LTS Termination Proof

Input

Integer Transition System

Initial Location: l5, l4, l6, l3, l0, l2

Transitions: (pre-variables and post-variables)

l0	1	l1:	x1 = ___const_19HAT0 ∧ x2 = _oldX0HAT0 ∧ x3 = _oldX1HAT0 ∧ x4 = _x0HAT0 ∧ x1 = ___const_19HATpost ∧ x2 = _oldX0HATpost ∧ x3 = _oldX1HATpost ∧ x4 = _x0HATpost ∧ ___const_19HAT0 = ___const_19HATpost ∧ _x0HATpost = _oldX1HATpost ∧ _oldX1HATpost = _oldX1HATpost ∧ _oldX0HATpost = _x0HAT0
l2	2	l3:	x1 = _x ∧ x2 = _x1 ∧ x3 = _x2 ∧ x4 = _x3 ∧ x1 = _x4 ∧ x2 = _x5 ∧ x3 = _x6 ∧ x4 = _x7 ∧ _x2 = _x6 ∧ _x = _x4 ∧ _x7 = 1 + _x5 ∧ _x5 = _x3
l3	3	l0:	x1 = _x8 ∧ x2 = _x9 ∧ x3 = _x10 ∧ x4 = _x11 ∧ x1 = _x12 ∧ x2 = _x13 ∧ x3 = _x14 ∧ x4 = _x15 ∧ _x10 = _x14 ∧ _x8 = _x12 ∧ _x15 = _x13 ∧ 1 + _x8 ≤ _x13 ∧ _x13 = _x11
l3	4	l2:	x1 = _x16 ∧ x2 = _x17 ∧ x3 = _x18 ∧ x4 = _x19 ∧ x1 = _x20 ∧ x2 = _x21 ∧ x3 = _x22 ∧ x4 = _x23 ∧ _x18 = _x22 ∧ _x16 = _x20 ∧ _x23 = _x21 ∧ _x21 ≤ _x16 ∧ _x21 = _x19
l4	5	l3:	x1 = _x24 ∧ x2 = _x25 ∧ x3 = _x26 ∧ x4 = _x27 ∧ x1 = _x28 ∧ x2 = _x29 ∧ x3 = _x30 ∧ x4 = _x31 ∧ _x26 = _x30 ∧ _x24 = _x28 ∧ _x31 = 0 ∧ _x29 = _x27
l5	6	l4:	x1 = _x32 ∧ x2 = _x33 ∧ x3 = _x34 ∧ x4 = _x35 ∧ x1 = _x36 ∧ x2 = _x37 ∧ x3 = _x38 ∧ x4 = _x39 ∧ _x32 = _x36 ∧ _x39 = _x38 ∧ _x38 = _x38 ∧ _x37 = _x35
l5	7	l1:	x1 = _x40 ∧ x2 = _x41 ∧ x3 = _x42 ∧ x4 = _x43 ∧ x1 = _x44 ∧ x2 = _x45 ∧ x3 = _x46 ∧ x4 = _x47 ∧ _x43 = _x47 ∧ _x42 = _x46 ∧ _x41 = _x45 ∧ _x40 = _x44
l5	8	l0:	x1 = _x48 ∧ x2 = _x49 ∧ x3 = _x50 ∧ x4 = _x51 ∧ x1 = _x52 ∧ x2 = _x53 ∧ x3 = _x54 ∧ x4 = _x55 ∧ _x51 = _x55 ∧ _x50 = _x54 ∧ _x49 = _x53 ∧ _x48 = _x52
l5	9	l2:	x1 = _x56 ∧ x2 = _x57 ∧ x3 = _x58 ∧ x4 = _x59 ∧ x1 = _x60 ∧ x2 = _x61 ∧ x3 = _x62 ∧ x4 = _x63 ∧ _x59 = _x63 ∧ _x58 = _x62 ∧ _x57 = _x61 ∧ _x56 = _x60
l5	10	l3:	x1 = _x64 ∧ x2 = _x65 ∧ x3 = _x66 ∧ x4 = _x67 ∧ x1 = _x68 ∧ x2 = _x69 ∧ x3 = _x70 ∧ x4 = _x71 ∧ _x67 = _x71 ∧ _x66 = _x70 ∧ _x65 = _x69 ∧ _x64 = _x68
l5	11	l4:	x1 = _x72 ∧ x2 = _x73 ∧ x3 = _x74 ∧ x4 = _x75 ∧ x1 = _x76 ∧ x2 = _x77 ∧ x3 = _x78 ∧ x4 = _x79 ∧ _x75 = _x79 ∧ _x74 = _x78 ∧ _x73 = _x77 ∧ _x72 = _x76
l6	12	l5:	x1 = _x80 ∧ x2 = _x81 ∧ x3 = _x82 ∧ x4 = _x83 ∧ x1 = _x84 ∧ x2 = _x85 ∧ x3 = _x86 ∧ x4 = _x87 ∧ _x83 = _x87 ∧ _x82 = _x86 ∧ _x81 = _x85 ∧ _x80 = _x84

Proof

1 Switch to Cooperation Termination Proof

We consider the following cutpoint-transitions:

l5	l5	l5:	x1 = x1 ∧ x2 = x2 ∧ x3 = x3 ∧ x4 = x4
l4	l4	l4:	x1 = x1 ∧ x2 = x2 ∧ x3 = x3 ∧ x4 = x4
l6	l6	l6:	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

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 { l3, l2 }.

2.1.1 Transition Removal

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

l2:	2⋅x1 − 2⋅x4 − 1
l3:	2⋅x1 − 2⋅x4

2.1.2 Transition Removal

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

l2:	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)