# LTS Termination Proof

by AProVE

## Input

Integer Transition System
• Initial Location: l5, l6, l1, l3, l0, l2
• Transitions: (pre-variables and post-variables)  l0 1 l1: x1 = ___const_8HAT0 ∧ x2 = _nI6HAT0 ∧ x3 = _nX4HAT0 ∧ x4 = _nX9HAT0 ∧ x5 = _nXHAT0 ∧ x6 = _res10HAT0 ∧ x7 = _res5HAT0 ∧ x8 = _ret_nBC18HAT0 ∧ x9 = _ret_nBC211HAT0 ∧ x10 = _tmp7HAT0 ∧ x11 = _tmpHAT0 ∧ x12 = _tmp___0HAT0 ∧ x1 = ___const_8HATpost ∧ x2 = _nI6HATpost ∧ x3 = _nX4HATpost ∧ x4 = _nX9HATpost ∧ x5 = _nXHATpost ∧ x6 = _res10HATpost ∧ x7 = _res5HATpost ∧ x8 = _ret_nBC18HATpost ∧ x9 = _ret_nBC211HATpost ∧ x10 = _tmp7HATpost ∧ x11 = _tmpHATpost ∧ x12 = _tmp___0HATpost ∧ _tmp___0HAT0 = _tmp___0HATpost ∧ _tmp7HAT0 = _tmp7HATpost ∧ _tmpHAT0 = _tmpHATpost ∧ _ret_nBC211HAT0 = _ret_nBC211HATpost ∧ _ret_nBC18HAT0 = _ret_nBC18HATpost ∧ _res10HAT0 = _res10HATpost ∧ _nX9HAT0 = _nX9HATpost ∧ _nX4HAT0 = _nX4HATpost ∧ _nXHAT0 = _nXHATpost ∧ ___const_8HAT0 = ___const_8HATpost ∧ _nI6HATpost = 1 + _nI6HAT0 ∧ _res5HATpost = _res5HAT0 + _tmp7HAT0 l2 2 l0: x1 = _x ∧ x2 = _x1 ∧ x3 = _x2 ∧ x4 = _x3 ∧ x5 = _x4 ∧ x6 = _x5 ∧ x7 = _x6 ∧ x8 = _x7 ∧ x9 = _x8 ∧ x10 = _x9 ∧ x11 = _x10 ∧ x12 = _x11 ∧ x1 = _x12 ∧ x2 = _x13 ∧ x3 = _x14 ∧ x4 = _x15 ∧ x5 = _x16 ∧ x6 = _x17 ∧ x7 = _x18 ∧ x8 = _x19 ∧ x9 = _x20 ∧ x10 = _x21 ∧ x11 = _x22 ∧ x12 = _x23 ∧ _x11 = _x23 ∧ _x10 = _x22 ∧ _x8 = _x20 ∧ _x7 = _x19 ∧ _x6 = _x18 ∧ _x5 = _x17 ∧ _x3 = _x15 ∧ _x2 = _x14 ∧ _x4 = _x16 ∧ _x1 = _x13 ∧ _x = _x12 ∧ _x21 = 0 l2 3 l0: x1 = _x24 ∧ x2 = _x25 ∧ x3 = _x26 ∧ x4 = _x27 ∧ x5 = _x28 ∧ x6 = _x29 ∧ x7 = _x30 ∧ x8 = _x31 ∧ x9 = _x32 ∧ x10 = _x33 ∧ x11 = _x34 ∧ x12 = _x35 ∧ x1 = _x36 ∧ x2 = _x37 ∧ x3 = _x38 ∧ x4 = _x39 ∧ x5 = _x40 ∧ x6 = _x41 ∧ x7 = _x42 ∧ x8 = _x43 ∧ x9 = _x44 ∧ x10 = _x45 ∧ x11 = _x46 ∧ x12 = _x47 ∧ _x35 = _x47 ∧ _x34 = _x46 ∧ _x32 = _x44 ∧ _x31 = _x43 ∧ _x30 = _x42 ∧ _x29 = _x41 ∧ _x27 = _x39 ∧ _x26 = _x38 ∧ _x28 = _x40 ∧ _x25 = _x37 ∧ _x24 = _x36 ∧ _x45 = 1 l3 4 l4: x1 = _x48 ∧ x2 = _x49 ∧ x3 = _x50 ∧ x4 = _x51 ∧ x5 = _x52 ∧ x6 = _x53 ∧ x7 = _x54 ∧ x8 = _x55 ∧ x9 = _x56 ∧ x10 = _x57 ∧ x11 = _x58 ∧ x12 = _x59 ∧ x1 = _x60 ∧ x2 = _x61 ∧ x3 = _x62 ∧ x4 = _x63 ∧ x5 = _x64 ∧ x6 = _x65 ∧ x7 = _x66 ∧ x8 = _x67 ∧ x9 = _x68 ∧ x10 = _x69 ∧ x11 = _x70 ∧ x12 = _x71 ∧ 4⋅_x48 ≤ _x49 ∧ _x67 = _x54 ∧ _x70 = _x67 ∧ _x63 = _x52 ∧ _x72 = _x63 ∧ _x73 = _x73 ∧ _x74 = _x74 ∧ _x75 = _x75 ∧ _x76 = _x76 ∧ _x65 = _x65 ∧ _x68 = _x65 ∧ _x71 = _x68 ∧ _x48 = _x60 ∧ _x49 = _x61 ∧ _x52 = _x64 ∧ _x50 = _x62 ∧ _x54 = _x66 ∧ _x57 = _x69 l3 5 l2: x1 = _x77 ∧ x2 = _x78 ∧ x3 = _x79 ∧ x4 = _x80 ∧ x5 = _x81 ∧ x6 = _x82 ∧ x7 = _x83 ∧ x8 = _x84 ∧ x9 = _x85 ∧ x10 = _x86 ∧ x11 = _x87 ∧ x12 = _x88 ∧ x1 = _x89 ∧ x2 = _x90 ∧ x3 = _x91 ∧ x4 = _x92 ∧ x5 = _x93 ∧ x6 = _x94 ∧ x7 = _x95 ∧ x8 = _x96 ∧ x9 = _x97 ∧ x10 = _x98 ∧ x11 = _x99 ∧ x12 = _x100 ∧ _x88 = _x100 ∧ _x86 = _x98 ∧ _x87 = _x99 ∧ _x85 = _x97 ∧ _x84 = _x96 ∧ _x83 = _x95 ∧ _x82 = _x94 ∧ _x80 = _x92 ∧ _x79 = _x91 ∧ _x81 = _x93 ∧ _x78 = _x90 ∧ _x77 = _x89 ∧ 1 + _x78 ≤ 4⋅_x77 l1 6 l3: x1 = _x101 ∧ x2 = _x102 ∧ x3 = _x103 ∧ x4 = _x104 ∧ x5 = _x105 ∧ x6 = _x106 ∧ x7 = _x107 ∧ x8 = _x108 ∧ x9 = _x109 ∧ x10 = _x110 ∧ x11 = _x111 ∧ x12 = _x112 ∧ x1 = _x113 ∧ x2 = _x114 ∧ x3 = _x115 ∧ x4 = _x116 ∧ x5 = _x117 ∧ x6 = _x118 ∧ x7 = _x119 ∧ x8 = _x120 ∧ x9 = _x121 ∧ x10 = _x122 ∧ x11 = _x123 ∧ x12 = _x124 ∧ _x112 = _x124 ∧ _x110 = _x122 ∧ _x111 = _x123 ∧ _x109 = _x121 ∧ _x108 = _x120 ∧ _x107 = _x119 ∧ _x106 = _x118 ∧ _x104 = _x116 ∧ _x103 = _x115 ∧ _x105 = _x117 ∧ _x102 = _x114 ∧ _x101 = _x113 l5 7 l1: x1 = _x125 ∧ x2 = _x126 ∧ x3 = _x127 ∧ x4 = _x128 ∧ x5 = _x129 ∧ x6 = _x130 ∧ x7 = _x131 ∧ x8 = _x132 ∧ x9 = _x133 ∧ x10 = _x134 ∧ x11 = _x135 ∧ x12 = _x136 ∧ x1 = _x137 ∧ x2 = _x138 ∧ x3 = _x139 ∧ x4 = _x140 ∧ x5 = _x141 ∧ x6 = _x142 ∧ x7 = _x143 ∧ x8 = _x144 ∧ x9 = _x145 ∧ x10 = _x146 ∧ x11 = _x147 ∧ x12 = _x148 ∧ _x136 = _x148 ∧ _x134 = _x146 ∧ _x135 = _x147 ∧ _x133 = _x145 ∧ _x132 = _x144 ∧ _x130 = _x142 ∧ _x128 = _x140 ∧ _x129 = _x141 ∧ _x125 = _x137 ∧ _x138 = 0 ∧ _x143 = 0 ∧ _x139 = _x129 l6 8 l5: x1 = _x149 ∧ x2 = _x150 ∧ x3 = _x151 ∧ x4 = _x152 ∧ x5 = _x153 ∧ x6 = _x154 ∧ x7 = _x155 ∧ x8 = _x156 ∧ x9 = _x157 ∧ x10 = _x158 ∧ x11 = _x159 ∧ x12 = _x160 ∧ x1 = _x161 ∧ x2 = _x162 ∧ x3 = _x163 ∧ x4 = _x164 ∧ x5 = _x165 ∧ x6 = _x166 ∧ x7 = _x167 ∧ x8 = _x168 ∧ x9 = _x169 ∧ x10 = _x170 ∧ x11 = _x171 ∧ x12 = _x172 ∧ _x160 = _x172 ∧ _x158 = _x170 ∧ _x159 = _x171 ∧ _x157 = _x169 ∧ _x156 = _x168 ∧ _x155 = _x167 ∧ _x154 = _x166 ∧ _x152 = _x164 ∧ _x151 = _x163 ∧ _x153 = _x165 ∧ _x150 = _x162 ∧ _x149 = _x161

