LTS Termination Proof

Input

Integer Transition System

Initial Location: f1220_0_create_Return, f2992_0_collectVariables_NULL, f2072_0_create_GE, f1_0_main_Load, f804_0_create_GT, f2311_0_getVariables_InvokeMethod, __init

Transitions: (pre-variables and post-variables)

f1_0_main_Load	1	f2311_0_getVariables_InvokeMethod:	x1 = _arg1 ∧ x2 = _arg2 ∧ x3 = _arg3 ∧ x4 = _arg4 ∧ x5 = _arg5 ∧ x1 = _arg1P ∧ x2 = _arg2P ∧ x3 = _arg3P ∧ x4 = _arg4P ∧ x5 = _arg5P ∧ −1 ≤ _x7 − 1 ∧ 0 ≤ _arg2 − 1 ∧ _arg1P ≤ _arg1 ∧ 0 ≤ _arg1 − 1 ∧ 0 ≤ _arg1P − 1 ∧ 3 ≤ _arg2P − 1
f1220_0_create_Return	2	f2311_0_getVariables_InvokeMethod:	x1 = _x ∧ x2 = _x1 ∧ x3 = _x2 ∧ x4 = _x3 ∧ x5 = _x4 ∧ x1 = _x5 ∧ x2 = _x6 ∧ x3 = _x8 ∧ x4 = _x9 ∧ x5 = _x10 ∧ _x4 = _x10 ∧ _x3 = _x9 ∧ _x2 = _x8 ∧ _x3 + 5 ≤ _x1 ∧ _x4 + 5 ≤ _x1 ∧ 3 ≤ _x6 − 1 ∧ 0 ≤ _x5 − 1 ∧ 3 ≤ _x1 − 1 ∧ 0 ≤ _x − 1 ∧ _x6 ≤ _x1 ∧ _x5 + 3 ≤ _x1 ∧ _x5 ≤ _x
f1_0_main_Load	3	f804_0_create_GT:	x1 = _x11 ∧ x2 = _x12 ∧ x3 = _x13 ∧ x4 = _x14 ∧ x5 = _x15 ∧ x1 = _x16 ∧ x2 = _x17 ∧ x3 = _x18 ∧ x4 = _x19 ∧ x5 = _x20 ∧ 1 = _x18 ∧ _x12 = _x17 ∧ 0 ≤ _x11 − 1 ∧ 0 ≤ _x12 − 1 ∧ −1 ≤ _x16 − 1
f804_0_create_GT	4	f2072_0_create_GE:	x1 = _x21 ∧ x2 = _x22 ∧ x3 = _x23 ∧ x4 = _x24 ∧ x5 = _x25 ∧ x1 = _x26 ∧ x2 = _x27 ∧ x3 = _x28 ∧ x4 = _x29 ∧ x5 = _x30 ∧ _x23 + 1 = _x30 ∧ _x22 = _x29 ∧ 0 = _x27 ∧ _x21 = _x26 ∧ −1 ≤ _x28 − 1 ∧ _x23 ≤ _x22 − 1 ∧ 0 ≤ _x23 − 1 ∧ 1 ≤ _x21 − 1 ∧ −1 ≤ _x22 − 1
f2072_0_create_GE	5	f804_0_create_GT:	x1 = _x31 ∧ x2 = _x32 ∧ x3 = _x33 ∧ x4 = _x35 ∧ x5 = _x36 ∧ x1 = _x37 ∧ x2 = _x38 ∧ x3 = _x39 ∧ x4 = _x40 ∧ x5 = _x41 ∧ _x36 = _x39 ∧ _x35 = _x38 ∧ _x31 − 1 = _x37 ∧ _x31 − 1 ≤ _x31 − 1 ∧ 1 ≤ _x36 − 1 ∧ 0 ≤ _x33 − 1 ∧ 1 ≤ _x31 − 1 ∧ _x32 ≤ _x33 − 1
f2072_0_create_GE	6	f2072_0_create_GE:	x1 = _x42 ∧ x2 = _x43 ∧ x3 = _x44 ∧ x4 = _x45 ∧ x5 = _x46 ∧ x1 = _x47 ∧ x2 = _x48 ∧ x3 = _x49 ∧ x4 = _x50 ∧ x5 = _x51 ∧ _x45 = _x50 ∧ _x44 = _x49 ∧ _x43 + 1 = _x48 ∧ _x42 = _x47 ∧ _x42 − 1 ≤ _x42 − 1 ∧ 1 ≤ _x46 − 1 ∧ 0 ≤ _x44 − 1 ∧ 1 ≤ _x42 − 1 ∧ _x43 ≤ _x44 − 1
f2311_0_getVariables_InvokeMethod	7	f2992_0_collectVariables_NULL:	x1 = _x52 ∧ x2 = _x53 ∧ x3 = _x54 ∧ x4 = _x55 ∧ x5 = _x56 ∧ x1 = _x57 ∧ x2 = _x58 ∧ x3 = _x59 ∧ x4 = _x60 ∧ x5 = _x61 ∧ 0 ≤ _x62 − 1 ∧ 1 ≤ _x54 − 1 ∧ _x57 − 1 ≤ _x52 ∧ _x57 + 2 ≤ _x53 ∧ _x58 + 3 ≤ _x53 ∧ 0 ≤ _x52 − 1 ∧ 3 ≤ _x53 − 1 ∧ 1 ≤ _x57 − 1 ∧ 0 ≤ _x58 − 1 ∧ _x56 + 5 ≤ _x53 ∧ _x55 + 5 ≤ _x53 ∧ _x54 = _x59
f2992_0_collectVariables_NULL	8	f2992_0_collectVariables_NULL:	x1 = _x63 ∧ x2 = _x64 ∧ x3 = _x65 ∧ x4 = _x66 ∧ x5 = _x67 ∧ x1 = _x68 ∧ x2 = _x69 ∧ x3 = _x70 ∧ x4 = _x71 ∧ x5 = _x72 ∧ _x65 = _x70 ∧ −1 ≤ _x69 − 1 ∧ 0 ≤ _x68 − 1 ∧ 2 ≤ _x64 − 1 ∧ 0 ≤ _x63 − 1 ∧ _x69 + 3 ≤ _x64 ∧ 1 ≤ _x65 − 1 ∧ _x68 ≤ _x63
f2992_0_collectVariables_NULL	9	f2992_0_collectVariables_NULL:	x1 = _x73 ∧ x2 = _x74 ∧ x3 = _x75 ∧ x4 = _x76 ∧ x5 = _x77 ∧ x1 = _x78 ∧ x2 = _x79 ∧ x3 = _x80 ∧ x4 = _x81 ∧ x5 = _x82 ∧ _x75 = _x80 ∧ −1 ≤ _x79 − 1 ∧ 0 ≤ _x78 − 1 ∧ 2 ≤ _x74 − 1 ∧ 0 ≤ _x73 − 1 ∧ 1 ≤ _x75 − 1 ∧ _x79 + 2 ≤ _x74
f2992_0_collectVariables_NULL	10	f2992_0_collectVariables_NULL:	x1 = _x83 ∧ x2 = _x84 ∧ x3 = _x85 ∧ x4 = _x86 ∧ x5 = _x87 ∧ x1 = _x88 ∧ x2 = _x89 ∧ x3 = _x90 ∧ x4 = _x91 ∧ x5 = _x92 ∧ _x85 = _x90 ∧ −1 ≤ _x89 − 1 ∧ 2 ≤ _x88 − 1 ∧ 2 ≤ _x84 − 1 ∧ 0 ≤ _x83 − 1 ∧ _x89 + 2 ≤ _x84
__init	11	f1_0_main_Load:	x1 = _x93 ∧ x2 = _x94 ∧ x3 = _x95 ∧ x4 = _x96 ∧ x5 = _x97 ∧ x1 = _x98 ∧ x2 = _x99 ∧ x3 = _x100 ∧ x4 = _x101 ∧ x5 = _x102 ∧ 0 ≤ 0

