LTS Termination Proof

Input

Integer Transition System

Initial Location: l5, l4, l6, l3, l0, l2

Transitions: (pre-variables and post-variables)

l0	1	l1:	x1 = _oldX0HAT0 ∧ x2 = _oldX1HAT0 ∧ x3 = _oldX2HAT0 ∧ x4 = _oldX3HAT0 ∧ x5 = _oldX4HAT0 ∧ x6 = _oldX5HAT0 ∧ x7 = _oldX6HAT0 ∧ x8 = _oldX7HAT0 ∧ x9 = _x0HAT0 ∧ x10 = _x1HAT0 ∧ x11 = _x2HAT0 ∧ x12 = _x3HAT0 ∧ x1 = _oldX0HATpost ∧ x2 = _oldX1HATpost ∧ x3 = _oldX2HATpost ∧ x4 = _oldX3HATpost ∧ x5 = _oldX4HATpost ∧ x6 = _oldX5HATpost ∧ x7 = _oldX6HATpost ∧ x8 = _oldX7HATpost ∧ x9 = _x0HATpost ∧ x10 = _x1HATpost ∧ x11 = _x2HATpost ∧ x12 = _x3HATpost ∧ _x3HATpost = _oldX7HATpost ∧ _x2HATpost = _oldX6HATpost ∧ _x1HATpost = _oldX5HATpost ∧ _x0HATpost = _oldX4HATpost ∧ _oldX7HATpost = _oldX7HATpost ∧ _oldX6HATpost = _oldX6HATpost ∧ _oldX5HATpost = _oldX5HATpost ∧ _oldX4HATpost = _oldX4HATpost ∧ _oldX3HATpost = _x3HAT0 ∧ _oldX2HATpost = _x2HAT0 ∧ _oldX1HATpost = _x1HAT0 ∧ _oldX0HATpost = _x0HAT0
l2	2	l1:	x1 = _x ∧ x2 = _x1 ∧ x3 = _x2 ∧ x4 = _x3 ∧ x5 = _x4 ∧ x6 = _x5 ∧ x7 = _x6 ∧ x8 = _x7 ∧ x9 = _x8 ∧ x10 = _x9 ∧ x11 = _x10 ∧ x12 = _x11 ∧ x1 = _x12 ∧ x2 = _x13 ∧ x3 = _x14 ∧ x4 = _x15 ∧ x5 = _x16 ∧ x6 = _x17 ∧ x7 = _x18 ∧ x8 = _x19 ∧ x9 = _x20 ∧ x10 = _x21 ∧ x11 = _x22 ∧ x12 = _x23 ∧ _x23 = _x19 ∧ _x22 = _x18 ∧ _x21 = _x17 ∧ _x20 = _x16 ∧ _x19 = _x19 ∧ _x18 = _x18 ∧ _x17 = _x17 ∧ _x16 = _x16 ∧ _x15 = _x11 ∧ _x14 = _x10 ∧ _x13 = _x9 ∧ _x12 = _x8
l2	3	l3:	x1 = _x24 ∧ x2 = _x25 ∧ x3 = _x26 ∧ x4 = _x27 ∧ x5 = _x28 ∧ x6 = _x29 ∧ x7 = _x30 ∧ x8 = _x31 ∧ x9 = _x32 ∧ x10 = _x33 ∧ x11 = _x34 ∧ x12 = _x35 ∧ x1 = _x36 ∧ x2 = _x37 ∧ x3 = _x38 ∧ x4 = _x39 ∧ x5 = _x40 ∧ x6 = _x41 ∧ x7 = _x42 ∧ x8 = _x43 ∧ x9 = _x44 ∧ x10 = _x45 ∧ x11 = _x46 ∧ x12 = _x47 ∧ _x31 = _x43 ∧ _x30 = _x42 ∧ _x29 = _x41 ∧ _x28 = _x40 ∧ _x47 = _x37 ∧ _x46 = _x38 ∧ _x45 = _x39 ∧ _x44 = −1 + _x36 ∧ _x39 = _x35 ∧ _x38 = _x34 ∧ _x37 = _x33 ∧ _x36 = _x32
l2	4	l3:	x1 = _x48 ∧ x2 = _x49 ∧ x3 = _x50 ∧ x4 = _x51 ∧ x5 = _x52 ∧ x6 = _x53 ∧ x7 = _x54 ∧ x8 = _x55 ∧ x9 = _x56 ∧ x10 = _x57 ∧ x11 = _x58 ∧ x12 = _x59 ∧ x1 = _x60 ∧ x2 = _x61 ∧ x3 = _x62 ∧ x4 = _x63 ∧ x5 = _x64 ∧ x6 = _x65 ∧ x7 = _x66 ∧ x8 = _x67 ∧ x9 = _x68 ∧ x10 = _x69 ∧ x11 = _x70 ∧ x12 = _x71 ∧ _x55 = _x67 ∧ _x54 = _x66 ∧ _x53 = _x65 ∧ _x52 = _x64 ∧ _x71 = _x62 ∧ _x70 = _x63 ∧ _x69 = _x61 ∧ _x68 = −1 + _x60 ∧ _x63 = _x59 ∧ _x62 = _x58 ∧ _x61 = _x57 ∧ _x60 = _x56
