YES
proof of prog.inttrs
# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty


Termination of the given IRSwT could be proven:

(0) IRSwT
(1) IRSFormatTransformerProof [EQUIVALENT, 0 ms]
(2) IRSwT
(3) IRSwTTerminationDigraphProof [EQUIVALENT, 1257 ms]
(4) AND
    (5) IRSwT
        (6) IntTRSCompressionProof [EQUIVALENT, 8 ms]
        (7) IRSwT
        (8) IntTRSUnneededArgumentFilterProof [EQUIVALENT, 0 ms]
        (9) IRSwT
        (10) TempFilterProof [SOUND, 67 ms]
        (11) IntTRS
        (12) PolynomialOrderProcessor [EQUIVALENT, 0 ms]
        (13) AND
            (14) IntTRS
                (15) RankingReductionPairProof [EQUIVALENT, 9 ms]
                (16) YES
            (17) IntTRS
                (18) RankingReductionPairProof [EQUIVALENT, 5 ms]
                (19) YES
    (20) IRSwT
        (21) IntTRSCompressionProof [EQUIVALENT, 29 ms]
        (22) IRSwT
        (23) IntTRSUnneededArgumentFilterProof [EQUIVALENT, 0 ms]
        (24) IRSwT
        (25) FilterProof [EQUIVALENT, 0 ms]
        (26) IntTRS
        (27) IntTRSCompressionProof [EQUIVALENT, 0 ms]
        (28) IntTRS
        (29) RankingReductionPairProof [EQUIVALENT, 3 ms]
        (30) YES


----------------------------------------

(0)
Obligation:
Rules:
f1_0_main_Load(arg1, arg2, arg3, arg4) -> f1317_0_mirror_NULL(arg1P, arg2P, arg3P, arg4P) :|: -1 <= arg1P - 1 && 0 <= arg1 - 1
f322_0_createTree_Return(x, x1, x2, x3) -> f1317_0_mirror_NULL(x4, x5, x6, x7) :|: -1 <= x4 - 1 && -1 <= x - 1 && x4 <= x
f1317_0_mirror_NULL(x8, x9, x10, x11) -> f1317_0_mirror_NULL(x12, x13, x14, x15) :|: -1 <= x12 - 1 && 0 <= x8 - 1 && x12 + 1 <= x8
f477_0_createNode_Return(x16, x17, x18, x19) -> f731_0_random_ArrayAccess(x20, x21, x23, x24) :|: 0 <= x20 - 1
f1_0_main_Load(x25, x26, x28, x29) -> f731_0_random_ArrayAccess(x30, x31, x32, x33) :|: 0 <= x30 - 1 && 0 <= x25 - 1 && x30 <= x25
f512_0_createNode_Return(x34, x35, x36, x37) -> f750_0_random_ArrayAccess(x38, x39, x40, x41) :|: 0 <= x38 - 1
f1_0_main_Load(x43, x44, x45, x46) -> f750_0_random_ArrayAccess(x47, x48, x49, x50) :|: 0 <= x47 - 1 && 0 <= x43 - 1 && x47 <= x43
f750_0_random_ArrayAccess(x51, x52, x53, x54) -> f1551_0_createTree_LE(x55, x57, x58, x59) :|: 0 <= x60 - 1 && -1 <= x57 - 1 && x55 - 3 <= x51 && 0 <= x51 - 1 && 3 <= x55 - 1 && x60 + 1 = x59
f731_0_random_ArrayAccess(x61, x62, x63, x64) -> f1551_0_createTree_LE(x65, x66, x67, x68) :|: 0 <= x69 - 1 && -1 <= x66 - 1 && x65 - 1 <= x61 && 0 <= x61 - 1 && 1 <= x65 - 1 && x69 + 1 = x68
f1551_0_createTree_LE(x70, x71, x72, x73) -> f1551_0_createTree_LE(x74, x75, x76, x77) :|: x73 = x77 && x72 = x76 && x71 - 1 = x75 && -1 <= x74 - 1 && 1 <= x70 - 1 && 0 <= x71 - 1 && x74 + 2 <= x70
f1551_0_createTree_LE(x78, x79, x80, x81) -> f1551_0_createTree_LE(x82, x83, x84, 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(x87, x88, x89, x90) -> f1551_0_createTree_LE(x91, x92, x93, 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(x96, x97, x98, x99) -> f973_0_random_ArrayAccess(x100, x101, x102, x103) :|: 0 = x102 && 0 <= x101 - 1 && 0 <= x100 - 1 && 0 <= x96 - 1 && x101 <= x96 && -1 <= x97 - 1 && x100 <= x96
f1551_0_createTree_LE(x104, x105, x106, x107) -> f973_0_random_ArrayAccess(x108, x109, x110, 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(x112, x113, x114, x115) -> f973_0_random_ArrayAccess(x116, x117, x118, x119) :|: 0 <= x117 - 1 && 2 <= x112 - 1 && x117 + 2 <= x112 && -1 <= x114 - 1 && 0 <= x118 - 1 && 0 <= x115 - 1 && 0 <= x113 - 1
__init(x120, x121, x122, x123) -> f1_0_main_Load(x124, x125, x126, x127) :|: 0 <= 0
Start term: __init(arg1, arg2, arg3, arg4)

