# LTS Termination Proof

by AProVE

## Input

Integer Transition System
• Initial Location: l5, l4, l6, l10, l8, l1, l3, l0, l2, l9
• Transitions: (pre-variables and post-variables)  l0 1 l1: x1 = ___const_1023HAT0 ∧ x2 = _bHAT0 ∧ x3 = _iHAT0 ∧ x4 = _jHAT0 ∧ x5 = _nHAT0 ∧ x6 = _tmpHAT0 ∧ x1 = ___const_1023HATpost ∧ x2 = _bHATpost ∧ x3 = _iHATpost ∧ x4 = _jHATpost ∧ x5 = _nHATpost ∧ x6 = _tmpHATpost ∧ _tmpHAT0 = _tmpHATpost ∧ _nHAT0 = _nHATpost ∧ _jHAT0 = _jHATpost ∧ _iHAT0 = _iHATpost ∧ _bHAT0 = _bHATpost ∧ ___const_1023HAT0 = ___const_1023HATpost ∧ 1 + _nHAT0 ≤ _iHAT0 l0 2 l2: x1 = _x ∧ x2 = _x1 ∧ x3 = _x2 ∧ x4 = _x3 ∧ x5 = _x4 ∧ x6 = _x5 ∧ x1 = _x6 ∧ x2 = _x7 ∧ x3 = _x8 ∧ x4 = _x9 ∧ x5 = _x10 ∧ x6 = _x11 ∧ _x5 = _x11 ∧ _x4 = _x10 ∧ _x1 = _x7 ∧ _x = _x6 ∧ _x9 = 2 + _x3 ∧ _x8 = 1 + _x2 ∧ _x2 ≤ _x4 l3 3 l2: x1 = _x12 ∧ x2 = _x13 ∧ x3 = _x14 ∧ x4 = _x15 ∧ x5 = _x16 ∧ x6 = _x17 ∧ x1 = _x18 ∧ x2 = _x19 ∧ x3 = _x20 ∧ x4 = _x21 ∧ x5 = _x22 ∧ x6 = _x23 ∧ _x17 = _x23 ∧ _x16 = _x22 ∧ _x15 = _x21 ∧ _x13 = _x19 ∧ _x12 = _x18 ∧ _x20 = 0 l4 4 l3: x1 = _x24 ∧ x2 = _x25 ∧ x3 = _x26 ∧ x4 = _x27 ∧ x5 = _x28 ∧ x6 = _x29 ∧ x1 = _x30 ∧ x2 = _x31 ∧ x3 = _x32 ∧ x4 = _x33 ∧ x5 = _x34 ∧ x6 = _x35 ∧ _x29 = _x35 ∧ _x27 = _x33 ∧ _x26 = _x32 ∧ _x25 = _x31 ∧ _x24 = _x30 ∧ _x34 = 0 l5 5 l3: x1 = _x36 ∧ x2 = _x37 ∧ x3 = _x38 ∧ x4 = _x39 ∧ x5 = _x40 ∧ x6 = _x41 ∧ x1 = _x42 ∧ x2 = _x43 ∧ x3 = _x44 ∧ x4 = _x45 ∧ x5 = _x46 ∧ x6 = _x47 ∧ _x41 = _x47 ∧ _x39 = _x45 ∧ _x38 = _x44 ∧ _x37 = _x43 ∧ _x36 = _x42 ∧ _x46 = _x36 ∧ 0 ≤ _x41 ∧ _x41 ≤ 0 l5 6 l4: x1 = _x48 ∧ x2 = _x49 ∧ x3 = _x50 ∧ x4 = _x51 ∧ x5 = _x52 ∧ x6 = _x53 ∧ x1 = _x54 ∧ x2 = _x55 ∧ x3 = _x56 ∧ x4 = _x57 ∧ x5 = _x58 ∧ x6 = _x59 ∧ _x53 = _x59 ∧ _x52 = _x58 ∧ _x51 = _x57 ∧ _x50 = _x56 ∧ _x49 = _x55 ∧ _x48 = _x54 ∧ 1 ≤ _x53 l5 7 l4: x1 = _x60 ∧ x2 = _x61 ∧ x3 = _x62 ∧ x4 = _x63 ∧ x5 = _x64 ∧ x6 = _x65 ∧ x1 = _x66 ∧ x2 = _x67 ∧ x3 = _x68 ∧ x4 = _x69 ∧ x5 = _x70 ∧ x6 = _x71 ∧ _x65 = _x71 ∧ _x64 = _x70 ∧ _x63 = _x69 ∧ _x62 = _x68 ∧ _x61 = _x67 ∧ _x60 = _x66 ∧ 1 + _x65 ≤ 0 l2 8 l0: x1 = _x72 ∧ x2 = _x73 ∧ x3 = _x74 ∧ x4 = _x75 ∧ x5 = _x76 ∧ x6 = _x77 ∧ x1 = _x78 ∧ x2 = _x79 ∧ x3 = _x80 ∧ x4 = _x81 ∧ x5 = _x82 ∧ x6 = _x83 ∧ _x77 = _x83 ∧ _x76 = _x82 ∧ _x75 = _x81 ∧ _x74 = _x80 ∧ _x73 = _x79 ∧ _x72 = _x78 l6 9 l7: x1 = _x84 ∧ x2 = _x85 ∧ x3 = _x86 ∧ x4 = _x87 ∧ x5 = _x88 ∧ x6 = _x89 ∧ x1 = _x90 ∧ x2 = _x91 ∧ x3 = _x92 ∧ x4 = _x93 ∧ x5 = _x94 ∧ x6 = _x95 ∧ _x89 = _x95 ∧ _x88 = _x94 ∧ _x87 = _x93 ∧ _x86 = _x92 ∧ _x85 = _x91 ∧ _x84 = _x90 l8 10 l6: x1 = _x96 ∧ x2 = _x97 ∧ x3 = _x98 ∧ x4 = _x99 ∧ x5 = _x100 ∧ x6 = _x101 ∧ x1 = _x102 ∧ x2 = _x103 ∧ x3 = _x104 ∧ x4 = _x105 ∧ x5 = _x106 ∧ x6 = _x107 ∧ _x101 = _x107 ∧ _x100 = _x106 ∧ _x99 = _x105 ∧ _x98 = _x104 ∧ _x97 = _x103 ∧ _x96 = _x102 ∧ _x96 ≤ _x97 l8 11 l6: x1 = _x108 ∧ x2 = _x109 ∧ x3 = _x110 ∧ x4 = _x111 ∧ x5 = _x112 ∧ x6 = _x113 ∧ x1 = _x114 ∧ x2 = _x115 ∧ x3 = _x116 ∧ x4 = _x117 ∧ x5 = _x118 ∧ x6 = _x119 ∧ _x113 = _x119 ∧ _x112 = _x118 ∧ _x111 = _x117 ∧ _x110 = _x116 ∧ _x109 = _x115 ∧ _x108 = _x114 ∧ 1 + _x109 ≤ _x108 l1 12 l6: x1 = _x120 ∧ x2 = _x121 ∧ x3 = _x122 ∧ x4 = _x123 ∧ x5 = _x124 ∧ x6 = _x125 ∧ x1 = _x126 ∧ x2 = _x127 ∧ x3 = _x128 ∧ x4 = _x129 ∧ x5 = _x130 ∧ x6 = _x131 ∧ _x125 = _x131 ∧ _x124 = _x130 ∧ _x123 = _x129 ∧ _x122 = _x128 ∧ _x121 = _x127 ∧ _x120 = _x126 ∧ 1 + _x121 ≤ 0 l1 13 l8: x1 = _x132 ∧ x2 = _x133 ∧ x3 = _x134 ∧ x4 = _x135 ∧ x5 = _x136 ∧ x6 = _x137 ∧ x1 = _x138 ∧ x2 = _x139 ∧ x3 = _x140 ∧ x4 = _x141 ∧ x5 = _x142 ∧ x6 = _x143 ∧ _x137 = _x143 ∧ _x136 = _x142 ∧ _x135 = _x141 ∧ _x134 = _x140 ∧ _x133 = _x139 ∧ _x132 = _x138 ∧ 0 ≤ _x133 l9 14 l5: x1 = _x144 ∧ x2 = _x145 ∧ x3 = _x146 ∧ x4 = _x147 ∧ x5 = _x148 ∧ x6 = _x149 ∧ x1 = _x150 ∧ x2 = _x151 ∧ x3 = _x152 ∧ x4 = _x153 ∧ x5 = _x154 ∧ x6 = _x155 ∧ _x148 = _x154 ∧ _x147 = _x153 ∧ _x146 = _x152 ∧ _x145 = _x151 ∧ _x144 = _x150 ∧ _x155 = _x155 l10 15 l9: x1 = _x156 ∧ x2 = _x157 ∧ x3 = _x158 ∧ x4 = _x159 ∧ x5 = _x160 ∧ x6 = _x161 ∧ x1 = _x162 ∧ x2 = _x163 ∧ x3 = _x164 ∧ x4 = _x165 ∧ x5 = _x166 ∧ x6 = _x167 ∧ _x161 = _x167 ∧ _x160 = _x166 ∧ _x159 = _x165 ∧ _x158 = _x164 ∧ _x157 = _x163 ∧ _x156 = _x162

