LTS Termination Proof

Input

Integer Transition System

Initial Location: f731_0_random_ArrayAccess, f322_0_createTree_Return, f750_0_random_ArrayAccess, f512_0_createNode_Return, f1317_0_mirror_NULL, f1551_0_createTree_LE, f1_0_main_Load, f477_0_createNode_Return, __init

Transitions: (pre-variables and post-variables)

f1_0_main_Load	1	f1317_0_mirror_NULL:	x1 = _arg1 ∧ x2 = _arg2 ∧ x3 = _arg3 ∧ x4 = _arg4 ∧ x1 = _arg1P ∧ x2 = _arg2P ∧ x3 = _arg3P ∧ x4 = _arg4P ∧ −1 ≤ _arg1P − 1 ∧ 0 ≤ _arg1 − 1
f322_0_createTree_Return	2	f1317_0_mirror_NULL:	x1 = _x ∧ x2 = _x1 ∧ x3 = _x2 ∧ x4 = _x3 ∧ x1 = _x4 ∧ x2 = _x5 ∧ x3 = _x6 ∧ x4 = _x7 ∧ −1 ≤ _x4 − 1 ∧ −1 ≤ _x − 1 ∧ _x4 ≤ _x
f1317_0_mirror_NULL	3	f1317_0_mirror_NULL:	x1 = _x8 ∧ x2 = _x9 ∧ x3 = _x10 ∧ x4 = _x11 ∧ x1 = _x12 ∧ x2 = _x13 ∧ x3 = _x14 ∧ x4 = _x15 ∧ −1 ≤ _x12 − 1 ∧ 0 ≤ _x8 − 1 ∧ _x12 + 1 ≤ _x8
f477_0_createNode_Return	4	f731_0_random_ArrayAccess:	x1 = _x16 ∧ x2 = _x17 ∧ x3 = _x18 ∧ x4 = _x19 ∧ x1 = _x20 ∧ x2 = _x21 ∧ x3 = _x23 ∧ x4 = _x24 ∧ 0 ≤ _x20 − 1
f1_0_main_Load	5	f731_0_random_ArrayAccess:	x1 = _x25 ∧ x2 = _x26 ∧ x3 = _x28 ∧ x4 = _x29 ∧ x1 = _x30 ∧ x2 = _x31 ∧ x3 = _x32 ∧ x4 = _x33 ∧ 0 ≤ _x30 − 1 ∧ 0 ≤ _x25 − 1 ∧ _x30 ≤ _x25
f512_0_createNode_Return	6	f750_0_random_ArrayAccess:	x1 = _x34 ∧ x2 = _x35 ∧ x3 = _x36 ∧ x4 = _x37 ∧ x1 = _x38 ∧ x2 = _x39 ∧ x3 = _x40 ∧ x4 = _x41 ∧ 0 ≤ _x38 − 1
f1_0_main_Load	7	f750_0_random_ArrayAccess:	x1 = _x43 ∧ x2 = _x44 ∧ x3 = _x45 ∧ x4 = _x46 ∧ x1 = _x47 ∧ x2 = _x48 ∧ x3 = _x49 ∧ x4 = _x50 ∧ 0 ≤ _x47 − 1 ∧ 0 ≤ _x43 − 1 ∧ _x47 ≤ _x43
f750_0_random_ArrayAccess	8	f1551_0_createTree_LE:	x1 = _x51 ∧ x2 = _x52 ∧ x3 = _x53 ∧ x4 = _x54 ∧ x1 = _x55 ∧ x2 = _x57 ∧ x3 = _x58 ∧ x4 = _x59 ∧ 0 ≤ _x60 − 1 ∧ −1 ≤ _x57 − 1 ∧ _x55 − 3 ≤ _x51 ∧ 0 ≤ _x51 − 1 ∧ 3 ≤ _x55 − 1 ∧ _x60 + 1 = _x59
f731_0_random_ArrayAccess	9	f1551_0_createTree_LE:	x1 = _x61 ∧ x2 = _x62 ∧ x3 = _x63 ∧ x4 = _x64 ∧ x1 = _x65 ∧ x2 = _x66 ∧ x3 = _x67 ∧ x4 = _x68 ∧ 0 ≤ _x69 − 1 ∧ −1 ≤ _x66 − 1 ∧ _x65 − 1 ≤ _x61 ∧ 0 ≤ _x61 − 1 ∧ 1 ≤ _x65 − 1 ∧ _x69 + 1 = _x68
f1551_0_createTree_LE	10	f1551_0_createTree_LE:	x1 = _x70 ∧ x2 = _x71 ∧ x3 = _x72 ∧ x4 = _x73 ∧ x1 = _x74 ∧ x2 = _x75 ∧ x3 = _x76 ∧ x4 = _x77 ∧ _x73 = _x77 ∧ _x72 = _x76 ∧ _x71 − 1 = _x75 ∧ −1 ≤ _x74 − 1 ∧ 1 ≤ _x70 − 1 ∧ 0 ≤ _x71 − 1 ∧ _x74 + 2 ≤ _x70
f1551_0_createTree_LE	11	f1551_0_createTree_LE:	x1 = _x78 ∧ x2 = _x79 ∧ x3 = _x80 ∧ x4 = _x81 ∧ x1 = _x82 ∧ x2 = _x83 ∧ x3 = _x84 ∧ x4 = _x85 ∧ 0 ≤ _x79 − 1 ∧ 0 ≤ _x81 − 1 ∧ 0 ≤ _x86 − 1 ∧ _x82 − 2 ≤ _x78 ∧ 2 ≤ _x78 − 1 ∧ 3 ≤ _x82 − 1 ∧ _x79 − 1 = _x83 ∧ _x80 = _x84
f1551_0_createTree_LE	12	f1551_0_createTree_LE:	x1 = _x87 ∧ x2 = _x88 ∧ x3 = _x89 ∧ x4 = _x90 ∧ x1 = _x91 ∧ x2 = _x92 ∧ x3 = _x93 ∧ x4 = _x94 ∧ 0 ≤ _x88 − 1 ∧ 0 ≤ _x90 − 1 ∧ 0 ≤ _x95 − 1 ∧ _x91 − 3 ≤ _x87 ∧ 2 ≤ _x87 − 1 ∧ 5 ≤ _x91 − 1 ∧ _x88 − 1 = _x92 ∧ _x89 = _x93
f1_0_main_Load	13	f973_0_random_ArrayAccess:	x1 = _x96 ∧ x2 = _x97 ∧ x3 = _x98 ∧ x4 = _x99 ∧ x1 = _x100 ∧ x2 = _x101 ∧ x3 = _x102 ∧ x4 = _x103 ∧ 0 = _x102 ∧ 0 ≤ _x101 − 1 ∧ 0 ≤ _x100 − 1 ∧ 0 ≤ _x96 − 1 ∧ _x101 ≤ _x96 ∧ −1 ≤ _x97 − 1 ∧ _x100 ≤ _x96
f1551_0_createTree_LE	14	f973_0_random_ArrayAccess:	x1 = _x104 ∧ x2 = _x105 ∧ x3 = _x106 ∧ x4 = _x107 ∧ x1 = _x108 ∧ x2 = _x109 ∧ x3 = _x110 ∧ x4 = _x111 ∧ _x107 = _x110 ∧ 0 ≤ _x109 − 1 ∧ 2 ≤ _x104 − 1 ∧ _x109 + 2 ≤ _x104 ∧ −1 ≤ _x106 − 1 ∧ 0 ≤ _x107 − 1 ∧ 0 ≤ _x105 − 1
f1551_0_createTree_LE	15	f973_0_random_ArrayAccess:	x1 = _x112 ∧ x2 = _x113 ∧ x3 = _x114 ∧ x4 = _x115 ∧ x1 = _x116 ∧ x2 = _x117 ∧ x3 = _x118 ∧ x4 = _x119 ∧ 0 ≤ _x117 − 1 ∧ 2 ≤ _x112 − 1 ∧ _x117 + 2 ≤ _x112 ∧ −1 ≤ _x114 − 1 ∧ 0 ≤ _x118 − 1 ∧ 0 ≤ _x115 − 1 ∧ 0 ≤ _x113 − 1
__init	16	f1_0_main_Load:	x1 = _x120 ∧ x2 = _x121 ∧ x3 = _x122 ∧ x4 = _x123 ∧ x1 = _x124 ∧ x2 = _x125 ∧ x3 = _x126 ∧ x4 = _x127 ∧ 0 ≤ 0

