LTS Termination Proof

Input

Integer Transition System

Initial Location: l7, l11, l1, l3, l13, l2, l9, l4, l6, l10, l8, l0, l12

Transitions: (pre-variables and post-variables)

l0	1	l1:	x1 = ___const_400HAT0 ∧ x2 = ___const_5HAT0 ∧ x3 = _i2HAT0 ∧ x4 = _j3HAT0 ∧ x5 = _k4HAT0 ∧ x6 = _l5HAT0 ∧ x7 = _x1HAT0 ∧ x1 = ___const_400HATpost ∧ x2 = ___const_5HATpost ∧ x3 = _i2HATpost ∧ x4 = _j3HATpost ∧ x5 = _k4HATpost ∧ x6 = _l5HATpost ∧ x7 = _x1HATpost ∧ _x1HAT0 = _x1HATpost ∧ _l5HAT0 = _l5HATpost ∧ _k4HAT0 = _k4HATpost ∧ _j3HAT0 = _j3HATpost ∧ _i2HAT0 = _i2HATpost ∧ ___const_5HAT0 = ___const_5HATpost ∧ ___const_400HAT0 = ___const_400HATpost ∧ ___const_5HAT0 ≤ _i2HAT0
l0	2	l2:	x1 = _x ∧ x2 = _x1 ∧ x3 = _x2 ∧ x4 = _x3 ∧ x5 = _x4 ∧ x6 = _x5 ∧ x7 = _x6 ∧ x1 = _x7 ∧ x2 = _x8 ∧ x3 = _x9 ∧ x4 = _x10 ∧ x5 = _x11 ∧ x6 = _x12 ∧ x7 = _x13 ∧ _x6 = _x13 ∧ _x5 = _x12 ∧ _x4 = _x11 ∧ _x2 = _x9 ∧ _x1 = _x8 ∧ _x = _x7 ∧ _x10 = 0 ∧ 1 + _x2 ≤ _x1
l3	3	l0:	x1 = _x14 ∧ x2 = _x15 ∧ x3 = _x16 ∧ x4 = _x17 ∧ x5 = _x18 ∧ x6 = _x19 ∧ x7 = _x20 ∧ x1 = _x21 ∧ x2 = _x22 ∧ x3 = _x23 ∧ x4 = _x24 ∧ x5 = _x25 ∧ x6 = _x26 ∧ x7 = _x27 ∧ _x20 = _x27 ∧ _x19 = _x26 ∧ _x18 = _x25 ∧ _x17 = _x24 ∧ _x16 = _x23 ∧ _x15 = _x22 ∧ _x14 = _x21
l2	4	l4:	x1 = _x28 ∧ x2 = _x29 ∧ x3 = _x30 ∧ x4 = _x31 ∧ x5 = _x32 ∧ x6 = _x33 ∧ x7 = _x34 ∧ x1 = _x35 ∧ x2 = _x36 ∧ x3 = _x37 ∧ x4 = _x38 ∧ x5 = _x39 ∧ x6 = _x40 ∧ x7 = _x41 ∧ _x34 = _x41 ∧ _x33 = _x40 ∧ _x32 = _x39 ∧ _x31 = _x38 ∧ _x30 = _x37 ∧ _x29 = _x36 ∧ _x28 = _x35
l1	5	l5:	x1 = _x42 ∧ x2 = _x43 ∧ x3 = _x44 ∧ x4 = _x45 ∧ x5 = _x46 ∧ x6 = _x47 ∧ x7 = _x48 ∧ x1 = _x49 ∧ x2 = _x50 ∧ x3 = _x51 ∧ x4 = _x52 ∧ x5 = _x53 ∧ x6 = _x54 ∧ x7 = _x55 ∧ _x48 = _x55 ∧ _x47 = _x54 ∧ _x46 = _x53 ∧ _x45 = _x52 ∧ _x44 = _x51 ∧ _x43 = _x50 ∧ _x42 = _x49
l6	6	l7:	x1 = _x56 ∧ x2 = _x57 ∧ x3 = _x58 ∧ x4 = _x59 ∧ x5 = _x60 ∧ x6 = _x61 ∧ x7 = _x62 ∧ x1 = _x63 ∧ x2 = _x64 ∧ x3 = _x65 ∧ x4 = _x66 ∧ x5 = _x67 ∧ x6 = _x68 ∧ x7 = _x69 ∧ _x62 = _x69 ∧ _x61 = _x68 ∧ _x60 = _x67 ∧ _x59 = _x66 ∧ _x58 = _x65 ∧ _x57 = _x64 ∧ _x56 = _x63
