LTS Termination Proof

Input

Integer Transition System

Initial Location: l4, l7, l6, l10, l8, l1, l3, l13, l0, l12, l2, l9

Transitions: (pre-variables and post-variables)

l0	1	l1:	x1 = _oldX0HAT0 ∧ x2 = _oldX1HAT0 ∧ x3 = _oldX2HAT0 ∧ x4 = _oldX3HAT0 ∧ x5 = _x0HAT0 ∧ x6 = _x1HAT0 ∧ x1 = _oldX0HATpost ∧ x2 = _oldX1HATpost ∧ x3 = _oldX2HATpost ∧ x4 = _oldX3HATpost ∧ x5 = _x0HATpost ∧ x6 = _x1HATpost ∧ _oldX3HAT0 = _oldX3HATpost ∧ _oldX2HAT0 = _oldX2HATpost ∧ _x1HATpost = _oldX1HATpost ∧ _x0HATpost = _oldX0HATpost ∧ _oldX1HATpost ≤ 0 ∧ _oldX1HATpost = _x1HAT0 ∧ _oldX0HATpost = _x0HAT0
l0	2	l1:	x1 = _x ∧ x2 = _x1 ∧ x3 = _x2 ∧ x4 = _x3 ∧ x5 = _x4 ∧ x6 = _x5 ∧ x1 = _x6 ∧ x2 = _x7 ∧ x3 = _x8 ∧ x4 = _x9 ∧ x5 = _x10 ∧ x6 = _x11 ∧ _x3 = _x9 ∧ _x2 = _x8 ∧ _x11 = _x7 ∧ _x10 = _x6 ∧ 0 ≤ _x7 ∧ _x7 ≤ 0 ∧ _x7 = _x5 ∧ _x6 = _x4
l0	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 ∧ _x15 = _x21 ∧ _x14 = _x20 ∧ _x23 = _x19 ∧ _x22 = _x18 ∧ 1 ≤ _x19 ∧ 1 ≤ _x19 ∧ _x19 = _x17 ∧ _x18 = _x16
l0	4	l2:	x1 = _x24 ∧ x2 = _x25 ∧ x3 = _x26 ∧ x4 = _x27 ∧ x5 = _x28 ∧ x6 = _x29 ∧ x1 = _x30 ∧ x2 = _x31 ∧ x3 = _x32 ∧ x4 = _x33 ∧ x5 = _x34 ∧ x6 = _x35 ∧ _x27 = _x33 ∧ _x26 = _x32 ∧ _x35 = _x31 ∧ _x34 = _x30 ∧ 1 ≤ _x31 ∧ 1 + _x31 ≤ 0 ∧ _x31 = _x29 ∧ _x30 = _x28
l3	5	l0:	x1 = _x36 ∧ x2 = _x37 ∧ x3 = _x38 ∧ x4 = _x39 ∧ x5 = _x40 ∧ x6 = _x41 ∧ x1 = _x42 ∧ x2 = _x43 ∧ x3 = _x44 ∧ x4 = _x45 ∧ x5 = _x46 ∧ x6 = _x47 ∧ _x39 = _x45 ∧ _x38 = _x44 ∧ _x47 = _x43 ∧ _x46 = _x42 ∧ _x43 = _x41 ∧ _x42 = _x40
l4	6	l5:	x1 = _x48 ∧ x2 = _x49 ∧ x3 = _x50 ∧ x4 = _x51 ∧ x5 = _x52 ∧ x6 = _x53 ∧ x1 = _x54 ∧ x2 = _x55 ∧ x3 = _x56 ∧ x4 = _x57 ∧ x5 = _x58 ∧ x6 = _x59 ∧ _x59 = _x57 ∧ _x58 = _x56 ∧ _x57 = _x57 ∧ _x56 = _x56 ∧ _x55 = _x53 ∧ _x54 = _x52
l6	7	l5:	x1 = _x60 ∧ x2 = _x61 ∧ x3 = _x62 ∧ x4 = _x63 ∧ x5 = _x64 ∧ x6 = _x65 ∧ x1 = _x66 ∧ x2 = _x67 ∧ x3 = _x68 ∧ x4 = _x69 ∧ x5 = _x70 ∧ x6 = _x71 ∧ _x71 = _x69 ∧ _x70 = _x68 ∧ _x69 = _x69 ∧ _x68 = _x68 ∧ _x67 = _x65 ∧ _x66 = _x64
l6	8	l7:	x1 = _x72 ∧ x2 = _x73 ∧ x3 = _x74 ∧ x4 = _x75 ∧ x5 = _x76 ∧ x6 = _x77 ∧ x1 = _x78 ∧ x2 = _x79 ∧ x3 = _x80 ∧ x4 = _x81 ∧ x5 = _x82 ∧ x6 = _x83 ∧ _x75 = _x81 ∧ _x74 = _x80 ∧ _x83 = _x79 ∧ _x82 = −1 + _x78 ∧ _x79 = _x77 ∧ _x78 = _x76
l8	9	l4:	x1 = _x84 ∧ x2 = _x85 ∧ x3 = _x86 ∧ x4 = _x87 ∧ x5 = _x88 ∧ x6 = _x89 ∧ x1 = _x90 ∧ x2 = _x91 ∧ x3 = _x92 ∧ x4 = _x93 ∧ x5 = _x94 ∧ x6 = _x95 ∧ _x87 = _x93 ∧ _x86 = _x92 ∧ _x95 = _x91 ∧ _x94 = _x90 ∧ _x90 ≤ 0 ∧ _x91 = _x89 ∧ _x90 = _x88
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 ∧ _x99 = _x105 ∧ _x98 = _x104 ∧ _x107 = _x103 ∧ _x106 = _x102 ∧ 1 ≤ _x102 ∧ _x103 = _x101 ∧ _x102 = _x100
