# LTS Termination Proof

by AProVE

## Input

Integer Transition System
• Initial Location: f1314_0_main_LT, f1_0_main_Load, __init
• Transitions: (pre-variables and post-variables)  f1_0_main_Load 1 f1314_0_main_LT: x1 = _arg1 ∧ x2 = _arg2 ∧ x3 = _arg3 ∧ x4 = _arg4 ∧ x5 = _arg5 ∧ x6 = _arg6 ∧ x7 = _arg7 ∧ x8 = _arg8 ∧ x1 = _arg1P ∧ x2 = _arg2P ∧ x3 = _arg3P ∧ x4 = _arg4P ∧ x5 = _arg5P ∧ x6 = _arg6P ∧ x7 = _arg7P ∧ x8 = _arg8P ∧ 0 = _arg8P ∧ 0 = _arg7P ∧ 0 = _arg6P ∧ 0 = _arg5P ∧ 0 = _arg4P ∧ 0 = _arg3P ∧ 0 = _arg2P ∧ 0 = _arg2 ∧ 0 ≤ _arg1P − 1 ∧ 0 ≤ _arg1 − 1 ∧ _arg1P ≤ _arg1 f1_0_main_Load 2 f1314_0_main_LT: x1 = _x ∧ x2 = _x1 ∧ x3 = _x2 ∧ x4 = _x3 ∧ x5 = _x4 ∧ x6 = _x5 ∧ x7 = _x6 ∧ x8 = _x7 ∧ x1 = _x8 ∧ x2 = _x9 ∧ x3 = _x10 ∧ x4 = _x11 ∧ x5 = _x12 ∧ x6 = _x13 ∧ x7 = _x14 ∧ x8 = _x15 ∧ 1 = _x15 ∧ 1 = _x14 ∧ 1 = _x13 ∧ 0 = _x12 ∧ 0 = _x11 ∧ 0 = _x10 ∧ 1 = _x1 ∧ 0 ≤ _x8 − 1 ∧ 0 ≤ _x − 1 ∧ −1 ≤ _x9 − 1 ∧ _x8 ≤ _x f1_0_main_Load 3 f1314_0_main_LT: x1 = _x16 ∧ x2 = _x17 ∧ x3 = _x18 ∧ x4 = _x19 ∧ x5 = _x20 ∧ x6 = _x21 ∧ x7 = _x22 ∧ x8 = _x23 ∧ x1 = _x24 ∧ x2 = _x25 ∧ x3 = _x26 ∧ x4 = _x27 ∧ x5 = _x28 ∧ x6 = _x29 ∧ x7 = _x30 ∧ x8 = _x31 ∧ 2 = _x31 ∧ 2 = _x30 ∧ 2 = _x29 ∧ 0 = _x28 ∧ 0 = _x27 ∧ 2 = _x17 ∧ 0 ≤ _x24 − 1 ∧ 0 ≤ _x16 − 1 ∧ _x24 ≤ _x16 ∧ −1 ≤ _x26 − 1 ∧ −1 ≤ _x25 − 1 f1_0_main_Load 4 f1314_0_main_LT: x1 = _x32 ∧ x2 = _x33 ∧ x3 = _x34 ∧ x4 = _x35 ∧ x5 = _x36 ∧ x6 = _x37 ∧ x7 = _x38 ∧ x8 = _x39 ∧ x1 = _x40 ∧ x2 = _x41 ∧ x3 = _x42 ∧ x4 = _x43 ∧ x5 = _x44 ∧ x6 = _x45 ∧ x7 = _x46 ∧ x8 = _x47 ∧ _x33 = _x47 ∧ 3 = _x46 ∧ _x33 = _x45 ∧ _x43 = _x44 ∧ 0 ≤ _x40 − 1 ∧ 0 ≤ _x32 − 1 ∧ _x40 ≤ _x32 ∧ −1 ≤ _x42 − 1 ∧ −1 ≤ _x43 − 1 ∧ 2 ≤ _x33 − 1 ∧ −1 ≤ _x41 − 1 f1314_0_main_LT 5 f1314_0_main_LT: x1 = _x48 ∧ x2 = _x49 ∧ x3 = _x50 ∧ x4 = _x51 ∧ x5 = _x52 ∧ x6 = _x53 ∧ x7 = _x54 ∧ x8 = _x55 ∧ x1 = _x56 ∧ x2 = _x57 ∧ x3 = _x58 ∧ x4 = _x59 ∧ x5 = _x60 ∧ x6 = _x61 ∧ x7 = _x62 ∧ x8 = _x63 ∧ _x53 = _x63 ∧ _x54 = _x62 ∧ _x53 = _x61 ∧ _x51 = _x60 ∧ _x51 = _x59 ∧ 0 = _x58 ∧ 0 = _x57 ∧ _x53 = _x55 ∧ _x51 = _x52 ∧ 0 ≤ _x56 − 1 ∧ 0 ≤ _x48 − 1 ∧ _x56 ≤ _x48 ∧ 3⋅_x50 − 2⋅_x49 ≤ −1 ∧ 0 ≤ 3⋅_x50 ∧ 0 ≤ 2⋅_x49 ∧ _x53 ≤ _x54 ∧ −1 ≤ _x53 − 1 ∧ −1 ≤ _x51 − 1 f1314_0_main_LT 6 f1314_0_main_LT: x1 = _x64 ∧ x2 = _x65 ∧ x3 = _x66 ∧ x4 = _x67 ∧ x5 = _x68 ∧ x6 = _x69 ∧ x7 = _x70 ∧ x8 = _x71 ∧ x1 = _x72 ∧ x2 = _x73 ∧ x3 = _x74 ∧ x4 = _x75 ∧ x5 = _x76 ∧ x6 = _x77 ∧ x7 = _x78 ∧ x8 = _x79 ∧ _x69 = _x79 ∧ _x70 + 1 = _x78 ∧ _x69 = _x77 ∧ _x67 − 0 − 2⋅_x73 = _x76 ∧ _x67 − 0 − 2⋅_x73 = _x75 ∧ 0 = _x74 ∧ _x69 = _x71 ∧ _x67 = _x68 ∧ 0 ≤ _x72 − 1 ∧ 0 ≤ _x64 − 1 ∧ _x72 ≤ _x64 ∧ 3⋅_x66 − 2⋅_x65 ≤ 0 − 2⋅_x73 − 1 ∧ 0 ≤ 2⋅_x73 ∧ 0 ≤ 3⋅_x66 ∧ 0 ≤ 2⋅_x65 ∧ _x69 ≤ _x70 + 1 ∧ −1 ≤ _x73 − 1 ∧ _x70 ≤ _x69 − 1 ∧ −1 ≤ _x70 − 1 ∧ −1 ≤ _x69 − 1 ∧ −1 ≤ _x67 − 1 f1314_0_main_LT 7 f1314_0_main_LT: x1 = _x80 ∧ x2 = _x81 ∧ x3 = _x82 ∧ x4 = _x83 ∧ x5 = _x84 ∧ x6 = _x85 ∧ x7 = _x86 ∧ x8 = _x87 ∧ x1 = _x88 ∧ x2 = _x89 ∧ x3 = _x90 ∧ x4 = _x91 ∧ x5 = _x92 ∧ x6 = _x93 ∧ x7 = _x94 ∧ x8 = _x95 ∧ _x85 = _x95 ∧ _x86 + 2 = _x94 ∧ _x85 = _x93 ∧ _x83 − 3⋅_x90 − 2⋅_x89 = _x92 ∧ _x83 − 3⋅_x90 − 2⋅_x89 = _x91 ∧ _x85 = _x87 ∧ _x83 = _x84 ∧ 0 ≤ _x88 − 1 ∧ 0 ≤ _x80 − 1 ∧ _x88 ≤ _x80 ∧ 3⋅_x82 − 2⋅_x81 ≤ 3⋅_x90 − 2⋅_x89 − 1 ∧ 0 ≤ 3⋅_x90 ∧ 0 ≤ 2⋅_x89 ∧ 0 ≤ 3⋅_x82 ∧ 0 ≤ 2⋅_x81 ∧ −1 ≤ _x90 − 1 ∧ −1 ≤ _x89 − 1 ∧ −1 ≤ _x86 − 1 ∧ −1 ≤ _x83 − 1 ∧ _x86 + 1 ≤ _x85 − 1 ∧ −1 ≤ _x85 − 1 __init 8 f1_0_main_Load: x1 = _x96 ∧ x2 = _x97 ∧ x3 = _x98 ∧ x4 = _x99 ∧ x5 = _x100 ∧ x6 = _x101 ∧ x7 = _x102 ∧ x8 = _x103 ∧ x1 = _x104 ∧ x2 = _x105 ∧ x3 = _x106 ∧ x4 = _x107 ∧ x5 = _x108 ∧ x6 = _x109 ∧ x7 = _x110 ∧ x8 = _x111 ∧ 0 ≤ 0

## Proof

### 1 Switch to Cooperation Termination Proof

We consider the following cutpoint-transitions:
 f1314_0_main_LT f1314_0_main_LT f1314_0_main_LT: x1 = x1 ∧ x2 = x2 ∧ x3 = x3 ∧ x4 = x4 ∧ x5 = x5 ∧ x6 = x6 ∧ x7 = x7 ∧ x8 = x8 f1_0_main_Load f1_0_main_Load f1_0_main_Load: x1 = x1 ∧ x2 = x2 ∧ x3 = x3 ∧ x4 = x4 ∧ x5 = x5 ∧ x6 = x6 ∧ x7 = x7 ∧ x8 = x8 __init __init __init: x1 = x1 ∧ x2 = x2 ∧ x3 = x3 ∧ x4 = x4 ∧ x5 = x5 ∧ x6 = x6 ∧ x7 = x7 ∧ x8 = x8
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 { f1314_0_main_LT }.

### 2.1.1 Transition Removal

We remove transitions 6, 7 using the following ranking functions, which are bounded by 0.

 f1314_0_main_LT: −1 − x7 + x8

### 2.1.2 Transition Removal

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

 f1314_0_main_LT: x2

### 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)