# LTS Termination Proof

by AProVE

## Input

Integer Transition System
• Initial Location: l5, l7, l11, l3, l13, l17, l2, l9, l14, l4, l10, l6, l8, l15, l16, l0, l12
• Transitions: (pre-variables and post-variables)  l0 1 l1: x1 = _edgecountHAT0 ∧ x2 = _iHAT0 ∧ x3 = _jHAT0 ∧ x4 = _nodecountHAT0 ∧ x5 = _sourceHAT0 ∧ x6 = _xHAT0 ∧ x7 = _yHAT0 ∧ x1 = _edgecountHATpost ∧ x2 = _iHATpost ∧ x3 = _jHATpost ∧ x4 = _nodecountHATpost ∧ x5 = _sourceHATpost ∧ x6 = _xHATpost ∧ x7 = _yHATpost ∧ _yHAT0 = _yHATpost ∧ _xHAT0 = _xHATpost ∧ _sourceHAT0 = _sourceHATpost ∧ _nodecountHAT0 = _nodecountHATpost ∧ _jHAT0 = _jHATpost ∧ _iHAT0 = _iHATpost ∧ _edgecountHAT0 = _edgecountHATpost ∧ _nodecountHAT0 ≤ _iHAT0 l0 2 l2: x1 = _x ∧ x2 = _x1 ∧ x3 = _x2 ∧ x4 = _x3 ∧ x5 = _x4 ∧ x6 = _x5 ∧ x7 = _x6 ∧ x1 = _x7 ∧ x2 = _x8 ∧ x3 = _x9 ∧ x4 = _x10 ∧ x5 = _x11 ∧ x6 = _x12 ∧ x7 = _x13 ∧ _x6 = _x13 ∧ _x5 = _x12 ∧ _x4 = _x11 ∧ _x3 = _x10 ∧ _x2 = _x9 ∧ _x = _x7 ∧ _x8 = 1 + _x1 ∧ 1 + _x1 ≤ _x3 l3 3 l4: x1 = _x14 ∧ x2 = _x15 ∧ x3 = _x16 ∧ x4 = _x17 ∧ x5 = _x18 ∧ x6 = _x19 ∧ x7 = _x20 ∧ x1 = _x21 ∧ x2 = _x22 ∧ x3 = _x23 ∧ x4 = _x24 ∧ x5 = _x25 ∧ x6 = _x26 ∧ x7 = _x27 ∧ _x20 = _x27 ∧ _x19 = _x26 ∧ _x18 = _x25 ∧ _x17 = _x24 ∧ _x16 = _x23 ∧ _x15 = _x22 ∧ _x14 = _x21 l5 4 l6: x1 = _x28 ∧ x2 = _x29 ∧ x3 = _x30 ∧ x4 = _x31 ∧ x5 = _x32 ∧ x6 = _x33 ∧ x7 = _x34 ∧ x1 = _x35 ∧ x2 = _x36 ∧ x3 = _x37 ∧ x4 = _x38 ∧ x5 = _x39 ∧ x6 = _x40 ∧ x7 = _x41 ∧ _x34 = _x41 ∧ _x33 = _x40 ∧ _x32 = _x39 ∧ _x31 = _x38 ∧ _x30 = _x37 ∧ _x28 = _x35 ∧ _x36 = 1 + _x29 l5 5 l1: x1 = _x42 ∧ x2 = _x43 ∧ x3 = _x44 ∧ x4 = _x45 ∧ x5 = _x46 ∧ x6 = _x47 ∧ x7 = _x48 ∧ x1 = _x49 ∧ x2 = _x50 ∧ x3 = _x51 ∧ x4 = _x52 ∧ x5 = _x53 ∧ x6 = _x54 ∧ x7 = _x55 ∧ _x48 = _x55 ∧ _x47 = _x54 ∧ _x46 = _x53 ∧ _x45 = _x52 ∧ _x44 = _x51 ∧ _x43 = _x50 ∧ _x42 = _x49 l7 6 l2: x1 = _x56 ∧ x2 = _x57 ∧ x3 = _x58 ∧ x4 = _x59 ∧ x5 = _x60 ∧ x6 = _x61 ∧ x7 = _x62 ∧ x1 = _x63 ∧ x2 = _x64 ∧ x3 = _x65 ∧ x4 = _x66 ∧ x5 = _x67 ∧ x6 = _x68 ∧ x7 = _x69 ∧ _x62 = _x69 ∧ _x61 = _x68 ∧ _x60 = _x67 ∧ _x59 = _x66 ∧ _x58 = _x65 ∧ _x56 = _x63 ∧ _x64 = 0 ∧ _x56 ≤ _x57 l7 7 l5: x1 = _x70 ∧ x2 = _x71 ∧ x3 = _x72 ∧ x4 = _x73 ∧ x5 = _x74 ∧ x6 = _x75 ∧ x7 = _x76 ∧ x1 = _x77 ∧ x2 = _x78 ∧ x3 = _x79 ∧ x4 = _x80 ∧ x5 = _x81 ∧ x6 = _x82 ∧ x7 = _x83 ∧ _x74 = _x81 ∧ _x73 = _x80 ∧ _x72 = _x79 ∧ _x71 = _x78 ∧ _x70 = _x77 ∧ _x83 = _x83 ∧ _x82 = _x82 ∧ 1 + _x71 ≤ _x70 l8 8 l9: x1 = _x84 ∧ x2 = _x85 ∧ x3 = _x86 ∧ x4 = _x87 ∧ x5 = _x88 ∧ x6 = _x89 ∧ x7 = _x90 ∧ x1 = _x91 ∧ x2 = _x92 ∧ x3 = _x93 ∧ x4 = _x94 ∧ x5 = _x95 ∧ x6 = _x96 ∧ x7 = _x97 ∧ _x90 = _x97 ∧ _x89 = _x96 ∧ _x88 = _x95 ∧ _x87 = _x94 ∧ _x86 = _x93 ∧ _x85 = _x92 ∧ _x84 = _x91 l10 9 l11: x1 = _x98 ∧ x2 = _x99 ∧ x3 = _x100 ∧ x4 = _x101 ∧ x5 = _x102 ∧ x6 = _x103 ∧ x7 = _x104 ∧ x1 = _x105 ∧ x2 = _x106 ∧ x3 = _x107 ∧ x4 = _x108 ∧ x5 = _x109 ∧ x6 = _x110 ∧ x7 = _x111 ∧ _x104 = _x111 ∧ _x103 = _x110 ∧ _x102 = _x109 ∧ _x101 = _x108 ∧ _x99 = _x106 ∧ _x98 = _x105 ∧ _x107 = 1 + _x100 l11 10 l12: x1 = _x112 ∧ x2 = _x113 ∧ x3 = _x114 ∧ x4 = _x115 ∧ x5 = _x116 ∧ x6 = _x117 ∧ x7 = _x118 ∧ x1 = _x119 ∧ x2 = _x120 ∧ x3 = _x121 ∧ x4 = _x122 ∧ x5 = _x123 ∧ x6 = _x124 ∧ x7 = _x125 ∧ _x118 = _x125 ∧ _x117 = _x124 ∧ _x116 = _x123 ∧ _x115 = _x122 ∧ _x114 = _x121 ∧ _x113 = _x120 ∧ _x112 = _x119 l12 11 l8: x1 = _x126 ∧ x2 = _x127 ∧ x3 = _x128 ∧ x4 = _x129 ∧ x5 = _x130 ∧ x6 = _x131 ∧ x7 = _x132 ∧ x1 = _x133 ∧ x2 = _x134 ∧ x3 = _x135 ∧ x4 = _x136 ∧ x5 = _x137 ∧ x6 = _x138 ∧ x7 = _x139 ∧ _x132 = _x139 ∧ _x131 = _x138 ∧ _x130 = _x137 ∧ _x129 = _x136 ∧ _x128 = _x135 ∧ _x126 = _x133 ∧ _x134 = 1 + _x127 ∧ _x126 ≤ _x128 l12 12 l10: x1 = _x140 ∧ x2 = _x141 ∧ x3 = _x142 ∧ x4 = _x143 ∧ x5 = _x144 ∧ x6 = _x145 ∧ x7 = _x146 ∧ x1 = _x147 ∧ x2 = _x148 ∧ x3 = _x149 ∧ x4 = _x150 ∧ x5 = _x151 ∧ x6 = _x152 ∧ x7 = _x153 ∧ _x144 = _x151 ∧ _x143 = _x150 ∧ _x142 = _x149 ∧ _x141 = _x148 ∧ _x140 = _x147 ∧ _x153 = _x153 ∧ _x152 = _x152 ∧ 1 + _x142 ≤ _x140 l9 13 l6: x1 = _x154 ∧ x2 = _x155 ∧ x3 = _x156 ∧ x4 = _x157 ∧ x5 = _x158 ∧ x6 = _x159 ∧ x7 = _x160 ∧ x1 = _x161 ∧ x2 = _x162 ∧ x3 = _x163 ∧ x4 = _x164 ∧ x5 = _x165 ∧ x6 = _x166 ∧ x7 = _x167 ∧ _x160 = _x167 ∧ _x159 = _x166 ∧ _x158 = _x165 ∧ _x157 = _x164 ∧ _x156 = _x163 ∧ _x154 = _x161 ∧ _x162 = 0 ∧ _x157 ≤ _x155 l9 14 l11: x1 = _x168 ∧ x2 = _x169 ∧ x3 = _x170 ∧ x4 = _x171 ∧ x5 = _x172 ∧ x6 = _x173 ∧ x7 = _x174 ∧ x1 = _x175 ∧ x2 = _x176 ∧ x3 = _x177 ∧ x4 = _x178 ∧ x5 = _x179 ∧ x6 = _x180 ∧ x7 = _x181 ∧ _x174 = _x181 ∧ _x173 = _x180 ∧ _x172 = _x179 ∧ _x171 = _x178 ∧ _x169 = _x176 ∧ _x168 = _x175 ∧ _x177 = 0 ∧ 1 + _x169 ≤ _x171 l13 15 l14: x1 = _x182 ∧ x2 = _x183 ∧ x3 = _x184 ∧ x4 = _x185 ∧ x5 = _x186 ∧ x6 = _x187 ∧ x7 = _x188 ∧ x1 = _x189 ∧ x2 = _x190 ∧ x3 = _x191 ∧ x4 = _x192 ∧ x5 = _x193 ∧ x6 = _x194 ∧ x7 = _x195 ∧ _x188 = _x195 ∧ _x187 = _x194 ∧ _x186 = _x193 ∧ _x185 = _x192 ∧ _x184 = _x191 ∧ _x183 = _x190 ∧ _x182 = _x189 l14 16 l3: x1 = _x196 ∧ x2 = _x197 ∧ x3 = _x198 ∧ x4 = _x199 ∧ x5 = _x200 ∧ x6 = _x201 ∧ x7 = _x202 ∧ x1 = _x203 ∧ x2 = _x204 ∧ x3 = _x205 ∧ x4 = _x206 ∧ x5 = _x207 ∧ x6 = _x208 ∧ x7 = _x209 ∧ _x202 = _x209 ∧ _x201 = _x208 ∧ _x200 = _x207 ∧ _x199 = _x206 ∧ _x198 = _x205 ∧ _x196 = _x203 ∧ _x204 = 1 + _x197 l6 17 l7: x1 = _x210 ∧ x2 = _x211 ∧ x3 = _x212 ∧ x4 = _x213 ∧ x5 = _x214 ∧ x6 = _x215 ∧ x7 = _x216 ∧ x1 = _x217 ∧ x2 = _x218 ∧ x3 = _x219 ∧ x4 = _x220 ∧ x5 = _x221 ∧ x6 = _x222 ∧ x7 = _x223 ∧ _x216 = _x223 ∧ _x215 = _x222 ∧ _x214 = _x221 ∧ _x213 = _x220 ∧ _x212 = _x219 ∧ _x211 = _x218 ∧ _x210 = _x217 l15 18 l13: x1 = _x224 ∧ x2 = _x225 ∧ x3 = _x226 ∧ x4 = _x227 ∧ x5 = _x228 ∧ x6 = _x229 ∧ x7 = _x230 ∧ x1 = _x231 ∧ x2 = _x232 ∧ x3 = _x233 ∧ x4 = _x234 ∧ x5 = _x235 ∧ x6 = _x236 ∧ x7 = _x237 ∧ _x230 = _x237 ∧ _x229 = _x236 ∧ _x228 = _x235 ∧ _x227 = _x234 ∧ _x226 = _x233 ∧ _x225 = _x232 ∧ _x224 = _x231 ∧ 1 + _x228 ≤ _x225 l15 19 l13: x1 = _x238 ∧ x2 = _x239 ∧ x3 = _x240 ∧ x4 = _x241 ∧ x5 = _x242 ∧ x6 = _x243 ∧ x7 = _x244 ∧ x1 = _x245 ∧ x2 = _x246 ∧ x3 = _x247 ∧ x4 = _x248 ∧ x5 = _x249 ∧ x6 = _x250 ∧ x7 = _x251 ∧ _x244 = _x251 ∧ _x243 = _x250 ∧ _x242 = _x249 ∧ _x241 = _x248 ∧ _x240 = _x247 ∧ _x239 = _x246 ∧ _x238 = _x245 ∧ 1 + _x239 ≤ _x242 l15 20 l14: x1 = _x252 ∧ x2 = _x253 ∧ x3 = _x254 ∧ x4 = _x255 ∧ x5 = _x256 ∧ x6 = _x257 ∧ x7 = _x258 ∧ x1 = _x259 ∧ x2 = _x260 ∧ x3 = _x261 ∧ x4 = _x262 ∧ x5 = _x263 ∧ x6 = _x264 ∧ x7 = _x265 ∧ _x258 = _x265 ∧ _x257 = _x264 ∧ _x256 = _x263 ∧ _x255 = _x262 ∧ _x254 = _x261 ∧ _x253 = _x260 ∧ _x252 = _x259 ∧ _x256 ≤ _x253 ∧ _x253 ≤ _x256 l4 21 l8: x1 = _x266 ∧ x2 = _x267 ∧ x3 = _x268 ∧ x4 = _x269 ∧ x5 = _x270 ∧ x6 = _x271 ∧ x7 = _x272 ∧ x1 = _x273 ∧ x2 = _x274 ∧ x3 = _x275 ∧ x4 = _x276 ∧ x5 = _x277 ∧ x6 = _x278 ∧ x7 = _x279 ∧ _x272 = _x279 ∧ _x271 = _x278 ∧ _x270 = _x277 ∧ _x269 = _x276 ∧ _x268 = _x275 ∧ _x266 = _x273 ∧ _x274 = 0 ∧ _x269 ≤ _x267 l4 22 l15: x1 = _x280 ∧ x2 = _x281 ∧ x3 = _x282 ∧ x4 = _x283 ∧ x5 = _x284 ∧ x6 = _x285 ∧ x7 = _x286 ∧ x1 = _x287 ∧ x2 = _x288 ∧ x3 = _x289 ∧ x4 = _x290 ∧ x5 = _x291 ∧ x6 = _x292 ∧ x7 = _x293 ∧ _x286 = _x293 ∧ _x285 = _x292 ∧ _x284 = _x291 ∧ _x283 = _x290 ∧ _x282 = _x289 ∧ _x281 = _x288 ∧ _x280 = _x287 ∧ 1 + _x281 ≤ _x283 l2 23 l0: x1 = _x294 ∧ x2 = _x295 ∧ x3 = _x296 ∧ x4 = _x297 ∧ x5 = _x298 ∧ x6 = _x299 ∧ x7 = _x300 ∧ x1 = _x301 ∧ x2 = _x302 ∧ x3 = _x303 ∧ x4 = _x304 ∧ x5 = _x305 ∧ x6 = _x306 ∧ x7 = _x307 ∧ _x300 = _x307 ∧ _x299 = _x306 ∧ _x298 = _x305 ∧ _x297 = _x304 ∧ _x296 = _x303 ∧ _x295 = _x302 ∧ _x294 = _x301 l16 24 l3: x1 = _x308 ∧ x2 = _x309 ∧ x3 = _x310 ∧ x4 = _x311 ∧ x5 = _x312 ∧ x6 = _x313 ∧ x7 = _x314 ∧ x1 = _x315 ∧ x2 = _x316 ∧ x3 = _x317 ∧ x4 = _x318 ∧ x5 = _x319 ∧ x6 = _x320 ∧ x7 = _x321 ∧ _x314 = _x321 ∧ _x313 = _x320 ∧ _x310 = _x317 ∧ _x316 = 0 ∧ _x319 = 0 ∧ _x315 = 6 ∧ _x318 = 5 l17 25 l16: x1 = _x322 ∧ x2 = _x323 ∧ x3 = _x324 ∧ x4 = _x325 ∧ x5 = _x326 ∧ x6 = _x327 ∧ x7 = _x328 ∧ x1 = _x329 ∧ x2 = _x330 ∧ x3 = _x331 ∧ x4 = _x332 ∧ x5 = _x333 ∧ x6 = _x334 ∧ x7 = _x335 ∧ _x328 = _x335 ∧ _x327 = _x334 ∧ _x326 = _x333 ∧ _x325 = _x332 ∧ _x324 = _x331 ∧ _x323 = _x330 ∧ _x322 = _x329

