# LTS Termination Proof

by AProVE

## Input

Integer Transition System
• Initial Location: f288_0_slide68_EQ, f1_0_main_Load, f196_0_create_LE, f234_0_slide68_FieldAccess, f288_0_slide68_EQ', __init
• Transitions: (pre-variables and post-variables)  f1_0_main_Load 1 f234_0_slide68_FieldAccess: x1 = _arg1 ∧ x2 = _arg2 ∧ x3 = _arg3 ∧ x4 = _arg4 ∧ x5 = _arg5 ∧ x6 = _arg6 ∧ x7 = _arg7 ∧ x1 = _arg1P ∧ x2 = _arg2P ∧ x3 = _arg3P ∧ x4 = _arg4P ∧ x5 = _arg5P ∧ x6 = _arg6P ∧ x7 = _arg7P ∧ −1 ≤ _x4 − 1 ∧ 1 ≤ _arg2 − 1 ∧ −1 ≤ _arg1P − 1 ∧ _arg2P ≤ _x5 − 1 ∧ −1 ≤ _x5 − 1 ∧ 0 ≤ _arg1 − 1 f1_0_main_Load 2 f234_0_slide68_FieldAccess: x1 = _x ∧ x2 = _x1 ∧ x3 = _x2 ∧ x4 = _x3 ∧ x5 = _x6 ∧ x6 = _x7 ∧ x7 = _x8 ∧ x1 = _x9 ∧ x2 = _x11 ∧ x3 = _x12 ∧ x4 = _x13 ∧ x5 = _x14 ∧ x6 = _x15 ∧ x7 = _x16 ∧ −1 ≤ _x17 − 1 ∧ 1 ≤ _x1 − 1 ∧ _x11 ≤ 0 ∧ −1 ≤ _x9 − 1 ∧ 0 ≤ _x − 1 f234_0_slide68_FieldAccess 3 f288_0_slide68_EQ: x1 = _x18 ∧ x2 = _x19 ∧ x3 = _x20 ∧ x4 = _x21 ∧ x5 = _x22 ∧ x6 = _x23 ∧ x7 = _x24 ∧ x1 = _x25 ∧ x2 = _x26 ∧ x3 = _x27 ∧ x4 = _x28 ∧ x5 = _x29 ∧ x6 = _x30 ∧ x7 = _x31 ∧ _x19 = _x31 ∧ _x19 = _x30 ∧ 0 = _x29 ∧ _x19 = _x28 ∧ _x19 = _x27 ∧ 0 = _x26 ∧ _x18 = _x25 ∧ 0 ≤ _x19 − 1 f288_0_slide68_EQ 4 f288_0_slide68_EQ': x1 = _x32 ∧ x2 = _x33 ∧ x3 = _x34 ∧ x4 = _x35 ∧ x5 = _x36 ∧ x6 = _x37 ∧ x7 = _x38 ∧ x1 = _x39 ∧ x2 = _x40 ∧ x3 = _x41 ∧ x4 = _x42 ∧ x5 = _x43 ∧ x6 = _x44 ∧ x7 = _x45 ∧ _x46 ≤ _x34 − 1 ∧ 0 ≤ _x34 − 1 ∧ _x47 ≤ _x37 − 1 ∧ −1 ≤ _x37 − 1 ∧ _x48 ≤ _x49 − 1 ∧ −1 ≤ _x47 − 1 ∧ _x47 ≤ _x48 − 1 ∧ −1 ≤ _x49 − 1 ∧ _x47 ≤ _x50 − 1 ∧ _x47 ≤ _x46 − 1 ∧ _x51 ≤ _x32 ∧ _x32 − 2⋅_x52 = 0 ∧ _x34 = _x35 ∧ _x37 = _x38 ∧ _x32 = _x39 ∧ _x33 = _x40 ∧ _x34 = _x41 ∧ _x34 = _x42 ∧ _x36 = _x43 ∧ _x37 = _x44 ∧ _x37 = _x45 f288_0_slide68_EQ' 5 f288_0_slide68_EQ: x1 = _x53 ∧ x2 = _x54 ∧ x3 = _x55 ∧ x4 = _x56 ∧ x5 = _x57 ∧ x6 = _x58 ∧ x7 = _x59 ∧ x1 = _x60 ∧ x2 = _x61 ∧ x3 = _x62 ∧ x4 = _x63 ∧ x5 = _x64 ∧ x6 = _x65 ∧ x7 = _x66 ∧ _x63 ≤ _x55 − 1 ∧ 0 ≤ _x55 − 1 ∧ _x67 ≤ _x58 − 1 ∧ −1 ≤ _x58 − 1 ∧ _x64 ≤ _x68 − 1 ∧ −1 ≤ _x67 − 1 ∧ _x67 ≤ _x64 − 1 ∧ −1 ≤ _x68 − 1 ∧ _x67 ≤ _x69 − 1 ∧ _x67 ≤ _x63 − 1 ∧ _x53 − 2⋅_x70 = 0 ∧ _x60 ≤ _x53 ∧ 0 ≤ _x53 − 2⋅_x70 ∧ _x53 − 2⋅_x70 ≤ 1 ∧ _x53 − 2⋅_x60 ≤ 1 ∧ 0 ≤ _x53 − 2⋅_x60 ∧ _x55 = _x56 ∧ _x58 = _x59 ∧ 1 = _x61 ∧ 0 = _x65 ∧ _x64 = _x66 f288_0_slide68_EQ 6 f288_0_slide68_EQ': x1 = _x71 ∧ x2 = _x72 ∧ x3 = _x73 ∧ x4 = _x74 ∧ x5 = _x75 ∧ x6 = _x76 ∧ x7 = _x77 ∧ x1 = _x78 ∧ x2 = _x79 ∧ x3 = _x80 ∧ x4 = _x81 ∧ x5 = _x82 ∧ x6 = _x83 ∧ x7 = _x84 ∧ 0 ≤ _x74 − 1 ∧ 0 ≤ _x75 − 1 ∧ −1 ≤ _x77 − 1 ∧ _x85 ≤ _x77 − 1 ∧ _x86 ≤ _x74 − 1 ∧ _x85 ≤ _x75 − 1 ∧ 0 ≤ _x72 − 1 ∧ _x71 − 2⋅_x87 = 1 ∧ −1 ≤ _x88 − 1 ∧ _x86 ≤ _x88 − 1 ∧ _x89 ≤ _x71 ∧ −1 ≤ _x86 − 1 ∧ _x90 ≤ _x86 ∧ 0 ≤ _x85 − 1 ∧ _x71 = _x78 ∧ _x72 = _x79 ∧ _x73 = _x80 ∧ _x74 = _x81 ∧ _x75 = _x82 ∧ _x76 = _x83 ∧ _x77 = _x84 f288_0_slide68_EQ' 7 f288_0_slide68_EQ: x1 = _x91 ∧ x2 = _x92 ∧ x3 = _x93 ∧ x4 = _x94 ∧ x5 = _x97 ∧ x6 = _x98 ∧ x7 = _x106 ∧ x1 = _x107 ∧ x2 = _x108 ∧ x3 = _x109 ∧ x4 = _x110 ∧ x5 = _x111 ∧ x6 = _x112 ∧ x7 = _x113 ∧ 0 ≤ _x94 − 1 ∧ 0 ≤ _x97 − 1 ∧ −1 ≤ _x106 − 1 ∧ _x111 ≤ _x106 − 1 ∧ _x114 ≤ _x94 − 1 ∧ _x111 ≤ _x97 − 1 ∧ 0 ≤ _x92 − 1 ∧ _x91 − 2⋅_x115 = 1 ∧ −1 ≤ _x116 − 1 ∧ _x114 ≤ _x116 − 1 ∧ _x107 ≤ _x91 ∧ −1 ≤ _x114 − 1 ∧ 0 ≤ _x111 − 1 ∧ _x110 ≤ _x114 ∧ 0 ≤ _x91 − 2⋅_x115 ∧ _x91 − 2⋅_x115 ≤ 1 ∧ _x91 − 2⋅_x107 ≤ 1 ∧ 0 ≤ _x91 − 2⋅_x107 ∧ 1 = _x108 ∧ 0 = _x112 ∧ _x111 = _x113 f288_0_slide68_EQ 8 f288_0_slide68_EQ': x1 = _x121 ∧ x2 = _x122 ∧ x3 = _x123 ∧ x4 = _x124 ∧ x5 = _x125 ∧ x6 = _x126 ∧ x7 = _x127 ∧ x1 = _x134 ∧ x2 = _x135 ∧ x3 = _x136 ∧ x4 = _x137 ∧ x5 = _x138 ∧ x6 = _x139 ∧ x7 = _x140 ∧ _x141 ≤ _x123 − 1 ∧ 0 ≤ _x123 − 1 ∧ _x141 ≤ _x126 − 1 ∧ −1 ≤ _x126 − 1 ∧ _x142 ≤ _x143 − 1 ∧ −1 ≤ _x143 − 1 ∧ _x121 − 2⋅_x144 = 1 ∧ 0 ≤ _x141 − 1 ∧ _x148 ≤ _x121 ∧ _x123 = _x124 ∧ _x126 = _x127 ∧ _x121 = _x134 ∧ _x122 = _x135 ∧ _x123 = _x136 ∧ _x123 = _x137 ∧ _x125 = _x138 ∧ _x126 = _x139 ∧ _x126 = _x140 f288_0_slide68_EQ' 9 f288_0_slide68_EQ: x1 = _x149 ∧ x2 = _x150 ∧ x3 = _x151 ∧ x4 = _x152 ∧ x5 = _x153 ∧ x6 = _x154 ∧ x7 = _x155 ∧ x1 = _x161 ∧ x2 = _x162 ∧ x3 = _x163 ∧ x4 = _x164 ∧ x5 = _x165 ∧ x6 = _x166 ∧ x7 = _x167 ∧ _x163 ≤ _x151 − 1 ∧ 0 ≤ _x151 − 1 ∧ _x163 ≤ _x154 − 1 ∧ −1 ≤ _x154 − 1 ∧ _x168 ≤ _x169 − 1 ∧ −1 ≤ _x169 − 1 ∧ _x149 − 2⋅_x171 = 1 ∧ _x161 ≤ _x149 ∧ 0 ≤ _x163 − 1 ∧ 0 ≤ _x149 − 2⋅_x171 ∧ _x149 − 2⋅_x171 ≤ 1 ∧ _x149 − 2⋅_x161 ≤ 1 ∧ 0 ≤ _x149 − 2⋅_x161 ∧ _x151 = _x152 ∧ _x154 = _x155 ∧ 0 = _x162 ∧ _x163 = _x164 ∧ 0 = _x165 ∧ _x163 = _x166 ∧ _x163 = _x167 f288_0_slide68_EQ 10 f288_0_slide68_EQ': x1 = _x174 ∧ x2 = _x175 ∧ x3 = _x176 ∧ x4 = _x177 ∧ x5 = _x178 ∧ x6 = _x179 ∧ x7 = _x180 ∧ x1 = _x190 ∧ x2 = _x191 ∧ x3 = _x192 ∧ x4 = _x193 ∧ x5 = _x194 ∧ x6 = _x195 ∧ x7 = _x196 ∧ 0 ≤ _x177 − 1 ∧ 0 ≤ _x178 − 1 ∧ −1 ≤ _x179 − 1 ∧ _x179 ≤ _x175 − 1 ∧ _x179 ≤ _x197 − 1 ∧ _x179 ≤ _x176 − 1 ∧ _x179 ≤ _x198 − 1 ∧ −1 ≤ _x180 − 1 ∧ _x199 ≤ _x180 − 1 ∧ _x199 ≤ _x200 − 1 ∧ _x201 ≤ _x177 − 1 ∧ −1 ≤ _x199 − 1 ∧ _x199 ≤ _x207 − 1 ∧ _x208 ≤ _x178 − 1 ∧ 0 ≤ _x175 − 1 ∧ _x199 ≤ _x201 − 1 ∧ _x199 ≤ _x208 − 1 ∧ _x209 ≤ _x174 ∧ _x174 − 2⋅_x210 = 0 ∧ _x174 = _x190 ∧ _x175 = _x191 ∧ _x176 = _x192 ∧ _x177 = _x193 ∧ _x178 = _x194 ∧ _x179 = _x195 ∧ _x180 = _x196 f288_0_slide68_EQ' 11 f288_0_slide68_EQ: x1 = _x211 ∧ x2 = _x212 ∧ x3 = _x213 ∧ x4 = _x222 ∧ x5 = _x223 ∧ x6 = _x224 ∧ x7 = _x225 ∧ x1 = _x226 ∧ x2 = _x227 ∧ x3 = _x228 ∧ x4 = _x229 ∧ x5 = _x230 ∧ x6 = _x231 ∧ x7 = _x232 ∧ 0 ≤ _x222 − 1 ∧ 0 ≤ _x223 − 1 ∧ −1 ≤ _x224 − 1 ∧ _x224 ≤ _x212 − 1 ∧ _x224 ≤ _x233 − 1 ∧ _x224 ≤ _x213 − 1 ∧ _x224 ≤ _x234 − 1 ∧ −1 ≤ _x225 − 1 ∧ _x238 ≤ _x225 − 1 ∧ _x238 ≤ _x232 − 1 ∧ _x229 ≤ _x222 − 1 ∧ −1 ≤ _x238 − 1 ∧ _x238 ≤ _x239 − 1 ∧ _x230 ≤ _x223 − 1 ∧ 0 ≤ _x212 − 1 ∧ _x238 ≤ _x229 − 1 ∧ _x238 ≤ _x230 − 1 ∧ _x211 − 2⋅_x240 = 0 ∧ _x226 ≤ _x211 ∧ 0 ≤ _x211 − 2⋅_x240 ∧ _x211 − 2⋅_x240 ≤ 1 ∧ _x211 − 2⋅_x226 ≤ 1 ∧ 0 ≤ _x211 − 2⋅_x226 ∧ _x213 = _x228 ∧ _x212 = _x231 f288_0_slide68_EQ 12 f288_0_slide68_EQ': x1 = _x241 ∧ x2 = _x242 ∧ x3 = _x243 ∧ x4 = _x244 ∧ x5 = _x252 ∧ x6 = _x253 ∧ x7 = _x254 ∧ x1 = _x255 ∧ x2 = _x256 ∧ x3 = _x257 ∧ x4 = _x258 ∧ x5 = _x259 ∧ x6 = _x260 ∧ x7 = _x261 ∧ 0 ≤ _x244 − 1 ∧ 0 ≤ _x252 − 1 ∧ −1 ≤ _x254 − 1 ∧ _x262 ≤ _x254 − 1 ∧ _x263 ≤ _x244 − 1 ∧ _x264 ≤ _x252 − 1 ∧ 0 ≤ _x242 − 1 ∧ _x241 − 2⋅_x265 = 1 ∧ −1 ≤ _x243 − 1 ∧ _x270 ≤ _x243 ∧ _x271 ≤ _x263 ∧ −1 ≤ _x263 − 1 ∧ _x272 ≤ _x264 ∧ 0 ≤ _x253 − 1 ∧ _x273 ≤ _x241 ∧ −1 ≤ _x264 − 1 ∧ 0 ≤ _x262 − 1 ∧ _x241 = _x255 ∧ _x242 = _x256 ∧ _x243 = _x257 ∧ _x244 = _x258 ∧ _x252 = _x259 ∧ _x253 = _x260 ∧ _x254 = _x261 f288_0_slide68_EQ' 13 f288_0_slide68_EQ: x1 = _x274 ∧ x2 = _x275 ∧ x3 = _x276 ∧ x4 = _x277 ∧ x5 = _x278 ∧ x6 = _x279 ∧ x7 = _x280 ∧ x1 = _x281 ∧ x2 = _x282 ∧ x3 = _x283 ∧ x4 = _x284 ∧ x5 = _x285 ∧ x6 = _x286 ∧ x7 = _x287 ∧ 0 ≤ _x277 − 1 ∧ 0 ≤ _x278 − 1 ∧ −1 ≤ _x280 − 1 ∧ _x287 ≤ _x280 − 1 ∧ _x288 ≤ _x277 − 1 ∧ _x289 ≤ _x278 − 1 ∧ 0 ≤ _x275 − 1 ∧ _x274 − 2⋅_x290 = 1 ∧ −1 ≤ _x276 − 1 ∧ _x283 ≤ _x276 ∧ _x284 ≤ _x288 ∧ −1 ≤ _x288 − 1 ∧ _x285 ≤ _x289 ∧ 0 ≤ _x279 − 1 ∧ _x281 ≤ _x274 ∧ 0 ≤ _x287 − 1 ∧ −1 ≤ _x289 − 1 ∧ 0 ≤ _x274 − 2⋅_x290 ∧ _x274 − 2⋅_x290 ≤ 1 ∧ _x274 − 2⋅_x281 ≤ 1 ∧ 0 ≤ _x274 − 2⋅_x281 ∧ _x279 = _x286 f288_0_slide68_EQ 14 f288_0_slide68_EQ': x1 = _x291 ∧ x2 = _x292 ∧ x3 = _x293 ∧ x4 = _x294 ∧ x5 = _x295 ∧ x6 = _x296 ∧ x7 = _x297 ∧ x1 = _x298 ∧ x2 = _x299 ∧ x3 = _x300 ∧ x4 = _x301 ∧ x5 = _x302 ∧ x6 = _x303 ∧ x7 = _x304 ∧ _x305 ≤ _x293 − 1 ∧ 0 ≤ _x293 − 1 ∧ _x306 ≤ _x296 − 1 ∧ −1 ≤ _x296 − 1 ∧ _x307 ≤ _x308 − 1 ∧ −1 ≤ _x308 − 1 ∧ _x291 − 2⋅_x309 = 1 ∧ _x310 ≤ _x291 ∧ −1 ≤ _x305 − 1 ∧ _x311 ≤ _x305 ∧ 0 ≤ _x306 − 1 ∧ _x293 = _x294 ∧ _x296 = _x297 ∧ _x291 = _x298 ∧ _x292 = _x299 ∧ _x293 = _x300 ∧ _x293 = _x301 ∧ _x295 = _x302 ∧ _x296 = _x303 ∧ _x296 = _x304 f288_0_slide68_EQ' 15 f288_0_slide68_EQ: x1 = _x312 ∧ x2 = _x313 ∧ x3 = _x314 ∧ x4 = _x315 ∧ x5 = _x316 ∧ x6 = _x317 ∧ x7 = _x318 ∧ x1 = _x319 ∧ x2 = _x320 ∧ x3 = _x321 ∧ x4 = _x322 ∧ x5 = _x323 ∧ x6 = _x324 ∧ x7 = _x325 ∧ _x326 ≤ _x314 − 1 ∧ 0 ≤ _x314 − 1 ∧ _x325 ≤ _x317 − 1 ∧ −1 ≤ _x317 − 1 ∧ _x327 ≤ _x328 − 1 ∧ −1 ≤ _x328 − 1 ∧ _x312 − 2⋅_x329 = 1 ∧ _x319 ≤ _x312 ∧ −1 ≤ _x326 − 1 ∧ 0 ≤ _x325 − 1 ∧ _x322 ≤ _x326 ∧ 0 ≤ _x312 − 2⋅_x329 ∧ _x312 − 2⋅_x329 ≤ 1 ∧ _x312 − 2⋅_x319 ≤ 1 ∧ 0 ≤ _x312 − 2⋅_x319 ∧ _x314 = _x315 ∧ _x317 = _x318 f1_0_main_Load 16 f196_0_create_LE: x1 = _x330 ∧ x2 = _x331 ∧ x3 = _x332 ∧ x4 = _x333 ∧ x5 = _x334 ∧ x6 = _x335 ∧ x7 = _x336 ∧ x1 = _x337 ∧ x2 = _x338 ∧ x3 = _x339 ∧ x4 = _x340 ∧ x5 = _x341 ∧ x6 = _x342 ∧ x7 = _x343 ∧ −1 ≤ _x344 − 1 ∧ 1 ≤ _x331 − 1 ∧ −1 ≤ _x345 − 1 ∧ 0 ≤ _x330 − 1 ∧ _x345 − 1 = _x337 f196_0_create_LE 17 f196_0_create_LE: x1 = _x346 ∧ x2 = _x347 ∧ x3 = _x348 ∧ x4 = _x349 ∧ x5 = _x350 ∧ x6 = _x351 ∧ x7 = _x352 ∧ x1 = _x353 ∧ x2 = _x354 ∧ x3 = _x355 ∧ x4 = _x356 ∧ x5 = _x357 ∧ x6 = _x358 ∧ x7 = _x359 ∧ _x346 − 1 = _x353 ∧ 0 ≤ _x346 − 1 __init 18 f1_0_main_Load: x1 = _x360 ∧ x2 = _x361 ∧ x3 = _x362 ∧ x4 = _x363 ∧ x5 = _x364 ∧ x6 = _x365 ∧ x7 = _x366 ∧ x1 = _x367 ∧ x2 = _x368 ∧ x3 = _x369 ∧ x4 = _x370 ∧ x5 = _x371 ∧ x6 = _x372 ∧ x7 = _x373 ∧ 0 ≤ 0

