LTS Termination Proof

Input

Integer Transition System

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

Transitions: (pre-variables and post-variables)

l0	1	l1:	x1 = _result_11HAT0 ∧ x2 = _temp0_14HAT0 ∧ x3 = _x_13HAT0 ∧ x4 = _x_27HAT0 ∧ x5 = _x_32HAT0 ∧ x6 = _y_16HAT0 ∧ x7 = _y_28HAT0 ∧ x8 = _y_33HAT0 ∧ x1 = _result_11HATpost ∧ x2 = _temp0_14HATpost ∧ x3 = _x_13HATpost ∧ x4 = _x_27HATpost ∧ x5 = _x_32HATpost ∧ x6 = _y_16HATpost ∧ x7 = _y_28HATpost ∧ x8 = _y_33HATpost ∧ _y_33HAT0 = _y_33HATpost ∧ _y_28HAT0 = _y_28HATpost ∧ _x_32HAT0 = _x_32HATpost ∧ _x_27HAT0 = _x_27HATpost ∧ _temp0_14HAT0 = _temp0_14HATpost ∧ _result_11HAT0 = _result_11HATpost ∧ _y_16HATpost = _y_16HATpost ∧ _x_13HATpost = _x_13HATpost
l1	2	l2:	x1 = _x ∧ x2 = _x1 ∧ x3 = _x2 ∧ x4 = _x3 ∧ x5 = _x4 ∧ x6 = _x5 ∧ x7 = _x6 ∧ x8 = _x7 ∧ x1 = _x8 ∧ x2 = _x9 ∧ x3 = _x10 ∧ x4 = _x11 ∧ x5 = _x12 ∧ x6 = _x13 ∧ x7 = _x14 ∧ x8 = _x15 ∧ _x7 = _x15 ∧ _x6 = _x14 ∧ _x4 = _x12 ∧ _x3 = _x11 ∧ _x2 = _x10 ∧ _x1 = _x9 ∧ _x = _x8 ∧ 1 ≤ _x2 ∧ 1 ≤ _x13 ∧ _x13 = _x13 ∧ 1 ≤ _x2
l1	3	l3:	x1 = _x16 ∧ x2 = _x17 ∧ x3 = _x18 ∧ x4 = _x19 ∧ x5 = _x20 ∧ x6 = _x21 ∧ x7 = _x22 ∧ x8 = _x23 ∧ x1 = _x24 ∧ x2 = _x25 ∧ x3 = _x26 ∧ x4 = _x27 ∧ x5 = _x28 ∧ x6 = _x29 ∧ x7 = _x30 ∧ x8 = _x31 ∧ _x23 = _x31 ∧ _x22 = _x30 ∧ _x21 = _x29 ∧ _x20 = _x28 ∧ _x19 = _x27 ∧ _x18 = _x26 ∧ _x17 = _x25 ∧ _x24 = _x17 ∧ _x18 ≤ 0
l4	4	l2:	x1 = _x32 ∧ x2 = _x33 ∧ x3 = _x34 ∧ x4 = _x35 ∧ x5 = _x36 ∧ x6 = _x37 ∧ x7 = _x38 ∧ x8 = _x39 ∧ x1 = _x40 ∧ x2 = _x41 ∧ x3 = _x42 ∧ x4 = _x43 ∧ x5 = _x44 ∧ x6 = _x45 ∧ x7 = _x46 ∧ x8 = _x47 ∧ _x39 = _x47 ∧ _x36 = _x44 ∧ _x33 = _x41 ∧ _x32 = _x40 ∧ 1 ≤ _x46 ∧ 1 ≤ _x43 ∧ −1 + _x46 ≤ _x45 ∧ _x45 ≤ −1 + _x46 ∧ −1 + _x43 ≤ _x42 ∧ _x42 ≤ −1 + _x43 ∧ _x45 = −1 + _x37 ∧ _x42 = −1 + _x34 ∧ 1 ≤ _x37 ∧ _x46 = _x46 ∧ _x43 = _x43
l2	5	l1:	x1 = _x48 ∧ x2 = _x49 ∧ x3 = _x50 ∧ x4 = _x51 ∧ x5 = _x52 ∧ x6 = _x53 ∧ x7 = _x54 ∧ x8 = _x55 ∧ x1 = _x56 ∧ x2 = _x57 ∧ x3 = _x58 ∧ x4 = _x59 ∧ x5 = _x60 ∧ x6 = _x61 ∧ x7 = _x62 ∧ x8 = _x63 ∧ _x55 = _x63 ∧ _x54 = _x62 ∧ _x53 = _x61 ∧ _x52 = _x60 ∧ _x51 = _x59 ∧ _x50 = _x58 ∧ _x49 = _x57 ∧ _x48 = _x56 ∧ _x53 ≤ 0 ∧ _x53 ≤ 0
l2	6	l5:	x1 = _x64 ∧ x2 = _x65 ∧ x3 = _x66 ∧ x4 = _x67 ∧ x5 = _x68 ∧ x6 = _x69 ∧ x7 = _x70 ∧ x8 = _x71 ∧ x1 = _x72 ∧ x2 = _x73 ∧ x3 = _x74 ∧ x4 = _x75 ∧ x5 = _x76 ∧ x6 = _x77 ∧ x7 = _x78 ∧ x8 = _x79 ∧ _x70 = _x78 ∧ _x67 = _x75 ∧ _x65 = _x73 ∧ _x64 = _x72 ∧ 1 ≤ _x79 ∧ −1 + _x79 ≤ _x77 ∧ _x77 ≤ −1 + _x79 ∧ −1 + _x76 ≤ _x74 ∧ _x74 ≤ −1 + _x76 ∧ _x77 = −1 + _x69 ∧ _x74 = −1 + _x66 ∧ 1 ≤ _x69 ∧ _x79 = _x79 ∧ _x76 = _x76
l5	7	l2:	x1 = _x80 ∧ x2 = _x81 ∧ x3 = _x82 ∧ x4 = _x83 ∧ x5 = _x84 ∧ x6 = _x85 ∧ x7 = _x86 ∧ x8 = _x87 ∧ x1 = _x88 ∧ x2 = _x89 ∧ x3 = _x90 ∧ x4 = _x91 ∧ x5 = _x92 ∧ x6 = _x93 ∧ x7 = _x94 ∧ x8 = _x95 ∧ _x87 = _x95 ∧ _x86 = _x94 ∧ _x85 = _x93 ∧ _x84 = _x92 ∧ _x83 = _x91 ∧ _x82 = _x90 ∧ _x81 = _x89 ∧ _x80 = _x88
l6	8	l0:	x1 = _x96 ∧ x2 = _x97 ∧ x3 = _x98 ∧ x4 = _x99 ∧ x5 = _x100 ∧ x6 = _x101 ∧ x7 = _x102 ∧ x8 = _x103 ∧ x1 = _x104 ∧ x2 = _x105 ∧ x3 = _x106 ∧ x4 = _x107 ∧ x5 = _x108 ∧ x6 = _x109 ∧ x7 = _x110 ∧ x8 = _x111 ∧ _x103 = _x111 ∧ _x102 = _x110 ∧ _x101 = _x109 ∧ _x100 = _x108 ∧ _x99 = _x107 ∧ _x98 = _x106 ∧ _x97 = _x105 ∧ _x96 = _x104