l9	11	l5:	x1 = _x108 ∧ x2 = _x109 ∧ x3 = _x110 ∧ x4 = _x111 ∧ x5 = _x112 ∧ x6 = _x113 ∧ x1 = _x114 ∧ x2 = _x115 ∧ x3 = _x116 ∧ x4 = _x117 ∧ x5 = _x118 ∧ x6 = _x119 ∧ _x119 = _x117 ∧ _x118 = _x116 ∧ _x117 = _x117 ∧ _x116 = _x116 ∧ _x115 = _x113 ∧ _x114 = _x112
l9	12	l7:	x1 = _x120 ∧ x2 = _x121 ∧ x3 = _x122 ∧ x4 = _x123 ∧ x5 = _x124 ∧ x6 = _x125 ∧ x1 = _x126 ∧ x2 = _x127 ∧ x3 = _x128 ∧ x4 = _x129 ∧ x5 = _x130 ∧ x6 = _x131 ∧ _x123 = _x129 ∧ _x122 = _x128 ∧ _x131 = −1 + _x127 ∧ _x130 = _x126 ∧ _x127 = _x125 ∧ _x126 = _x124
l10	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 ∧ _x135 = _x141 ∧ _x134 = _x140 ∧ _x143 = _x139 ∧ _x142 = _x138 ∧ _x139 ≤ 0 ∧ _x139 = _x137 ∧ _x138 = _x136
l10	14	l9:	x1 = _x144 ∧ x2 = _x145 ∧ x3 = _x146 ∧ x4 = _x147 ∧ x5 = _x148 ∧ x6 = _x149 ∧ x1 = _x150 ∧ x2 = _x151 ∧ x3 = _x152 ∧ x4 = _x153 ∧ x5 = _x154 ∧ x6 = _x155 ∧ _x147 = _x153 ∧ _x146 = _x152 ∧ _x155 = _x151 ∧ _x154 = _x150 ∧ 1 ≤ _x151 ∧ _x151 = _x149 ∧ _x150 = _x148
l1	15	l11:	x1 = _x156 ∧ x2 = _x157 ∧ x3 = _x158 ∧ x4 = _x159 ∧ x5 = _x160 ∧ x6 = _x161 ∧ x1 = _x162 ∧ x2 = _x163 ∧ x3 = _x164 ∧ x4 = _x165 ∧ x5 = _x166 ∧ x6 = _x167 ∧ _x167 = _x165 ∧ _x166 = _x164 ∧ _x165 = _x165 ∧ _x164 = _x164 ∧ _x163 = _x161 ∧ _x162 = _x160
l2	16	l11:	x1 = _x168 ∧ x2 = _x169 ∧ x3 = _x170 ∧ x4 = _x171 ∧ x5 = _x172 ∧ x6 = _x173 ∧ x1 = _x174 ∧ x2 = _x175 ∧ x3 = _x176 ∧ x4 = _x177 ∧ x5 = _x178 ∧ x6 = _x179 ∧ _x179 = _x177 ∧ _x178 = _x176 ∧ _x177 = _x177 ∧ _x176 = _x176 ∧ _x175 = _x173 ∧ _x174 = _x172
l2	17	l3:	x1 = _x180 ∧ x2 = _x181 ∧ x3 = _x182 ∧ x4 = _x183 ∧ x5 = _x184 ∧ x6 = _x185 ∧ x1 = _x186 ∧ x2 = _x187 ∧ x3 = _x188 ∧ x4 = _x189 ∧ x5 = _x190 ∧ x6 = _x191 ∧ _x183 = _x189 ∧ _x182 = _x188 ∧ _x191 = −1 + _x187 ∧ _x190 = _x186 ∧ _x187 = _x185 ∧ _x186 = _x184
l7	18	l10:	x1 = _x192 ∧ x2 = _x193 ∧ x3 = _x194 ∧ x4 = _x195 ∧ x5 = _x196 ∧ x6 = _x197 ∧ x1 = _x198 ∧ x2 = _x199 ∧ x3 = _x200 ∧ x4 = _x201 ∧ x5 = _x202 ∧ x6 = _x203 ∧ _x195 = _x201 ∧ _x194 = _x200 ∧ _x203 = _x199 ∧ _x202 = _x198 ∧ _x199 = _x197 ∧ _x198 = _x196
l12	19	l0:	x1 = _x204 ∧ x2 = _x205 ∧ x3 = _x206 ∧ x4 = _x207 ∧ x5 = _x208 ∧ x6 = _x209 ∧ x1 = _x210 ∧ x2 = _x211 ∧ x3 = _x212 ∧ x4 = _x213 ∧ x5 = _x214 ∧ x6 = _x215 ∧ _x209 = _x215 ∧ _x208 = _x214 ∧ _x207 = _x213 ∧ _x206 = _x212 ∧ _x205 = _x211 ∧ _x204 = _x210
l12	20	l3:	x1 = _x216 ∧ x2 = _x217 ∧ x3 = _x218 ∧ x4 = _x219 ∧ x5 = _x220 ∧ x6 = _x221 ∧ x1 = _x222 ∧ x2 = _x223 ∧ x3 = _x224 ∧ x4 = _x225 ∧ x5 = _x226 ∧ x6 = _x227 ∧ _x221 = _x227 ∧ _x220 = _x226 ∧ _x219 = _x225 ∧ _x218 = _x224 ∧ _x217 = _x223 ∧ _x216 = _x222
l12	21	l4:	x1 = _x228 ∧ x2 = _x229 ∧ x3 = _x230 ∧ x4 = _x231 ∧ x5 = _x232 ∧ x6 = _x233 ∧ x1 = _x234 ∧ x2 = _x235 ∧ x3 = _x236 ∧ x4 = _x237 ∧ x5 = _x238 ∧ x6 = _x239 ∧ _x233 = _x239 ∧ _x232 = _x238 ∧ _x231 = _x237 ∧ _x230 = _x236 ∧ _x229 = _x235 ∧ _x228 = _x234