Proof

1 Switch to Cooperation Termination Proof

We consider the following cutpoint-transitions:

f1220_0_create_Return	f1220_0_create_Return	f1220_0_create_Return:	x1 = x1 ∧ x2 = x2 ∧ x3 = x3 ∧ x4 = x4 ∧ x5 = x5
f2992_0_collectVariables_NULL	f2992_0_collectVariables_NULL	f2992_0_collectVariables_NULL:	x1 = x1 ∧ x2 = x2 ∧ x3 = x3 ∧ x4 = x4 ∧ x5 = x5
f2072_0_create_GE	f2072_0_create_GE	f2072_0_create_GE:	x1 = x1 ∧ x2 = x2 ∧ x3 = x3 ∧ x4 = x4 ∧ x5 = x5
f1_0_main_Load	f1_0_main_Load	f1_0_main_Load:	x1 = x1 ∧ x2 = x2 ∧ x3 = x3 ∧ x4 = x4 ∧ x5 = x5
f804_0_create_GT	f804_0_create_GT	f804_0_create_GT:	x1 = x1 ∧ x2 = x2 ∧ x3 = x3 ∧ x4 = x4 ∧ x5 = x5
f2311_0_getVariables_InvokeMethod	f2311_0_getVariables_InvokeMethod	f2311_0_getVariables_InvokeMethod:	x1 = x1 ∧ x2 = x2 ∧ x3 = x3 ∧ x4 = x4 ∧ x5 = x5
__init	__init	__init:	x1 = x1 ∧ x2 = x2 ∧ x3 = x3 ∧ x4 = x4 ∧ x5 = x5

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 { f2072_0_create_GE, f804_0_create_GT }.

2.1.1 Transition Removal

We remove transitions 4, 5 using the following ranking functions, which are bounded by 0.

f804_0_create_GT:	1 + 2⋅x1
f2072_0_create_GE:	2⋅x1

2.1.2 Transition Removal

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

f2072_0_create_GE:

− x2 + x3

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 2/2

Here we consider the SCC { f2992_0_collectVariables_NULL }.

2.2.1 Transition Removal

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

f2992_0_collectVariables_NULL:

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 % (9 real / 0 unknown / 0 assumptions / 9 total proof steps)