# LTS Termination Proof

by AProVE

## Input

Integer Transition System
• Initial Location: f292_0_main_LT, f101_0_main_GE, f422_0_main_GE, f1_0_main_ConstantStackPush, f255_0_main_GE, f448_0_main_LT, __init
• Transitions: (pre-variables and post-variables)  f1_0_main_ConstantStackPush 1 f101_0_main_GE: x1 = _arg1 ∧ x2 = _arg2 ∧ x3 = _arg3 ∧ x4 = _arg4 ∧ x5 = _arg5 ∧ x6 = _arg6 ∧ x7 = _arg7 ∧ x8 = _arg8 ∧ x9 = _arg9 ∧ x1 = _arg1P ∧ x2 = _arg2P ∧ x3 = _arg3P ∧ x4 = _arg4P ∧ x5 = _arg5P ∧ x6 = _arg6P ∧ x7 = _arg7P ∧ x8 = _arg8P ∧ x9 = _arg9P ∧ _arg2 = _arg3P ∧ 0 = _arg2P ∧ 0 ≤ _arg1P − 1 ∧ 0 ≤ _arg1 − 1 ∧ −1 ≤ _arg2 − 1 ∧ _arg1P ≤ _arg1 f101_0_main_GE 2 f255_0_main_GE: x1 = _x ∧ x2 = _x1 ∧ x3 = _x2 ∧ x4 = _x3 ∧ x5 = _x4 ∧ x6 = _x5 ∧ x7 = _x6 ∧ x8 = _x7 ∧ x9 = _x8 ∧ x1 = _x9 ∧ x2 = _x10 ∧ x3 = _x11 ∧ x4 = _x12 ∧ x5 = _x13 ∧ x6 = _x14 ∧ x7 = _x15 ∧ x8 = _x16 ∧ x9 = _x17 ∧ _x2 = _x13 ∧ 2⋅_x1 = _x12 ∧ 0 = _x11 ∧ _x1 = _x10 ∧ 0 ≤ _x9 − 1 ∧ 0 ≤ _x − 1 ∧ _x1 ≤ _x2 − 1 ∧ _x9 ≤ _x f255_0_main_GE 3 f101_0_main_GE: x1 = _x18 ∧ x2 = _x19 ∧ x3 = _x20 ∧ x4 = _x21 ∧ x5 = _x22 ∧ x6 = _x23 ∧ x7 = _x24 ∧ x8 = _x25 ∧ x9 = _x26 ∧ x1 = _x27 ∧ x2 = _x28 ∧ x3 = _x29 ∧ x4 = _x30 ∧ x5 = _x31 ∧ x6 = _x32 ∧ x7 = _x33 ∧ x8 = _x34 ∧ x9 = _x35 ∧ _x22 = _x29 ∧ _x19 + 1 = _x28 ∧ 0 ≤ _x27 − 1 ∧ 0 ≤ _x18 − 1 ∧ _x27 ≤ _x18 ∧ −1 ≤ _x22 − 1 ∧ _x21 ≤ _x20 f255_0_main_GE 4 f292_0_main_LT: x1 = _x36 ∧ x2 = _x37 ∧ x3 = _x38 ∧ x4 = _x39 ∧ x5 = _x40 ∧ x6 = _x41 ∧ x7 = _x42 ∧ x8 = _x43 ∧ x9 = _x44 ∧ x1 = _x45 ∧ x2 = _x46 ∧ x3 = _x47 ∧ x4 = _x48 ∧ x5 = _x49 ∧ x6 = _x50 ∧ x7 = _x51 ∧ x8 = _x52 ∧ x9 = _x53 ∧ _x40 = _x50 ∧ _x37 + _x38 = _x49 ∧ _x38 = _x48 ∧ _x39 = _x47 ∧ _x37 = _x46 ∧ 0 ≤ _x45 − 1 ∧ 0 ≤ _x36 − 1 ∧ _x45 ≤ _x36 ∧ −1 ≤ _x37 − 1 ∧ −1 ≤ _x38 − 1 ∧ _x38 ≤ _x39 − 1 f292_0_main_LT 5 f255_0_main_GE: x1 = _x54 ∧ x2 = _x55 ∧ x3 = _x56 ∧ x4 = _x57 ∧ x5 = _x58 ∧ x6 = _x59 ∧ x7 = _x60 ∧ x8 = _x61 ∧ x9 = _x62 ∧ x1 = _x63 ∧ x2 = _x64 ∧ x3 = _x65 ∧ x4 = _x66 ∧ x5 = _x67 ∧ x6 = _x68 ∧ x7 = _x69 ∧ x8 = _x70 ∧ x9 = _x71 ∧ _x59 = _x67 ∧ _x56 = _x66 ∧ _x57 + 1 = _x65 ∧ _x55 = _x64 ∧ 0 ≤ _x63 − 1 ∧ 0 ≤ _x54 − 1 ∧ _x58 ≤ −1 ∧ _x63 ≤ _x54 f292_0_main_LT 6 f422_0_main_GE: x1 = _x72 ∧ x2 = _x73 ∧ x3 = _x74 ∧ x4 = _x75 ∧ x5 = _x76 ∧ x6 = _x77 ∧ x7 = _x78 ∧ x8 = _x79 ∧ x9 = _x80 ∧ x1 = _x81 ∧ x2 = _x82 ∧ x3 = _x83 ∧ x4 = _x84 ∧ x5 = _x85 ∧ x6 = _x86 ∧ x7 = _x87 ∧ x8 = _x88 ∧ x9 = _x89 ∧ _x77 = _x88 ∧ 2⋅_x73 + 3⋅_x75 + 4⋅_x76 = _x87 ∧ 0 = _x86 ∧ _x76 = _x85 ∧ _x75 = _x84 ∧ _x74 = _x83 ∧ _x73 = _x82 ∧ 0 ≤ _x81 − 1 ∧ 0 ≤ _x72 − 1 ∧ _x81 ≤ _x72 ∧ 0 ≤ 4⋅_x76 ∧ 0 ≤ 2⋅_x73 + 3⋅_x75 ∧ 0 ≤ 2⋅_x73 ∧ −1 ≤ _x76 − 1 ∧ 0 ≤ 3⋅_x75 f422_0_main_GE 7 f292_0_main_LT: x1 = _x90 ∧ x2 = _x91 ∧ x3 = _x92 ∧ x4 = _x93 ∧ x5 = _x94 ∧ x6 = _x95 ∧ x7 = _x96 ∧ x8 = _x97 ∧ x9 = _x98 ∧ x1 = _x99 ∧ x2 = _x100 ∧ x3 = _x101 ∧ x4 = _x102 ∧ x5 = _x103 ∧ x6 = _x104 ∧ x7 = _x105 ∧ x8 = _x106 ∧ x9 = _x107 ∧ _x97 = _x104 ∧ _x94 − 1 = _x103 ∧ _x93 = _x102 ∧ _x92 = _x101 ∧ _x91 = _x100 ∧ 0 ≤ _x99 − 1 ∧ 0 ≤ _x90 − 1 ∧ _x96 ≤ _x95 ∧ _x99 ≤ _x90 f422_0_main_GE 8 f448_0_main_LT: x1 = _x108 ∧ x2 = _x109 ∧ x3 = _x110 ∧ x4 = _x111 ∧ x5 = _x112 ∧ x6 = _x113 ∧ x7 = _x114 ∧ x8 = _x115 ∧ x9 = _x116 ∧ x1 = _x117 ∧ x2 = _x118 ∧ x3 = _x119 ∧ x4 = _x120 ∧ x5 = _x121 ∧ x6 = _x122 ∧ x7 = _x123 ∧ x8 = _x124 ∧ x9 = _x125 ∧ _x115 = _x125 ∧ 1000⋅_x109 + 100⋅_x111 + 10⋅_x112 + _x113 = _x124 ∧ _x113 = _x123 ∧ _x114 = _x122 ∧ _x112 = _x121 ∧ _x111 = _x120 ∧ _x110 = _x119 ∧ _x109 = _x118 ∧ 0 ≤ _x117 − 1 ∧ 0 ≤ _x108 − 1 ∧ _x117 ≤ _x108 ∧ −1 ≤ _x113 − 1 ∧ 0 ≤ 1000⋅_x109 + 100⋅_x111 + 10⋅_x112 ∧ 0 ≤ 1000⋅_x109 + 100⋅_x111 ∧ 0 ≤ 10⋅_x112 ∧ 0 ≤ 1000⋅_x109 ∧ _x113 ≤ _x114 − 1 ∧ 0 ≤ 100⋅_x111 f448_0_main_LT 9 f422_0_main_GE: x1 = _x126 ∧ x2 = _x127 ∧ x3 = _x128 ∧ x4 = _x129 ∧ x5 = _x130 ∧ x6 = _x131 ∧ x7 = _x132 ∧ x8 = _x133 ∧ x9 = _x134 ∧ x1 = _x135 ∧ x2 = _x136 ∧ x3 = _x137 ∧ x4 = _x138 ∧ x5 = _x139 ∧ x6 = _x140 ∧ x7 = _x141 ∧ x8 = _x142 ∧ x9 = _x143 ∧ _x134 = _x142 ∧ _x131 = _x141 ∧ _x132 + 1 = _x140 ∧ _x130 = _x139 ∧ _x129 = _x138 ∧ _x128 = _x137 ∧ _x127 = _x136 ∧ 0 ≤ _x135 − 1 ∧ 0 ≤ _x126 − 1 ∧ _x133 ≤ −1 ∧ _x135 ≤ _x126 f448_0_main_LT 10 f448_0_main_LT: x1 = _x144 ∧ x2 = _x145 ∧ x3 = _x146 ∧ x4 = _x147 ∧ x5 = _x148 ∧ x6 = _x149 ∧ x7 = _x150 ∧ x8 = _x151 ∧ x9 = _x152 ∧ x1 = _x153 ∧ x2 = _x154 ∧ x3 = _x155 ∧ x4 = _x156 ∧ x5 = _x157 ∧ x6 = _x158 ∧ x7 = _x159 ∧ x8 = _x160 ∧ x9 = _x161 ∧ _x152 = _x161 ∧ _x151 − 1 = _x160 ∧ _x150 = _x159 ∧ _x149 = _x158 ∧ _x148 = _x157 ∧ _x147 = _x156 ∧ _x146 = _x155 ∧ _x145 = _x154 ∧ 0 ≤ _x153 − 1 ∧ 0 ≤ _x144 − 1 ∧ −1 ≤ _x151 − 1 ∧ _x153 ≤ _x144 __init 11 f1_0_main_ConstantStackPush: x1 = _x162 ∧ x2 = _x163 ∧ x3 = _x164 ∧ x4 = _x165 ∧ x5 = _x166 ∧ x6 = _x167 ∧ x7 = _x168 ∧ x8 = _x169 ∧ x9 = _x170 ∧ x1 = _x171 ∧ x2 = _x172 ∧ x3 = _x173 ∧ x4 = _x174 ∧ x5 = _x175 ∧ x6 = _x176 ∧ x7 = _x177 ∧ x8 = _x178 ∧ x9 = _x179 ∧ 0 ≤ 0

