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 = _i2HAT0 ∧ x2 = _j3HAT0 ∧ x3 = _k4HAT0 ∧ x4 = _l5HAT0 ∧ x5 = _x1HAT0 ∧ x1 = _i2HATpost ∧ x2 = _j3HATpost ∧ x3 = _k4HATpost ∧ x4 = _l5HATpost ∧ x5 = _x1HATpost ∧ _x1HAT0 = _x1HATpost ∧ _l5HAT0 = _l5HATpost ∧ _k4HAT0 = _k4HATpost ∧ _j3HAT0 = _j3HATpost ∧ _i2HAT0 = _i2HATpost ∧ 5 ≤ _i2HAT0
l0	2	l2:	x1 = _x ∧ x2 = _x1 ∧ x3 = _x2 ∧ x4 = _x3 ∧ x5 = _x4 ∧ x1 = _x5 ∧ x2 = _x6 ∧ x3 = _x7 ∧ x4 = _x8 ∧ x5 = _x9 ∧ _x4 = _x9 ∧ _x3 = _x8 ∧ _x2 = _x7 ∧ _x = _x5 ∧ _x6 = 0 ∧ 1 + _x ≤ 5
l3	3	l0:	x1 = _x10 ∧ x2 = _x11 ∧ x3 = _x12 ∧ x4 = _x13 ∧ x5 = _x14 ∧ x1 = _x15 ∧ x2 = _x16 ∧ x3 = _x17 ∧ x4 = _x18 ∧ x5 = _x19 ∧ _x14 = _x19 ∧ _x13 = _x18 ∧ _x12 = _x17 ∧ _x11 = _x16 ∧ _x10 = _x15
l2	4	l4:	x1 = _x20 ∧ x2 = _x21 ∧ x3 = _x22 ∧ x4 = _x23 ∧ x5 = _x24 ∧ x1 = _x25 ∧ x2 = _x26 ∧ x3 = _x27 ∧ x4 = _x28 ∧ x5 = _x29 ∧ _x24 = _x29 ∧ _x23 = _x28 ∧ _x22 = _x27 ∧ _x21 = _x26 ∧ _x20 = _x25
l1	5	l5:	x1 = _x30 ∧ x2 = _x31 ∧ x3 = _x32 ∧ x4 = _x33 ∧ x5 = _x34 ∧ x1 = _x35 ∧ x2 = _x36 ∧ x3 = _x37 ∧ x4 = _x38 ∧ x5 = _x39 ∧ _x34 = _x39 ∧ _x33 = _x38 ∧ _x32 = _x37 ∧ _x31 = _x36 ∧ _x30 = _x35
l6	6	l7:	x1 = _x40 ∧ x2 = _x41 ∧ x3 = _x42 ∧ x4 = _x43 ∧ x5 = _x44 ∧ x1 = _x45 ∧ x2 = _x46 ∧ x3 = _x47 ∧ x4 = _x48 ∧ x5 = _x49 ∧ _x44 = _x49 ∧ _x43 = _x48 ∧ _x42 = _x47 ∧ _x41 = _x46 ∧ _x40 = _x45
l8	7	l9:	x1 = _x50 ∧ x2 = _x51 ∧ x3 = _x52 ∧ x4 = _x53 ∧ x5 = _x54 ∧ x1 = _x55 ∧ x2 = _x56 ∧ x3 = _x57 ∧ x4 = _x58 ∧ x5 = _x59 ∧ _x54 = _x59 ∧ _x52 = _x57 ∧ _x51 = _x56 ∧ _x50 = _x55 ∧ _x58 = 1 + _x53
l9	8	l10:	x1 = _x60 ∧ x2 = _x61 ∧ x3 = _x62 ∧ x4 = _x63 ∧ x5 = _x64 ∧ x1 = _x65 ∧ x2 = _x66 ∧ x3 = _x67 ∧ x4 = _x68 ∧ x5 = _x69 ∧ _x64 = _x69 ∧ _x63 = _x68 ∧ _x62 = _x67 ∧ _x61 = _x66 ∧ _x60 = _x65
l11	9	l8:	x1 = _x70 ∧ x2 = _x71 ∧ x3 = _x72 ∧ x4 = _x73 ∧ x5 = _x74 ∧ x1 = _x75 ∧ x2 = _x76 ∧ x3 = _x77 ∧ x4 = _x78 ∧ x5 = _x79 ∧ _x74 = _x79 ∧ _x73 = _x78 ∧ _x72 = _x77 ∧ _x71 = _x76 ∧ _x70 = _x75
l11	10	l1:	x1 = _x80 ∧ x2 = _x81 ∧ x3 = _x82 ∧ x4 = _x83 ∧ x5 = _x84 ∧ x1 = _x85 ∧ x2 = _x86 ∧ x3 = _x87 ∧ x4 = _x88 ∧ x5 = _x89 ∧ _x84 = _x89 ∧ _x83 = _x88 ∧ _x82 = _x87 ∧ _x81 = _x86 ∧ _x80 = _x85
l10	11	l6:	x1 = _x90 ∧ x2 = _x91 ∧ x3 = _x92 ∧ x4 = _x93 ∧ x5 = _x94 ∧ x1 = _x95 ∧ x2 = _x96 ∧ x3 = _x97 ∧ x4 = _x98 ∧ x5 = _x99 ∧ _x94 = _x99 ∧ _x93 = _x98 ∧ _x91 = _x96 ∧ _x90 = _x95 ∧ _x97 = 1 + _x92 ∧ 5 ≤ _x93
l10	12	l11:	x1 = _x100 ∧ x2 = _x101 ∧ x3 = _x102 ∧ x4 = _x103 ∧ x5 = _x104 ∧ x1 = _x105 ∧ x2 = _x106 ∧ x3 = _x107 ∧ x4 = _x108 ∧ x5 = _x109 ∧ _x104 = _x109 ∧ _x103 = _x108 ∧ _x102 = _x107 ∧ _x101 = _x106 ∧ _x100 = _x105 ∧ 1 + _x103 ≤ 5
l7	13	l2:	x1 = _x110 ∧ x2 = _x111 ∧ x3 = _x112 ∧ x4 = _x113 ∧ x5 = _x114 ∧ x1 = _x115 ∧ x2 = _x116 ∧ x3 = _x117 ∧ x4 = _x118 ∧ x5 = _x119 ∧ _x114 = _x119 ∧ _x113 = _x118 ∧ _x112 = _x117 ∧ _x110 = _x115 ∧ _x116 = 1 + _x111 ∧ 5 ≤ _x112
l7	14	l9:	x1 = _x120 ∧ x2 = _x121 ∧ x3 = _x122 ∧ x4 = _x123 ∧ x5 = _x124 ∧ x1 = _x125 ∧ x2 = _x126 ∧ x3 = _x127 ∧ x4 = _x128 ∧ x5 = _x129 ∧ _x124 = _x129 ∧ _x122 = _x127 ∧ _x121 = _x126 ∧ _x120 = _x125 ∧ _x128 = 0 ∧ 1 + _x122 ≤ 5
l4	15	l3:	x1 = _x130 ∧ x2 = _x131 ∧ x3 = _x132 ∧ x4 = _x133 ∧ x5 = _x134 ∧ x1 = _x135 ∧ x2 = _x136 ∧ x3 = _x137 ∧ x4 = _x138 ∧ x5 = _x139 ∧ _x134 = _x139 ∧ _x133 = _x138 ∧ _x132 = _x137 ∧ _x131 = _x136 ∧ _x135 = 1 + _x130 ∧ 5 ≤ _x131
l4	16	l6:	x1 = _x140 ∧ x2 = _x141 ∧ x3 = _x142 ∧ x4 = _x143 ∧ x5 = _x144 ∧ x1 = _x145 ∧ x2 = _x146 ∧ x3 = _x147 ∧ x4 = _x148 ∧ x5 = _x149 ∧ _x144 = _x149 ∧ _x143 = _x148 ∧ _x141 = _x146 ∧ _x140 = _x145 ∧ _x147 = 0 ∧ 1 + _x141 ≤ 5
l12	17	l3:	x1 = _x150 ∧ x2 = _x151 ∧ x3 = _x152 ∧ x4 = _x153 ∧ x5 = _x154 ∧ x1 = _x155 ∧ x2 = _x156 ∧ x3 = _x157 ∧ x4 = _x158 ∧ x5 = _x159 ∧ _x153 = _x158 ∧ _x152 = _x157 ∧ _x151 = _x156 ∧ _x155 = 0 ∧ _x159 = 400
l13	18	l12:	x1 = _x160 ∧ x2 = _x161 ∧ x3 = _x162 ∧ x4 = _x163 ∧ x5 = _x164 ∧ x1 = _x165 ∧ x2 = _x166 ∧ x3 = _x167 ∧ x4 = _x168 ∧ x5 = _x169 ∧ _x164 = _x169 ∧ _x163 = _x168 ∧ _x162 = _x167 ∧ _x161 = _x166 ∧ _x160 = _x165

