# LTS Termination Proof

by AProVE

## Input

Integer Transition System
• Initial Location: f152_0_createList_LE, f196_0_reverse_NULL, f1_0_main_Load, __init
• Transitions: (pre-variables and post-variables)  f1_0_main_Load 1 f152_0_createList_LE: x1 = _arg1 ∧ x2 = _arg2 ∧ x1 = _arg1P ∧ x2 = _arg2P ∧ 0 ≤ _arg2 − 1 ∧ −1 ≤ _x3 − 1 ∧ _arg1P + 1 ≤ _arg1 ∧ 0 ≤ _arg1 − 1 ∧ −1 ≤ _arg1P − 1 ∧ _x3 − 1 = _arg2P f152_0_createList_LE 2 f196_0_reverse_NULL: x1 = _x ∧ x2 = _x1 ∧ x1 = _x2 ∧ x2 = _x4 ∧ −1 ≤ _x2 − 1 ∧ −1 ≤ _x − 1 ∧ _x1 ≤ 0 ∧ _x2 ≤ _x f152_0_createList_LE 3 f152_0_createList_LE: x1 = _x5 ∧ x2 = _x6 ∧ x1 = _x7 ∧ x2 = _x8 ∧ _x6 − 1 = _x8 ∧ 0 ≤ _x7 − 1 ∧ −1 ≤ _x5 − 1 ∧ 0 ≤ _x6 − 1 ∧ _x7 − 2 ≤ _x5 f196_0_reverse_NULL 4 f196_0_reverse_NULL: x1 = _x9 ∧ x2 = _x10 ∧ x1 = _x11 ∧ x2 = _x12 ∧ −1 ≤ _x11 − 1 ∧ 0 ≤ _x9 − 1 ∧ _x11 + 1 ≤ _x9 __init 5 f1_0_main_Load: x1 = _x13 ∧ x2 = _x14 ∧ x1 = _x15 ∧ x2 = _x16 ∧ 0 ≤ 0

## Proof

### 1 Switch to Cooperation Termination Proof

We consider the following cutpoint-transitions:
 f152_0_createList_LE f152_0_createList_LE f152_0_createList_LE: x1 = x1 ∧ x2 = x2 f196_0_reverse_NULL f196_0_reverse_NULL f196_0_reverse_NULL: x1 = x1 ∧ x2 = x2 f1_0_main_Load f1_0_main_Load f1_0_main_Load: 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 2 SCC(s) of the program graph.

### 2.1 SCC Subproblem 1/2

Here we consider the SCC { f152_0_createList_LE }.

### 2.1.1 Transition Removal

We remove transition 3 using the following ranking functions, which are bounded by 0.

 f152_0_createList_LE: 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 { f196_0_reverse_NULL }.

### 2.2.1 Transition Removal

We remove transition 4 using the following ranking functions, which are bounded by 0.

 f196_0_reverse_NULL: x1

### 2.2.2 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 % (7 real / 0 unknown / 0 assumptions / 7 total proof steps)