690_0_SOLVE_CONSTANTSTACKPUSH(-(x0[4], 1)) which results in the following constraint:

(29)    (COND_746_1_SOLVE_INVOKEMETHOD(TRUE, 723_0_solve_Return, x0[4], 1)≥NonInfC∧COND_746_1_SOLVE_INVOKEMETHOD(TRUE, 723_0_solve_Return, x0[4], 1)≥690_0_SOLVE_CONSTANTSTACKPUSH(-(x0[4], 1))∧(UIncreasing(690_0_SOLVE_CONSTANTSTACKPUSH(-(x0[4], 1))), ≥))

We simplified constraint (29) using rule (POLY_CONSTRAINTS) which results in the following new constraint:

(30)    ((UIncreasing(690_0_SOLVE_CONSTANTSTACKPUSH(-(x0[4], 1))), ≥)∧[(-1)bso_26] ≥ 0)

We simplified constraint (30) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:

(31)    ((UIncreasing(690_0_SOLVE_CONSTANTSTACKPUSH(-(x0[4], 1))), ≥)∧[(-1)bso_26] ≥ 0)

We simplified constraint (31) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:

(32)    ((UIncreasing(690_0_SOLVE_CONSTANTSTACKPUSH(-(x0[4], 1))), ≥)∧[(-1)bso_26] ≥ 0)

We simplified constraint (32) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint:

(33)    ((UIncreasing(690_0_SOLVE_CONSTANTSTACKPUSH(-(x0[4], 1))), ≥)∧0 = 0∧[(-1)bso_26] ≥ 0)

For Pair 746_1_SOLVE_INVOKEMETHOD(886_0_solve_Return, x0, x1) → COND_746_1_SOLVE_INVOKEMETHOD1(>(x0, 0), 886_0_solve_Return, x0, x1) the following chains were created:
• We consider the chain 746_1_SOLVE_INVOKEMETHOD(886_0_solve_Return, x0[5], x1[5]) → COND_746_1_SOLVE_INVOKEMETHOD1(>(x0[5], 0), 886_0_solve_Return, x0[5], x1[5]), COND_746_1_SOLVE_INVOKEMETHOD1(TRUE, 886_0_solve_Return, x0[6], x1[6]) → 690_0_SOLVE_CONSTANTSTACKPUSH(-(x0[6], 1)) which results in the following constraint:

(34)    (>(x0[5], 0)=TRUEx0[5]=x0[6]x1[5]=x1[6]746_1_SOLVE_INVOKEMETHOD(886_0_solve_Return, x0[5], x1[5])≥NonInfC∧746_1_SOLVE_INVOKEMETHOD(886_0_solve_Return, x0[5], x1[5])≥COND_746_1_SOLVE_INVOKEMETHOD1(>(x0[5], 0), 886_0_solve_Return, x0[5], x1[5])∧(UIncreasing(COND_746_1_SOLVE_INVOKEMETHOD1(>(x0[5], 0), 886_0_solve_Return, x0[5], x1[5])), ≥))

We simplified constraint (34) using rule (IV) which results in the following new constraint:

(35)    (>(x0[5], 0)=TRUE746_1_SOLVE_INVOKEMETHOD(886_0_solve_Return, x0[5], x1[5])≥NonInfC∧746_1_SOLVE_INVOKEMETHOD(886_0_solve_Return, x0[5], x1[5])≥COND_746_1_SOLVE_INVOKEMETHOD1(>(x0[5], 0), 886_0_solve_Return, x0[5], x1[5])∧(UIncreasing(COND_746_1_SOLVE_INVOKEMETHOD1(>(x0[5], 0), 886_0_solve_Return, x0[5], x1[5])), ≥))

We simplified constraint (35) using rule (POLY_CONSTRAINTS) which results in the following new constraint:

(36)    (x0[5] + [-1] ≥ 0 ⇒ (UIncreasing(COND_746_1_SOLVE_INVOKEMETHOD1(>(x0[5], 0), 886_0_solve_Return, x0[5], x1[5])), ≥)∧[bni_27 + (-1)Bound*bni_27] + [bni_27]x0[5] ≥ 0∧[(-1)bso_28] ≥ 0)

We simplified constraint (36) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:

(37)    (x0[5] + [-1] ≥ 0 ⇒ (UIncreasing(COND_746_1_SOLVE_INVOKEMETHOD1(>(x0[5], 0), 886_0_solve_Return, x0[5], x1[5])), ≥)∧[bni_27 + (-1)Bound*bni_27] + [bni_27]x0[5] ≥ 0∧[(-1)bso_28] ≥ 0)

We simplified constraint (37) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:

(38)    (x0[5] + [-1] ≥ 0 ⇒ (UIncreasing(COND_746_1_SOLVE_INVOKEMETHOD1(>(x0[5], 0), 886_0_solve_Return, x0[5], x1[5])), ≥)∧[bni_27 + (-1)Bound*bni_27] + [bni_27]x0[5] ≥ 0∧[(-1)bso_28] ≥ 0)

We simplified constraint (38) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint:

(39)    (x0[5] + [-1] ≥ 0 ⇒ (UIncreasing(COND_746_1_SOLVE_INVOKEMETHOD1(>(x0[5], 0), 886_0_solve_Return, x0[5], x1[5])), ≥)∧0 = 0∧[bni_27 + (-1)Bound*bni_27] + [bni_27]x0[5] ≥ 0∧0 = 0∧[(-1)bso_28] ≥ 0)

We simplified constraint (39) using rule (IDP_SMT_SPLIT) which results in the following new constraint:

(40)    (x0[5] ≥ 0 ⇒ (UIncreasing(COND_746_1_SOLVE_INVOKEMETHOD1(>(x0[5], 0), 886_0_solve_Return, x0[5], x1[5])), ≥)∧0 = 0∧[(2)bni_27 + (-1)Bound*bni_27] + [bni_27]x0[5] ≥ 0∧0 = 0∧[(-1)bso_28] ≥ 0)

For Pair COND_746_1_SOLVE_INVOKEMETHOD1(TRUE, 886_0_solve_Return, x0, x1) → 690_0_SOLVE_CONSTANTSTACKPUSH(-(x0, 1)) the following chains were created:
• We consider the chain COND_746_1_SOLVE_INVOKEMETHOD1(TRUE, 886_0_solve_Return, x0[6], x1[6]) → 690_0_SOLVE_CONSTANTSTACKPUSH(-(x0[6], 1)) which results in the following constraint:

(41)    (COND_746_1_SOLVE_INVOKEMETHOD1(TRUE, 886_0_solve_Return, x0[6], x1[6])≥NonInfC∧COND_746_1_SOLVE_INVOKEMETHOD1(TRUE, 886_0_solve_Return, x0[6], x1[6])≥690_0_SOLVE_CONSTANTSTACKPUSH(-(x0[6], 1))∧(UIncreasing(690_0_SOLVE_CONSTANTSTACKPUSH(-(x0[6], 1))), ≥))

We simplified constraint (41) using rule (POLY_CONSTRAINTS) which results in the following new constraint:

(42)    ((UIncreasing(690_0_SOLVE_CONSTANTSTACKPUSH(-(x0[6], 1))), ≥)∧[1 + (-1)bso_30] ≥ 0)

We simplified constraint (42) using rule (IDP_POLY_SIMPLIFY) which results in the following new constraint:

(43)    ((UIncreasing(690_0_SOLVE_CONSTANTSTACKPUSH(-(x0[6], 1))), ≥)∧[1 + (-1)bso_30] ≥ 0)

We simplified constraint (43) using rule (POLY_REMOVE_MIN_MAX) which results in the following new constraint:

(44)    ((UIncreasing(690_0_SOLVE_CONSTANTSTACKPUSH(-(x0[6], 1))), ≥)∧[1 + (-1)bso_30] ≥ 0)

We simplified constraint (44) using rule (IDP_UNRESTRICTED_VARS) which results in the following new constraint:

(45)    ((UIncreasing(690_0_SOLVE_CONSTANTSTACKPUSH(-(x0[6], 1))), ≥)∧0 = 0∧0 = 0∧[1 + (-1)bso_30] ≥ 0)

To summarize, we get the following constraints P for the following pairs.
• 690_0_SOLVE_CONSTANTSTACKPUSH(x0) → COND_690_0_SOLVE_CONSTANTSTACKPUSH(>(x0, 1), x0)
• (x0[0] ≥ 0 ⇒ (UIncreasing(COND_690_0_SOLVE_CONSTANTSTACKPUSH(>(x0[0], 1), x0[0])), ≥)∧[(3)bni_17 + (-1)Bound*bni_17] + [bni_17]x0[0] ≥ 0∧[(-1)bso_18] ≥ 0)
• (x0[0] ≥ 0 ⇒ (UIncreasing(COND_690_0_SOLVE_CONSTANTSTACKPUSH(>(x0[0], 1), x0[0])), ≥)∧[(3)bni_17 + (-1)Bound*bni_17] + [bni_17]x0[0] ≥ 0∧[(-1)bso_18] ≥ 0)