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
l11	l11	l11:	x1 = x1 ∧ x2 = x2 ∧ x3 = x3 ∧ x4 = x4 ∧ x5 = x5
l1	l1	l1:	x1 = x1 ∧ x2 = x2 ∧ x3 = x3 ∧ x4 = x4 ∧ x5 = x5
l3	l3	l3:	x1 = x1 ∧ x2 = x2 ∧ x3 = x3 ∧ x4 = x4 ∧ x5 = x5
l13	l13	l13:	x1 = x1 ∧ x2 = x2 ∧ x3 = x3 ∧ x4 = x4 ∧ x5 = x5
l2	l2	l2:	x1 = x1 ∧ x2 = x2 ∧ x3 = x3 ∧ x4 = x4 ∧ x5 = x5
l9	l9	l9:	x1 = x1 ∧ x2 = x2 ∧ x3 = x3 ∧ x4 = x4 ∧ x5 = x5
l4	l4	l4:	x1 = x1 ∧ x2 = x2 ∧ x3 = x3 ∧ x4 = x4 ∧ x5 = x5
l6	l6	l6:	x1 = x1 ∧ x2 = x2 ∧ x3 = x3 ∧ x4 = x4 ∧ x5 = x5
l10	l10	l10:	x1 = x1 ∧ x2 = x2 ∧ x3 = x3 ∧ x4 = x4 ∧ x5 = x5
l8	l8	l8:	x1 = x1 ∧ x2 = x2 ∧ x3 = x3 ∧ x4 = x4 ∧ x5 = x5
l0	l0	l0:	x1 = x1 ∧ x2 = x2 ∧ x3 = x3 ∧ x4 = x4 ∧ x5 = x5
l12	l12	l12:	x1 = x1 ∧ x2 = x2 ∧ x3 = x3 ∧ x4 = x4 ∧ x5 = x5

