LTS Termination Proof

Input

Integer Transition System

Initial Location: l5, l4, l7, l6, l1, l8, l0, l2

Transitions: (pre-variables and post-variables)

l0	1	l1:	x1 = ___const_10HAT0 ∧ x2 = _i2HAT0 ∧ x3 = _size1010HAT0 ∧ x4 = _size1HAT0 ∧ x5 = _size77HAT0 ∧ x6 = _tmp1111HAT0 ∧ x7 = _tmp88HAT0 ∧ x1 = ___const_10HATpost ∧ x2 = _i2HATpost ∧ x3 = _size1010HATpost ∧ x4 = _size1HATpost ∧ x5 = _size77HATpost ∧ x6 = _tmp1111HATpost ∧ x7 = _tmp88HATpost ∧ _tmp88HAT0 = _tmp88HATpost ∧ _tmp1111HAT0 = _tmp1111HATpost ∧ _size77HAT0 = _size77HATpost ∧ _size1010HAT0 = _size1010HATpost ∧ _size1HAT0 = _size1HATpost ∧ _i2HAT0 = _i2HATpost ∧ ___const_10HAT0 = ___const_10HATpost
l2	2	l3:	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 ∧ _x3 = _x10 ∧ _x1 = _x8 ∧ _x = _x7 ∧ _x3 ≤ _x1
l2	3	l4:	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 ∧ _x16 = _x23 ∧ _x17 = _x24 ∧ _x14 = _x21 ∧ _x22 = 1 + _x15 ∧ 1 + _x15 ≤ _x17
l5	4	l6:	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 ∧ _x30 = _x37 ∧ _x31 = _x38 ∧ _x29 = _x36 ∧ _x28 = _x35
l6	5	l4:	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 ∧ _x44 = _x51 ∧ _x45 = _x52 ∧ _x42 = _x49 ∧ _x50 = 0 ∧ _x45 ≤ _x43
l6	6	l5:	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 ∧ _x58 = _x65 ∧ _x59 = _x66 ∧ _x56 = _x63 ∧ _x64 = 1 + _x57 ∧ 1 + _x57 ≤ _x59
l4	7	l2:	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 ∧ _x72 = _x79 ∧ _x73 = _x80 ∧ _x71 = _x78 ∧ _x70 = _x77
l1	8	l5:	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 ∧ _x86 = _x93 ∧ _x87 = _x94 ∧ _x84 = _x91 ∧ _x92 = 0 ∧ _x87 ≤ _x85
l1	9	l0:	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 ∧ _x100 = _x107 ∧ _x101 = _x108 ∧ _x98 = _x105 ∧ _x106 = 1 + _x99 ∧ 1 + _x99 ≤ _x101
l7	10	l0:	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 ∧ _x112 = _x119 ∧ _x120 = 0 ∧ _x124 = _x124 ∧ _x121 = _x122 ∧ _x125 = _x125 ∧ _x123 = _x122 ∧ _x122 = _x112
l8	11	l7:	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 ∧ _x128 = _x135 ∧ _x129 = _x136 ∧ _x127 = _x134 ∧ _x126 = _x133

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
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
l1	l1	l1:	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
l2	l2	l2:	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 3 SCC(s) of the program graph.

2.1 SCC Subproblem 1/3

Here we consider the SCC { l1, l0 }.

2.1.1 Transition Removal

We remove transition 9 using the following ranking functions, which are bounded by 0.

l0:	−1 − x2 + x4
l1:	−1 − x2 + x4

2.1.2 Transition Removal

We remove transition 1 using the following ranking functions, which are bounded by 0.

l0:	0
l1:	−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 2/3

Here we consider the SCC { l5, l6 }.

2.2.1 Transition Removal

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

l5:	−2⋅x2 + 2⋅x4 + 1
l6:	−2⋅x2 + 2⋅x4

2.2.2 Transition Removal

We remove transition 4 using the following ranking functions, which are bounded by 0.

l5:	0
l6:	−1

2.2.3 Trivial Cooperation Program

There are no more "sharp" transitions in the cooperation program. Hence the cooperation termination is proved.

2.3 SCC Subproblem 3/3

Here we consider the SCC { l4, l2 }.

2.3.1 Transition Removal

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

l4:	−1 − x2 + x4
l2:	−1 − x2 + x4

2.3.2 Transition Removal

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

l4:	0
l2:	−1

2.3.3 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 % (11 real / 0 unknown / 0 assumptions / 11 total proof steps)