LTS Termination Proof

Input

Integer Transition System

Initial Location: l5, l7, l6, l1, l8, l3, l0, l2, l9

Transitions: (pre-variables and post-variables)

l0	1	l1:	x1 = _guardHAT0 ∧ x2 = _iHAT0 ∧ x3 = _jHAT0 ∧ x4 = _nHAT0 ∧ x1 = _guardHATpost ∧ x2 = _iHATpost ∧ x3 = _jHATpost ∧ x4 = _nHATpost ∧ _nHAT0 = _nHATpost ∧ _jHAT0 = _jHATpost ∧ _guardHAT0 = _guardHATpost ∧ _iHATpost = 1 + _iHAT0 ∧ _nHAT0 ≤ _jHAT0
l0	2	l2:	x1 = _x ∧ x2 = _x1 ∧ x3 = _x2 ∧ x4 = _x3 ∧ x1 = _x4 ∧ x2 = _x5 ∧ x3 = _x6 ∧ x4 = _x7 ∧ _x3 = _x7 ∧ _x2 = _x6 ∧ _x1 = _x5 ∧ _x4 = _x4 ∧ 1 + _x2 ≤ _x3
l3	3	l4:	x1 = _x8 ∧ x2 = _x9 ∧ x3 = _x10 ∧ x4 = _x11 ∧ x1 = _x12 ∧ x2 = _x13 ∧ x3 = _x14 ∧ x4 = _x15 ∧ _x11 = _x15 ∧ _x10 = _x14 ∧ _x9 = _x13 ∧ _x8 = _x12 ∧ _x11 ≤ _x9
l3	4	l5:	x1 = _x16 ∧ x2 = _x17 ∧ x3 = _x18 ∧ x4 = _x19 ∧ x1 = _x20 ∧ x2 = _x21 ∧ x3 = _x22 ∧ x4 = _x23 ∧ _x19 = _x23 ∧ _x17 = _x21 ∧ _x16 = _x20 ∧ _x22 = 1 + _x17 ∧ 1 + _x17 ≤ _x19
l1	5	l3:	x1 = _x24 ∧ x2 = _x25 ∧ x3 = _x26 ∧ x4 = _x27 ∧ x1 = _x28 ∧ x2 = _x29 ∧ x3 = _x30 ∧ x4 = _x31 ∧ _x27 = _x31 ∧ _x26 = _x30 ∧ _x25 = _x29 ∧ _x24 = _x28
l5	6	l0:	x1 = _x32 ∧ x2 = _x33 ∧ x3 = _x34 ∧ x4 = _x35 ∧ x1 = _x36 ∧ x2 = _x37 ∧ x3 = _x38 ∧ x4 = _x39 ∧ _x35 = _x39 ∧ _x34 = _x38 ∧ _x33 = _x37 ∧ _x32 = _x36
l6	7	l5:	x1 = _x40 ∧ x2 = _x41 ∧ x3 = _x42 ∧ x4 = _x43 ∧ x1 = _x44 ∧ x2 = _x45 ∧ x3 = _x46 ∧ x4 = _x47 ∧ _x43 = _x47 ∧ _x41 = _x45 ∧ _x40 = _x44 ∧ _x46 = 1 + _x42
l7	8	l6:	x1 = _x48 ∧ x2 = _x49 ∧ x3 = _x50 ∧ x4 = _x51 ∧ x1 = _x52 ∧ x2 = _x53 ∧ x3 = _x54 ∧ x4 = _x55 ∧ _x49 = _x53 ∧ _x48 = _x52 ∧ _x55 = −1 + _x51 ∧ _x54 = −1 + _x50
l2	9	l6:	x1 = _x56 ∧ x2 = _x57 ∧ x3 = _x58 ∧ x4 = _x59 ∧ x1 = _x60 ∧ x2 = _x61 ∧ x3 = _x62 ∧ x4 = _x63 ∧ _x59 = _x63 ∧ _x58 = _x62 ∧ _x57 = _x61 ∧ _x56 = _x60 ∧ 0 ≤ _x56 ∧ _x56 ≤ 0
l2	10	l7:	x1 = _x64 ∧ x2 = _x65 ∧ x3 = _x66 ∧ x4 = _x67 ∧ x1 = _x68 ∧ x2 = _x69 ∧ x3 = _x70 ∧ x4 = _x71 ∧ _x67 = _x71 ∧ _x66 = _x70 ∧ _x65 = _x69 ∧ _x64 = _x68 ∧ 1 ≤ _x64
l2	11	l7:	x1 = _x72 ∧ x2 = _x73 ∧ x3 = _x74 ∧ x4 = _x75 ∧ x1 = _x76 ∧ x2 = _x77 ∧ x3 = _x78 ∧ x4 = _x79 ∧ _x75 = _x79 ∧ _x74 = _x78 ∧ _x73 = _x77 ∧ _x72 = _x76 ∧ 1 + _x72 ≤ 0
l8	12	l1:	x1 = _x80 ∧ x2 = _x81 ∧ x3 = _x82 ∧ x4 = _x83 ∧ x1 = _x84 ∧ x2 = _x85 ∧ x3 = _x86 ∧ x4 = _x87 ∧ _x83 = _x87 ∧ _x82 = _x86 ∧ _x80 = _x84 ∧ _x85 = 0
l9	13	l8:	x1 = _x88 ∧ x2 = _x89 ∧ x3 = _x90 ∧ x4 = _x91 ∧ x1 = _x92 ∧ x2 = _x93 ∧ x3 = _x94 ∧ x4 = _x95 ∧ _x91 = _x95 ∧ _x90 = _x94 ∧ _x89 = _x93 ∧ _x88 = _x92