## 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 l7 l7 l7: x1 = x1 ∧ x2 = x2 ∧ x3 = x3 ∧ x4 = x4 ∧ x5 = x5 ∧ x6 = x6 ∧ x7 = x7 l11 l11 l11: x1 = x1 ∧ x2 = x2 ∧ x3 = x3 ∧ x4 = x4 ∧ x5 = x5 ∧ x6 = x6 ∧ x7 = x7 l3 l3 l3: x1 = x1 ∧ x2 = x2 ∧ x3 = x3 ∧ x4 = x4 ∧ x5 = x5 ∧ x6 = x6 ∧ x7 = x7 l13 l13 l13: x1 = x1 ∧ x2 = x2 ∧ x3 = x3 ∧ x4 = x4 ∧ x5 = x5 ∧ x6 = x6 ∧ x7 = x7 l17 l17 l17: x1 = x1 ∧ x2 = x2 ∧ x3 = x3 ∧ x4 = x4 ∧ x5 = x5 ∧ x6 = x6 ∧ x7 = x7 l2 l2 l2: x1 = x1 ∧ x2 = x2 ∧ x3 = x3 ∧ x4 = x4 ∧ x5 = x5 ∧ x6 = x6 ∧ x7 = x7 l9 l9 l9: x1 = x1 ∧ x2 = x2 ∧ x3 = x3 ∧ x4 = x4 ∧ x5 = x5 ∧ x6 = x6 ∧ x7 = x7 l14 l14 l14: x1 = x1 ∧ x2 = x2 ∧ x3 = x3 ∧ x4 = x4 ∧ x5 = x5 ∧ x6 = x6 ∧ x7 = x7 l4 l4 l4: x1 = x1 ∧ x2 = x2 ∧ x3 = x3 ∧ x4 = x4 ∧ x5 = x5 ∧ x6 = x6 ∧ x7 = x7 l10 l10 l10: x1 = x1 ∧ x2 = x2 ∧ x3 = x3 ∧ x4 = x4 ∧ x5 = x5 ∧ x6 = x6 ∧ x7 = x7 l6 l6 l6: x1 = x1 ∧ x2 = x2 ∧ x3 = x3 ∧ x4 = x4 ∧ x5 = x5 ∧ x6 = x6 ∧ x7 = x7 l8 l8 l8: x1 = x1 ∧ x2 = x2 ∧ x3 = x3 ∧ x4 = x4 ∧ x5 = x5 ∧ x6 = x6 ∧ x7 = x7 l15 l15 l15: x1 = x1 ∧ x2 = x2 ∧ x3 = x3 ∧ x4 = x4 ∧ x5 = x5 ∧ x6 = x6 ∧ x7 = x7 l16 l16 l16: x1 = x1 ∧ x2 = x2 ∧ x3 = x3 ∧ x4 = x4 ∧ x5 = x5 ∧ x6 = x6 ∧ x7 = x7 l0 l0 l0: x1 = x1 ∧ x2 = x2 ∧ x3 = x3 ∧ x4 = x4 ∧ x5 = x5 ∧ x6 = x6 ∧ x7 = x7 l12 l12 l12: x1 = x1 ∧ x2 = x2 ∧ x3 = x3 ∧ x4 = x4 ∧ x5 = x5 ∧ x6 = x6 ∧ x7 = x7
and for every transition t, a duplicate t is considered.