l8	7	l9:	x1 = _x70 ∧ x2 = _x71 ∧ x3 = _x72 ∧ x4 = _x73 ∧ x5 = _x74 ∧ x6 = _x75 ∧ x7 = _x76 ∧ x1 = _x77 ∧ x2 = _x78 ∧ x3 = _x79 ∧ x4 = _x80 ∧ x5 = _x81 ∧ x6 = _x82 ∧ x7 = _x83 ∧ _x76 = _x83 ∧ _x74 = _x81 ∧ _x73 = _x80 ∧ _x72 = _x79 ∧ _x71 = _x78 ∧ _x70 = _x77 ∧ _x82 = 1 + _x75
l9	8	l10:	x1 = _x84 ∧ x2 = _x85 ∧ x3 = _x86 ∧ x4 = _x87 ∧ x5 = _x88 ∧ x6 = _x89 ∧ x7 = _x90 ∧ x1 = _x91 ∧ x2 = _x92 ∧ x3 = _x93 ∧ x4 = _x94 ∧ x5 = _x95 ∧ x6 = _x96 ∧ x7 = _x97 ∧ _x90 = _x97 ∧ _x89 = _x96 ∧ _x88 = _x95 ∧ _x87 = _x94 ∧ _x86 = _x93 ∧ _x85 = _x92 ∧ _x84 = _x91
l11	9	l8:	x1 = _x98 ∧ x2 = _x99 ∧ x3 = _x100 ∧ x4 = _x101 ∧ x5 = _x102 ∧ x6 = _x103 ∧ x7 = _x104 ∧ x1 = _x105 ∧ x2 = _x106 ∧ x3 = _x107 ∧ x4 = _x108 ∧ x5 = _x109 ∧ x6 = _x110 ∧ x7 = _x111 ∧ _x104 = _x111 ∧ _x103 = _x110 ∧ _x102 = _x109 ∧ _x101 = _x108 ∧ _x100 = _x107 ∧ _x99 = _x106 ∧ _x98 = _x105
l11	10	l1:	x1 = _x112 ∧ x2 = _x113 ∧ x3 = _x114 ∧ x4 = _x115 ∧ x5 = _x116 ∧ x6 = _x117 ∧ x7 = _x118 ∧ x1 = _x119 ∧ x2 = _x120 ∧ x3 = _x121 ∧ x4 = _x122 ∧ x5 = _x123 ∧ x6 = _x124 ∧ x7 = _x125 ∧ _x118 = _x125 ∧ _x117 = _x124 ∧ _x116 = _x123 ∧ _x115 = _x122 ∧ _x114 = _x121 ∧ _x113 = _x120 ∧ _x112 = _x119
l10	11	l6:	x1 = _x126 ∧ x2 = _x127 ∧ x3 = _x128 ∧ x4 = _x129 ∧ x5 = _x130 ∧ x6 = _x131 ∧ x7 = _x132 ∧ x1 = _x133 ∧ x2 = _x134 ∧ x3 = _x135 ∧ x4 = _x136 ∧ x5 = _x137 ∧ x6 = _x138 ∧ x7 = _x139 ∧ _x132 = _x139 ∧ _x131 = _x138 ∧ _x129 = _x136 ∧ _x128 = _x135 ∧ _x127 = _x134 ∧ _x126 = _x133 ∧ _x137 = 1 + _x130 ∧ _x127 ≤ _x131
l10	12	l11:	x1 = _x140 ∧ x2 = _x141 ∧ x3 = _x142 ∧ x4 = _x143 ∧ x5 = _x144 ∧ x6 = _x145 ∧ x7 = _x146 ∧ x1 = _x147 ∧ x2 = _x148 ∧ x3 = _x149 ∧ x4 = _x150 ∧ x5 = _x151 ∧ x6 = _x152 ∧ x7 = _x153 ∧ _x146 = _x153 ∧ _x145 = _x152 ∧ _x144 = _x151 ∧ _x143 = _x150 ∧ _x142 = _x149 ∧ _x141 = _x148 ∧ _x140 = _x147 ∧ 1 + _x145 ≤ _x141