l12	22	l6:	x1 = _x240 ∧ x2 = _x241 ∧ x3 = _x242 ∧ x4 = _x243 ∧ x5 = _x244 ∧ x6 = _x245 ∧ x1 = _x246 ∧ x2 = _x247 ∧ x3 = _x248 ∧ x4 = _x249 ∧ x5 = _x250 ∧ x6 = _x251 ∧ _x245 = _x251 ∧ _x244 = _x250 ∧ _x243 = _x249 ∧ _x242 = _x248 ∧ _x241 = _x247 ∧ _x240 = _x246
l12	23	l5:	x1 = _x252 ∧ x2 = _x253 ∧ x3 = _x254 ∧ x4 = _x255 ∧ x5 = _x256 ∧ x6 = _x257 ∧ x1 = _x258 ∧ x2 = _x259 ∧ x3 = _x260 ∧ x4 = _x261 ∧ x5 = _x262 ∧ x6 = _x263 ∧ _x257 = _x263 ∧ _x256 = _x262 ∧ _x255 = _x261 ∧ _x254 = _x260 ∧ _x253 = _x259 ∧ _x252 = _x258
l12	24	l8:	x1 = _x264 ∧ x2 = _x265 ∧ x3 = _x266 ∧ x4 = _x267 ∧ x5 = _x268 ∧ x6 = _x269 ∧ x1 = _x270 ∧ x2 = _x271 ∧ x3 = _x272 ∧ x4 = _x273 ∧ x5 = _x274 ∧ x6 = _x275 ∧ _x269 = _x275 ∧ _x268 = _x274 ∧ _x267 = _x273 ∧ _x266 = _x272 ∧ _x265 = _x271 ∧ _x264 = _x270
l12	25	l9:	x1 = _x276 ∧ x2 = _x277 ∧ x3 = _x278 ∧ x4 = _x279 ∧ x5 = _x280 ∧ x6 = _x281 ∧ x1 = _x282 ∧ x2 = _x283 ∧ x3 = _x284 ∧ x4 = _x285 ∧ x5 = _x286 ∧ x6 = _x287 ∧ _x281 = _x287 ∧ _x280 = _x286 ∧ _x279 = _x285 ∧ _x278 = _x284 ∧ _x277 = _x283 ∧ _x276 = _x282
l12	26	l10:	x1 = _x288 ∧ x2 = _x289 ∧ x3 = _x290 ∧ x4 = _x291 ∧ x5 = _x292 ∧ x6 = _x293 ∧ x1 = _x294 ∧ x2 = _x295 ∧ x3 = _x296 ∧ x4 = _x297 ∧ x5 = _x298 ∧ x6 = _x299 ∧ _x293 = _x299 ∧ _x292 = _x298 ∧ _x291 = _x297 ∧ _x290 = _x296 ∧ _x289 = _x295 ∧ _x288 = _x294
l12	27	l11:	x1 = _x300 ∧ x2 = _x301 ∧ x3 = _x302 ∧ x4 = _x303 ∧ x5 = _x304 ∧ x6 = _x305 ∧ x1 = _x306 ∧ x2 = _x307 ∧ x3 = _x308 ∧ x4 = _x309 ∧ x5 = _x310 ∧ x6 = _x311 ∧ _x305 = _x311 ∧ _x304 = _x310 ∧ _x303 = _x309 ∧ _x302 = _x308 ∧ _x301 = _x307 ∧ _x300 = _x306
l12	28	l1:	x1 = _x312 ∧ x2 = _x313 ∧ x3 = _x314 ∧ x4 = _x315 ∧ x5 = _x316 ∧ x6 = _x317 ∧ x1 = _x318 ∧ x2 = _x319 ∧ x3 = _x320 ∧ x4 = _x321 ∧ x5 = _x322 ∧ x6 = _x323 ∧ _x317 = _x323 ∧ _x316 = _x322 ∧ _x315 = _x321 ∧ _x314 = _x320 ∧ _x313 = _x319 ∧ _x312 = _x318
l12	29	l2:	x1 = _x324 ∧ x2 = _x325 ∧ x3 = _x326 ∧ x4 = _x327 ∧ x5 = _x328 ∧ x6 = _x329 ∧ x1 = _x330 ∧ x2 = _x331 ∧ x3 = _x332 ∧ x4 = _x333 ∧ x5 = _x334 ∧ x6 = _x335 ∧ _x329 = _x335 ∧ _x328 = _x334 ∧ _x327 = _x333 ∧ _x326 = _x332 ∧ _x325 = _x331 ∧ _x324 = _x330
l12	30	l7:	x1 = _x336 ∧ x2 = _x337 ∧ x3 = _x338 ∧ x4 = _x339 ∧ x5 = _x340 ∧ x6 = _x341 ∧ x1 = _x342 ∧ x2 = _x343 ∧ x3 = _x344 ∧ x4 = _x345 ∧ x5 = _x346 ∧ x6 = _x347 ∧ _x341 = _x347 ∧ _x340 = _x346 ∧ _x339 = _x345 ∧ _x338 = _x344 ∧ _x337 = _x343 ∧ _x336 = _x342
l13	31	l12:	x1 = _x348 ∧ x2 = _x349 ∧ x3 = _x350 ∧ x4 = _x351 ∧ x5 = _x352 ∧ x6 = _x353 ∧ x1 = _x354 ∧ x2 = _x355 ∧ x3 = _x356 ∧ x4 = _x357 ∧ x5 = _x358 ∧ x6 = _x359 ∧ _x353 = _x359 ∧ _x352 = _x358 ∧ _x351 = _x357 ∧ _x350 = _x356 ∧ _x349 = _x355 ∧ _x348 = _x354

