# LTS Termination Proof

by AProVE

## Input

Integer Transition System
• Initial Location: f81_0_fib_Return, f1_0_main_Load, f142_0_fib_InvokeMethod, f31_0_fib_NE, __init
• Transitions: (pre-variables and post-variables)  f1_0_main_Load 1 f31_0_fib_NE: x1 = _arg1 ∧ x2 = _arg2 ∧ x1 = _arg1P ∧ x2 = _arg2P ∧ _arg2 = _arg1P ∧ −1 ≤ _arg2 − 1 ∧ 0 ≤ _arg1 − 1 f31_0_fib_NE 2 f31_0_fib_NE: x1 = _x ∧ x2 = _x1 ∧ x1 = _x2 ∧ x2 = _x3 ∧ _x − 1 = _x2 ∧ _x − 1 ≤ _x − 1 ∧ 1 ≤ _x − 1 f31_0_fib_NE 3 f142_0_fib_InvokeMethod: x1 = _x4 ∧ x2 = _x5 ∧ x1 = _x6 ∧ x2 = _x7 ∧ 0 = _x7 ∧ 2 = _x6 ∧ 2 = _x4 f31_0_fib_NE 4 f142_0_fib_InvokeMethod: x1 = _x8 ∧ x2 = _x9 ∧ x1 = _x10 ∧ x2 = _x11 ∧ _x8 − 2 = _x11 ∧ _x8 = _x10 ∧ _x8 − 1 ≤ _x8 − 1 ∧ 1 ≤ _x8 − 1 f81_0_fib_Return 5 f142_0_fib_InvokeMethod: x1 = _x12 ∧ x2 = _x13 ∧ x1 = _x14 ∧ x2 = _x15 ∧ _x12 − 2 = _x15 ∧ _x12 = _x14 ∧ 0 ≤ _x12 − 1 f142_0_fib_InvokeMethod 6 f31_0_fib_NE: x1 = _x16 ∧ x2 = _x17 ∧ x1 = _x18 ∧ x2 = _x19 ∧ _x17 = _x18 ∧ _x17 ≤ _x16 − 1 ∧ 1 ≤ _x16 − 1 __init 7 f1_0_main_Load: x1 = _x20 ∧ x2 = _x21 ∧ x1 = _x22 ∧ x2 = _x23 ∧ 0 ≤ 0

## Proof

### 1 Switch to Cooperation Termination Proof

We consider the following cutpoint-transitions:
 f81_0_fib_Return f81_0_fib_Return f81_0_fib_Return: x1 = x1 ∧ x2 = x2 f1_0_main_Load f1_0_main_Load f1_0_main_Load: x1 = x1 ∧ x2 = x2 f142_0_fib_InvokeMethod f142_0_fib_InvokeMethod f142_0_fib_InvokeMethod: x1 = x1 ∧ x2 = x2 f31_0_fib_NE f31_0_fib_NE f31_0_fib_NE: x1 = x1 ∧ x2 = x2 __init __init __init: x1 = x1 ∧ x2 = x2
and for every transition t, a duplicate t is considered.

### 2 SCC Decomposition

We consider subproblems for each of the 1 SCC(s) of the program graph.

### 2.1 SCC Subproblem 1/1

Here we consider the SCC { f142_0_fib_InvokeMethod, f31_0_fib_NE }.

### 2.1.1 Transition Removal

We remove transitions 2, 6, 4, 3 using the following ranking functions, which are bounded by 0.

 f31_0_fib_NE: 2⋅x1 f142_0_fib_InvokeMethod: −1 + 2⋅x1

### 2.1.2 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 % (5 real / 0 unknown / 0 assumptions / 5 total proof steps)