l7	13	l2:	x1 = _x154 ∧ x2 = _x155 ∧ x3 = _x156 ∧ x4 = _x157 ∧ x5 = _x158 ∧ x6 = _x159 ∧ x7 = _x160 ∧ x1 = _x161 ∧ x2 = _x162 ∧ x3 = _x163 ∧ x4 = _x164 ∧ x5 = _x165 ∧ x6 = _x166 ∧ x7 = _x167 ∧ _x160 = _x167 ∧ _x159 = _x166 ∧ _x158 = _x165 ∧ _x156 = _x163 ∧ _x155 = _x162 ∧ _x154 = _x161 ∧ _x164 = 1 + _x157 ∧ _x155 ≤ _x158
l7	14	l9:	x1 = _x168 ∧ x2 = _x169 ∧ x3 = _x170 ∧ x4 = _x171 ∧ x5 = _x172 ∧ x6 = _x173 ∧ x7 = _x174 ∧ x1 = _x175 ∧ x2 = _x176 ∧ x3 = _x177 ∧ x4 = _x178 ∧ x5 = _x179 ∧ x6 = _x180 ∧ x7 = _x181 ∧ _x174 = _x181 ∧ _x172 = _x179 ∧ _x171 = _x178 ∧ _x170 = _x177 ∧ _x169 = _x176 ∧ _x168 = _x175 ∧ _x180 = 0 ∧ 1 + _x172 ≤ _x169
l4	15	l3:	x1 = _x182 ∧ x2 = _x183 ∧ x3 = _x184 ∧ x4 = _x185 ∧ x5 = _x186 ∧ x6 = _x187 ∧ x7 = _x188 ∧ x1 = _x189 ∧ x2 = _x190 ∧ x3 = _x191 ∧ x4 = _x192 ∧ x5 = _x193 ∧ x6 = _x194 ∧ x7 = _x195 ∧ _x188 = _x195 ∧ _x187 = _x194 ∧ _x186 = _x193 ∧ _x185 = _x192 ∧ _x183 = _x190 ∧ _x182 = _x189 ∧ _x191 = 1 + _x184 ∧ _x183 ≤ _x185
l4	16	l6:	x1 = _x196 ∧ x2 = _x197 ∧ x3 = _x198 ∧ x4 = _x199 ∧ x5 = _x200 ∧ x6 = _x201 ∧ x7 = _x202 ∧ x1 = _x203 ∧ x2 = _x204 ∧ x3 = _x205 ∧ x4 = _x206 ∧ x5 = _x207 ∧ x6 = _x208 ∧ x7 = _x209 ∧ _x202 = _x209 ∧ _x201 = _x208 ∧ _x199 = _x206 ∧ _x198 = _x205 ∧ _x197 = _x204 ∧ _x196 = _x203 ∧ _x207 = 0 ∧ 1 + _x199 ≤ _x197
l12	17	l3:	x1 = _x210 ∧ x2 = _x211 ∧ x3 = _x212 ∧ x4 = _x213 ∧ x5 = _x214 ∧ x6 = _x215 ∧ x7 = _x216 ∧ x1 = _x217 ∧ x2 = _x218 ∧ x3 = _x219 ∧ x4 = _x220 ∧ x5 = _x221 ∧ x6 = _x222 ∧ x7 = _x223 ∧ _x215 = _x222 ∧ _x214 = _x221 ∧ _x213 = _x220 ∧ _x211 = _x218 ∧ _x210 = _x217 ∧ _x219 = 0 ∧ _x223 = _x210
l13	18	l12:	x1 = _x224 ∧ x2 = _x225 ∧ x3 = _x226 ∧ x4 = _x227 ∧ x5 = _x228 ∧ x6 = _x229 ∧ x7 = _x230 ∧ x1 = _x231 ∧ x2 = _x232 ∧ x3 = _x233 ∧ x4 = _x234 ∧ x5 = _x235 ∧ x6 = _x236 ∧ x7 = _x237 ∧ _x230 = _x237 ∧ _x229 = _x236 ∧ _x228 = _x235 ∧ _x227 = _x234 ∧ _x226 = _x233 ∧ _x225 = _x232 ∧ _x224 = _x231

