# LTS Termination Proof

by AProVE

## Input

Integer Transition System
• Initial Location: f218_0_times_Return, f469_0_times_InvokeMethod, f484_0_plus_LE, f127_0_times_NE, f1_0_main_Load, __init
• Transitions: (pre-variables and post-variables)  f1_0_main_Load 1 f127_0_times_NE: x1 = _arg1 ∧ x2 = _arg2 ∧ x3 = _arg3 ∧ x1 = _arg1P ∧ x2 = _arg2P ∧ x3 = _arg3P ∧ 0 ≤ _arg1 − 1 ∧ −1 ≤ _arg1P − 1 ∧ 1 ≤ _arg2 − 1 ∧ −1 ≤ _arg2P − 1 f127_0_times_NE 2 f127_0_times_NE: x1 = _x ∧ x2 = _x1 ∧ x3 = _x2 ∧ x1 = _x3 ∧ x2 = _x4 ∧ x3 = _x5 ∧ _x1 − 1 = _x4 ∧ _x = _x3 ∧ _x1 − 1 ≤ _x1 − 1 ∧ 0 ≤ _x1 − 1 f127_0_times_NE 3 f469_0_times_InvokeMethod: x1 = _x6 ∧ x2 = _x7 ∧ x3 = _x8 ∧ x1 = _x9 ∧ x2 = _x10 ∧ x3 = _x11 ∧ _x6 = _x11 ∧ 0 = _x10 ∧ 1 = _x9 ∧ 1 = _x7 f127_0_times_NE 4 f469_0_times_InvokeMethod: x1 = _x12 ∧ x2 = _x13 ∧ x3 = _x14 ∧ x1 = _x15 ∧ x2 = _x16 ∧ x3 = _x17 ∧ 0 = _x17 ∧ 0 = _x16 ∧ _x13 = _x15 ∧ 0 = _x12 ∧ _x13 − 1 ≤ _x13 − 1 ∧ 0 ≤ _x13 − 1 f127_0_times_NE 5 f469_0_times_InvokeMethod: x1 = _x18 ∧ x2 = _x19 ∧ x3 = _x20 ∧ x1 = _x21 ∧ x2 = _x22 ∧ x3 = _x23 ∧ _x18 = _x23 ∧ _x19 = _x21 ∧ _x19 − 1 ≤ _x19 − 1 ∧ 0 ≤ _x19 − 1 f127_0_times_NE 6 f469_0_times_InvokeMethod: x1 = _x24 ∧ x2 = _x25 ∧ x3 = _x26 ∧ x1 = _x27 ∧ x2 = _x28 ∧ x3 = _x29 ∧ 0 = _x29 ∧ _x25 = _x27 ∧ 0 = _x24 ∧ _x25 − 1 ≤ _x25 − 1 ∧ 0 ≤ _x25 − 1 f218_0_times_Return 7 f469_0_times_InvokeMethod: x1 = _x30 ∧ x2 = _x31 ∧ x3 = _x32 ∧ x1 = _x33 ∧ x2 = _x34 ∧ x3 = _x35 ∧ _x31 = _x35 ∧ _x32 = _x34 ∧ _x30 = _x33 f469_0_times_InvokeMethod 8 f484_0_plus_LE: x1 = _x36 ∧ x2 = _x37 ∧ x3 = _x38 ∧ x1 = _x39 ∧ x2 = _x40 ∧ x3 = _x41 ∧ _x38 = _x40 ∧ _x37 = _x39 ∧ 0 ≤ _x36 − 1 f484_0_plus_LE 9 f484_0_plus_LE: x1 = _x42 ∧ x2 = _x43 ∧ x3 = _x44 ∧ x1 = _x45 ∧ x2 = _x46 ∧ x3 = _x47 ∧ _x43 − 1 = _x46 ∧ _x42 = _x45 ∧ _x43 − 1 ≤ _x43 − 1 ∧ 0 ≤ _x43 − 1 f484_0_plus_LE 10 f484_0_plus_LE: x1 = _x48 ∧ x2 = _x49 ∧ x3 = _x50 ∧ x1 = _x51 ∧ x2 = _x52 ∧ x3 = _x53 ∧ _x49 = _x52 ∧ _x48 − 1 = _x51 ∧ 0 ≤ _x48 − 1 ∧ _x48 − 1 ≤ _x48 − 1 ∧ _x49 ≤ 0 __init 11 f1_0_main_Load: x1 = _x54 ∧ x2 = _x55 ∧ x3 = _x56 ∧ x1 = _x57 ∧ x2 = _x58 ∧ x3 = _x59 ∧ 0 ≤ 0

## Proof

### 1 Switch to Cooperation Termination Proof

We consider the following cutpoint-transitions:
 f218_0_times_Return f218_0_times_Return f218_0_times_Return: x1 = x1 ∧ x2 = x2 ∧ x3 = x3 f469_0_times_InvokeMethod f469_0_times_InvokeMethod f469_0_times_InvokeMethod: x1 = x1 ∧ x2 = x2 ∧ x3 = x3 f484_0_plus_LE f484_0_plus_LE f484_0_plus_LE: x1 = x1 ∧ x2 = x2 ∧ x3 = x3 f127_0_times_NE f127_0_times_NE f127_0_times_NE: x1 = x1 ∧ x2 = x2 ∧ x3 = x3 f1_0_main_Load f1_0_main_Load f1_0_main_Load: x1 = x1 ∧ x2 = x2 ∧ x3 = x3 __init __init __init: x1 = x1 ∧ x2 = x2 ∧ x3 = x3
and for every transition t, a duplicate t is considered.

### 2 SCC Decomposition

We consider subproblems for each of the 2 SCC(s) of the program graph.

### 2.1 SCC Subproblem 1/2

Here we consider the SCC { f127_0_times_NE }.

### 2.1.1 Transition Removal

We remove transition 2 using the following ranking functions, which are bounded by 0.

 f127_0_times_NE: x2

### 2.1.2 Trivial Cooperation Program

There are no more "sharp" transitions in the cooperation program. Hence the cooperation termination is proved.

### 2.2 SCC Subproblem 2/2

Here we consider the SCC { f484_0_plus_LE }.

### 2.2.1 Transition Removal

We remove transition 10 using the following ranking functions, which are bounded by 0.

 f484_0_plus_LE: −1 + x1

### 2.2.2 Transition Removal

We remove transition 9 using the following ranking functions, which are bounded by 0.

 f484_0_plus_LE: x2

### 2.2.3 Trivial Cooperation Program

There are no more "sharp" transitions in the cooperation program. Hence the cooperation termination is proved.

## Tool configuration

AProVE

• version: AProVE Commit ID: unknown
• strategy: Statistics for single proof: 100.00 % (9 real / 0 unknown / 0 assumptions / 9 total proof steps)