# LTS Termination Proof

by AProVE

## Input

Integer Transition System
• Initial Location: l4, l3, l0, l2
• Transitions: (pre-variables and post-variables)  l0 1 l1: x1 = ___const_20HAT0 ∧ x2 = _iHAT0 ∧ x3 = _jHAT0 ∧ x1 = ___const_20HATpost ∧ x2 = _iHATpost ∧ x3 = _jHATpost ∧ _jHAT0 = _jHATpost ∧ _iHAT0 = _iHATpost ∧ ___const_20HAT0 = ___const_20HATpost ∧ ___const_20HAT0 ≤ _jHAT0 l0 2 l2: x1 = _x ∧ x2 = _x1 ∧ x3 = _x2 ∧ x1 = _x3 ∧ x2 = _x4 ∧ x3 = _x5 ∧ _x2 = _x5 ∧ _x1 = _x4 ∧ _x = _x3 ∧ 1 + _x2 ≤ _x l2 3 l0: x1 = _x6 ∧ x2 = _x7 ∧ x3 = _x8 ∧ x1 = _x9 ∧ x2 = _x10 ∧ x3 = _x11 ∧ _x6 = _x9 ∧ _x10 = 2 + _x11 ∧ _x11 = 2 + _x8 l3 4 l2: x1 = _x12 ∧ x2 = _x13 ∧ x3 = _x14 ∧ x1 = _x15 ∧ x2 = _x16 ∧ x3 = _x17 ∧ _x13 = _x16 ∧ _x12 = _x15 ∧ _x17 = 0 l4 5 l3: x1 = _x18 ∧ x2 = _x19 ∧ x3 = _x20 ∧ x1 = _x21 ∧ x2 = _x22 ∧ x3 = _x23 ∧ _x20 = _x23 ∧ _x19 = _x22 ∧ _x18 = _x21

## Proof

### 1 Switch to Cooperation Termination Proof

We consider the following cutpoint-transitions:
 l4 l4 l4: x1 = x1 ∧ x2 = x2 ∧ x3 = x3 l3 l3 l3: x1 = x1 ∧ x2 = x2 ∧ x3 = x3 l0 l0 l0: x1 = x1 ∧ x2 = x2 ∧ x3 = x3 l2 l2 l2: 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 1 SCC(s) of the program graph.

### 2.1 SCC Subproblem 1/1

Here we consider the SCC { l0, l2 }.

### 2.1.1 Transition Removal

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

 l0: −2⋅x3 + 2⋅x1 l2: 2⋅x1 − 2⋅x3 − 1

### 2.1.2 Transition Removal

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

 l2: 0 l0: −1

### 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 % (6 real / 0 unknown / 0 assumptions / 6 total proof steps)