# 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 = _constant22HAT0 ∧ x2 = _i20HAT0 ∧ x3 = _lx2HAT0 ∧ x4 = _tmp03HAT0 ∧ x5 = _tmp1011HAT0 ∧ x6 = _tmp1112HAT0 ∧ x7 = _tmp1213HAT0 ∧ x8 = _tmp1314HAT0 ∧ x9 = _tmp14HAT0 ∧ x10 = _tmp25HAT0 ∧ x11 = _tmp36HAT0 ∧ x12 = _tmp47HAT0 ∧ x13 = _tmp58HAT0 ∧ x14 = _tmp69HAT0 ∧ x15 = _tmp710HAT0 ∧ x16 = _z115HAT0 ∧ x17 = _z216HAT0 ∧ x18 = _z317HAT0 ∧ x19 = _z418HAT0 ∧ x20 = _z519HAT0 ∧ x1 = _constant22HATpost ∧ x2 = _i20HATpost ∧ x3 = _lx2HATpost ∧ x4 = _tmp03HATpost ∧ x5 = _tmp1011HATpost ∧ x6 = _tmp1112HATpost ∧ x7 = _tmp1213HATpost ∧ x8 = _tmp1314HATpost ∧ x9 = _tmp14HATpost ∧ x10 = _tmp25HATpost ∧ x11 = _tmp36HATpost ∧ x12 = _tmp47HATpost ∧ x13 = _tmp58HATpost ∧ x14 = _tmp69HATpost ∧ x15 = _tmp710HATpost ∧ x16 = _z115HATpost ∧ x17 = _z216HATpost ∧ x18 = _z317HATpost ∧ x19 = _z418HATpost ∧ x20 = _z519HATpost ∧ _z519HAT0 = _z519HATpost ∧ _z418HAT0 = _z418HATpost ∧ _z317HAT0 = _z317HATpost ∧ _z216HAT0 = _z216HATpost ∧ _z115HAT0 = _z115HATpost ∧ _tmp710HAT0 = _tmp710HATpost ∧ _tmp69HAT0 = _tmp69HATpost ∧ _tmp58HAT0 = _tmp58HATpost ∧ _tmp47HAT0 = _tmp47HATpost ∧ _tmp36HAT0 = _tmp36HATpost ∧ _tmp25HAT0 = _tmp25HATpost ∧ _tmp14HAT0 = _tmp14HATpost ∧ _tmp1314HAT0 = _tmp1314HATpost ∧ _tmp1213HAT0 = _tmp1213HATpost ∧ _tmp1112HAT0 = _tmp1112HATpost ∧ _tmp1011HAT0 = _tmp1011HATpost ∧ _tmp03HAT0 = _tmp03HATpost ∧ _lx2HAT0 = _lx2HATpost ∧ _constant22HAT0 = _constant22HATpost ∧ _i20HATpost = 0 ∧ 8 ≤ _i20HAT0 l0 2 l2: x1 = _x ∧ x2 = _x1 ∧ x3 = _x2 ∧ x4 = _x3 ∧ x5 = _x4 ∧ x6 = _x5 ∧ x7 = _x6 ∧ x8 = _x7 ∧ x9 = _x8 ∧ x10 = _x9 ∧ x11 = _x10 ∧ x12 = _x11 ∧ x13 = _x12 ∧ x14 = _x13 ∧ x15 = _x14 ∧ x16 = _x15 ∧ x17 = _x16 ∧ x18 = _x17 ∧ x19 = _x18 ∧ x20 = _x19 ∧ x1 = _x20 ∧ x2 = _x21 ∧ x3 = _x22 ∧ x4 = _x23 ∧ x5 = _x24 ∧ x6 = _x25 ∧ x7 = _x26 ∧ x8 = _x27 ∧ x9 = _x28 ∧ x10 = _x29 ∧ x11 = _x30 ∧ x12 = _x31 ∧ x13 = _x32 ∧ x14 = _x33 ∧ x15 = _x34 ∧ x16 = _x35 ∧ x17 = _x36 ∧ x18 = _x37 ∧ x19 = _x38 ∧ x20 = _x39 ∧ 1 + _x1 ≤ 8 ∧ _x23 = _x23 ∧ _x40 = _x40 ∧ _x28 = _x28 ∧ _x41 = _x41 ∧ _x29 = _x29 ∧ _x42 = _x42 ∧ _x30 = _x30 ∧ _x43 = _x43 ∧ _x24 = _x23 + _x30 ∧ _x27 = _x23 − _x30 ∧ _x25 = _x28 + _x29 ∧ _x26 = _x28 − _x29 ∧ _x44 = 4433 ∧ _x45 = _x45 ∧ _x46 = 6270 ∧ _x47 = −15137 ∧ _x48 = _x43 + _x40 ∧ _x49 = _x42 + _x41 ∧ _x50 = _x43 + _x41 ∧ _x51 = _x42 + _x40 ∧ _x52 = 9633 ∧ _x39 = _x39 ∧ _x53 = 2446 ∧ _x31 = _x31 ∧ _x54 = 16819 ∧ _x32 = _x32 ∧ _x55 = 25172 ∧ _x33 = _x33 ∧ _x56 = 12299 ∧ _x34 = _x34 ∧ _x57 = −7373 ∧ _x35 = _x35 ∧ _x58 = −20995 ∧ _x36 = _x36 ∧ _x59 = −16069 ∧ _x60 = _x60 ∧ _x20 = −3196 ∧ _x61 = _x61 ∧ _x37 = _x60 + _x39 ∧ _x38 = _x61 + _x39 ∧ _x21 = 1 + _x1 ∧ _x2 = _x22 l3 3 l4: x1 = _x62 ∧ x2 = _x63 ∧ x3 = _x64 ∧ x4 = _x65 ∧ x5 = _x66 ∧ x6 = _x67 ∧ x7 = _x68 ∧ x8 = _x69 ∧ x9 = _x70 ∧ x10 = _x71 ∧ x11 = _x72 ∧ x12 = _x73 ∧ x13 = _x74 ∧ x14 = _x75 ∧ x15 = _x76 ∧ x16 = _x77 ∧ x17 = _x78 ∧ x18 = _x79 ∧ x19 = _x80 ∧ x20 = _x81 ∧ x1 = _x82 ∧ x2 = _x83 ∧ x3 = _x84 ∧ x4 = _x85 ∧ x5 = _x86 ∧ x6 = _x87 ∧ x7 = _x88 ∧ x8 = _x89 ∧ x9 = _x90 ∧ x10 = _x91 ∧ x11 = _x92 ∧ x12 = _x93 ∧ x13 = _x94 ∧ x14 = _x95 ∧ x15 = _x96 ∧ x16 = _x97 ∧ x17 = _x98 ∧ x18 = _x99 ∧ x19 = _x100 ∧ x20 = _x101 ∧ _x81 = _x101 ∧ _x80 = _x100 ∧ _x79 = _x99 ∧ _x78 = _x98 ∧ _x77 = _x97 ∧ _x76 = _x96 ∧ _x75 = _x95 ∧ _x74 = _x94 ∧ _x73 = _x93 ∧ _x72 = _x92 ∧ _x71 = _x91 ∧ _x70 = _x90 ∧ _x69 = _x89 ∧ _x68 = _x88 ∧ _x67 = _x87 ∧ _x66 = _x86 ∧ _x65 = _x85 ∧ _x64 = _x84 ∧ _x63 = _x83 ∧ _x62 = _x82 ∧ 8 ≤ _x63 l3 4 l1: x1 = _x102 ∧ x2 = _x103 ∧ x3 = _x104 ∧ x4 = _x105 ∧ x5 = _x106 ∧ x6 = _x107 ∧ x7 = _x108 ∧ x8 = _x109 ∧ x9 = _x110 ∧ x10 = _x111 ∧ x11 = _x112 ∧ x12 = _x113 ∧ x13 = _x114 ∧ x14 = _x115 ∧ x15 = _x116 ∧ x16 = _x117 ∧ x17 = _x118 ∧ x18 = _x119 ∧ x19 = _x120 ∧ x20 = _x121 ∧ x1 = _x122 ∧ x2 = _x123 ∧ x3 = _x124 ∧ x4 = _x125 ∧ x5 = _x126 ∧ x6 = _x127 ∧ x7 = _x128 ∧ x8 = _x129 ∧ x9 = _x130 ∧ x10 = _x131 ∧ x11 = _x132 ∧ x12 = _x133 ∧ x13 = _x134 ∧ x14 = _x135 ∧ x15 = _x136 ∧ x16 = _x137 ∧ x17 = _x138 ∧ x18 = _x139 ∧ x19 = _x140 ∧ x20 = _x141 ∧ 1 + _x103 ≤ 8 ∧ _x125 = _x125 ∧ _x142 = _x142 ∧ _x130 = _x130 ∧ _x143 = _x143 ∧ _x131 = _x131 ∧ _x144 = _x144 ∧ _x132 = _x132 ∧ _x145 = _x145 ∧ _x126 = _x125 + _x132 ∧ _x129 = _x125 − _x132 ∧ _x127 = _x130 + _x131 ∧ _x128 = _x130 − _x131 ∧ _x146 = 4433 ∧ _x147 = _x147 ∧ _x148 = 6270 ∧ _x149 = −15137 ∧ _x150 = _x145 + _x142 ∧ _x151 = _x144 + _x143 ∧ _x152 = _x145 + _x143 ∧ _x153 = _x144 + _x142 ∧ _x154 = 9633 ∧ _x141 = _x141 ∧ _x155 = 2446 ∧ _x133 = _x133 ∧ _x156 = 16819 ∧ _x134 = _x134 ∧ _x157 = 25172 ∧ _x135 = _x135 ∧ _x158 = 12299 ∧ _x136 = _x136 ∧ _x159 = −7373 ∧ _x137 = _x137 ∧ _x160 = −20995 ∧ _x138 = _x138 ∧ _x161 = −16069 ∧ _x162 = _x162 ∧ _x122 = −3196 ∧ _x163 = _x163 ∧ _x139 = _x162 + _x141 ∧ _x140 = _x163 + _x141 ∧ _x123 = 1 + _x103 ∧ _x104 = _x124 l2 5 l0: x1 = _x164 ∧ x2 = _x165 ∧ x3 = _x166 ∧ x4 = _x167 ∧ x5 = _x168 ∧ x6 = _x169 ∧ x7 = _x170 ∧ x8 = _x171 ∧ x9 = _x172 ∧ x10 = _x173 ∧ x11 = _x174 ∧ x12 = _x175 ∧ x13 = _x176 ∧ x14 = _x177 ∧ x15 = _x178 ∧ x16 = _x179 ∧ x17 = _x180 ∧ x18 = _x181 ∧ x19 = _x182 ∧ x20 = _x183 ∧ x1 = _x184 ∧ x2 = _x185 ∧ x3 = _x186 ∧ x4 = _x187 ∧ x5 = _x188 ∧ x6 = _x189 ∧ x7 = _x190 ∧ x8 = _x191 ∧ x9 = _x192 ∧ x10 = _x193 ∧ x11 = _x194 ∧ x12 = _x195 ∧ x13 = _x196 ∧ x14 = _x197 ∧ x15 = _x198 ∧ x16 = _x199 ∧ x17 = _x200 ∧ x18 = _x201 ∧ x19 = _x202 ∧ x20 = _x203 ∧ _x183 = _x203 ∧ _x182 = _x202 ∧ _x181 = _x201 ∧ _x180 = _x200 ∧ _x179 = _x199 ∧ _x178 = _x198 ∧ _x177 = _x197 ∧ _x176 = _x196 ∧ _x175 = _x195 ∧ _x174 = _x194 ∧ _x173 = _x193 ∧ _x172 = _x192 ∧ _x171 = _x191 ∧ _x170 = _x190 ∧ _x169 = _x189 ∧ _x168 = _x188 ∧ _x167 = _x187 ∧ _x166 = _x186 ∧ _x165 = _x185 ∧ _x164 = _x184 l1 6 l3: x1 = _x204 ∧ x2 = _x205 ∧ x3 = _x206 ∧ x4 = _x207 ∧ x5 = _x208 ∧ x6 = _x209 ∧ x7 = _x210 ∧ x8 = _x211 ∧ x9 = _x212 ∧ x10 = _x213 ∧ x11 = _x214 ∧ x12 = _x215 ∧ x13 = _x216 ∧ x14 = _x217 ∧ x15 = _x218 ∧ x16 = _x219 ∧ x17 = _x220 ∧ x18 = _x221 ∧ x19 = _x222 ∧ x20 = _x223 ∧ x1 = _x224 ∧ x2 = _x225 ∧ x3 = _x226 ∧ x4 = _x227 ∧ x5 = _x228 ∧ x6 = _x229 ∧ x7 = _x230 ∧ x8 = _x231 ∧ x9 = _x232 ∧ x10 = _x233 ∧ x11 = _x234 ∧ x12 = _x235 ∧ x13 = _x236 ∧ x14 = _x237 ∧ x15 = _x238 ∧ x16 = _x239 ∧ x17 = _x240 ∧ x18 = _x241 ∧ x19 = _x242 ∧ x20 = _x243 ∧ _x223 = _x243 ∧ _x222 = _x242 ∧ _x221 = _x241 ∧ _x220 = _x240 ∧ _x219 = _x239 ∧ _x218 = _x238 ∧ _x217 = _x237 ∧ _x216 = _x236 ∧ _x215 = _x235 ∧ _x214 = _x234 ∧ _x213 = _x233 ∧ _x212 = _x232 ∧ _x211 = _x231 ∧ _x210 = _x230 ∧ _x209 = _x229 ∧ _x208 = _x228 ∧ _x207 = _x227 ∧ _x206 = _x226 ∧ _x205 = _x225 ∧ _x204 = _x224 l5 7 l2: x1 = _x244 ∧ x2 = _x245 ∧ x3 = _x246 ∧ x4 = _x247 ∧ x5 = _x248 ∧ x6 = _x249 ∧ x7 = _x250 ∧ x8 = _x251 ∧ x9 = _x252 ∧ x10 = _x253 ∧ x11 = _x254 ∧ x12 = _x255 ∧ x13 = _x256 ∧ x14 = _x257 ∧ x15 = _x258 ∧ x16 = _x259 ∧ x17 = _x260 ∧ x18 = _x261 ∧ x19 = _x262 ∧ x20 = _x263 ∧ x1 = _x264 ∧ x2 = _x265 ∧ x3 = _x266 ∧ x4 = _x267 ∧ x5 = _x268 ∧ x6 = _x269 ∧ x7 = _x270 ∧ x8 = _x271 ∧ x9 = _x272 ∧ x10 = _x273 ∧ x11 = _x274 ∧ x12 = _x275 ∧ x13 = _x276 ∧ x14 = _x277 ∧ x15 = _x278 ∧ x16 = _x279 ∧ x17 = _x280 ∧ x18 = _x281 ∧ x19 = _x282 ∧ x20 = _x283 ∧ _x263 = _x283 ∧ _x262 = _x282 ∧ _x261 = _x281 ∧ _x260 = _x280 ∧ _x259 = _x279 ∧ _x258 = _x278 ∧ _x257 = _x277 ∧ _x256 = _x276 ∧ _x255 = _x275 ∧ _x254 = _x274 ∧ _x253 = _x273 ∧ _x252 = _x272 ∧ _x251 = _x271 ∧ _x250 = _x270 ∧ _x249 = _x269 ∧ _x248 = _x268 ∧ _x247 = _x267 ∧ _x244 = _x264 ∧ _x265 = 0 ∧ _x266 = 8 l6 8 l5: x1 = _x284 ∧ x2 = _x285 ∧ x3 = _x286 ∧ x4 = _x287 ∧ x5 = _x288 ∧ x6 = _x289 ∧ x7 = _x290 ∧ x8 = _x291 ∧ x9 = _x292 ∧ x10 = _x293 ∧ x11 = _x294 ∧ x12 = _x295 ∧ x13 = _x296 ∧ x14 = _x297 ∧ x15 = _x298 ∧ x16 = _x299 ∧ x17 = _x300 ∧ x18 = _x301 ∧ x19 = _x302 ∧ x20 = _x303 ∧ x1 = _x304 ∧ x2 = _x305 ∧ x3 = _x306 ∧ x4 = _x307 ∧ x5 = _x308 ∧ x6 = _x309 ∧ x7 = _x310 ∧ x8 = _x311 ∧ x9 = _x312 ∧ x10 = _x313 ∧ x11 = _x314 ∧ x12 = _x315 ∧ x13 = _x316 ∧ x14 = _x317 ∧ x15 = _x318 ∧ x16 = _x319 ∧ x17 = _x320 ∧ x18 = _x321 ∧ x19 = _x322 ∧ x20 = _x323 ∧ _x303 = _x323 ∧ _x302 = _x322 ∧ _x301 = _x321 ∧ _x300 = _x320 ∧ _x299 = _x319 ∧ _x298 = _x318 ∧ _x297 = _x317 ∧ _x296 = _x316 ∧ _x295 = _x315 ∧ _x294 = _x314 ∧ _x293 = _x313 ∧ _x292 = _x312 ∧ _x291 = _x311 ∧ _x290 = _x310 ∧ _x289 = _x309 ∧ _x288 = _x308 ∧ _x287 = _x307 ∧ _x286 = _x306 ∧ _x285 = _x305 ∧ _x284 = _x304