• COND_690_0_SOLVE_CONSTANTSTACKPUSH(TRUE, x0) → 746_1_SOLVE_INVOKEMETHOD(690_0_solve_ConstantStackPush(-(x0, 1)), x0, -(x0, 1))
• ((UIncreasing(746_1_SOLVE_INVOKEMETHOD(690_0_solve_ConstantStackPush(-(x0[1], 1)), x0[1], -(x0[1], 1))), ≥)∧0 = 0∧[(-1)bso_20] ≥ 0)

• COND_690_0_SOLVE_CONSTANTSTACKPUSH(TRUE, x0) → 690_0_SOLVE_CONSTANTSTACKPUSH(-(x0, 1))
• ((UIncreasing(690_0_SOLVE_CONSTANTSTACKPUSH(-(x0[2], 1))), ≥)∧0 = 0∧[1 + (-1)bso_22] ≥ 0)

• 746_1_SOLVE_INVOKEMETHOD(723_0_solve_Return, x0, 1) → COND_746_1_SOLVE_INVOKEMETHOD(>(x0, 0), 723_0_solve_Return, x0, 1)
• (x0[3] ≥ 0 ⇒ (UIncreasing(COND_746_1_SOLVE_INVOKEMETHOD(>(x0[3], 0), 723_0_solve_Return, x0[3], 1)), ≥)∧[(2)bni_23 + (-1)Bound*bni_23] + [bni_23]x0[3] ≥ 0∧[1 + (-1)bso_24] ≥ 0)

• COND_746_1_SOLVE_INVOKEMETHOD(TRUE, 723_0_solve_Return, x0, 1) → 690_0_SOLVE_CONSTANTSTACKPUSH(-(x0, 1))
• ((UIncreasing(690_0_SOLVE_CONSTANTSTACKPUSH(-(x0[4], 1))), ≥)∧0 = 0∧[(-1)bso_26] ≥ 0)

• 746_1_SOLVE_INVOKEMETHOD(886_0_solve_Return, x0, x1) → COND_746_1_SOLVE_INVOKEMETHOD1(>(x0, 0), 886_0_solve_Return, x0, x1)
• (x0[5] ≥ 0 ⇒ (UIncreasing(COND_746_1_SOLVE_INVOKEMETHOD1(>(x0[5], 0), 886_0_solve_Return, x0[5], x1[5])), ≥)∧0 = 0∧[(2)bni_27 + (-1)Bound*bni_27] + [bni_27]x0[5] ≥ 0∧0 = 0∧[(-1)bso_28] ≥ 0)

• COND_746_1_SOLVE_INVOKEMETHOD1(TRUE, 886_0_solve_Return, x0, x1) → 690_0_SOLVE_CONSTANTSTACKPUSH(-(x0, 1))
• ((UIncreasing(690_0_SOLVE_CONSTANTSTACKPUSH(-(x0[6], 1))), ≥)∧0 = 0∧0 = 0∧[1 + (-1)bso_30] ≥ 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(836_1_solve_InvokeMethod(x1, x2)) = [-1]
POL(723_0_solve_Return) = [-1]
POL(1) = [1]
POL(886_0_solve_Return) = [1]
POL(690_0_SOLVE_CONSTANTSTACKPUSH(x1)) = [1] + x1
POL(COND_690_0_SOLVE_CONSTANTSTACKPUSH(x1, x2)) = [1] + x2
POL(>(x1, x2)) = [-1]
POL(746_1_SOLVE_INVOKEMETHOD(x1, x2, x3)) = [1] + x2
POL(690_0_solve_ConstantStackPush(x1)) = x1
POL(-(x1, x2)) = x1 + [-1]x2
POL(COND_746_1_SOLVE_INVOKEMETHOD(x1, x2, x3, x4)) = x3
POL(0) = 0
POL(COND_746_1_SOLVE_INVOKEMETHOD1(x1, x2, x3, x4)) = [1] + x3

The following pairs are in P>:

COND_690_0_SOLVE_CONSTANTSTACKPUSH(TRUE, x0[2]) → 690_0_SOLVE_CONSTANTSTACKPUSH(-(x0[2], 1))
746_1_SOLVE_INVOKEMETHOD(723_0_solve_Return, x0[3], 1) → COND_746_1_SOLVE_INVOKEMETHOD(>(x0[3], 0), 723_0_solve_Return, x0[3], 1)
COND_746_1_SOLVE_INVOKEMETHOD1(TRUE, 886_0_solve_Return, x0[6], x1[6]) → 690_0_SOLVE_CONSTANTSTACKPUSH(-(x0[6], 1))

The following pairs are in Pbound:

