# LTS Termination Proof

by AProVE

## Input

Integer Transition System
• Initial Location: l5, l7, l11, l1, l13, l3, l2, l9, l14, l4, l6, l10, l8, l15, l16, l0
• Transitions: (pre-variables and post-variables)  l0 1 l1: x1 = _Index15HAT0 ∧ x2 = _Inner14HAT0 ∧ x3 = _InnerIndex6HAT0 ∧ x4 = _InnerIndex9HAT0 ∧ x5 = _Outer13HAT0 ∧ x6 = _OuterIndex5HAT0 ∧ x7 = _OuterIndex8HAT0 ∧ x8 = _SeedHAT0 ∧ x9 = _ret_RandomInteger16HAT0 ∧ x10 = _ret_RandomInteger17HAT0 ∧ x1 = _Index15HATpost ∧ x2 = _Inner14HATpost ∧ x3 = _InnerIndex6HATpost ∧ x4 = _InnerIndex9HATpost ∧ x5 = _Outer13HATpost ∧ x6 = _OuterIndex5HATpost ∧ x7 = _OuterIndex8HATpost ∧ x8 = _SeedHATpost ∧ x9 = _ret_RandomInteger16HATpost ∧ x10 = _ret_RandomInteger17HATpost ∧ _ret_RandomInteger17HAT0 = _ret_RandomInteger17HATpost ∧ _ret_RandomInteger16HAT0 = _ret_RandomInteger16HATpost ∧ _SeedHAT0 = _SeedHATpost ∧ _OuterIndex8HAT0 = _OuterIndex8HATpost ∧ _OuterIndex5HAT0 = _OuterIndex5HATpost ∧ _Outer13HAT0 = _Outer13HATpost ∧ _InnerIndex9HAT0 = _InnerIndex9HATpost ∧ _InnerIndex6HAT0 = _InnerIndex6HATpost ∧ _Inner14HAT0 = _Inner14HATpost ∧ _Index15HAT0 = _Index15HATpost l2 2 l3: x1 = _x ∧ x2 = _x1 ∧ x3 = _x2 ∧ x4 = _x3 ∧ x5 = _x4 ∧ x6 = _x5 ∧ x7 = _x6 ∧ x8 = _x7 ∧ x9 = _x8 ∧ x10 = _x9 ∧ x1 = _x10 ∧ x2 = _x11 ∧ x3 = _x12 ∧ x4 = _x13 ∧ x5 = _x14 ∧ x6 = _x15 ∧ x7 = _x16 ∧ x8 = _x17 ∧ x9 = _x18 ∧ x10 = _x19 ∧ _x9 = _x19 ∧ _x8 = _x18 ∧ _x7 = _x17 ∧ _x6 = _x16 ∧ _x5 = _x15 ∧ _x4 = _x14 ∧ _x3 = _x13 ∧ _x2 = _x12 ∧ _x1 = _x11 ∧ _x = _x10 l4 3 l5: x1 = _x20 ∧ x2 = _x21 ∧ x3 = _x22 ∧ x4 = _x23 ∧ x5 = _x24 ∧ x6 = _x25 ∧ x7 = _x26 ∧ x8 = _x27 ∧ x9 = _x28 ∧ x10 = _x29 ∧ x1 = _x30 ∧ x2 = _x31 ∧ x3 = _x32 ∧ x4 = _x33 ∧ x5 = _x34 ∧ x6 = _x35 ∧ x7 = _x36 ∧ x8 = _x37 ∧ x9 = _x38 ∧ x10 = _x39 ∧ _x29 = _x39 ∧ _x28 = _x38 ∧ _x27 = _x37 ∧ _x26 = _x36 ∧ _x25 = _x35 ∧ _x24 = _x34 ∧ _x23 = _x33 ∧ _x22 = _x32 ∧ _x21 = _x31 ∧ _x20 = _x30 l6 4 l7: x1 = _x40 ∧ x2 = _x41 ∧ x3 = _x42 ∧ x4 = _x43 ∧ x5 = _x44 ∧ x6 = _x45 ∧ x7 = _x46 ∧ x8 = _x47 ∧ x9 = _x48 ∧ x10 = _x49 ∧ x1 = _x50 ∧ x2 = _x51 ∧ x3 = _x52 ∧ x4 = _x53 ∧ x5 = _x54 ∧ x6 = _x55 ∧ x7 = _x56 ∧ x8 = _x57 ∧ x9 = _x58 ∧ x10 = _x59 ∧ _x49 = _x59 ∧ _x48 = _x58 ∧ _x47 = _x57 ∧ _x46 = _x56 ∧ _x45 = _x55 ∧ _x44 = _x54 ∧ _x43 = _x53 ∧ _x42 = _x52 ∧ _x40 = _x50 ∧ _x51 = 1 + _x41 ∧ 20 ≤ _x40 l6 5 l8: x1 = _x60 ∧ x2 = _x61 ∧ x3 = _x62 ∧ x4 = _x63 ∧ x5 = _x64 ∧ x6 = _x65 ∧ x7 = _x66 ∧ x8 = _x67 ∧ x9 = _x68 ∧ x10 = _x69 ∧ x1 = _x70 ∧ x2 = _x71 ∧ x3 = _x72 ∧ x4 = _x73 ∧ x5 = _x74 ∧ x6 = _x75 ∧ x7 = _x76 ∧ x8 = _x77 ∧ x9 = _x78 ∧ x10 = _x79 ∧ _x69 = _x79 ∧ _x68 = _x78 ∧ _x67 = _x77 ∧ _x66 = _x76 ∧ _x65 = _x75 ∧ _x64 = _x74 ∧ _x63 = _x73 ∧ _x62 = _x72 ∧ _x61 = _x71 ∧ _x70 = 1 + _x60 ∧ 1 + _x60 ≤ 20 l9 6 l10: x1 = _x80 ∧ x2 = _x81 ∧ x3 = _x82 ∧ x4 = _x83 ∧ x5 = _x84 ∧ x6 = _x85 ∧ x7 = _x86 ∧ x8 = _x87 ∧ x9 = _x88 ∧ x10 = _x89 ∧ x1 = _x90 ∧ x2 = _x91 ∧ x3 = _x92 ∧ x4 = _x93 ∧ x5 = _x94 ∧ x6 = _x95 ∧ x7 = _x96 ∧ x8 = _x97 ∧ x9 = _x98 ∧ x10 = _x99 ∧ _x89 = _x99 ∧ _x88 = _x98 ∧ _x87 = _x97 ∧ _x86 = _x96 ∧ _x85 = _x95 ∧ _x83 = _x93 ∧ _x82 = _x92 ∧ _x81 = _x91 ∧ _x80 = _x90 ∧ _x94 = 1 + _x84 ∧ 20 ≤ _x81 l9 7 l8: x1 = _x100 ∧ x2 = _x101 ∧ x3 = _x102 ∧ x4 = _x103 ∧ x5 = _x104 ∧ x6 = _x105 ∧ x7 = _x106 ∧ x8 = _x107 ∧ x9 = _x108 ∧ x10 = _x109 ∧ x1 = _x110 ∧ x2 = _x111 ∧ x3 = _x112 ∧ x4 = _x113 ∧ x5 = _x114 ∧ x6 = _x115 ∧ x7 = _x116 ∧ x8 = _x117 ∧ x9 = _x118 ∧ x10 = _x119 ∧ _x109 = _x119 ∧ _x108 = _x118 ∧ _x107 = _x117 ∧ _x106 = _x116 ∧ _x105 = _x115 ∧ _x104 = _x114 ∧ _x103 = _x113 ∧ _x102 = _x112 ∧ _x101 = _x111 ∧ _x110 = 0 ∧ 1 + _x101 ≤ 20 l11 8 l12: x1 = _x120 ∧ x2 = _x121 ∧ x3 = _x122 ∧ x4 = _x123 ∧ x5 = _x124 ∧ x6 = _x125 ∧ x7 = _x126 ∧ x8 = _x127 ∧ x9 = _x128 ∧ x10 = _x129 ∧ x1 = _x130 ∧ x2 = _x131 ∧ x3 = _x132 ∧ x4 = _x133 ∧ x5 = _x134 ∧ x6 = _x135 ∧ x7 = _x136 ∧ x8 = _x137 ∧ x9 = _x138 ∧ x10 = _x139 ∧ _x129 = _x139 ∧ _x128 = _x138 ∧ _x127 = _x137 ∧ _x126 = _x136 ∧ _x125 = _x135 ∧ _x124 = _x134 ∧ _x123 = _x133 ∧ _x122 = _x132 ∧ _x121 = _x131 ∧ _x120 = _x130 ∧ 20 ≤ _x124 l11 9 l7: x1 = _x140 ∧ x2 = _x141 ∧ x3 = _x142 ∧ x4 = _x143 ∧ x5 = _x144 ∧ x6 = _x145 ∧ x7 = _x146 ∧ x8 = _x147 ∧ x9 = _x148 ∧ x10 = _x149 ∧ x1 = _x150 ∧ x2 = _x151 ∧ x3 = _x152 ∧ x4 = _x153 ∧ x5 = _x154 ∧ x6 = _x155 ∧ x7 = _x156 ∧ x8 = _x157 ∧ x9 = _x158 ∧ x10 = _x159 ∧ _x149 = _x159 ∧ _x148 = _x158 ∧ _x147 = _x157 ∧ _x146 = _x156 ∧ _x145 = _x155 ∧ _x144 = _x154 ∧ _x143 = _x153 ∧ _x142 = _x152 ∧ _x140 = _x150 ∧ _x151 = 0 ∧ 1 + _x144 ≤ 20 l13 10 l14: x1 = _x160 ∧ x2 = _x161 ∧ x3 = _x162 ∧ x4 = _x163 ∧ x5 = _x164 ∧ x6 = _x165 ∧ x7 = _x166 ∧ x8 = _x167 ∧ x9 = _x168 ∧ x10 = _x169 ∧ x1 = _x170 ∧ x2 = _x171 ∧ x3 = _x172 ∧ x4 = _x173 ∧ x5 = _x174 ∧ x6 = _x175 ∧ x7 = _x176 ∧ x8 = _x177 ∧ x9 = _x178 ∧ x10 = _x179 ∧ _x169 = _x179 ∧ _x168 = _x178 ∧ _x167 = _x177 ∧ _x166 = _x176 ∧ _x165 = _x175 ∧ _x164 = _x174 ∧ _x163 = _x173 ∧ _x162 = _x172 ∧ _x161 = _x171 ∧ _x160 = _x170 l10 11 l11: x1 = _x180 ∧ x2 = _x181 ∧ x3 = _x182 ∧ x4 = _x183 ∧ x5 = _x184 ∧ x6 = _x185 ∧ x7 = _x186 ∧ x8 = _x187 ∧ x9 = _x188 ∧ x10 = _x189 ∧ x1 = _x190 ∧ x2 = _x191 ∧ x3 = _x192 ∧ x4 = _x193 ∧ x5 = _x194 ∧ x6 = _x195 ∧ x7 = _x196 ∧ x8 = _x197 ∧ x9 = _x198 ∧ x10 = _x199 ∧ _x189 = _x199 ∧ _x188 = _x198 ∧ _x187 = _x197 ∧ _x186 = _x196 ∧ _x185 = _x195 ∧ _x184 = _x194 ∧ _x183 = _x193 ∧ _x182 = _x192 ∧ _x181 = _x191 ∧ _x180 = _x190 l14 12 l4: x1 = _x200 ∧ x2 = _x201 ∧ x3 = _x202 ∧ x4 = _x203 ∧ x5 = _x204 ∧ x6 = _x205 ∧ x7 = _x206 ∧ x8 = _x207 ∧ x9 = _x208 ∧ x10 = _x209 ∧ x1 = _x210 ∧ x2 = _x211 ∧ x3 = _x212 ∧ x4 = _x213 ∧ x5 = _x214 ∧ x6 = _x215 ∧ x7 = _x216 ∧ x8 = _x217 ∧ x9 = _x218 ∧ x10 = _x219 ∧ _x209 = _x219 ∧ _x208 = _x218 ∧ _x207 = _x217 ∧ _x205 = _x215 ∧ _x204 = _x214 ∧ _x203 = _x213 ∧ _x202 = _x212 ∧ _x201 = _x211 ∧ _x200 = _x210 ∧ _x216 = 1 + _x206 ∧ 20 ≤ _x203 l14 13 l13: x1 = _x220 ∧ x2 = _x221 ∧ x3 = _x222 ∧ x4 = _x223 ∧ x5 = _x224 ∧ x6 = _x225 ∧ x7 = _x226 ∧ x8 = _x227 ∧ x9 = _x228 ∧ x10 = _x229 ∧ x1 = _x230 ∧ x2 = _x231 ∧ x3 = _x232 ∧ x4 = _x233 ∧ x5 = _x234 ∧ x6 = _x235 ∧ x7 = _x236 ∧ x8 = _x237 ∧ x9 = _x238 ∧ x10 = _x239 ∧ _x228 = _x238 ∧ _x226 = _x236 ∧ _x225 = _x235 ∧ _x224 = _x234 ∧ _x222 = _x232 ∧ _x221 = _x231 ∧ _x220 = _x230 ∧ _x233 = 1 + _x223 ∧ _x239 = _x237 ∧ _x237 = _x237 ∧ 1 + _x223 ≤ 20 l5 14 l10: x1 = _x240 ∧ x2 = _x241 ∧ x3 = _x242 ∧ x4 = _x243 ∧ x5 = _x244 ∧ x6 = _x245 ∧ x7 = _x246 ∧ x8 = _x247 ∧ x9 = _x248 ∧ x10 = _x249 ∧ x1 = _x250 ∧ x2 = _x251 ∧ x3 = _x252 ∧ x4 = _x253 ∧ x5 = _x254 ∧ x6 = _x255 ∧ x7 = _x256 ∧ x8 = _x257 ∧ x9 = _x258 ∧ x10 = _x259 ∧ _x249 = _x259 ∧ _x248 = _x258 ∧ _x247 = _x257 ∧ _x246 = _x256 ∧ _x245 = _x255 ∧ _x243 = _x253 ∧ _x242 = _x252 ∧ _x241 = _x251 ∧ _x240 = _x250 ∧ _x254 = 0 ∧ 20 ≤ _x246 l5 15 l13: x1 = _x260 ∧ x2 = _x261 ∧ x3 = _x262 ∧ x4 = _x263 ∧ x5 = _x264 ∧ x6 = _x265 ∧ x7 = _x266 ∧ x8 = _x267 ∧ x9 = _x268 ∧ x10 = _x269 ∧ x1 = _x270 ∧ x2 = _x271 ∧ x3 = _x272 ∧ x4 = _x273 ∧ x5 = _x274 ∧ x6 = _x275 ∧ x7 = _x276 ∧ x8 = _x277 ∧ x9 = _x278 ∧ x10 = _x279 ∧ _x269 = _x279 ∧ _x268 = _x278 ∧ _x267 = _x277 ∧ _x266 = _x276 ∧ _x265 = _x275 ∧ _x264 = _x274 ∧ _x262 = _x272 ∧ _x261 = _x271 ∧ _x260 = _x270 ∧ _x273 = 0 ∧ 1 + _x266 ≤ 20 l7 16 l9: x1 = _x280 ∧ x2 = _x281 ∧ x3 = _x282 ∧ x4 = _x283 ∧ x5 = _x284 ∧ x6 = _x285 ∧ x7 = _x286 ∧ x8 = _x287 ∧ x9 = _x288 ∧ x10 = _x289 ∧ x1 = _x290 ∧ x2 = _x291 ∧ x3 = _x292 ∧ x4 = _x293 ∧ x5 = _x294 ∧ x6 = _x295 ∧ x7 = _x296 ∧ x8 = _x297 ∧ x9 = _x298 ∧ x10 = _x299 ∧ _x289 = _x299 ∧ _x288 = _x298 ∧ _x287 = _x297 ∧ _x286 = _x296 ∧ _x285 = _x295 ∧ _x284 = _x294 ∧ _x283 = _x293 ∧ _x282 = _x292 ∧ _x281 = _x291 ∧ _x280 = _x290 l8 17 l6: x1 = _x300 ∧ x2 = _x301 ∧ x3 = _x302 ∧ x4 = _x303 ∧ x5 = _x304 ∧ x6 = _x305 ∧ x7 = _x306 ∧ x8 = _x307 ∧ x9 = _x308 ∧ x10 = _x309 ∧ x1 = _x310 ∧ x2 = _x311 ∧ x3 = _x312 ∧ x4 = _x313 ∧ x5 = _x314 ∧ x6 = _x315 ∧ x7 = _x316 ∧ x8 = _x317 ∧ x9 = _x318 ∧ x10 = _x319 ∧ _x309 = _x319 ∧ _x308 = _x318 ∧ _x307 = _x317 ∧ _x306 = _x316 ∧ _x305 = _x315 ∧ _x304 = _x314 ∧ _x303 = _x313 ∧ _x302 = _x312 ∧ _x301 = _x311 ∧ _x300 = _x310 l3 18 l0: x1 = _x320 ∧ x2 = _x321 ∧ x3 = _x322 ∧ x4 = _x323 ∧ x5 = _x324 ∧ x6 = _x325 ∧ x7 = _x326 ∧ x8 = _x327 ∧ x9 = _x328 ∧ x10 = _x329 ∧ x1 = _x330 ∧ x2 = _x331 ∧ x3 = _x332 ∧ x4 = _x333 ∧ x5 = _x334 ∧ x6 = _x335 ∧ x7 = _x336 ∧ x8 = _x337 ∧ x9 = _x338 ∧ x10 = _x339 ∧ _x329 = _x339 ∧ _x328 = _x338 ∧ _x327 = _x337 ∧ _x326 = _x336 ∧ _x324 = _x334 ∧ _x323 = _x333 ∧ _x322 = _x332 ∧ _x321 = _x331 ∧ _x320 = _x330 ∧ _x335 = 1 + _x325 ∧ 20 ≤ _x322 l3 19 l2: x1 = _x340 ∧ x2 = _x341 ∧ x3 = _x342 ∧ x4 = _x343 ∧ x5 = _x344 ∧ x6 = _x345 ∧ x7 = _x346 ∧ x8 = _x347 ∧ x9 = _x348 ∧ x10 = _x349 ∧ x1 = _x350 ∧ x2 = _x351 ∧ x3 = _x352 ∧ x4 = _x353 ∧ x5 = _x354 ∧ x6 = _x355 ∧ x7 = _x356 ∧ x8 = _x357 ∧ x9 = _x358 ∧ x10 = _x359 ∧ _x349 = _x359 ∧ _x346 = _x356 ∧ _x345 = _x355 ∧ _x344 = _x354 ∧ _x343 = _x353 ∧ _x341 = _x351 ∧ _x340 = _x350 ∧ _x352 = 1 + _x342 ∧ _x358 = _x357 ∧ _x357 = _x357 ∧ 1 + _x342 ≤ 20 l1 20 l4: x1 = _x360 ∧ x2 = _x361 ∧ x3 = _x362 ∧ x4 = _x363 ∧ x5 = _x364 ∧ x6 = _x365 ∧ x7 = _x366 ∧ x8 = _x367 ∧ x9 = _x368 ∧ x10 = _x369 ∧ x1 = _x370 ∧ x2 = _x371 ∧ x3 = _x372 ∧ x4 = _x373 ∧ x5 = _x374 ∧ x6 = _x375 ∧ x7 = _x376 ∧ x8 = _x377 ∧ x9 = _x378 ∧ x10 = _x379 ∧ _x369 = _x379 ∧ _x368 = _x378 ∧ _x367 = _x377 ∧ _x365 = _x375 ∧ _x364 = _x374 ∧ _x363 = _x373 ∧ _x362 = _x372 ∧ _x361 = _x371 ∧ _x360 = _x370 ∧ _x376 = 0 ∧ 20 ≤ _x365 l1 21 l2: x1 = _x380 ∧ x2 = _x381 ∧ x3 = _x382 ∧ x4 = _x383 ∧ x5 = _x384 ∧ x6 = _x385 ∧ x7 = _x386 ∧ x8 = _x387 ∧ x9 = _x388 ∧ x10 = _x389 ∧ x1 = _x390 ∧ x2 = _x391 ∧ x3 = _x392 ∧ x4 = _x393 ∧ x5 = _x394 ∧ x6 = _x395 ∧ x7 = _x396 ∧ x8 = _x397 ∧ x9 = _x398 ∧ x10 = _x399 ∧ _x389 = _x399 ∧ _x388 = _x398 ∧ _x387 = _x397 ∧ _x386 = _x396 ∧ _x385 = _x395 ∧ _x384 = _x394 ∧ _x383 = _x393 ∧ _x381 = _x391 ∧ _x380 = _x390 ∧ _x392 = 0 ∧ 1 + _x385 ≤ 20 l15 22 l0: x1 = _x400 ∧ x2 = _x401 ∧ x3 = _x402 ∧ x4 = _x403 ∧ x5 = _x404 ∧ x6 = _x405 ∧ x7 = _x406 ∧ x8 = _x407 ∧ x9 = _x408 ∧ x10 = _x409 ∧ x1 = _x410 ∧ x2 = _x411 ∧ x3 = _x412 ∧ x4 = _x413 ∧ x5 = _x414 ∧ x6 = _x415 ∧ x7 = _x416 ∧ x8 = _x417 ∧ x9 = _x418 ∧ x10 = _x419 ∧ _x409 = _x419 ∧ _x408 = _x418 ∧ _x406 = _x416 ∧ _x404 = _x414 ∧ _x403 = _x413 ∧ _x402 = _x412 ∧ _x401 = _x411 ∧ _x400 = _x410 ∧ _x415 = 0 ∧ _x417 = 0 l16 23 l15: x1 = _x420 ∧ x2 = _x421 ∧ x3 = _x422 ∧ x4 = _x423 ∧ x5 = _x424 ∧ x6 = _x425 ∧ x7 = _x426 ∧ x8 = _x427 ∧ x9 = _x428 ∧ x10 = _x429 ∧ x1 = _x430 ∧ x2 = _x431 ∧ x3 = _x432 ∧ x4 = _x433 ∧ x5 = _x434 ∧ x6 = _x435 ∧ x7 = _x436 ∧ x8 = _x437 ∧ x9 = _x438 ∧ x10 = _x439 ∧ _x429 = _x439 ∧ _x428 = _x438 ∧ _x427 = _x437 ∧ _x426 = _x436 ∧ _x425 = _x435 ∧ _x424 = _x434 ∧ _x423 = _x433 ∧ _x422 = _x432 ∧ _x421 = _x431 ∧ _x420 = _x430

