# LTS Termination Proof

by AProVE

## Input

Integer Transition System
• Initial Location: f1_0_main_New, f83_0_doSum_NONNULL, f160_0_factorial_GT, __init
• Transitions: (pre-variables and post-variables)  f1_0_main_New 1 f83_0_doSum_NONNULL: x1 = _arg1 ∧ x2 = _arg2 ∧ x3 = _arg3 ∧ x4 = _arg4 ∧ x5 = _arg5 ∧ x6 = _arg6 ∧ x1 = _arg1P ∧ x2 = _arg2P ∧ x3 = _arg3P ∧ x4 = _arg4P ∧ x5 = _arg5P ∧ x6 = _arg6P ∧ 3 ≤ _arg1P − 1 f83_0_doSum_NONNULL 2 f160_0_factorial_GT: x1 = _x ∧ x2 = _x1 ∧ x3 = _x2 ∧ x4 = _x3 ∧ x5 = _x4 ∧ x6 = _x5 ∧ x1 = _x6 ∧ x2 = _x7 ∧ x3 = _x8 ∧ x4 = _x9 ∧ x5 = _x10 ∧ x6 = _x11 ∧ 1 = _x9 ∧ 1 = _x8 ∧ 1 = _x7 ∧ _x10 + 2 ≤ _x ∧ −1 ≤ _x11 − 1 ∧ 0 ≤ _x6 − 1 ∧ 0 ≤ _x − 1 ∧ _x11 + 1 ≤ _x ∧ _x6 ≤ _x f160_0_factorial_GT 3 f83_0_doSum_NONNULL: x1 = _x12 ∧ x2 = _x13 ∧ x3 = _x14 ∧ x4 = _x15 ∧ x5 = _x16 ∧ x6 = _x17 ∧ x1 = _x18 ∧ x2 = _x19 ∧ x3 = _x20 ∧ x4 = _x21 ∧ x5 = _x22 ∧ x6 = _x23 ∧ _x14 = _x15 ∧ _x16 + 2 ≤ _x12 ∧ −1 ≤ _x18 − 1 ∧ −1 ≤ _x17 − 1 ∧ 0 ≤ _x12 − 1 ∧ _x18 ≤ _x17 ∧ _x18 + 1 ≤ _x12 ∧ _x16 ≤ _x14 − 1 ∧ 0 ≤ _x13 − 1 f160_0_factorial_GT 4 f160_0_factorial_GT: x1 = _x24 ∧ x2 = _x25 ∧ x3 = _x26 ∧ x4 = _x27 ∧ x5 = _x28 ∧ x6 = _x29 ∧ x1 = _x30 ∧ x2 = _x31 ∧ x3 = _x32 ∧ x4 = _x33 ∧ x5 = _x34 ∧ x6 = _x35 ∧ _x28 = _x34 ∧ _x26 + 1 = _x33 ∧ _x26 + 1 = _x32 ∧ _x25⋅_x26 = _x31 ∧ _x26 = _x27 ∧ _x28 + 2 ≤ _x24 ∧ −1 ≤ _x35 − 1 ∧ 0 ≤ _x30 − 1 ∧ −1 ≤ _x29 − 1 ∧ 0 ≤ _x24 − 1 ∧ _x35 ≤ _x29 ∧ _x35 + 1 ≤ _x24 ∧ _x30 ≤ _x24 ∧ 0 ≤ _x25 − 1 ∧ 0 ≤ _x26 − 1 ∧ _x26 ≤ _x28 __init 5 f1_0_main_New: x1 = _x36 ∧ x2 = _x37 ∧ x3 = _x38 ∧ x4 = _x39 ∧ x5 = _x40 ∧ x6 = _x41 ∧ x1 = _x42 ∧ x2 = _x43 ∧ x3 = _x44 ∧ x4 = _x45 ∧ x5 = _x46 ∧ x6 = _x47 ∧ 0 ≤ 0

## Proof

### 1 Switch to Cooperation Termination Proof

We consider the following cutpoint-transitions:
 f1_0_main_New f1_0_main_New f1_0_main_New: x1 = x1 ∧ x2 = x2 ∧ x3 = x3 ∧ x4 = x4 ∧ x5 = x5 ∧ x6 = x6 f83_0_doSum_NONNULL f83_0_doSum_NONNULL f83_0_doSum_NONNULL: x1 = x1 ∧ x2 = x2 ∧ x3 = x3 ∧ x4 = x4 ∧ x5 = x5 ∧ x6 = x6 f160_0_factorial_GT f160_0_factorial_GT f160_0_factorial_GT: x1 = x1 ∧ x2 = x2 ∧ x3 = x3 ∧ x4 = x4 ∧ x5 = x5 ∧ x6 = x6 __init __init __init: x1 = x1 ∧ x2 = x2 ∧ x3 = x3 ∧ x4 = x4 ∧ x5 = x5 ∧ x6 = x6
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 { f83_0_doSum_NONNULL, f160_0_factorial_GT }.

### 2.1.1 Transition Removal

We remove transitions 2, 3 using the following ranking functions, which are bounded by 0.

 f83_0_doSum_NONNULL: 1 + 2⋅x1 f160_0_factorial_GT: 2⋅x1

### 2.1.2 Transition Removal

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

 f160_0_factorial_GT: − x4 + x5

### 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)