l4	5	l0:	x1 = _x72 ∧ x2 = _x73 ∧ x3 = _x74 ∧ x4 = _x75 ∧ x5 = _x76 ∧ x6 = _x77 ∧ x7 = _x78 ∧ x8 = _x79 ∧ x9 = _x80 ∧ x10 = _x81 ∧ x11 = _x82 ∧ x12 = _x83 ∧ x1 = _x84 ∧ x2 = _x85 ∧ x3 = _x86 ∧ x4 = _x87 ∧ x5 = _x88 ∧ x6 = _x89 ∧ x7 = _x90 ∧ x8 = _x91 ∧ x9 = _x92 ∧ x10 = _x93 ∧ x11 = _x94 ∧ x12 = _x95 ∧ _x79 = _x91 ∧ _x78 = _x90 ∧ _x77 = _x89 ∧ _x76 = _x88 ∧ _x95 = _x87 ∧ _x94 = _x86 ∧ _x93 = _x85 ∧ _x92 = _x84 ∧ 1 ≤ _x84 ∧ _x84 ≤ 1 ∧ _x87 = _x83 ∧ _x86 = _x82 ∧ _x85 = _x81 ∧ _x84 = _x80
l4	6	l0:	x1 = _x96 ∧ x2 = _x97 ∧ x3 = _x98 ∧ x4 = _x99 ∧ x5 = _x100 ∧ x6 = _x101 ∧ x7 = _x102 ∧ x8 = _x103 ∧ x9 = _x104 ∧ x10 = _x105 ∧ x11 = _x106 ∧ x12 = _x107 ∧ x1 = _x108 ∧ x2 = _x109 ∧ x3 = _x110 ∧ x4 = _x111 ∧ x5 = _x112 ∧ x6 = _x113 ∧ x7 = _x114 ∧ x8 = _x115 ∧ x9 = _x116 ∧ x10 = _x117 ∧ x11 = _x118 ∧ x12 = _x119 ∧ _x103 = _x115 ∧ _x102 = _x114 ∧ _x101 = _x113 ∧ _x100 = _x112 ∧ _x119 = _x111 ∧ _x118 = _x110 ∧ _x117 = _x109 ∧ _x116 = _x108 ∧ _x108 ≤ 0 ∧ _x111 = _x107 ∧ _x110 = _x106 ∧ _x109 = _x105 ∧ _x108 = _x104
l4	7	l2:	x1 = _x120 ∧ x2 = _x121 ∧ x3 = _x122 ∧ x4 = _x123 ∧ x5 = _x124 ∧ x6 = _x125 ∧ x7 = _x126 ∧ x8 = _x127 ∧ x9 = _x128 ∧ x10 = _x129 ∧ x11 = _x130 ∧ x12 = _x131 ∧ x1 = _x132 ∧ x2 = _x133 ∧ x3 = _x134 ∧ x4 = _x135 ∧ x5 = _x136 ∧ x6 = _x137 ∧ x7 = _x138 ∧ x8 = _x139 ∧ x9 = _x140 ∧ x10 = _x141 ∧ x11 = _x142 ∧ x12 = _x143 ∧ _x127 = _x139 ∧ _x126 = _x138 ∧ _x125 = _x137 ∧ _x124 = _x136 ∧ _x143 = _x135 ∧ _x142 = _x134 ∧ _x141 = _x133 ∧ _x140 = _x132 ∧ 2 ≤ _x132 ∧ 1 ≤ _x132 ∧ _x135 = _x131 ∧ _x134 = _x130 ∧ _x133 = _x129 ∧ _x132 = _x128
l4	8	l2:	x1 = _x144 ∧ x2 = _x145 ∧ x3 = _x146 ∧ x4 = _x147 ∧ x5 = _x148 ∧ x6 = _x149 ∧ x7 = _x150 ∧ x8 = _x151 ∧ x9 = _x152 ∧ x10 = _x153 ∧ x11 = _x154 ∧ x12 = _x155 ∧ x1 = _x156 ∧ x2 = _x157 ∧ x3 = _x158 ∧ x4 = _x159 ∧ x5 = _x160 ∧ x6 = _x161 ∧ x7 = _x162 ∧ x8 = _x163 ∧ x9 = _x164 ∧ x10 = _x165 ∧ x11 = _x166 ∧ x12 = _x167 ∧ _x151 = _x163 ∧ _x150 = _x162 ∧ _x149 = _x161 ∧ _x148 = _x160 ∧ _x167 = _x159 ∧ _x166 = _x158 ∧ _x165 = _x157 ∧ _x164 = _x156 ∧ 1 + _x156 ≤ 1 ∧ 1 ≤ _x156 ∧ _x159 = _x155 ∧ _x158 = _x154 ∧ _x157 = _x153 ∧ _x156 = _x152
