LTS Termination Proof

by AProVE

Input

Integer Transition System
• Initial Location: l5, l4, l6, l10, l1, l8, l3, l0, l2, l9
• Transitions: (pre-variables and post-variables)  l0 1 l1: x1 = _oldX0HAT0 ∧ x2 = _oldX1HAT0 ∧ x3 = _oldX2HAT0 ∧ x4 = _oldX3HAT0 ∧ x5 = _oldX4HAT0 ∧ x6 = _oldX5HAT0 ∧ x7 = _x0HAT0 ∧ x8 = _x1HAT0 ∧ x9 = _x2HAT0 ∧ x1 = _oldX0HATpost ∧ x2 = _oldX1HATpost ∧ x3 = _oldX2HATpost ∧ x4 = _oldX3HATpost ∧ x5 = _oldX4HATpost ∧ x6 = _oldX5HATpost ∧ x7 = _x0HATpost ∧ x8 = _x1HATpost ∧ x9 = _x2HATpost ∧ _oldX5HAT0 = _oldX5HATpost ∧ _oldX4HAT0 = _oldX4HATpost ∧ _oldX3HAT0 = _oldX3HATpost ∧ _x2HATpost = 1 + _oldX2HATpost ∧ _x1HATpost = _oldX1HATpost ∧ _x0HATpost = _oldX0HATpost ∧ _oldX2HATpost = _x2HAT0 ∧ _oldX1HATpost = _x1HAT0 ∧ _oldX0HATpost = _x0HAT0 l2 2 l0: 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 ∧ _x5 = _x14 ∧ _x4 = _x13 ∧ _x3 = _x12 ∧ _x17 = _x11 ∧ _x16 = _x10 ∧ _x15 = _x9 ∧ _x11 = _x8 ∧ _x10 = _x7 ∧ _x9 = _x6 l3 3 l4: 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 ∧ _x23 = _x32 ∧ _x22 = _x31 ∧ _x35 = _x30 ∧ _x34 = 1 + _x28 ∧ _x33 = _x27 ∧ _x30 = _x30 ∧ _x29 = _x26 ∧ _x28 = _x25 ∧ _x27 = _x24 l5 4 l0: 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 ∧ _x41 = _x50 ∧ _x53 = _x47 ∧ _x52 = _x46 ∧ _x51 = _x45 ∧ _x48 ≤ _x49 ∧ _x49 = _x49 ∧ _x48 = _x48 ∧ _x47 = _x44 ∧ _x46 = _x43 ∧ _x45 = _x42 l5 5 l2: 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 = _x68 ∧ _x71 = _x65 ∧ _x70 = _x64 ∧ _x69 = _x63 ∧ 1 + _x67 ≤ _x66 ∧ _x67 = _x67 ∧ _x66 = _x66 ∧ _x65 = _x62 ∧ _x64 = _x61 ∧ _x63 = _x60 l6 6 l7: 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 ∧ _x89 = _x86 ∧ _x88 = _x85 ∧ _x87 = _x84 ∧ _x86 = _x86 ∧ _x85 = _x85 ∧ _x84 = _x84 ∧ _x83 = _x80 ∧ _x82 = _x79 ∧ _x81 = _x78 l1 7 l3: 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 ∧ _x95 = _x104 ∧ _x94 = _x103 ∧ _x93 = _x102 ∧ _x107 = _x101 ∧ _x106 = _x100 ∧ _x105 = _x99 ∧ _x99 − _x100 ≤ _x101 ∧ _x101 = _x98 ∧ _x100 = _x97 ∧ _x99 = _x96 l1 8 l5: 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 ∧ _x113 = _x122 ∧ _x112 = _x121 ∧ _x111 = _x120 ∧ _x125 = _x119 ∧ _x124 = _x118 ∧ _x123 = _x117 ∧ 1 + _x119 ≤ _x117 − _x118 ∧ _x119 = _x116 ∧ _x118 = _x115 ∧ _x117 = _x114 l4 9 l6: 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 ∧ _x131 = _x140 ∧ _x130 = _x139 ∧ _x143 = _x138 ∧ _x142 = _x136 ∧ _x141 = _x135 ∧ _x135 ≤ _x136 ∧ _x138 = _x138 ∧ _x137 = _x134 ∧ _x136 = _x133 ∧ _x135 = _x132 l4 10 l1: 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 ∧ _x149 = _x158 ∧ _x148 = _x157 ∧ _x147 = _x156 ∧ _x161 = 0 ∧ _x160 = _x154 ∧ _x159 = _x153 ∧ 1 + _x154 ≤ _x153 ∧ _x155 = _x152 ∧ _x154 = _x151 ∧ _x153 = _x150 l8 11 l4: 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 ∧ _x167 = _x176 ∧ _x166 = _x175 ∧ _x179 = _x174 ∧ _x178 = 1 ∧ _x177 = _x171 ∧ _x174 = _x174 ∧ _x173 = _x170 ∧ _x172 = _x169 ∧ _x171 = _x168 l9 12 l8: x1 = _x180 ∧ x2 = _x181 ∧ x3 = _x182 ∧ x4 = _x183 ∧ x5 = _x184 ∧ x6 = _x185 ∧ x7 = _x186 ∧ x8 = _x187 ∧ x9 = _x188 ∧ x1 = _x189 ∧ x2 = _x190 ∧ x3 = _x191 ∧ x4 = _x192 ∧ x5 = _x193 ∧ x6 = _x194 ∧ x7 = _x195 ∧ x8 = _x196 ∧ x9 = _x197 ∧ _x185 = _x194 ∧ _x197 = _x193 ∧ _x196 = _x192 ∧ _x195 = _x189 ∧ _x193 = _x193 ∧ _x192 = _x192 ∧ _x191 = _x188 ∧ _x190 = _x187 ∧ _x189 = _x186 l9 13 l0: x1 = _x198 ∧ x2 = _x199 ∧ x3 = _x200 ∧ x4 = _x201 ∧ x5 = _x202 ∧ x6 = _x203 ∧ x7 = _x204 ∧ x8 = _x205 ∧ x9 = _x206 ∧ x1 = _x207 ∧ x2 = _x208 ∧ x3 = _x209 ∧ x4 = _x210 ∧ x5 = _x211 ∧ x6 = _x212 ∧ x7 = _x213 ∧ x8 = _x214 ∧ x9 = _x215 ∧ _x206 = _x215 ∧ _x205 = _x214 ∧ _x204 = _x213 ∧ _x203 = _x212 ∧ _x202 = _x211 ∧ _x201 = _x210 ∧ _x200 = _x209 ∧ _x199 = _x208 ∧ _x198 = _x207 l9 14 l2: x1 = _x216 ∧ x2 = _x217 ∧ x3 = _x218 ∧ x4 = _x219 ∧ x5 = _x220 ∧ x6 = _x221 ∧ x7 = _x222 ∧ x8 = _x223 ∧ x9 = _x224 ∧ x1 = _x225 ∧ x2 = _x226 ∧ x3 = _x227 ∧ x4 = _x228 ∧ x5 = _x229 ∧ x6 = _x230 ∧ x7 = _x231 ∧ x8 = _x232 ∧ x9 = _x233 ∧ _x224 = _x233 ∧ _x223 = _x232 ∧ _x222 = _x231 ∧ _x221 = _x230 ∧ _x220 = _x229 ∧ _x219 = _x228 ∧ _x218 = _x227 ∧ _x217 = _x226 ∧ _x216 = _x225 l9 15 l3: x1 = _x234 ∧ x2 = _x235 ∧ x3 = _x236 ∧ x4 = _x237 ∧ x5 = _x238 ∧ x6 = _x239 ∧ x7 = _x240 ∧ x8 = _x241 ∧ x9 = _x242 ∧ x1 = _x243 ∧ x2 = _x244 ∧ x3 = _x245 ∧ x4 = _x246 ∧ x5 = _x247 ∧ x6 = _x248 ∧ x7 = _x249 ∧ x8 = _x250 ∧ x9 = _x251 ∧ _x242 = _x251 ∧ _x241 = _x250 ∧ _x240 = _x249 ∧ _x239 = _x248 ∧ _x238 = _x247 ∧ _x237 = _x246 ∧ _x236 = _x245 ∧ _x235 = _x244 ∧ _x234 = _x243 l9 16 l5: x1 = _x252 ∧ x2 = _x253 ∧ x3 = _x254 ∧ x4 = _x255 ∧ x5 = _x256 ∧ x6 = _x257 ∧ x7 = _x258 ∧ x8 = _x259 ∧ x9 = _x260 ∧ x1 = _x261 ∧ x2 = _x262 ∧ x3 = _x263 ∧ x4 = _x264 ∧ x5 = _x265 ∧ x6 = _x266 ∧ x7 = _x267 ∧ x8 = _x268 ∧ x9 = _x269 ∧ _x260 = _x269 ∧ _x259 = _x268 ∧ _x258 = _x267 ∧ _x257 = _x266 ∧ _x256 = _x265 ∧ _x255 = _x264 ∧ _x254 = _x263 ∧ _x253 = _x262 ∧ _x252 = _x261 l9 17 l6: x1 = _x270 ∧ x2 = _x271 ∧ x3 = _x272 ∧ x4 = _x273 ∧ x5 = _x274 ∧ x6 = _x275 ∧ x7 = _x276 ∧ x8 = _x277 ∧ x9 = _x278 ∧ x1 = _x279 ∧ x2 = _x280 ∧ x3 = _x281 ∧ x4 = _x282 ∧ x5 = _x283 ∧ x6 = _x284 ∧ x7 = _x285 ∧ x8 = _x286 ∧ x9 = _x287 ∧ _x278 = _x287 ∧ _x277 = _x286 ∧ _x276 = _x285 ∧ _x275 = _x284 ∧ _x274 = _x283 ∧ _x273 = _x282 ∧ _x272 = _x281 ∧ _x271 = _x280 ∧ _x270 = _x279 l9 18 l1: x1 = _x288 ∧ x2 = _x289 ∧ x3 = _x290 ∧ x4 = _x291 ∧ x5 = _x292 ∧ x6 = _x293 ∧ x7 = _x294 ∧ x8 = _x295 ∧ x9 = _x296 ∧ x1 = _x297 ∧ x2 = _x298 ∧ x3 = _x299 ∧ x4 = _x300 ∧ x5 = _x301 ∧ x6 = _x302 ∧ x7 = _x303 ∧ x8 = _x304 ∧ x9 = _x305 ∧ _x296 = _x305 ∧ _x295 = _x304 ∧ _x294 = _x303 ∧ _x293 = _x302 ∧ _x292 = _x301 ∧ _x291 = _x300 ∧ _x290 = _x299 ∧ _x289 = _x298 ∧ _x288 = _x297 l9 19 l4: x1 = _x306 ∧ x2 = _x307 ∧ x3 = _x308 ∧ x4 = _x309 ∧ x5 = _x310 ∧ x6 = _x311 ∧ x7 = _x312 ∧ x8 = _x313 ∧ x9 = _x314 ∧ x1 = _x315 ∧ x2 = _x316 ∧ x3 = _x317 ∧ x4 = _x318 ∧ x5 = _x319 ∧ x6 = _x320 ∧ x7 = _x321 ∧ x8 = _x322 ∧ x9 = _x323 ∧ _x314 = _x323 ∧ _x313 = _x322 ∧ _x312 = _x321 ∧ _x311 = _x320 ∧ _x310 = _x319 ∧ _x309 = _x318 ∧ _x308 = _x317 ∧ _x307 = _x316 ∧ _x306 = _x315 l9 20 l8: x1 = _x324 ∧ x2 = _x325 ∧ x3 = _x326 ∧ x4 = _x327 ∧ x5 = _x328 ∧ x6 = _x329 ∧ x7 = _x330 ∧ x8 = _x331 ∧ x9 = _x332 ∧ x1 = _x333 ∧ x2 = _x334 ∧ x3 = _x335 ∧ x4 = _x336 ∧ x5 = _x337 ∧ x6 = _x338 ∧ x7 = _x339 ∧ x8 = _x340 ∧ x9 = _x341 ∧ _x332 = _x341 ∧ _x331 = _x340 ∧ _x330 = _x339 ∧ _x329 = _x338 ∧ _x328 = _x337 ∧ _x327 = _x336 ∧ _x326 = _x335 ∧ _x325 = _x334 ∧ _x324 = _x333 l9 21 l7: x1 = _x342 ∧ x2 = _x343 ∧ x3 = _x344 ∧ x4 = _x345 ∧ x5 = _x346 ∧ x6 = _x347 ∧ x7 = _x348 ∧ x8 = _x349 ∧ x9 = _x350 ∧ x1 = _x351 ∧ x2 = _x352 ∧ x3 = _x353 ∧ x4 = _x354 ∧ x5 = _x355 ∧ x6 = _x356 ∧ x7 = _x357 ∧ x8 = _x358 ∧ x9 = _x359 ∧ _x350 = _x359 ∧ _x349 = _x358 ∧ _x348 = _x357 ∧ _x347 = _x356 ∧ _x346 = _x355 ∧ _x345 = _x354 ∧ _x344 = _x353 ∧ _x343 = _x352 ∧ _x342 = _x351 l10 22 l9: x1 = _x360 ∧ x2 = _x361 ∧ x3 = _x362 ∧ x4 = _x363 ∧ x5 = _x364 ∧ x6 = _x365 ∧ x7 = _x366 ∧ x8 = _x367 ∧ x9 = _x368 ∧ x1 = _x369 ∧ x2 = _x370 ∧ x3 = _x371 ∧ x4 = _x372 ∧ x5 = _x373 ∧ x6 = _x374 ∧ x7 = _x375 ∧ x8 = _x376 ∧ x9 = _x377 ∧ _x368 = _x377 ∧ _x367 = _x376 ∧ _x366 = _x375 ∧ _x365 = _x374 ∧ _x364 = _x373 ∧ _x363 = _x372 ∧ _x362 = _x371 ∧ _x361 = _x370 ∧ _x360 = _x369