Proof

1 Switch to Cooperation Termination Proof

We consider the following cutpoint-transitions:

f731_0_random_ArrayAccess	f731_0_random_ArrayAccess	f731_0_random_ArrayAccess:	x1 = x1 ∧ x2 = x2 ∧ x3 = x3 ∧ x4 = x4
f322_0_createTree_Return	f322_0_createTree_Return	f322_0_createTree_Return:	x1 = x1 ∧ x2 = x2 ∧ x3 = x3 ∧ x4 = x4
f750_0_random_ArrayAccess	f750_0_random_ArrayAccess	f750_0_random_ArrayAccess:	x1 = x1 ∧ x2 = x2 ∧ x3 = x3 ∧ x4 = x4
f512_0_createNode_Return	f512_0_createNode_Return	f512_0_createNode_Return:	x1 = x1 ∧ x2 = x2 ∧ x3 = x3 ∧ x4 = x4
f1317_0_mirror_NULL	f1317_0_mirror_NULL	f1317_0_mirror_NULL:	x1 = x1 ∧ x2 = x2 ∧ x3 = x3 ∧ x4 = x4
f1551_0_createTree_LE	f1551_0_createTree_LE	f1551_0_createTree_LE:	x1 = x1 ∧ x2 = x2 ∧ x3 = x3 ∧ x4 = x4
f1_0_main_Load	f1_0_main_Load	f1_0_main_Load:	x1 = x1 ∧ x2 = x2 ∧ x3 = x3 ∧ x4 = x4
f477_0_createNode_Return	f477_0_createNode_Return	f477_0_createNode_Return:	x1 = x1 ∧ x2 = x2 ∧ x3 = x3 ∧ x4 = x4
__init	__init	__init:	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 2 SCC(s) of the program graph.

2.1 SCC Subproblem 1/2

Here we consider the SCC { f1551_0_createTree_LE }.

2.1.1 Transition Removal

We remove transitions 10, 11, 12 using the following ranking functions, which are bounded by 0.

f1551_0_createTree_LE:

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 { f1317_0_mirror_NULL }.

2.2.1 Transition Removal

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

f1317_0_mirror_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 % (7 real / 0 unknown / 0 assumptions / 7 total proof steps)