# LTS Termination Proof

by AProVE

## Input

Integer Transition System
• Initial Location: f1_0_main_Load, f180_0_ack_GT, __init
• Transitions: (pre-variables and post-variables)  f1_0_main_Load 1 f180_0_ack_GT: x1 = _arg1 ∧ x2 = _arg2 ∧ x1 = _arg1P ∧ x2 = _arg2P ∧ 0 = _arg2P ∧ 0 = _arg1P ∧ 0 = _arg2 ∧ 0 ≤ _arg1 − 1 f1_0_main_Load 2 f180_0_ack_GT: x1 = _x ∧ x2 = _x1 ∧ x1 = _x2 ∧ x2 = _x3 ∧ 0 = _x2 ∧ 1 = _x1 ∧ −1 ≤ _x3 − 1 ∧ 0 ≤ _x − 1 f1_0_main_Load 3 f180_0_ack_GT: x1 = _x4 ∧ x2 = _x5 ∧ x1 = _x6 ∧ x2 = _x7 ∧ 0 ≤ _x4 − 1 ∧ −1 ≤ _x7 − 1 ∧ 1 ≤ _x5 − 1 ∧ −1 ≤ _x6 − 1 f180_0_ack_GT 4 f180_0_ack_GT: x1 = _x8 ∧ x2 = _x9 ∧ x1 = _x10 ∧ x2 = _x11 ∧ _x9 − 1 = _x11 ∧ 1 = _x10 ∧ 0 = _x8 ∧ _x9 − 1 ≤ _x9 − 1 ∧ 0 ≤ _x9 − 1 f180_0_ack_GT 5 f180_0_ack_GT: x1 = _x12 ∧ x2 = _x13 ∧ x1 = _x14 ∧ x2 = _x15 ∧ _x13 = _x15 ∧ _x12 − 1 = _x14 ∧ 0 ≤ _x13 − 1 ∧ _x13 − 1 ≤ _x13 − 1 ∧ 0 ≤ _x12 − 1 f180_0_ack_GT 6 f180_0_ack_GT: x1 = _x16 ∧ x2 = _x17 ∧ x1 = _x18 ∧ x2 = _x19 ∧ _x17 − 1 = _x19 ∧ _x17 − 1 ≤ _x17 − 1 ∧ 0 ≤ _x18 − 1 ∧ 0 ≤ _x17 − 1 ∧ 0 ≤ _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:
 f1_0_main_Load f1_0_main_Load f1_0_main_Load: x1 = x1 ∧ x2 = x2 f180_0_ack_GT f180_0_ack_GT f180_0_ack_GT: 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 { f180_0_ack_GT }.

### 2.1.1 Transition Removal

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

 f180_0_ack_GT: x2

### 2.1.2 Transition Removal

We remove transition 5 using the following ranking functions, which are bounded by 0.

 f180_0_ack_GT: x1

### 2.1.3 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 % (5 real / 0 unknown / 0 assumptions / 5 total proof steps)