## 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 ∧ x13 = x13 ∧ x14 = x14 ∧ x15 = x15 ∧ x16 = x16 ∧ x17 = x17 ∧ x18 = x18 ∧ x19 = x19 ∧ x20 = x20 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 ∧ x13 = x13 ∧ x14 = x14 ∧ x15 = x15 ∧ x16 = x16 ∧ x17 = x17 ∧ x18 = x18 ∧ x19 = x19 ∧ x20 = x20 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 ∧ x13 = x13 ∧ x14 = x14 ∧ x15 = x15 ∧ x16 = x16 ∧ x17 = x17 ∧ x18 = x18 ∧ x19 = x19 ∧ x20 = x20 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 ∧ x13 = x13 ∧ x14 = x14 ∧ x15 = x15 ∧ x16 = x16 ∧ x17 = x17 ∧ x18 = x18 ∧ x19 = x19 ∧ x20 = x20 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 ∧ x13 = x13 ∧ x14 = x14 ∧ x15 = x15 ∧ x16 = x16 ∧ x17 = x17 ∧ x18 = x18 ∧ x19 = x19 ∧ x20 = x20 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 ∧ x13 = x13 ∧ x14 = x14 ∧ x15 = x15 ∧ x16 = x16 ∧ x17 = x17 ∧ x18 = x18 ∧ x19 = x19 ∧ x20 = x20
and for every transition t, a duplicate t is considered.

### 2 SCC Decomposition

We consider subproblems for each of the 2 SCC(s) of the program graph.

### 2.1 SCC Subproblem 1/2

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: 7 − x2 l2: 7 − x2

### 2.1.2 Transition Removal

We remove transition 5 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.

### 2.2 SCC Subproblem 2/2

Here we consider the SCC { l1, l3 }.

### 2.2.1 Transition Removal

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

 l1: 7 − x2 l3: 7 − x2

### 2.2.2 Transition Removal

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

 l1: 0 l3: −1

### 2.2.3 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 % (10 real / 0 unknown / 0 assumptions / 10 total proof steps)