## Proof

### 1 Switch to Cooperation Termination Proof

We consider the following cutpoint-transitions:
 l5 l5 l5: x1 = x1 ∧ x2 = x2 ∧ x3 = x3 ∧ x4 = x4 ∧ x5 = x5 ∧ x6 = x6 ∧ x7 = x7 ∧ x8 = x8 ∧ x9 = x9 ∧ x10 = x10 ∧ x11 = x11 ∧ x12 = x12 l6 l6 l6: x1 = x1 ∧ x2 = x2 ∧ x3 = x3 ∧ x4 = x4 ∧ x5 = x5 ∧ x6 = x6 ∧ x7 = x7 ∧ x8 = x8 ∧ x9 = x9 ∧ x10 = x10 ∧ x11 = x11 ∧ x12 = x12 l1 l1 l1: x1 = x1 ∧ x2 = x2 ∧ x3 = x3 ∧ x4 = x4 ∧ x5 = x5 ∧ x6 = x6 ∧ x7 = x7 ∧ x8 = x8 ∧ x9 = x9 ∧ x10 = x10 ∧ x11 = x11 ∧ x12 = x12 l3 l3 l3: x1 = x1 ∧ x2 = x2 ∧ x3 = x3 ∧ x4 = x4 ∧ x5 = x5 ∧ x6 = x6 ∧ x7 = x7 ∧ x8 = x8 ∧ x9 = x9 ∧ x10 = x10 ∧ x11 = x11 ∧ x12 = x12 l0 l0 l0: x1 = x1 ∧ x2 = x2 ∧ x3 = x3 ∧ x4 = x4 ∧ x5 = x5 ∧ x6 = x6 ∧ x7 = x7 ∧ x8 = x8 ∧ x9 = x9 ∧ x10 = x10 ∧ x11 = x11 ∧ x12 = x12 l2 l2 l2: x1 = x1 ∧ x2 = x2 ∧ x3 = x3 ∧ x4 = x4 ∧ x5 = x5 ∧ x6 = x6 ∧ x7 = x7 ∧ x8 = x8 ∧ x9 = x9 ∧ x10 = x10 ∧ x11 = x11 ∧ x12 = x12
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 { l1, l3, l0, l2 }.

### 2.1.1 Transition Removal

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

 l0: −2 + 4⋅x1 − x2 l1: −1 + 4⋅x1 − x2 l2: −2 + 4⋅x1 − x2 l3: −1 + 4⋅x1 − x2

### 2.1.2 Transition Removal

We remove transitions 1, 3, 2, 6 using the following ranking functions, which are bounded by 0.

 l0: 2 l1: 1 l2: 3 l3: 0

### 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)