----------------------------------------

(1) IRSFormatTransformerProof (EQUIVALENT)
Reformatted IRS to match normalized format (transformed away non-linear left-hand sides, !=, / and %).
----------------------------------------

(2)
Obligation:
Rules:
f1_0_main_Load(arg1, arg2, arg3, arg4) -> f1317_0_mirror_NULL(arg1P, arg2P, arg3P, arg4P) :|: -1 <= arg1P - 1 && 0 <= arg1 - 1
f322_0_createTree_Return(x, x1, x2, x3) -> f1317_0_mirror_NULL(x4, x5, x6, x7) :|: -1 <= x4 - 1 && -1 <= x - 1 && x4 <= x
f1317_0_mirror_NULL(x8, x9, x10, x11) -> f1317_0_mirror_NULL(x12, x13, x14, x15) :|: -1 <= x12 - 1 && 0 <= x8 - 1 && x12 + 1 <= x8
f477_0_createNode_Return(x16, x17, x18, x19) -> f731_0_random_ArrayAccess(x20, x21, x23, x24) :|: 0 <= x20 - 1
f1_0_main_Load(x25, x26, x28, x29) -> f731_0_random_ArrayAccess(x30, x31, x32, x33) :|: 0 <= x30 - 1 && 0 <= x25 - 1 && x30 <= x25
f512_0_createNode_Return(x34, x35, x36, x37) -> f750_0_random_ArrayAccess(x38, x39, x40, x41) :|: 0 <= x38 - 1
f1_0_main_Load(x43, x44, x45, x46) -> f750_0_random_ArrayAccess(x47, x48, x49, x50) :|: 0 <= x47 - 1 && 0 <= x43 - 1 && x47 <= x43
f750_0_random_ArrayAccess(x51, x52, x53, x54) -> f1551_0_createTree_LE(x55, x57, x58, x59) :|: 0 <= x60 - 1 && -1 <= x57 - 1 && x55 - 3 <= x51 && 0 <= x51 - 1 && 3 <= x55 - 1 && x60 + 1 = x59
f731_0_random_ArrayAccess(x61, x62, x63, x64) -> f1551_0_createTree_LE(x65, x66, x67, x68) :|: 0 <= x69 - 1 && -1 <= x66 - 1 && x65 - 1 <= x61 && 0 <= x61 - 1 && 1 <= x65 - 1 && x69 + 1 = x68
f1551_0_createTree_LE(x70, x71, x72, x73) -> f1551_0_createTree_LE(x74, x75, x76, x77) :|: x73 = x77 && x72 = x76 && x71 - 1 = x75 && -1 <= x74 - 1 && 1 <= x70 - 1 && 0 <= x71 - 1 && x74 + 2 <= x70
f1551_0_createTree_LE(x78, x79, x80, x81) -> f1551_0_createTree_LE(x82, x83, x84, 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(x87, x88, x89, x90) -> f1551_0_createTree_LE(x91, x92, x93, 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(x96, x97, x98, x99) -> f973_0_random_ArrayAccess(x100, x101, x102, x103) :|: 0 = x102 && 0 <= x101 - 1 && 0 <= x100 - 1 && 0 <= x96 - 1 && x101 <= x96 && -1 <= x97 - 1 && x100 <= x96
f1551_0_createTree_LE(x104, x105, x106, x107) -> f973_0_random_ArrayAccess(x108, x109, x110, 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(x112, x113, x114, x115) -> f973_0_random_ArrayAccess(x116, x117, x118, x119) :|: 0 <= x117 - 1 && 2 <= x112 - 1 && x117 + 2 <= x112 && -1 <= x114 - 1 && 0 <= x118 - 1 && 0 <= x115 - 1 && 0 <= x113 - 1
__init(x120, x121, x122, x123) -> f1_0_main_Load(x124, x125, x126, x127) :|: 0 <= 0
Start term: __init(arg1, arg2, arg3, arg4)