## Proof

### 1 Switch to Cooperation Termination Proof

We consider the following cutpoint-transitions:
 f288_0_slide68_EQ f288_0_slide68_EQ f288_0_slide68_EQ: x1 = x1 ∧ x2 = x2 ∧ x3 = x3 ∧ x4 = x4 ∧ x5 = x5 ∧ x6 = x6 ∧ x7 = x7 f1_0_main_Load f1_0_main_Load f1_0_main_Load: x1 = x1 ∧ x2 = x2 ∧ x3 = x3 ∧ x4 = x4 ∧ x5 = x5 ∧ x6 = x6 ∧ x7 = x7 f196_0_create_LE f196_0_create_LE f196_0_create_LE: x1 = x1 ∧ x2 = x2 ∧ x3 = x3 ∧ x4 = x4 ∧ x5 = x5 ∧ x6 = x6 ∧ x7 = x7 f234_0_slide68_FieldAccess f234_0_slide68_FieldAccess f234_0_slide68_FieldAccess: x1 = x1 ∧ x2 = x2 ∧ x3 = x3 ∧ x4 = x4 ∧ x5 = x5 ∧ x6 = x6 ∧ x7 = x7 f288_0_slide68_EQ' f288_0_slide68_EQ' f288_0_slide68_EQ': x1 = x1 ∧ x2 = x2 ∧ x3 = x3 ∧ x4 = x4 ∧ x5 = x5 ∧ x6 = x6 ∧ x7 = x7 __init __init __init: x1 = x1 ∧ x2 = x2 ∧ x3 = x3 ∧ x4 = x4 ∧ x5 = x5 ∧ x6 = x6 ∧ x7 = x7
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 { f196_0_create_LE }.

### 2.1.1 Transition Removal

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

 f196_0_create_LE: x1

### 2.1.2 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 { f288_0_slide68_EQ, f288_0_slide68_EQ' }.

### 2.2.1 Transition Removal

We remove transitions 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15 using the following ranking functions, which are bounded by 0.

 f288_0_slide68_EQ: 2⋅x4 + 1 f288_0_slide68_EQ': 2⋅x4

### 2.2.2 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 % (7 real / 0 unknown / 0 assumptions / 7 total proof steps)