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:	4 − x1
l2:	3 − x1
l3:	4 − x1
l4:	3 − x1
l7:	3 − x1
l6:	3 − x1
l10:	3 − x1
l9:	3 − x1
l8:	3 − x1
l11:	3 − x1

2.1.2 Transition Removal

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

l3:	−2
l0:	−2
l4:	0
l2:	0
l7:	0
l6:	0
l10:	0
l9:	0
l8:	0
l11:	0

2.1.3 Transition Removal

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

l3:	1
l0:	0
l2:	−1 + 2⋅x1 − x2 + 3⋅x5
l4:	−2 + 2⋅x1 − x2 + 3⋅x5
l7:	−2 + 2⋅x1 − x2 + 3⋅x5
l6:	−2 + 2⋅x1 − x2 + 3⋅x5
l10:	−2 + 2⋅x1 − x2 + 3⋅x5
l9:	−2 + 2⋅x1 − x2 + 3⋅x5
l8:	−2 + 2⋅x1 − x2 + 3⋅x5
l11:	−2 + 2⋅x1 − x2 + 3⋅x5

2.1.4 Transition Removal

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

l2:	4 − x2
l4:	4 − x2
l7:	3 − x2
l6:	3 − x2
l10:	3 − x2
l9:	3 − x2
l8:	3 − x2
l11:	3 − x2

2.1.5 Transition Removal

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

l2:	4 − x3
l4:	4 − x3
l7:	4 − x3
l6:	4 − x3
l10:	3 − x3
l9:	3 − x3
l8:	3 − x3
l11:	3 − x3

2.1.6 Transition Removal

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

l2:	−1 − x3 − x4
l4:	−1 − x3 − x4
l7:	−6 − x4
l6:	4 − x4
l10:	4 − x4
l9:	4 − x4
l8:	3 − x4
l11:	3 − x4

2.1.7 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

2.1.8 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 % (11 real / 0 unknown / 0 assumptions / 11 total proof steps)