LTS Termination Proof

Input

Integer Transition System

Initial Location: f765_0_insert_GT, f1_0_main_Load, f672_0_main_GE, __init

Transitions: (pre-variables and post-variables)

f1_0_main_Load	1	f672_0_main_GE:	x1 = _arg1 ∧ x2 = _arg2 ∧ x3 = _arg3 ∧ x4 = _arg4 ∧ x5 = _arg5 ∧ x6 = _arg6 ∧ x1 = _arg1P ∧ x2 = _arg2P ∧ x3 = _arg3P ∧ x4 = _arg4P ∧ x5 = _arg5P ∧ x6 = _arg6P ∧ 2 = _arg5P ∧ 0 = _arg3P ∧ 1 ≤ _arg2P − 1 ∧ 0 ≤ _arg1P − 1 ∧ 0 ≤ _arg1 − 1 ∧ _arg1P ≤ _arg1 ∧ −1 ≤ _arg6P − 1 ∧ 1 ≤ _arg2 − 1 ∧ −1 ≤ _arg4P − 1
f672_0_main_GE	2	f672_0_main_GE:	x1 = _x ∧ x2 = _x1 ∧ x3 = _x2 ∧ x4 = _x3 ∧ x5 = _x4 ∧ x6 = _x5 ∧ x1 = _x7 ∧ x2 = _x8 ∧ x3 = _x9 ∧ x4 = _x10 ∧ x5 = _x11 ∧ x6 = _x12 ∧ −1 ≤ _x4 − 1 ∧ _x2 ≤ _x3 − 1 ∧ 0 ≤ _x3 − 1 ∧ 1 ≤ _x13 − 1 ∧ −1 ≤ _x14 − 1 ∧ _x4 ≤ _x13 − 1 ∧ _x7 ≤ _x ∧ _x7 ≤ _x1 ∧ 0 ≤ _x − 1 ∧ 0 ≤ _x1 − 1 ∧ 0 ≤ _x7 − 1 ∧ 3 ≤ _x8 − 1 ∧ _x5 + 2 ≤ _x1 ∧ _x2 + 1 = _x9 ∧ _x3 = _x10 ∧ _x4 + 1 = _x11
f672_0_main_GE	3	f672_0_main_GE:	x1 = _x15 ∧ x2 = _x17 ∧ x3 = _x18 ∧ x4 = _x19 ∧ x5 = _x20 ∧ x6 = _x21 ∧ x1 = _x22 ∧ x2 = _x23 ∧ x3 = _x24 ∧ x4 = _x25 ∧ x5 = _x26 ∧ x6 = _x27 ∧ −1 ≤ _x20 − 1 ∧ _x18 ≤ _x19 − 1 ∧ 0 ≤ _x19 − 1 ∧ 1 ≤ _x29 − 1 ∧ −1 ≤ _x30 − 1 ∧ _x20 ≤ _x29 − 1 ∧ _x22 ≤ _x15 ∧ _x22 ≤ _x17 ∧ 0 ≤ _x15 − 1 ∧ 0 ≤ _x17 − 1 ∧ 0 ≤ _x22 − 1 ∧ 2 ≤ _x23 − 1 ∧ _x21 + 2 ≤ _x17 ∧ _x18 + 1 = _x24 ∧ _x19 = _x25 ∧ _x20 + 1 = _x26
f672_0_main_GE	4	f765_0_insert_GT:	x1 = _x31 ∧ x2 = _x32 ∧ x3 = _x33 ∧ x4 = _x34 ∧ x5 = _x35 ∧ x6 = _x36 ∧ x1 = _x37 ∧ x2 = _x39 ∧ x3 = _x40 ∧ x4 = _x41 ∧ x5 = _x42 ∧ x6 = _x43 ∧ −1 ≤ _x35 − 1 ∧ _x33 ≤ _x34 − 1 ∧ 0 ≤ _x34 − 1 ∧ 1 ≤ _x44 − 1 ∧ −1 ≤ _x39 − 1 ∧ _x35 ≤ _x44 − 1 ∧ _x37 ≤ _x32 ∧ 0 ≤ _x31 − 1 ∧ 0 ≤ _x32 − 1 ∧ 0 ≤ _x37 − 1 ∧ _x36 + 2 ≤ _x32 ∧ _x36 = _x40
f765_0_insert_GT	5	f765_0_insert_GT:	x1 = _x45 ∧ x2 = _x46 ∧ x3 = _x47 ∧ x4 = _x48 ∧ x5 = _x49 ∧ x6 = _x51 ∧ x1 = _x52 ∧ x2 = _x53 ∧ x3 = _x54 ∧ x4 = _x55 ∧ x5 = _x56 ∧ x6 = _x57 ∧ _x46 = _x53 ∧ _x54 + 4 ≤ _x45 ∧ _x47 + 2 ≤ _x45 ∧ 0 ≤ _x52 − 1 ∧ 2 ≤ _x45 − 1 ∧ _x47 ≤ _x46 − 1 ∧ _x52 + 2 ≤ _x45
f765_0_insert_GT	6	f765_0_insert_GT:	x1 = _x58 ∧ x2 = _x59 ∧ x3 = _x60 ∧ x4 = _x61 ∧ x5 = _x62 ∧ x6 = _x63 ∧ x1 = _x64 ∧ x2 = _x65 ∧ x3 = _x66 ∧ x4 = _x67 ∧ x5 = _x68 ∧ x6 = _x69 ∧ _x59 = _x65 ∧ _x66 + 4 ≤ _x58 ∧ _x60 + 2 ≤ _x58 ∧ 0 ≤ _x64 − 1 ∧ 2 ≤ _x58 − 1 ∧ _x59 ≤ _x60 ∧ _x64 + 2 ≤ _x58
__init	7	f1_0_main_Load:	x1 = _x70 ∧ x2 = _x71 ∧ x3 = _x72 ∧ x4 = _x73 ∧ x5 = _x74 ∧ x6 = _x75 ∧ x1 = _x76 ∧ x2 = _x77 ∧ x3 = _x78 ∧ x4 = _x79 ∧ x5 = _x80 ∧ x6 = _x81 ∧ 0 ≤ 0