l3	9	l4:	x1 = _x168 ∧ x2 = _x169 ∧ x3 = _x170 ∧ x4 = _x171 ∧ x5 = _x172 ∧ x6 = _x173 ∧ x7 = _x174 ∧ x8 = _x175 ∧ x9 = _x176 ∧ x10 = _x177 ∧ x11 = _x178 ∧ x12 = _x179 ∧ x1 = _x180 ∧ x2 = _x181 ∧ x3 = _x182 ∧ x4 = _x183 ∧ x5 = _x184 ∧ x6 = _x185 ∧ x7 = _x186 ∧ x8 = _x187 ∧ x9 = _x188 ∧ x10 = _x189 ∧ x11 = _x190 ∧ x12 = _x191 ∧ _x175 = _x187 ∧ _x174 = _x186 ∧ _x173 = _x185 ∧ _x172 = _x184 ∧ _x191 = _x183 ∧ _x190 = _x182 ∧ _x189 = _x181 ∧ _x188 = _x180 ∧ _x183 = _x179 ∧ _x182 = _x178 ∧ _x181 = _x177 ∧ _x180 = _x176
l5	10	l1:	x1 = _x192 ∧ x2 = _x193 ∧ x3 = _x194 ∧ x4 = _x195 ∧ x5 = _x196 ∧ x6 = _x197 ∧ x7 = _x198 ∧ x8 = _x199 ∧ x9 = _x200 ∧ x10 = _x201 ∧ x11 = _x202 ∧ x12 = _x203 ∧ x1 = _x204 ∧ x2 = _x205 ∧ x3 = _x206 ∧ x4 = _x207 ∧ x5 = _x208 ∧ x6 = _x209 ∧ x7 = _x210 ∧ x8 = _x211 ∧ x9 = _x212 ∧ x10 = _x213 ∧ x11 = _x214 ∧ x12 = _x215 ∧ _x203 = _x215 ∧ _x202 = _x214 ∧ _x201 = _x213 ∧ _x200 = _x212 ∧ _x199 = _x211 ∧ _x198 = _x210 ∧ _x197 = _x209 ∧ _x196 = _x208 ∧ _x195 = _x207 ∧ _x194 = _x206 ∧ _x193 = _x205 ∧ _x192 = _x204
l5	11	l0:	x1 = _x216 ∧ x2 = _x217 ∧ x3 = _x218 ∧ x4 = _x219 ∧ x5 = _x220 ∧ x6 = _x221 ∧ x7 = _x222 ∧ x8 = _x223 ∧ x9 = _x224 ∧ x10 = _x225 ∧ x11 = _x226 ∧ x12 = _x227 ∧ x1 = _x228 ∧ x2 = _x229 ∧ x3 = _x230 ∧ x4 = _x231 ∧ x5 = _x232 ∧ x6 = _x233 ∧ x7 = _x234 ∧ x8 = _x235 ∧ x9 = _x236 ∧ x10 = _x237 ∧ x11 = _x238 ∧ x12 = _x239 ∧ _x227 = _x239 ∧ _x226 = _x238 ∧ _x225 = _x237 ∧ _x224 = _x236 ∧ _x223 = _x235 ∧ _x222 = _x234 ∧ _x221 = _x233 ∧ _x220 = _x232 ∧ _x219 = _x231 ∧ _x218 = _x230 ∧ _x217 = _x229 ∧ _x216 = _x228
l5	12	l2:	x1 = _x240 ∧ x2 = _x241 ∧ x3 = _x242 ∧ x4 = _x243 ∧ x5 = _x244 ∧ x6 = _x245 ∧ x7 = _x246 ∧ x8 = _x247 ∧ x9 = _x248 ∧ x10 = _x249 ∧ x11 = _x250 ∧ x12 = _x251 ∧ x1 = _x252 ∧ x2 = _x253 ∧ x3 = _x254 ∧ x4 = _x255 ∧ x5 = _x256 ∧ x6 = _x257 ∧ x7 = _x258 ∧ x8 = _x259 ∧ x9 = _x260 ∧ x10 = _x261 ∧ x11 = _x262 ∧ x12 = _x263 ∧ _x251 = _x263 ∧ _x250 = _x262 ∧ _x249 = _x261 ∧ _x248 = _x260 ∧ _x247 = _x259 ∧ _x246 = _x258 ∧ _x245 = _x257 ∧ _x244 = _x256 ∧ _x243 = _x255 ∧ _x242 = _x254 ∧ _x241 = _x253 ∧ _x240 = _x252