Proof

1 Switch to Cooperation Termination Proof

We consider the following cutpoint-transitions:

l7	l7	l7:	x1 = x1 ∧ x2 = x2 ∧ x3 = x3 ∧ x4 = x4 ∧ x5 = x5 ∧ x6 = x6 ∧ x7 = x7
l11	l11	l11:	x1 = x1 ∧ x2 = x2 ∧ x3 = x3 ∧ x4 = x4 ∧ x5 = x5 ∧ x6 = x6 ∧ x7 = x7
l1	l1	l1:	x1 = x1 ∧ x2 = x2 ∧ x3 = x3 ∧ x4 = x4 ∧ x5 = x5 ∧ x6 = x6 ∧ x7 = x7
l3	l3	l3:	x1 = x1 ∧ x2 = x2 ∧ x3 = x3 ∧ x4 = x4 ∧ x5 = x5 ∧ x6 = x6 ∧ x7 = x7
l13	l13	l13:	x1 = x1 ∧ x2 = x2 ∧ x3 = x3 ∧ x4 = x4 ∧ x5 = x5 ∧ x6 = x6 ∧ x7 = x7
l2	l2	l2:	x1 = x1 ∧ x2 = x2 ∧ x3 = x3 ∧ x4 = x4 ∧ x5 = x5 ∧ x6 = x6 ∧ x7 = x7
l9	l9	l9:	x1 = x1 ∧ x2 = x2 ∧ x3 = x3 ∧ x4 = x4 ∧ x5 = x5 ∧ x6 = x6 ∧ x7 = x7
l4	l4	l4:	x1 = x1 ∧ x2 = x2 ∧ x3 = x3 ∧ x4 = x4 ∧ x5 = x5 ∧ x6 = x6 ∧ x7 = x7
l6	l6	l6:	x1 = x1 ∧ x2 = x2 ∧ x3 = x3 ∧ x4 = x4 ∧ x5 = x5 ∧ x6 = x6 ∧ x7 = x7
l10	l10	l10:	x1 = x1 ∧ x2 = x2 ∧ x3 = x3 ∧ x4 = x4 ∧ x5 = x5 ∧ x6 = x6 ∧ x7 = x7
l8	l8	l8:	x1 = x1 ∧ x2 = x2 ∧ x3 = x3 ∧ x4 = x4 ∧ x5 = x5 ∧ x6 = x6 ∧ x7 = x7
l0	l0	l0:	x1 = x1 ∧ x2 = x2 ∧ x3 = x3 ∧ x4 = x4 ∧ x5 = x5 ∧ x6 = x6 ∧ x7 = x7
l12	l12	l12:	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 1 SCC(s) of the program graph.

2.1 SCC Subproblem 1/1

Here we consider the SCC { l4, l7, l6, l10, l11, l8, l3, l0, l2, l9 }.

2.1.1 Transition Removal

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

l0:	−1 + x2 − x3
l2:	−2 + x2 − x3
l3:	−1 + x2 − x3
l4:	−2 + x2 − x3
l7:	−2 + x2 − x3
l6:	−2 + x2 − x3
l10:	−2 + x2 − x3
l9:	−2 + x2 − x3
l8:	−2 + x2 − x3
l11:	−2 + x2 − x3

2.1.2 Transition Removal

We remove transitions 3, 15 using the following ranking functions, which are bounded by 0.

l3:	0
l0:	−1
l4:	1
l2:	1
l7:	1
l6:	1
l10:	1
l9:	1
l8:	1
l11:	1

2.1.3 Transition Removal

We remove transition 16 using the following ranking functions, which are bounded by 0.

l2:	−1 + x2 − x4
l4:	−1 + x2 − x4
l7:	−2 + x2 − x4
l6:	−2 + x2 − x4
l10:	−2 + x2 − x4
l9:	−2 + x2 − x4
l8:	−2 + x2 − x4
l11:	−2 + x2 − x4

2.1.4 Transition Removal

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

l2:	−1 + x2 − x5
l4:	−1 + x2 − x5
l7:	−1 + x2 − x5
l6:	−1 + x2 − x5
l10:	−2 + x2 − x5
l9:	−2 + x2 − x5
l8:	−2 + x2 − x5
l11:	−2 + x2 − x5

2.1.5 Transition Removal

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

l2:	4⋅x2 − 4⋅x6 − 3
l4:	4⋅x2 − 4⋅x6 − 4
l7:	4⋅x2 − 4⋅x6 − 2
l6:	4⋅x2 − 4⋅x6 − 1
l10:	4⋅x2 − 4⋅x6
l9:	4⋅x2 − 4⋅x6 + 1
l8:	4⋅x2 − 4⋅x6 − 2
l11:	4⋅x2 − 4⋅x6 − 1

2.1.6 Transition Removal

We remove transitions 4, 13, 6, 11, 8, 7, 9 using the following ranking functions, which are bounded by −5.

l2:	−5
l4:	−6
l7:	−4
l6:	−3
l10:	−2
l9:	−1
l8:	0
l11:	1

2.1.7 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)