Proof

1 Switch to Cooperation Termination Proof

We consider the following cutpoint-transitions:

f765_0_insert_GT	f765_0_insert_GT	f765_0_insert_GT:	x1 = x1 ∧ x2 = x2 ∧ x3 = x3 ∧ x4 = x4 ∧ x5 = x5 ∧ x6 = x6
f1_0_main_Load	f1_0_main_Load	f1_0_main_Load:	x1 = x1 ∧ x2 = x2 ∧ x3 = x3 ∧ x4 = x4 ∧ x5 = x5 ∧ x6 = x6
f672_0_main_GE	f672_0_main_GE	f672_0_main_GE:	x1 = x1 ∧ x2 = x2 ∧ x3 = x3 ∧ x4 = x4 ∧ x5 = x5 ∧ x6 = x6
__init	__init	__init:	x1 = x1 ∧ x2 = x2 ∧ x3 = x3 ∧ x4 = x4 ∧ x5 = x5 ∧ x6 = x6

and for every transition t, a duplicate t is considered.

2 SCC Decomposition

We consider subproblems for each of the 2 SCC(s) of the program graph.

2.1 SCC Subproblem 1/2

Here we consider the SCC { f672_0_main_GE }.

2.1.1 Transition Removal

We remove transitions 2, 3 using the following ranking functions, which are bounded by 0.

f672_0_main_GE:

− x3 + x4

2.1.2 Trivial Cooperation Program

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

2.2 SCC Subproblem 2/2

Here we consider the SCC { f765_0_insert_GT }.

2.2.1 Transition Removal

We remove transitions 5, 6 using the following ranking functions, which are bounded by 0.

f765_0_insert_GT:

2.2.2 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 % (7 real / 0 unknown / 0 assumptions / 7 total proof steps)