LTS Termination Proof

Input

Integer Transition System

Initial Location: l5, l7, l1, l8, l3, l0, l2

Transitions: (pre-variables and post-variables)

l0	1	l1:	x1 = _Result_4HAT0 ∧ x2 = ___cil_tmp2_6HAT0 ∧ x3 = ___cil_tmp6_12HAT0 ∧ x4 = ___const_400HAT0 ∧ x5 = ___disjvr_0HAT0 ∧ x6 = _maxRetries_9HAT0 ∧ x7 = _retryCount_10HAT0 ∧ x8 = _selected_11HAT0 ∧ x9 = _x_5HAT0 ∧ x1 = _Result_4HATpost ∧ x2 = ___cil_tmp2_6HATpost ∧ x3 = ___cil_tmp6_12HATpost ∧ x4 = ___const_400HATpost ∧ x5 = ___disjvr_0HATpost ∧ x6 = _maxRetries_9HATpost ∧ x7 = _retryCount_10HATpost ∧ x8 = _selected_11HATpost ∧ x9 = _x_5HATpost ∧ ___cil_tmp2_6HATpost = _x_5HAT0 ∧ _Result_4HAT1 = ___cil_tmp2_6HATpost ∧ _selected_11HATpost = _Result_4HAT1 ∧ _Result_4HATpost = _Result_4HATpost ∧ ___cil_tmp6_12HAT0 = ___cil_tmp6_12HATpost ∧ ___const_400HAT0 = ___const_400HATpost ∧ ___disjvr_0HAT0 = ___disjvr_0HATpost ∧ _maxRetries_9HAT0 = _maxRetries_9HATpost ∧ _retryCount_10HAT0 = _retryCount_10HATpost ∧ _x_5HAT0 = _x_5HATpost
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 ∧ _x10 = _x8 ∧ _x18 = _x10 ∧ _x16 = _x18 ∧ _x9 = _x9 ∧ _x15 = 1 + _x6 ∧ _x2 = _x11 ∧ _x3 = _x12 ∧ _x4 = _x13 ∧ _x5 = _x14 ∧ _x8 = _x17
l3	3	l5:	x1 = _x19 ∧ x2 = _x20 ∧ x3 = _x21 ∧ x4 = _x22 ∧ x5 = _x23 ∧ x6 = _x24 ∧ x7 = _x25 ∧ x8 = _x26 ∧ x9 = _x27 ∧ x1 = _x28 ∧ x2 = _x29 ∧ x3 = _x30 ∧ x4 = _x31 ∧ x5 = _x32 ∧ x6 = _x33 ∧ x7 = _x34 ∧ x8 = _x35 ∧ x9 = _x36 ∧ _x27 = _x36 ∧ _x26 = _x35 ∧ _x25 = _x34 ∧ _x24 = _x33 ∧ _x23 = _x32 ∧ _x22 = _x31 ∧ _x21 = _x30 ∧ _x20 = _x29 ∧ _x19 = _x28 ∧ _x32 = _x23
l5	4	l4:	x1 = _x37 ∧ x2 = _x38 ∧ x3 = _x39 ∧ x4 = _x40 ∧ x5 = _x41 ∧ x6 = _x42 ∧ x7 = _x43 ∧ x8 = _x44 ∧ x9 = _x45 ∧ x1 = _x46 ∧ x2 = _x47 ∧ x3 = _x48 ∧ x4 = _x49 ∧ x5 = _x50 ∧ x6 = _x51 ∧ x7 = _x52 ∧ x8 = _x53 ∧ x9 = _x54 ∧ _x45 = _x54 ∧ _x44 = _x53 ∧ _x43 = _x52 ∧ _x42 = _x51 ∧ _x41 = _x50 ∧ _x40 = _x49 ∧ _x38 = _x47 ∧ _x46 = _x48 ∧ _x48 = _x44
l3	5	l6:	x1 = _x55 ∧ x2 = _x56 ∧ x3 = _x57 ∧ x4 = _x58 ∧ x5 = _x59 ∧ x6 = _x60 ∧ x7 = _x61 ∧ x8 = _x62 ∧ x9 = _x63 ∧ x1 = _x64 ∧ x2 = _x65 ∧ x3 = _x66 ∧ x4 = _x67 ∧ x5 = _x68 ∧ x6 = _x69 ∧ x7 = _x70 ∧ x8 = _x71 ∧ x9 = _x72 ∧ _x63 = _x72 ∧ _x62 = _x71 ∧ _x61 = _x70 ∧ _x60 = _x69 ∧ _x59 = _x68 ∧ _x58 = _x67 ∧ _x56 = _x65 ∧ _x64 = _x66 ∧ _x66 = _x62 ∧ _x60 − _x61 ≤ 0 ∧ 0 ≤ _x62 ∧ _x62 ≤ 0