Proof

1 Switch to Cooperation Termination Proof

We consider the following cutpoint-transitions:

l4	l4	l4:	x1 = x1 ∧ x2 = x2 ∧ x3 = x3 ∧ x4 = x4 ∧ x5 = x5 ∧ x6 = x6
l7	l7	l7:	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
l13	l13	l13:	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
l12	l12	l12:	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 2 SCC(s) of the program graph.

2.1 SCC Subproblem 1/2

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

2.1.1 Transition Removal

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

l6:	4⋅x6
l7:	4⋅x6
l8:	4⋅x6
l10:	4⋅x6
l9:	4⋅x6 − 1

2.1.2 Transition Removal

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

l6:	−1
l7:	−1
l8:	−1
l10:	−1
l9:	0

2.1.3 Transition Removal

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

l6:	−2 + x5
l7:	−1 + x5
l8:	−1 + x5
l10:	−1 + x5

2.1.4 Transition Removal

We remove transitions 8, 13, 18 using the following ranking functions, which are bounded by 0.

l6:	3
l7:	2
l10:	1
l8:	0

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 2/2

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

2.2.1 Transition Removal

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

l0:	x3 + 2⋅x4
l2:	x3 + 2⋅x4
l3:	x3 + 2⋅x4

2.2.2 Transition Removal

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

l0:	−1 + x6
l2:	−2 + x6
l3:	−1 + x6

2.2.3 Transition Removal

We remove transitions 5, 17 using the following ranking functions, which are bounded by 0.

l3:	0
l0:	−1
l2:	1

2.2.4 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 % (13 real / 0 unknown / 0 assumptions / 13 total proof steps)