----------------------------------------

(3) IRSwTTerminationDigraphProof (EQUIVALENT)
Constructed termination digraph!
Nodes:
(1) f1_0_main_Load(arg1, arg2, arg3, arg4) -> f1317_0_mirror_NULL(arg1P, arg2P, arg3P, arg4P) :|: -1 <= arg1P - 1 && 0 <= arg1 - 1
(2) f322_0_createTree_Return(x, x1, x2, x3) -> f1317_0_mirror_NULL(x4, x5, x6, x7) :|: -1 <= x4 - 1 && -1 <= x - 1 && x4 <= x
(3) f1317_0_mirror_NULL(x8, x9, x10, x11) -> f1317_0_mirror_NULL(x12, x13, x14, x15) :|: -1 <= x12 - 1 && 0 <= x8 - 1 && x12 + 1 <= x8
(4) f477_0_createNode_Return(x16, x17, x18, x19) -> f731_0_random_ArrayAccess(x20, x21, x23, x24) :|: 0 <= x20 - 1
(5) f1_0_main_Load(x25, x26, x28, x29) -> f731_0_random_ArrayAccess(x30, x31, x32, x33) :|: 0 <= x30 - 1 && 0 <= x25 - 1 && x30 <= x25
(6) f512_0_createNode_Return(x34, x35, x36, x37) -> f750_0_random_ArrayAccess(x38, x39, x40, x41) :|: 0 <= x38 - 1
(7) f1_0_main_Load(x43, x44, x45, x46) -> f750_0_random_ArrayAccess(x47, x48, x49, x50) :|: 0 <= x47 - 1 && 0 <= x43 - 1 && x47 <= x43
(8) f750_0_random_ArrayAccess(x51, x52, x53, x54) -> f1551_0_createTree_LE(x55, x57, x58, x59) :|: 0 <= x60 - 1 && -1 <= x57 - 1 && x55 - 3 <= x51 && 0 <= x51 - 1 && 3 <= x55 - 1 && x60 + 1 = x59
(9) f731_0_random_ArrayAccess(x61, x62, x63, x64) -> f1551_0_createTree_LE(x65, x66, x67, x68) :|: 0 <= x69 - 1 && -1 <= x66 - 1 && x65 - 1 <= x61 && 0 <= x61 - 1 && 1 <= x65 - 1 && x69 + 1 = x68
(10) f1551_0_createTree_LE(x70, x71, x72, x73) -> f1551_0_createTree_LE(x74, x75, x76, x77) :|: x73 = x77 && x72 = x76 && x71 - 1 = x75 && -1 <= x74 - 1 && 1 <= x70 - 1 && 0 <= x71 - 1 && x74 + 2 <= x70
(11) f1551_0_createTree_LE(x78, x79, x80, x81) -> f1551_0_createTree_LE(x82, x83, x84, x85) :|: 0 <= x79 - 1 && 0 <= x81 - 1 && 0 <= x86 - 1 && x82 - 2 <= x78 && 2 <= x78 - 1 && 3 <= x82 - 1 && x79 - 1 = x83 && x80 = x84
(12) f1551_0_createTree_LE(x87, x88, x89, x90) -> f1551_0_createTree_LE(x91, x92, x93, x94) :|: 0 <= x88 - 1 && 0 <= x90 - 1 && 0 <= x95 - 1 && x91 - 3 <= x87 && 2 <= x87 - 1 && 5 <= x91 - 1 && x88 - 1 = x92 && x89 = x93
(13) f1_0_main_Load(x96, x97, x98, x99) -> f973_0_random_ArrayAccess(x100, x101, x102, x103) :|: 0 = x102 && 0 <= x101 - 1 && 0 <= x100 - 1 && 0 <= x96 - 1 && x101 <= x96 && -1 <= x97 - 1 && x100 <= x96
(14) f1551_0_createTree_LE(x104, x105, x106, x107) -> f973_0_random_ArrayAccess(x108, x109, x110, x111) :|: x107 = x110 && 0 <= x109 - 1 && 2 <= x104 - 1 && x109 + 2 <= x104 && -1 <= x106 - 1 && 0 <= x107 - 1 && 0 <= x105 - 1
(15) f1551_0_createTree_LE(x112, x113, x114, x115) -> f973_0_random_ArrayAccess(x116, x117, x118, x119) :|: 0 <= x117 - 1 && 2 <= x112 - 1 && x117 + 2 <= x112 && -1 <= x114 - 1 && 0 <= x118 - 1 && 0 <= x115 - 1 && 0 <= x113 - 1
(16) __init(x120, x121, x122, x123) -> f1_0_main_Load(x124, x125, x126, x127) :|: 0 <= 0

