# LTS Termination Proof

by AProVE

## Input

Integer Transition System
• Initial Location: l5, l4, l1, l0, l2
• Transitions: (pre-variables and post-variables)  l0 1 l1: x1 = ___const_5HAT0 ∧ x2 = _xHAT0 ∧ x1 = ___const_5HATpost ∧ x2 = _xHATpost ∧ ___const_5HAT0 = ___const_5HATpost ∧ _xHATpost = 1 ∧ _xHAT0 ≤ 0 l0 2 l1: x1 = _x ∧ x2 = _x1 ∧ x1 = _x2 ∧ x2 = _x3 ∧ _x = _x2 ∧ _x3 = 1 + _x1 ∧ 1 ≤ _x1 l2 3 l3: x1 = _x4 ∧ x2 = _x5 ∧ x1 = _x6 ∧ x2 = _x7 ∧ _x5 = _x7 ∧ _x4 = _x6 ∧ 4 ≤ _x5 l2 4 l0: x1 = _x8 ∧ x2 = _x9 ∧ x1 = _x10 ∧ x2 = _x11 ∧ _x9 = _x11 ∧ _x8 = _x10 ∧ 1 + _x9 ≤ 4 l1 5 l2: x1 = _x12 ∧ x2 = _x13 ∧ x1 = _x14 ∧ x2 = _x15 ∧ _x13 = _x15 ∧ _x12 = _x14 l4 6 l1: x1 = _x16 ∧ x2 = _x17 ∧ x1 = _x18 ∧ x2 = _x19 ∧ _x20 = _x16 ∧ _x19 = _x19 ∧ _x16 = _x18 l5 7 l4: x1 = _x21 ∧ x2 = _x22 ∧ x1 = _x23 ∧ x2 = _x24 ∧ _x22 = _x24 ∧ _x21 = _x23

## Proof

### 1 Switch to Cooperation Termination Proof

We consider the following cutpoint-transitions:
 l5 l5 l5: x1 = x1 ∧ x2 = x2 l4 l4 l4: x1 = x1 ∧ x2 = x2 l1 l1 l1: x1 = x1 ∧ x2 = x2 l0 l0 l0: x1 = x1 ∧ x2 = x2 l2 l2 l2: 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 { l1, l0, l2 }.

### 2.1.1 Transition Removal

We remove transition 4 using the following ranking functions, which are bounded by −7.

 l0: −3⋅x2 + 1 l1: −3⋅x2 + 3 l2: −3⋅x2 + 2

### 2.1.2 Transition Removal

We remove transitions 1, 5, 2 using the following ranking functions, which are bounded by −1.

 l0: 0 l1: −1 l2: −2

### 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)