LTS Termination Proof

Input

Integer Transition System

Initial Location: l5, l4, l7, l6, l8, l3, l0, l2

Transitions: (pre-variables and post-variables)

l0	1	l1:	x1 = ___const_20HAT0 ∧ x2 = ___const_5HAT0 ∧ x3 = ___const_99HAT0 ∧ x4 = _oldX0HAT0 ∧ x5 = _oldX1HAT0 ∧ x6 = _oldX2HAT0 ∧ x7 = _oldX3HAT0 ∧ x8 = _x0HAT0 ∧ x9 = _x1HAT0 ∧ x1 = ___const_20HATpost ∧ x2 = ___const_5HATpost ∧ x3 = ___const_99HATpost ∧ x4 = _oldX0HATpost ∧ x5 = _oldX1HATpost ∧ x6 = _oldX2HATpost ∧ x7 = _oldX3HATpost ∧ x8 = _x0HATpost ∧ x9 = _x1HATpost ∧ ___const_99HAT0 = ___const_99HATpost ∧ ___const_5HAT0 = ___const_5HATpost ∧ ___const_20HAT0 = ___const_20HATpost ∧ _x1HATpost = _oldX3HATpost ∧ _x0HATpost = _oldX2HATpost ∧ _oldX3HATpost = _oldX3HATpost ∧ _oldX2HATpost = _oldX2HATpost ∧ _oldX1HATpost = _x1HAT0 ∧ _oldX0HATpost = _x0HAT0
l2	2	l3:	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 ∧ _x6 = _x15 ∧ _x5 = _x14 ∧ _x2 = _x11 ∧ _x1 = _x10 ∧ _x = _x9 ∧ _x17 = 3 + _x13 ∧ _x16 = _x12 ∧ _x13 = _x8 ∧ _x12 = _x7
l3	3	l0:	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 ∧ _x24 = _x33 ∧ _x23 = _x32 ∧ _x20 = _x29 ∧ _x19 = _x28 ∧ _x18 = _x27 ∧ _x35 = _x31 ∧ _x34 = _x30 ∧ 1 + _x18 ≤ _x31 ∧ _x31 = _x26 ∧ _x30 = _x25
l3	4	l2:	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 ∧ _x42 = _x51 ∧ _x41 = _x50 ∧ _x38 = _x47 ∧ _x37 = _x46 ∧ _x36 = _x45 ∧ _x53 = _x49 ∧ _x52 = _x48 ∧ _x49 ≤ _x36 ∧ _x49 = _x44 ∧ _x48 = _x43
l4	5	l5:	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 ∧ _x60 = _x69 ∧ _x56 = _x65 ∧ _x55 = _x64 ∧ _x54 = _x63 ∧ _x71 = _x68 ∧ _x70 = 1 + _x66 ∧ _x68 = _x68 ∧ _x67 = _x62 ∧ _x66 = _x61
l5	6	l3:	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 ∧ _x78 = _x87 ∧ _x77 = _x86 ∧ _x74 = _x83 ∧ _x73 = _x82 ∧ _x72 = _x81 ∧ _x89 = _x73 ∧ _x88 = _x84 ∧ 1 + _x74 ≤ _x84 ∧ _x85 = _x80 ∧ _x84 = _x79
l5	7	l4:	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 ∧ _x96 = _x105 ∧ _x92 = _x101 ∧ _x91 = _x100 ∧ _x90 = _x99 ∧ _x107 = _x104 ∧ _x106 = _x102 ∧ _x102 ≤ _x92 ∧ _x104 = _x104 ∧ _x103 = _x98 ∧ _x102 = _x97
l6	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 ∧ _x114 = _x123 ∧ _x110 = _x119 ∧ _x109 = _x118 ∧ _x108 = _x117 ∧ _x125 = _x122 ∧ _x124 = 0 ∧ _x122 = _x122 ∧ _x121 = _x116 ∧ _x120 = _x115
l7	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 ∧ _x128 = _x137 ∧ _x127 = _x136 ∧ _x126 = _x135 ∧ _x143 = _x141 ∧ _x142 = _x140 ∧ _x141 = _x141 ∧ _x140 = _x140 ∧ _x139 = _x134 ∧ _x138 = _x133
l7	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 ∧ _x152 = _x161 ∧ _x151 = _x160 ∧ _x150 = _x159 ∧ _x149 = _x158 ∧ _x148 = _x157 ∧ _x147 = _x156 ∧ _x146 = _x155 ∧ _x145 = _x154 ∧ _x144 = _x153
l7	11	l0:	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 ∧ _x170 = _x179 ∧ _x169 = _x178 ∧ _x168 = _x177 ∧ _x167 = _x176 ∧ _x166 = _x175 ∧ _x165 = _x174 ∧ _x164 = _x173 ∧ _x163 = _x172 ∧ _x162 = _x171
l7	12	l2:	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 ∧ _x188 = _x197 ∧ _x187 = _x196 ∧ _x186 = _x195 ∧ _x185 = _x194 ∧ _x184 = _x193 ∧ _x183 = _x192 ∧ _x182 = _x191 ∧ _x181 = _x190 ∧ _x180 = _x189
l7	13	l3:	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
l7	14	l4:	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
l7	15	l5:	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
l7	16	l6:	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
l8	17	l7:	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

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
l7	l7	l7:	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
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

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 { l5, l4 }.

2.1.1 Transition Removal

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

l4:	2⋅x3 − 2⋅x8 − 1
l5:	2⋅x3 − 2⋅x8

2.1.2 Transition Removal

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

l4:	0
l5:	−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 { l3, l2 }.

2.2.1 Transition Removal

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

l2:	2⋅x1 − 2⋅x9 − 1
l3:	2⋅x1 − 2⋅x9

2.2.2 Transition Removal

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

l2:	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)