Arcs:
(1) -> (3)
(2) -> (3)
(3) -> (3)
(4) -> (9)
(5) -> (9)
(6) -> (8)
(7) -> (8)
(8) -> (10), (11), (12), (14), (15)
(9) -> (10), (11), (12), (14), (15)
(10) -> (10), (11), (12), (14), (15)
(11) -> (10), (11), (12), (14), (15)
(12) -> (10), (11), (12), (14), (15)
(16) -> (1), (5), (7), (13)

This digraph is fully evaluated!
----------------------------------------

(4)
Complex Obligation (AND)

----------------------------------------

(5)
Obligation:

Termination digraph:
Nodes:
(1) f1551_0_createTree_LE(x70, x71, x72, x73) -> f1551_0_createTree_LE(x74, x75, x76, x77) :|: x73 = x77 && x72 = x76 && x71 - 1 = x75 && -1 <= x74 - 1 && 1 <= x70 - 1 && 0 <= x71 - 1 && x74 + 2 <= x70
(2) f1551_0_createTree_LE(x78, x79, x80, x81) -> f1551_0_createTree_LE(x82, x83, x84, x85) :|: 0 <= x79 - 1 && 0 <= x81 - 1 && 0 <= x86 - 1 && x82 - 2 <= x78 && 2 <= x78 - 1 && 3 <= x82 - 1 && x79 - 1 = x83 && x80 = x84
(3) f1551_0_createTree_LE(x87, x88, x89, x90) -> f1551_0_createTree_LE(x91, x92, x93, x94) :|: 0 <= x88 - 1 && 0 <= x90 - 1 && 0 <= x95 - 1 && x91 - 3 <= x87 && 2 <= x87 - 1 && 5 <= x91 - 1 && x88 - 1 = x92 && x89 = x93

Arcs:
(1) -> (1), (2), (3)
(2) -> (1), (2), (3)
(3) -> (1), (2), (3)

This digraph is fully evaluated!

----------------------------------------

(6) IntTRSCompressionProof (EQUIVALENT)
Compressed rules.
----------------------------------------

(7)
Obligation:
Rules:
f1551_0_createTree_LE(x78:0, x79:0, x80:0, x81:0) -> f1551_0_createTree_LE(x82:0, x79:0 - 1, x80:0, x85:0) :|: x78:0 > 2 && x82:0 > 3 && x82:0 - 2 <= x78:0 && x86:0 > 0 && x81:0 > 0 && x79:0 > 0
f1551_0_createTree_LE(x87:0, x88:0, x89:0, x90:0) -> f1551_0_createTree_LE(x91:0, x88:0 - 1, x89:0, x94:0) :|: x87:0 > 2 && x91:0 > 5 && x91:0 - 3 <= x87:0 && x95:0 > 0 && x90:0 > 0 && x88:0 > 0
f1551_0_createTree_LE(x70:0, x71:0, x72:0, x73:0) -> f1551_0_createTree_LE(x74:0, x71:0 - 1, x72:0, x73:0) :|: x71:0 > 0 && x74:0 + 2 <= x70:0 && x74:0 > -1 && x70:0 > 1

----------------------------------------

(8) IntTRSUnneededArgumentFilterProof (EQUIVALENT)
Some arguments are removed because they cannot influence termination. We removed arguments according to the following replacements:

   f1551_0_createTree_LE(x1, x2, x3, x4) -> f1551_0_createTree_LE(x1, x2, x4)