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 l4 l4 l4: x1 = x1 ∧ x2 = x2 ∧ x3 = x3 ∧ x4 = x4 ∧ x5 = x5 ∧ x6 = x6 ∧ x7 = x7 ∧ x8 = x8 ∧ x9 = x9 l6 l6 l6: x1 = x1 ∧ x2 = x2 ∧ x3 = x3 ∧ x4 = x4 ∧ x5 = x5 ∧ x6 = x6 ∧ x7 = x7 ∧ x8 = x8 ∧ x9 = x9 l10 l10 l10: x1 = x1 ∧ x2 = x2 ∧ x3 = x3 ∧ x4 = x4 ∧ x5 = x5 ∧ x6 = x6 ∧ x7 = x7 ∧ x8 = x8 ∧ x9 = x9 l1 l1 l1: x1 = x1 ∧ x2 = x2 ∧ x3 = x3 ∧ x4 = x4 ∧ x5 = x5 ∧ x6 = x6 ∧ x7 = x7 ∧ x8 = x8 ∧ x9 = x9 l8 l8 l8: x1 = x1 ∧ x2 = x2 ∧ x3 = x3 ∧ x4 = x4 ∧ x5 = x5 ∧ x6 = x6 ∧ x7 = x7 ∧ x8 = x8 ∧ x9 = x9 l3 l3 l3: x1 = x1 ∧ x2 = x2 ∧ x3 = x3 ∧ x4 = x4 ∧ x5 = x5 ∧ x6 = x6 ∧ x7 = x7 ∧ x8 = x8 ∧ x9 = x9 l0 l0 l0: x1 = x1 ∧ x2 = x2 ∧ x3 = x3 ∧ x4 = x4 ∧ x5 = x5 ∧ x6 = x6 ∧ x7 = x7 ∧ x8 = x8 ∧ x9 = x9 l2 l2 l2: x1 = x1 ∧ x2 = x2 ∧ x3 = x3 ∧ x4 = x4 ∧ x5 = x5 ∧ x6 = x6 ∧ x7 = x7 ∧ x8 = x8 ∧ x9 = x9 l9 l9 l9: 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 { l5, l4, l1, l3, l0, l2 }.

