LTS Termination Proof

Input

Integer Transition System

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

Transitions: (pre-variables and post-variables)

l0	1	l1:	x1 = _ilHAT0 ∧ x2 = _ret_foo5HAT0 ∧ x3 = _ret_foo7HAT0 ∧ x4 = _tmpHAT0 ∧ x5 = _tmp___0HAT0 ∧ x6 = _x4HAT0 ∧ x7 = _x6HAT0 ∧ x1 = _ilHATpost ∧ x2 = _ret_foo5HATpost ∧ x3 = _ret_foo7HATpost ∧ x4 = _tmpHATpost ∧ x5 = _tmp___0HATpost ∧ x6 = _x4HATpost ∧ x7 = _x6HATpost ∧ _x6HAT0 = _x6HATpost ∧ _x4HAT0 = _x4HATpost ∧ _tmp___0HAT0 = _tmp___0HATpost ∧ _tmpHAT0 = _tmpHATpost ∧ _ret_foo7HAT0 = _ret_foo7HATpost ∧ _ret_foo5HAT0 = _ret_foo5HATpost ∧ _ilHAT0 = _ilHATpost ∧ 0 ≤ _tmpHAT0 ∧ _tmpHAT0 ≤ 0
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 ∧ _x3 = _x10 ∧ _x2 = _x9 ∧ _x1 = _x8 ∧ _x = _x7 ∧ 1 ≤ _x3
l0	3	l2:	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 ∧ 1 + _x17 ≤ 0
l3	4	l0:	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 ∧ _x32 = _x39 ∧ _x30 = _x37 ∧ _x28 = _x35 ∧ _x38 = _x36 ∧ _x36 = _x36 ∧ _x40 = _x28
l4	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 ∧ _x63 = 1 + _x56
l8	7	l4:	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 ∧ _x75 = _x82 ∧ _x74 = _x81 ∧ _x73 = _x80 ∧ _x72 = _x79 ∧ _x71 = _x78 ∧ _x70 = _x77 ∧ 0 ≤ _x74 ∧ _x74 ≤ 0
l8	8	l6:	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 ∧ 1 ≤ _x88
l8	9	l6:	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 ∧ 1 + _x102 ≤ 0
l7	10	l9:	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
l9	11	l4:	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 ∧ _x130 = _x137 ∧ _x129 = _x136 ∧ _x128 = _x135 ∧ _x127 = _x134 ∧ _x126 = _x133 ∧ 10 ≤ _x126
l9	12	l8:	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 ∧ _x145 = _x152 ∧ _x143 = _x150 ∧ _x141 = _x148 ∧ _x140 = _x147 ∧ _x151 = _x149 ∧ _x149 = _x149 ∧ _x153 = _x140 ∧ 1 + _x140 ≤ 10
l1	13	l7:	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 ∧ _x157 = _x164 ∧ _x156 = _x163 ∧ _x155 = _x162 ∧ _x161 = 0
l2	14	l1:	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 ∧ _x173 = _x180 ∧ _x172 = _x179 ∧ _x171 = _x178 ∧ _x170 = _x177 ∧ _x169 = _x176 ∧ _x168 = _x175 ∧ 10 ≤ _x168
l2	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 ∧ _x184 = _x191 ∧ _x183 = _x190 ∧ _x189 = 1 + _x182 ∧ 1 + _x182 ≤ 10
l10	16	l3:	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 ∧ _x200 = _x207 ∧ _x199 = _x206 ∧ _x198 = _x205 ∧ _x197 = _x204 ∧ _x203 = 0
l11	17	l10:	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 ∧ _x216 = _x223 ∧ _x215 = _x222 ∧ _x214 = _x221 ∧ _x213 = _x220 ∧ _x212 = _x219 ∧ _x211 = _x218 ∧ _x210 = _x217

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 ∧ x7 = x7
l7	l7	l7:	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
l11	l11	l11:	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
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
l0	l0	l0:	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

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 { l3, l0, l2 }.

2.1.1 Transition Removal

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

l0:	9 − x1
l2:	9 − x1
l3:	9 − x1

2.1.2 Transition Removal

We remove transitions 2, 4, 3 using the following ranking functions, which are bounded by 0.

l0:	0
l2:	−1
l3:	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 { l7, l6, l8, l9 }.

2.2.1 Transition Removal

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

l7:	9 − x1
l9:	9 − x1
l6:	8 − x1
l8:	8 − x1

2.2.2 Transition Removal

We remove transitions 10, 6, 9, 8 using the following ranking functions, which are bounded by 0.

l7:	0
l9:	−1
l6:	1
l8:	2

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)