Proof

1 Switch to Cooperation Termination Proof

We consider the following cutpoint-transitions:

l5	l5	l5:	x1 = x1 ∧ x2 = x2 ∧ x3 = x3 ∧ x4 = x4
l7	l7	l7:	x1 = x1 ∧ x2 = x2 ∧ x3 = x3 ∧ x4 = x4
l6	l6	l6:	x1 = x1 ∧ x2 = x2 ∧ x3 = x3 ∧ x4 = x4
l1	l1	l1:	x1 = x1 ∧ x2 = x2 ∧ x3 = x3 ∧ x4 = x4
l8	l8	l8:	x1 = x1 ∧ x2 = x2 ∧ x3 = x3 ∧ x4 = x4
l3	l3	l3:	x1 = x1 ∧ x2 = x2 ∧ x3 = x3 ∧ x4 = x4
l0	l0	l0:	x1 = x1 ∧ x2 = x2 ∧ x3 = x3 ∧ x4 = x4
l2	l2	l2:	x1 = x1 ∧ x2 = x2 ∧ x3 = x3 ∧ x4 = x4
l9	l9	l9:	x1 = x1 ∧ x2 = x2 ∧ x3 = x3 ∧ x4 = x4

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 { l5, l7, l6, l1, l3, l0, l2 }.

2.1.1 Transition Removal

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

l0:	−2 − x2 + x4
l1:	−1 − x2 + x4
l5:	−2 − x2 + x4
l6:	−2 − x2 + x4
l2:	−2 − x2 + x4
l7:	−2 − x2 + x4
l3:	−1 − x2 + x4

2.1.2 Transition Removal

We remove transitions 1, 5 using the following ranking functions, which are bounded by 0.

l0:	1
l1:	0
l5:	1
l6:	1
l2:	1
l7:	1
l3:	−1

2.1.3 Transition Removal

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

l5:	−1 − x3 + x4
l0:	−1 − x3 + x4
l6:	−2 − x3 + x4
l2:	−2 − x3 + x4
l7:	−2 − x3 + x4

2.1.4 Transition Removal

We remove transitions 6, 7, 9, 8, 11, 10 using the following ranking functions, which are bounded by 0.

l5:	0
l0:	−1
l6:	1
l2:	3
l7:	2

2.1.5 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 % (8 real / 0 unknown / 0 assumptions / 8 total proof steps)