## 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 l4 l4 l4: x1 = x1 ∧ x2 = x2 ∧ x3 = x3 ∧ x4 = x4 ∧ x5 = x5 ∧ x6 = x6 l6 l6 l6: x1 = x1 ∧ x2 = x2 ∧ x3 = x3 ∧ x4 = x4 ∧ x5 = x5 ∧ x6 = x6 l10 l10 l10: x1 = x1 ∧ x2 = x2 ∧ x3 = x3 ∧ x4 = x4 ∧ x5 = x5 ∧ x6 = x6 l8 l8 l8: x1 = x1 ∧ x2 = x2 ∧ x3 = x3 ∧ x4 = x4 ∧ x5 = x5 ∧ x6 = x6 l1 l1 l1: x1 = x1 ∧ x2 = x2 ∧ x3 = x3 ∧ x4 = x4 ∧ x5 = x5 ∧ x6 = x6 l3 l3 l3: x1 = x1 ∧ x2 = x2 ∧ x3 = x3 ∧ x4 = x4 ∧ x5 = x5 ∧ x6 = x6 l0 l0 l0: x1 = x1 ∧ x2 = x2 ∧ x3 = x3 ∧ x4 = x4 ∧ x5 = x5 ∧ x6 = x6 l2 l2 l2: x1 = x1 ∧ x2 = x2 ∧ x3 = x3 ∧ x4 = x4 ∧ x5 = x5 ∧ x6 = x6 l9 l9 l9: 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 { l0, l2 }.

### 2.1.1 Transition Removal

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

 l0: −2⋅x3 + 2⋅x5 l2: −2⋅x3 + 2⋅x5 + 1

### 2.1.2 Transition Removal

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

 l2: 0 l0: −1

### 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)