## 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 l7 l7 l7: x1 = x1 ∧ x2 = x2 ∧ x3 = x3 ∧ x4 = x4 ∧ x5 = x5 ∧ x6 = x6 ∧ x7 = x7 ∧ x8 = x8 ∧ x9 = x9 ∧ x10 = x10 l11 l11 l11: x1 = x1 ∧ x2 = x2 ∧ x3 = x3 ∧ x4 = x4 ∧ x5 = x5 ∧ x6 = x6 ∧ x7 = x7 ∧ x8 = x8 ∧ x9 = x9 ∧ x10 = x10 l1 l1 l1: x1 = x1 ∧ x2 = x2 ∧ x3 = x3 ∧ x4 = x4 ∧ x5 = x5 ∧ x6 = x6 ∧ x7 = x7 ∧ x8 = x8 ∧ x9 = x9 ∧ x10 = x10 l13 l13 l13: x1 = x1 ∧ x2 = x2 ∧ x3 = x3 ∧ x4 = x4 ∧ x5 = x5 ∧ x6 = x6 ∧ x7 = x7 ∧ x8 = x8 ∧ x9 = x9 ∧ x10 = x10 l3 l3 l3: x1 = x1 ∧ x2 = x2 ∧ x3 = x3 ∧ x4 = x4 ∧ x5 = x5 ∧ x6 = x6 ∧ x7 = x7 ∧ x8 = x8 ∧ x9 = x9 ∧ x10 = x10 l2 l2 l2: x1 = x1 ∧ x2 = x2 ∧ x3 = x3 ∧ x4 = x4 ∧ x5 = x5 ∧ x6 = x6 ∧ x7 = x7 ∧ x8 = x8 ∧ x9 = x9 ∧ x10 = x10 l9 l9 l9: x1 = x1 ∧ x2 = x2 ∧ x3 = x3 ∧ x4 = x4 ∧ x5 = x5 ∧ x6 = x6 ∧ x7 = x7 ∧ x8 = x8 ∧ x9 = x9 ∧ x10 = x10 l14 l14 l14: x1 = x1 ∧ x2 = x2 ∧ x3 = x3 ∧ x4 = x4 ∧ x5 = x5 ∧ x6 = x6 ∧ x7 = x7 ∧ x8 = x8 ∧ x9 = x9 ∧ x10 = x10 l4 l4 l4: x1 = x1 ∧ x2 = x2 ∧ x3 = x3 ∧ x4 = x4 ∧ x5 = x5 ∧ x6 = x6 ∧ x7 = x7 ∧ x8 = x8 ∧ x9 = x9 ∧ x10 = x10 l6 l6 l6: x1 = x1 ∧ x2 = x2 ∧ x3 = x3 ∧ x4 = x4 ∧ x5 = x5 ∧ x6 = x6 ∧ x7 = x7 ∧ x8 = x8 ∧ x9 = x9 ∧ x10 = x10 l10 l10 l10: x1 = x1 ∧ x2 = x2 ∧ x3 = x3 ∧ x4 = x4 ∧ x5 = x5 ∧ x6 = x6 ∧ x7 = x7 ∧ x8 = x8 ∧ x9 = x9 ∧ x10 = x10 l8 l8 l8: x1 = x1 ∧ x2 = x2 ∧ x3 = x3 ∧ x4 = x4 ∧ x5 = x5 ∧ x6 = x6 ∧ x7 = x7 ∧ x8 = x8 ∧ x9 = x9 ∧ x10 = x10 l15 l15 l15: x1 = x1 ∧ x2 = x2 ∧ x3 = x3 ∧ x4 = x4 ∧ x5 = x5 ∧ x6 = x6 ∧ x7 = x7 ∧ x8 = x8 ∧ x9 = x9 ∧ x10 = x10 l16 l16 l16: x1 = x1 ∧ x2 = x2 ∧ x3 = x3 ∧ x4 = x4 ∧ x5 = x5 ∧ x6 = x6 ∧ x7 = x7 ∧ x8 = x8 ∧ x9 = x9 ∧ x10 = x10 l0 l0 l0: x1 = x1 ∧ x2 = x2 ∧ x3 = x3 ∧ x4 = x4 ∧ x5 = x5 ∧ x6 = x6 ∧ x7 = x7 ∧ x8 = x8 ∧ x9 = x9 ∧ x10 = x10
and for every transition t, a duplicate t is considered.