### 2 SCC Decomposition

We consider subproblems for each of the 4 SCC(s) of the program graph.

### 2.1 SCC Subproblem 1/4

Here we consider the SCC { l4, l3, l13, l15, l14 }.

### 2.1.1 Transition Removal

We remove transition 22 using the following ranking functions, which are bounded by 0.

 l3: −1 − x2 + x4 l4: −1 − x2 + x4 l14: −2 − x2 + x4 l15: −2 − x2 + x4 l13: −2 − x2 + x4

### 2.1.2 Transition Removal

We remove transitions 3, 16, 20, 15, 19, 18 using the following ranking functions, which are bounded by 0.

 l3: 0 l4: −1 l14: 1 l15: 3 l13: 2

### 2.1.3 Trivial Cooperation Program

There are no more "sharp" transitions in the cooperation program. Hence the cooperation termination is proved.

### 2.2 SCC Subproblem 2/4

Here we consider the SCC { l10, l11, l8, l12, l9 }.

### 2.2.1 Transition Removal

We remove transition 14 using the following ranking functions, which are bounded by 0.

 l8: −1 − x2 + x4 l9: −1 − x2 + x4 l12: −2 − x2 + x4 l11: −2 − x2 + x4 l10: −2 − x2 + x4

### 2.2.2 Transition Removal

We remove transitions 8, 11 using the following ranking functions, which are bounded by 0.

 l8: 0 l9: −1 l12: 1 l11: 1 l10: 1

### 2.2.3 Transition Removal

We remove transition 12 using the following ranking functions, which are bounded by 0.

 l11: −1 + x1 − x3 l12: −1 + x1 − x3 l10: −2 + x1 − x3

### 2.2.4 Transition Removal

We remove transitions 10, 9 using the following ranking functions, which are bounded by 0.

 l11: 0 l12: −1 l10: 1

### 2.2.5 Trivial Cooperation Program

There are no more "sharp" transitions in the cooperation program. Hence the cooperation termination is proved.

### 2.3 SCC Subproblem 3/4

Here we consider the SCC { l5, l7, l6 }.

### 2.3.1 Transition Removal

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

 l6: −1 + x1 − x2 l7: −1 + x1 − x2 l5: −2 + x1 − x2

### 2.3.2 Transition Removal

We remove transitions 17, 4 using the following ranking functions, which are bounded by 0.

 l6: 0 l7: −1 l5: 1

### 2.3.3 Trivial Cooperation Program

There are no more "sharp" transitions in the cooperation program. Hence the cooperation termination is proved.

### 2.4 SCC Subproblem 4/4

Here we consider the SCC { l0, l2 }.

### 2.4.1 Transition Removal

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

 l0: −2⋅x2 + 2⋅x4 l2: −2⋅x2 + 2⋅x4 + 1

### 2.4.2 Transition Removal

We remove transition 23 using the following ranking functions, which are bounded by 0.

 l2: 0 l0: −1

### 2.4.3 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 % (18 real / 0 unknown / 0 assumptions / 18 total proof steps)