l5	13	l4:	x1 = _x264 ∧ x2 = _x265 ∧ x3 = _x266 ∧ x4 = _x267 ∧ x5 = _x268 ∧ x6 = _x269 ∧ x7 = _x270 ∧ x8 = _x271 ∧ x9 = _x272 ∧ x10 = _x273 ∧ x11 = _x274 ∧ x12 = _x275 ∧ x1 = _x276 ∧ x2 = _x277 ∧ x3 = _x278 ∧ x4 = _x279 ∧ x5 = _x280 ∧ x6 = _x281 ∧ x7 = _x282 ∧ x8 = _x283 ∧ x9 = _x284 ∧ x10 = _x285 ∧ x11 = _x286 ∧ x12 = _x287 ∧ _x275 = _x287 ∧ _x274 = _x286 ∧ _x273 = _x285 ∧ _x272 = _x284 ∧ _x271 = _x283 ∧ _x270 = _x282 ∧ _x269 = _x281 ∧ _x268 = _x280 ∧ _x267 = _x279 ∧ _x266 = _x278 ∧ _x265 = _x277 ∧ _x264 = _x276
l5	14	l3:	x1 = _x288 ∧ x2 = _x289 ∧ x3 = _x290 ∧ x4 = _x291 ∧ x5 = _x292 ∧ x6 = _x293 ∧ x7 = _x294 ∧ x8 = _x295 ∧ x9 = _x296 ∧ x10 = _x297 ∧ x11 = _x298 ∧ x12 = _x299 ∧ x1 = _x300 ∧ x2 = _x301 ∧ x3 = _x302 ∧ x4 = _x303 ∧ x5 = _x304 ∧ x6 = _x305 ∧ x7 = _x306 ∧ x8 = _x307 ∧ x9 = _x308 ∧ x10 = _x309 ∧ x11 = _x310 ∧ x12 = _x311 ∧ _x299 = _x311 ∧ _x298 = _x310 ∧ _x297 = _x309 ∧ _x296 = _x308 ∧ _x295 = _x307 ∧ _x294 = _x306 ∧ _x293 = _x305 ∧ _x292 = _x304 ∧ _x291 = _x303 ∧ _x290 = _x302 ∧ _x289 = _x301 ∧ _x288 = _x300
l6	15	l5:	x1 = _x312 ∧ x2 = _x313 ∧ x3 = _x314 ∧ x4 = _x315 ∧ x5 = _x316 ∧ x6 = _x317 ∧ x7 = _x318 ∧ x8 = _x319 ∧ x9 = _x320 ∧ x10 = _x321 ∧ x11 = _x322 ∧ x12 = _x323 ∧ x1 = _x324 ∧ x2 = _x325 ∧ x3 = _x326 ∧ x4 = _x327 ∧ x5 = _x328 ∧ x6 = _x329 ∧ x7 = _x330 ∧ x8 = _x331 ∧ x9 = _x332 ∧ x10 = _x333 ∧ x11 = _x334 ∧ x12 = _x335 ∧ _x323 = _x335 ∧ _x322 = _x334 ∧ _x321 = _x333 ∧ _x320 = _x332 ∧ _x319 = _x331 ∧ _x318 = _x330 ∧ _x317 = _x329 ∧ _x316 = _x328 ∧ _x315 = _x327 ∧ _x314 = _x326 ∧ _x313 = _x325 ∧ _x312 = _x324

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
l4	l4	l4:	x1 = x1 ∧ x2 = x2 ∧ x3 = x3 ∧ x4 = x4 ∧ x5 = x5 ∧ x6 = x6 ∧ x7 = x7 ∧ x8 = x8 ∧ x9 = x9 ∧ x10 = x10 ∧ x11 = x11 ∧ x12 = x12
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
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
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
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

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 { l4, l3, l2 }.

2.1.1 Transition Removal

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

l2:	−2 + x9
l3:	−1 + x9
l4:	−1 + x9

2.1.2 Transition Removal

We remove transitions 3, 9, 4, 8 using the following ranking functions, which are bounded by −2.

l2:	0
l3:	−1
l4:	−2

2.1.3 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 % (6 real / 0 unknown / 0 assumptions / 6 total proof steps)