### 2 SCC Decomposition

We consider subproblems for each of the 3 SCC(s) of the program graph.

### 2.1 SCC Subproblem 1/3

Here we consider the SCC { l1, l3, l0, l2 }.

### 2.1.1 Transition Removal

We remove transition 21 using the following ranking functions, which are bounded by 0.

 l0: 19 − x6 l1: 19 − x6 l3: 18 − x6 l2: 18 − x6

### 2.1.2 Transition Removal

We remove transitions 1, 18 using the following ranking functions, which are bounded by 0.

 l0: 0 l1: −1 l3: 1 l2: 1

### 2.1.3 Transition Removal

We remove transition 19 using the following ranking functions, which are bounded by 0.

 l2: 19 − x3 l3: 19 − x3

### 2.1.4 Transition Removal

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

 l2: 0 l3: −1

### 2.1.5 Trivial Cooperation Program

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

### 2.2 SCC Subproblem 2/3

Here we consider the SCC { l5, l4, l13, l14 }.

### 2.2.1 Transition Removal

We remove transition 15 using the following ranking functions, which are bounded by −57.

 l4: −3⋅x7 + 1 l5: −3⋅x7 l14: −3⋅x7 − 1 l13: −3⋅x7 − 1

### 2.2.2 Transition Removal

We remove transitions 3, 12 using the following ranking functions, which are bounded by 0.

 l4: 0 l5: −1 l14: 1 l13: 1

### 2.2.3 Transition Removal

We remove transition 13 using the following ranking functions, which are bounded by −38.

 l13: −2⋅x4 + 1 l14: −2⋅x4

### 2.2.4 Transition Removal

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

 l13: 0 l14: −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 3/3

Here we consider the SCC { l7, l10, l6, l11, l8, l9 }.

### 2.3.1 Transition Removal

We remove transition 9 using the following ranking functions, which are bounded by −55.

 l10: −3⋅x5 + 3 l11: −3⋅x5 + 2 l9: −3⋅x5 + 1 l7: −3⋅x5 + 1 l6: −3⋅x5 + 1 l8: −3⋅x5 + 1

### 2.3.2 Transition Removal

We remove transitions 11, 6 using the following ranking functions, which are bounded by 0.

 l10: 0 l11: −1 l9: 1 l7: 1 l6: 1 l8: 1

### 2.3.3 Transition Removal

We remove transition 7 using the following ranking functions, which are bounded by −57.

 l7: −3⋅x2 + 1 l9: −3⋅x2 l6: −3⋅x2 − 1 l8: −3⋅x2 − 1

### 2.3.4 Transition Removal

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

 l7: 19 − x1 l9: 19 − x1 l6: 19 − x1 l8: 19 − x1

### 2.3.5 Transition Removal

We remove transitions 16, 4, 17 using the following ranking functions, which are bounded by 0.

 l7: 0 l9: −1 l6: 1 l8: 2

### 2.3.6 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 % (21 real / 0 unknown / 0 assumptions / 21 total proof steps)