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
l4	l4	l4:	x1 = x1 ∧ x2 = x2 ∧ x3 = x3 ∧ x4 = x4 ∧ x5 = x5 ∧ x6 = x6 ∧ x7 = x7 ∧ x8 = x8
l6	l6	l6:	x1 = x1 ∧ x2 = x2 ∧ x3 = x3 ∧ x4 = x4 ∧ x5 = x5 ∧ x6 = x6 ∧ x7 = x7 ∧ x8 = x8
l1	l1	l1:	x1 = x1 ∧ x2 = x2 ∧ x3 = x3 ∧ x4 = x4 ∧ x5 = x5 ∧ x6 = x6 ∧ x7 = x7 ∧ x8 = x8
l0	l0	l0:	x1 = x1 ∧ x2 = x2 ∧ x3 = x3 ∧ x4 = x4 ∧ x5 = x5 ∧ x6 = x6 ∧ x7 = x7 ∧ x8 = x8
l2	l2	l2:	x1 = x1 ∧ x2 = x2 ∧ x3 = x3 ∧ x4 = x4 ∧ x5 = x5 ∧ x6 = x6 ∧ x7 = x7 ∧ x8 = x8

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, l1, l2 }.

2.1.1 Transition Removal

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

l1:	−1 + x3
l2:	−1 + x3 − x6
l5:	−1 + x3 − x6

2.1.2 Transition Removal

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

l2:	0
l1:	−1
l5:	0

2.1.3 Transition Removal

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

l5:	−1 + x6
l2:	−1 + x6

2.1.4 Transition Removal

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

l5:	0
l2:	−1

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)