690_0_SOLVE_CONSTANTSTACKPUSH(x0[0]) → COND_690_0_SOLVE_CONSTANTSTACKPUSH(>(x0[0], 1), x0[0])
746_1_SOLVE_INVOKEMETHOD(723_0_solve_Return, x0[3], 1) → COND_746_1_SOLVE_INVOKEMETHOD(>(x0[3], 0), 723_0_solve_Return, x0[3], 1)
746_1_SOLVE_INVOKEMETHOD(886_0_solve_Return, x0[5], x1[5]) → COND_746_1_SOLVE_INVOKEMETHOD1(>(x0[5], 0), 886_0_solve_Return, x0[5], x1[5])

The following pairs are in P:

690_0_SOLVE_CONSTANTSTACKPUSH(x0[0]) → COND_690_0_SOLVE_CONSTANTSTACKPUSH(>(x0[0], 1), x0[0])
COND_690_0_SOLVE_CONSTANTSTACKPUSH(TRUE, x0[1]) → 746_1_SOLVE_INVOKEMETHOD(690_0_solve_ConstantStackPush(-(x0[1], 1)), x0[1], -(x0[1], 1))
COND_746_1_SOLVE_INVOKEMETHOD(TRUE, 723_0_solve_Return, x0[4], 1) → 690_0_SOLVE_CONSTANTSTACKPUSH(-(x0[4], 1))
746_1_SOLVE_INVOKEMETHOD(886_0_solve_Return, x0[5], x1[5]) → COND_746_1_SOLVE_INVOKEMETHOD1(>(x0[5], 0), 886_0_solve_Return, x0[5], x1[5])

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:
836_1_solve_InvokeMethod(723_0_solve_Return, 1) → 886_0_solve_Return
836_1_solve_InvokeMethod(886_0_solve_Return, x0) → 886_0_solve_Return

The integer pair graph contains the following rules and edges:
(0): 690_0_SOLVE_CONSTANTSTACKPUSH(x0[0]) → COND_690_0_SOLVE_CONSTANTSTACKPUSH(x0[0] > 1, x0[0])
(1): COND_690_0_SOLVE_CONSTANTSTACKPUSH(TRUE, x0[1]) → 746_1_SOLVE_INVOKEMETHOD(690_0_solve_ConstantStackPush(x0[1] - 1), x0[1], x0[1] - 1)
(4): COND_746_1_SOLVE_INVOKEMETHOD(TRUE, 723_0_solve_Return, x0[4], 1) → 690_0_SOLVE_CONSTANTSTACKPUSH(x0[4] - 1)
(5): 746_1_SOLVE_INVOKEMETHOD(886_0_solve_Return, x0[5], x1[5]) → COND_746_1_SOLVE_INVOKEMETHOD1(x0[5] > 0, 886_0_solve_Return, x0[5], x1[5])

(4) -> (0), if ((x0[4] - 1* x0[0]))

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

(1) -> (5), if ((690_0_solve_ConstantStackPush(x0[1] - 1) →* 886_0_solve_Return)∧(x0[1]* x0[5])∧(x0[1] - 1* x1[5]))

The set Q consists of the following terms:
836_1_solve_InvokeMethod(723_0_solve_Return, 1)
836_1_solve_InvokeMethod(886_0_solve_Return, x0)

### (8) IDependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 4 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

The ITRS R consists of the following rules:
836_1_solve_InvokeMethod(723_0_solve_Return, 1) → 886_0_solve_Return
836_1_solve_InvokeMethod(886_0_solve_Return, x0) → 886_0_solve_Return

The integer pair graph contains the following rules and edges:
(1): COND_690_0_SOLVE_CONSTANTSTACKPUSH(TRUE, x0[1]) → 746_1_SOLVE_INVOKEMETHOD(690_0_solve_ConstantStackPush(x0[1] - 1), x0[1], x0[1] - 1)
(2): COND_690_0_SOLVE_CONSTANTSTACKPUSH(TRUE, x0[2]) → 690_0_SOLVE_CONSTANTSTACKPUSH(x0[2] - 1)
(4): COND_746_1_SOLVE_INVOKEMETHOD(TRUE, 723_0_solve_Return, x0[4], 1) → 690_0_SOLVE_CONSTANTSTACKPUSH(x0[4] - 1)
(6): COND_746_1_SOLVE_INVOKEMETHOD1(TRUE, 886_0_solve_Return, x0[6], x1[6]) → 690_0_SOLVE_CONSTANTSTACKPUSH(x0[6] - 1)

The set Q consists of the following terms:
836_1_solve_InvokeMethod(723_0_solve_Return, 1)
836_1_solve_InvokeMethod(886_0_solve_Return, x0)

### (11) IDependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 0 SCCs with 4 less nodes.