## Proof

### 1 Switch to Cooperation Termination Proof

We consider the following cutpoint-transitions:
 f292_0_main_LT f292_0_main_LT f292_0_main_LT: x1 = x1 ∧ x2 = x2 ∧ x3 = x3 ∧ x4 = x4 ∧ x5 = x5 ∧ x6 = x6 ∧ x7 = x7 ∧ x8 = x8 ∧ x9 = x9 f101_0_main_GE f101_0_main_GE f101_0_main_GE: x1 = x1 ∧ x2 = x2 ∧ x3 = x3 ∧ x4 = x4 ∧ x5 = x5 ∧ x6 = x6 ∧ x7 = x7 ∧ x8 = x8 ∧ x9 = x9 f422_0_main_GE f422_0_main_GE f422_0_main_GE: x1 = x1 ∧ x2 = x2 ∧ x3 = x3 ∧ x4 = x4 ∧ x5 = x5 ∧ x6 = x6 ∧ x7 = x7 ∧ x8 = x8 ∧ x9 = x9 f1_0_main_ConstantStackPush f1_0_main_ConstantStackPush f1_0_main_ConstantStackPush: x1 = x1 ∧ x2 = x2 ∧ x3 = x3 ∧ x4 = x4 ∧ x5 = x5 ∧ x6 = x6 ∧ x7 = x7 ∧ x8 = x8 ∧ x9 = x9 f255_0_main_GE f255_0_main_GE f255_0_main_GE: x1 = x1 ∧ x2 = x2 ∧ x3 = x3 ∧ x4 = x4 ∧ x5 = x5 ∧ x6 = x6 ∧ x7 = x7 ∧ x8 = x8 ∧ x9 = x9 f448_0_main_LT f448_0_main_LT f448_0_main_LT: x1 = x1 ∧ x2 = x2 ∧ x3 = x3 ∧ x4 = x4 ∧ x5 = x5 ∧ x6 = x6 ∧ x7 = x7 ∧ x8 = x8 ∧ x9 = x9 __init __init __init: x1 = x1 ∧ x2 = x2 ∧ x3 = x3 ∧ x4 = x4 ∧ x5 = x5 ∧ x6 = x6 ∧ x7 = x7 ∧ x8 = x8 ∧ x9 = x9
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 { f292_0_main_LT, f101_0_main_GE, f422_0_main_GE, f255_0_main_GE, f448_0_main_LT }.

### 2.1.1 Transition Removal

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

 f101_0_main_GE: −2⋅x2 + 2⋅x3 + 1 f255_0_main_GE: −2⋅x2 + 2⋅x5 f292_0_main_LT: −2⋅x2 + 2⋅x6 f422_0_main_GE: −2⋅x2 + 2⋅x8 f448_0_main_LT: −2⋅x2 + 2⋅x9

### 2.1.2 Transition Removal

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

 f255_0_main_GE: x5 f101_0_main_GE: −1 + x3 f292_0_main_LT: x6 f422_0_main_GE: x8 f448_0_main_LT: x9

### 2.1.3 Transition Removal

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

 f292_0_main_LT: −1 + x2 + x3 − x4 f255_0_main_GE: x2 − x3 + x4 f422_0_main_GE: −1 + x2 + x3 − x4 f448_0_main_LT: −1 + x2 + x3 − x4

### 2.1.4 Transition Removal

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

 f292_0_main_LT: 0 f255_0_main_GE: −1 f422_0_main_GE: 0 f448_0_main_LT: 0

### 2.1.5 Transition Removal

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

 f422_0_main_GE: −1 + 9⋅x2 + x4 + x5 f292_0_main_LT: 9⋅x2 + x4 + x5 f448_0_main_LT: −1 + 9⋅x2 + x4 + x5

### 2.1.6 Transition Removal

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

 f422_0_main_GE: 0 f292_0_main_LT: −1 f448_0_main_LT: 0

### 2.1.7 Transition Removal

We remove transition 8 using the following ranking functions, which are bounded by 0.

 f448_0_main_LT: −1 + 100⋅x2 + 10⋅x4 + x5 + x6 − x7 f422_0_main_GE: 100⋅x2 + 10⋅x4 + x5 − x6 + x7

### 2.1.8 Transition Removal

We remove transition 9 using the following ranking functions, which are bounded by 0.

 f448_0_main_LT: 0 f422_0_main_GE: −1

### 2.1.9 Transition Removal

We remove transition 10 using the following ranking functions, which are bounded by 0.

 f448_0_main_LT: x8

### 2.1.10 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 % (13 real / 0 unknown / 0 assumptions / 13 total proof steps)