----------------------------------------

(9)
Obligation:
Rules:
f1551_0_createTree_LE(x78:0, x79:0, x81:0) -> f1551_0_createTree_LE(x82:0, x79:0 - 1, x85:0) :|: x78:0 > 2 && x82:0 > 3 && x82:0 - 2 <= x78:0 && x86:0 > 0 && x81:0 > 0 && x79:0 > 0
f1551_0_createTree_LE(x87:0, x88:0, x90:0) -> f1551_0_createTree_LE(x91:0, x88:0 - 1, x94:0) :|: x87:0 > 2 && x91:0 > 5 && x91:0 - 3 <= x87:0 && x95:0 > 0 && x90:0 > 0 && x88:0 > 0
f1551_0_createTree_LE(x70:0, x71:0, x73:0) -> f1551_0_createTree_LE(x74:0, x71:0 - 1, x73:0) :|: x71:0 > 0 && x74:0 + 2 <= x70:0 && x74:0 > -1 && x70:0 > 1

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(10) TempFilterProof (SOUND)
Used the following sort dictionary for filtering: 
f1551_0_createTree_LE(INTEGER, INTEGER, VARIABLE)
Replaced non-predefined constructor symbols by 0.
----------------------------------------

(11)
Obligation:
Rules:
f1551_0_createTree_LE(x78:0, x79:0, x81:0) -> f1551_0_createTree_LE(x82:0, c, x85:0) :|: c = x79:0 - 1 && (x78:0 > 2 && x82:0 > 3 && x82:0 - 2 <= x78:0 && x86:0 > 0 && x81:0 > 0 && x79:0 > 0)
f1551_0_createTree_LE(x87:0, x88:0, x90:0) -> f1551_0_createTree_LE(x91:0, c1, x94:0) :|: c1 = x88:0 - 1 && (x87:0 > 2 && x91:0 > 5 && x91:0 - 3 <= x87:0 && x95:0 > 0 && x90:0 > 0 && x88:0 > 0)
f1551_0_createTree_LE(x70:0, x71:0, x73:0) -> f1551_0_createTree_LE(x74:0, c2, x73:0) :|: c2 = x71:0 - 1 && (x71:0 > 0 && x74:0 + 2 <= x70:0 && x74:0 > -1 && x70:0 > 1)

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(12) PolynomialOrderProcessor (EQUIVALENT)
Found the following polynomial interpretation:
[f1551_0_createTree_LE(x, x1, x2)] = -6 + x + 3*x1

The following rules are decreasing:
f1551_0_createTree_LE(x78:0, x79:0, x81:0) -> f1551_0_createTree_LE(x82:0, c, x85:0) :|: c = x79:0 - 1 && (x78:0 > 2 && x82:0 > 3 && x82:0 - 2 <= x78:0 && x86:0 > 0 && x81:0 > 0 && x79:0 > 0)
f1551_0_createTree_LE(x70:0, x71:0, x73:0) -> f1551_0_createTree_LE(x74:0, c2, x73:0) :|: c2 = x71:0 - 1 && (x71:0 > 0 && x74:0 + 2 <= x70:0 && x74:0 > -1 && x70:0 > 1)
The following rules are bounded:
f1551_0_createTree_LE(x78:0, x79:0, x81:0) -> f1551_0_createTree_LE(x82:0, c, x85:0) :|: c = x79:0 - 1 && (x78:0 > 2 && x82:0 > 3 && x82:0 - 2 <= x78:0 && x86:0 > 0 && x81:0 > 0 && x79:0 > 0)
f1551_0_createTree_LE(x87:0, x88:0, x90:0) -> f1551_0_createTree_LE(x91:0, c1, x94:0) :|: c1 = x88:0 - 1 && (x87:0 > 2 && x91:0 > 5 && x91:0 - 3 <= x87:0 && x95:0 > 0 && x90:0 > 0 && x88:0 > 0)

----------------------------------------

(13)
Complex Obligation (AND)

----------------------------------------

(14)
Obligation:
Rules:
f1551_0_createTree_LE(x87:0, x88:0, x90:0) -> f1551_0_createTree_LE(x91:0, c1, x94:0) :|: c1 = x88:0 - 1 && (x87:0 > 2 && x91:0 > 5 && x91:0 - 3 <= x87:0 && x95:0 > 0 && x90:0 > 0 && x88:0 > 0)

----------------------------------------

(15) RankingReductionPairProof (EQUIVALENT)
Interpretation:
[ f1551_0_createTree_LE ] = f1551_0_createTree_LE_2

The following rules are decreasing:
f1551_0_createTree_LE(x87:0, x88:0, x90:0) -> f1551_0_createTree_LE(x91:0, c1, x94:0) :|: c1 = x88:0 - 1 && (x87:0 > 2 && x91:0 > 5 && x91:0 - 3 <= x87:0 && x95:0 > 0 && x90:0 > 0 && x88:0 > 0)