2.1.1 Transition Removal

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

 l0: 6⋅x7 − 6⋅x8 + 1 l1: 6⋅x7 − 6⋅x8 + 1 l5: 6⋅x7 − 6⋅x8 + 1 l2: 6⋅x7 − 6⋅x8 + 1 l4: 6⋅x7 − 6⋅x8 + 2 l3: 6⋅x7 − 6⋅x8

2.1.2 Transition Removal

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

 l0: 2 l1: 2 l5: 2 l2: 2 l3: 3⋅x7 − 3⋅x8 − 3⋅x9 + 1 l4: 3⋅x1 − 3⋅x3 − 3⋅x8 + 3

2.1.3 Transition Removal

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

 l0: −1 + 2⋅x6 + x7 + x8 − x9 l1: −1 + 2⋅x6 + x7 + x8 − x9 l5: −1 + 2⋅x6 + x7 + x8 − x9 l2: −1 + 2⋅x6 + x7 + x8 − x9 l3: 0 l4: −1

2.1.4 Transition Removal

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

 l0: 4⋅x7 − 4⋅x8 − 4⋅x9 + 1 l1: 4⋅x7 − 4⋅x8 − 4⋅x9 + 4 l5: 4⋅x7 − 4⋅x8 − 4⋅x9 + 3 l2: 4⋅x7 − 4⋅x8 − 4⋅x9 + 2

2.1.5 Transition Removal

We remove transitions 1, 4, 2, 5 using the following ranking functions, which are bounded by 0.

 l0: 0 l1: x3 − x9 l5: 2 l2: 1

2.1.6 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 % (9 real / 0 unknown / 0 assumptions / 9 total proof steps)