# LTS Termination Proof

by AProVE

## Input

Integer Transition System
• Initial Location: f163_0_factorial_GT, f149_0_doSum_LT, f1_0_main_ConstantStackPush, __init
• Transitions: (pre-variables and post-variables)  f1_0_main_ConstantStackPush 1 f149_0_doSum_LT: x1 = _arg1 ∧ x1 = _arg1P ∧ 10 = _arg1P f149_0_doSum_LT 2 f163_0_factorial_GT: x1 = _x ∧ x1 = _x1 ∧ _x = _x1 ∧ −1 ≤ _x − 1 f149_0_doSum_LT 3 f149_0_doSum_LT: x1 = _x2 ∧ x1 = _x3 ∧ _x2 − 1 = _x3 ∧ −1 ≤ _x2 − 1 f163_0_factorial_GT 4 f163_0_factorial_GT: x1 = _x4 ∧ x1 = _x5 ∧ _x4 − 1 = _x5 ∧ _x4 − 1 ≤ _x4 − 1 ∧ 0 ≤ _x4 − 1 __init 5 f1_0_main_ConstantStackPush: x1 = _x6 ∧ x1 = _x7 ∧ 0 ≤ 0

## Proof

### 1 Switch to Cooperation Termination Proof

We consider the following cutpoint-transitions:
 f163_0_factorial_GT f163_0_factorial_GT f163_0_factorial_GT: x1 = x1 f149_0_doSum_LT f149_0_doSum_LT f149_0_doSum_LT: x1 = x1 f1_0_main_ConstantStackPush f1_0_main_ConstantStackPush f1_0_main_ConstantStackPush: x1 = x1 __init __init __init: x1 = x1
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 { f149_0_doSum_LT }.

### 2.1.1 Transition Removal

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

 f149_0_doSum_LT: x1

### 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 { f163_0_factorial_GT }.

### 2.2.1 Transition Removal

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

 f163_0_factorial_GT: 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 % (6 real / 0 unknown / 0 assumptions / 6 total proof steps)