The following rules are bounded:
f1551_0_createTree_LE(x87:0, x88:0, x90:0) -> f1551_0_createTree_LE(x91:0, c1, x94:0) :|: c1 = x88:0 - 1 && (x87:0 > 2 && x91:0 > 5 && x91:0 - 3 <= x87:0 && x95:0 > 0 && x90:0 > 0 && x88:0 > 0)


----------------------------------------

(16)
YES

----------------------------------------

(17)
Obligation:
Rules:
f1551_0_createTree_LE(x70:0, x71:0, x73:0) -> f1551_0_createTree_LE(x74:0, c2, x73:0) :|: c2 = x71:0 - 1 && (x71:0 > 0 && x74:0 + 2 <= x70:0 && x74:0 > -1 && x70:0 > 1)

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(18) RankingReductionPairProof (EQUIVALENT)
Interpretation:
[ f1551_0_createTree_LE ] = 1/2*f1551_0_createTree_LE_1

The following rules are decreasing:
f1551_0_createTree_LE(x70:0, x71:0, x73:0) -> f1551_0_createTree_LE(x74:0, c2, x73:0) :|: c2 = x71:0 - 1 && (x71:0 > 0 && x74:0 + 2 <= x70:0 && x74:0 > -1 && x70:0 > 1)

The following rules are bounded:
f1551_0_createTree_LE(x70:0, x71:0, x73:0) -> f1551_0_createTree_LE(x74:0, c2, x73:0) :|: c2 = x71:0 - 1 && (x71:0 > 0 && x74:0 + 2 <= x70:0 && x74:0 > -1 && x70:0 > 1)


----------------------------------------

(19)
YES

----------------------------------------

(20)
Obligation:

Termination digraph:
Nodes:
(1) f1317_0_mirror_NULL(x8, x9, x10, x11) -> f1317_0_mirror_NULL(x12, x13, x14, x15) :|: -1 <= x12 - 1 && 0 <= x8 - 1 && x12 + 1 <= x8

Arcs:
(1) -> (1)

This digraph is fully evaluated!

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(21) IntTRSCompressionProof (EQUIVALENT)
Compressed rules.
----------------------------------------

(22)
Obligation:
Rules:
f1317_0_mirror_NULL(x8:0, x9:0, x10:0, x11:0) -> f1317_0_mirror_NULL(x12:0, x13:0, x14:0, x15:0) :|: x12:0 > -1 && x8:0 > 0 && x8:0 >= x12:0 + 1

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(23) IntTRSUnneededArgumentFilterProof (EQUIVALENT)
Some arguments are removed because they cannot influence termination. We removed arguments according to the following replacements:

   f1317_0_mirror_NULL(x1, x2, x3, x4) -> f1317_0_mirror_NULL(x1)

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(24)
Obligation:
Rules:
f1317_0_mirror_NULL(x8:0) -> f1317_0_mirror_NULL(x12:0) :|: x12:0 > -1 && x8:0 > 0 && x8:0 >= x12:0 + 1

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(25) FilterProof (EQUIVALENT)
Used the following sort dictionary for filtering: 
f1317_0_mirror_NULL(INTEGER)
Replaced non-predefined constructor symbols by 0.
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(26)
Obligation:
Rules:
f1317_0_mirror_NULL(x8:0) -> f1317_0_mirror_NULL(x12:0) :|: x12:0 > -1 && x8:0 > 0 && x8:0 >= x12:0 + 1

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(27) IntTRSCompressionProof (EQUIVALENT)
Compressed rules.
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(28)
Obligation:
Rules:
f1317_0_mirror_NULL(x8:0:0) -> f1317_0_mirror_NULL(x12:0:0) :|: x12:0:0 > -1 && x8:0:0 > 0 && x8:0:0 >= x12:0:0 + 1

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(29) RankingReductionPairProof (EQUIVALENT)
Interpretation:
[ f1317_0_mirror_NULL ] = f1317_0_mirror_NULL_1

The following rules are decreasing:
f1317_0_mirror_NULL(x8:0:0) -> f1317_0_mirror_NULL(x12:0:0) :|: x12:0:0 > -1 && x8:0:0 > 0 && x8:0:0 >= x12:0:0 + 1

The following rules are bounded:
f1317_0_mirror_NULL(x8:0:0) -> f1317_0_mirror_NULL(x12:0:0) :|: x12:0:0 > -1 && x8:0:0 > 0 && x8:0:0 >= x12:0:0 + 1


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(30)
YES