l3	6	l1:	x1 = _x73 ∧ x2 = _x74 ∧ x3 = _x75 ∧ x4 = _x76 ∧ x5 = _x77 ∧ x6 = _x78 ∧ x7 = _x79 ∧ x8 = _x80 ∧ x9 = _x81 ∧ x1 = _x82 ∧ x2 = _x83 ∧ x3 = _x84 ∧ x4 = _x85 ∧ x5 = _x86 ∧ x6 = _x87 ∧ x7 = _x88 ∧ x8 = _x89 ∧ x9 = _x90 ∧ _x80 ≤ 0 ∧ 0 ≤ _x80 ∧ 0 ≤ −1 + _x78 − _x79 ∧ _x83 = _x81 ∧ _x91 = _x83 ∧ _x89 = _x91 ∧ _x82 = _x82 ∧ _x75 = _x84 ∧ _x76 = _x85 ∧ _x77 = _x86 ∧ _x78 = _x87 ∧ _x79 = _x88 ∧ _x81 = _x90
l1	7	l3:	x1 = _x92 ∧ x2 = _x93 ∧ x3 = _x94 ∧ x4 = _x95 ∧ x5 = _x96 ∧ x6 = _x97 ∧ x7 = _x98 ∧ x8 = _x99 ∧ x9 = _x100 ∧ x1 = _x101 ∧ x2 = _x102 ∧ x3 = _x103 ∧ x4 = _x104 ∧ x5 = _x105 ∧ x6 = _x106 ∧ x7 = _x107 ∧ x8 = _x108 ∧ x9 = _x109 ∧ _x100 = _x109 ∧ _x99 = _x108 ∧ _x97 = _x106 ∧ _x96 = _x105 ∧ _x95 = _x104 ∧ _x94 = _x103 ∧ _x93 = _x102 ∧ _x92 = _x101 ∧ _x107 = 1 + _x98
l7	8	l3:	x1 = _x110 ∧ x2 = _x111 ∧ x3 = _x112 ∧ x4 = _x113 ∧ x5 = _x114 ∧ x6 = _x115 ∧ x7 = _x116 ∧ x8 = _x117 ∧ x9 = _x118 ∧ x1 = _x119 ∧ x2 = _x120 ∧ x3 = _x121 ∧ x4 = _x122 ∧ x5 = _x123 ∧ x6 = _x124 ∧ x7 = _x125 ∧ x8 = _x126 ∧ x9 = _x127 ∧ _x118 = _x127 ∧ _x114 = _x123 ∧ _x113 = _x122 ∧ _x112 = _x121 ∧ _x111 = _x120 ∧ _x110 = _x119 ∧ _x126 = 0 ∧ _x125 = 0 ∧ _x124 = _x113
l8	9	l7:	x1 = _x128 ∧ x2 = _x129 ∧ x3 = _x130 ∧ x4 = _x131 ∧ x5 = _x132 ∧ x6 = _x133 ∧ x7 = _x134 ∧ x8 = _x135 ∧ x9 = _x136 ∧ x1 = _x137 ∧ x2 = _x138 ∧ x3 = _x139 ∧ x4 = _x140 ∧ x5 = _x141 ∧ x6 = _x142 ∧ x7 = _x143 ∧ x8 = _x144 ∧ x9 = _x145 ∧ _x136 = _x145 ∧ _x135 = _x144 ∧ _x134 = _x143 ∧ _x133 = _x142 ∧ _x132 = _x141 ∧ _x131 = _x140 ∧ _x130 = _x139 ∧ _x129 = _x138 ∧ _x128 = _x137

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
l7	l7	l7:	x1 = x1 ∧ x2 = x2 ∧ x3 = x3 ∧ x4 = x4 ∧ x5 = x5 ∧ x6 = x6 ∧ x7 = x7 ∧ x8 = x8 ∧ x9 = x9
l1	l1	l1:	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 1 SCC(s) of the program graph.

2.1 SCC Subproblem 1/1

Here we consider the SCC { l1, l3 }.

2.1.1 Transition Removal

We remove transition 6 using the following ranking functions, which are bounded by 0.

l1:	−1 + x6 − x7
l3:	x6 − x7

2.1.2 Transition Removal

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

l1:	0
l3:	−1

2.1.3 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 % (6 real / 0 unknown / 0 assumptions / 6 total proof steps)