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# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty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, 150 ms]
(4) AND
    (5) IRSwT
        (6) IntTRSCompressionProof [EQUIVALENT, 30 ms]
        (7) IRSwT
        (8) IntTRSUnneededArgumentFilterProof [EQUIVALENT, 0 ms]
        (9) IRSwT
        (10) TempFilterProof [SOUND, 28 ms]
        (11) IntTRS
        (12) PolynomialOrderProcessor [EQUIVALENT, 0 ms]
        (13) YES
    (14) IRSwT
        (15) IntTRSCompressionProof [EQUIVALENT, 7 ms]
        (16) IRSwT
        (17) FilterProof [EQUIVALENT, 0 ms]
        (18) IntTRS
        (19) IntTRSCompressionProof [EQUIVALENT, 0 ms]
        (20) IntTRS
        (21) PolynomialOrderProcessor [EQUIVALENT, 11 ms]
        (22) YES


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(0)
Obligation:
Rules:
f161_0_createList_Return(arg1, arg2) -> f227_0_isCyclic_NONNULL(arg1P, arg2P) :|: -1 <= arg1P - 1 && -1 <= arg1 - 1 && arg1P <= arg1
f1_0_main_Load(x, x1) -> f227_0_isCyclic_NONNULL(x2, x3) :|: -1 <= x2 - 1 && 0 <= x - 1
f227_0_isCyclic_NONNULL(x4, x5) -> f331_0_isCyclic_NULL(x6, x7) :|: -1 <= x7 - 1 && 0 <= x6 - 1 && 0 <= x4 - 1 && x7 + 1 <= x4 && x6 <= x4
f331_0_isCyclic_NULL(x8, x9) -> f331_0_isCyclic_NULL(x10, x11) :|: -1 <= x11 - 1 && -1 <= x10 - 1 && 2 <= x9 - 1 && 0 <= x8 - 1 && x11 + 3 <= x9 && x10 + 1 <= x8
f1_0_main_Load(x12, x13) -> f201_0_createList_LE(x14, x15) :|: 0 <= x12 - 1 && -1 <= x14 - 1 && -1 <= x13 - 1
f201_0_createList_LE(x16, x17) -> f201_0_createList_LE(x18, x19) :|: x16 - 1 = x18 && 0 <= x16 - 1
__init(x20, x21) -> f1_0_main_Load(x22, x23) :|: 0 <= 0
Start term: __init(arg1, arg2)

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(1) IRSFormatTransformerProof (EQUIVALENT)
Reformatted IRS to match normalized format (transformed away non-linear left-hand sides, !=, / and %).
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(2)
Obligation:
Rules:
f161_0_createList_Return(arg1, arg2) -> f227_0_isCyclic_NONNULL(arg1P, arg2P) :|: -1 <= arg1P - 1 && -1 <= arg1 - 1 && arg1P <= arg1
f1_0_main_Load(x, x1) -> f227_0_isCyclic_NONNULL(x2, x3) :|: -1 <= x2 - 1 && 0 <= x - 1
f227_0_isCyclic_NONNULL(x4, x5) -> f331_0_isCyclic_NULL(x6, x7) :|: -1 <= x7 - 1 && 0 <= x6 - 1 && 0 <= x4 - 1 && x7 + 1 <= x4 && x6 <= x4
f331_0_isCyclic_NULL(x8, x9) -> f331_0_isCyclic_NULL(x10, x11) :|: -1 <= x11 - 1 && -1 <= x10 - 1 && 2 <= x9 - 1 && 0 <= x8 - 1 && x11 + 3 <= x9 && x10 + 1 <= x8
f1_0_main_Load(x12, x13) -> f201_0_createList_LE(x14, x15) :|: 0 <= x12 - 1 && -1 <= x14 - 1 && -1 <= x13 - 1
f201_0_createList_LE(x16, x17) -> f201_0_createList_LE(x18, x19) :|: x16 - 1 = x18 && 0 <= x16 - 1
__init(x20, x21) -> f1_0_main_Load(x22, x23) :|: 0 <= 0
Start term: __init(arg1, arg2)

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(3) IRSwTTerminationDigraphProof (EQUIVALENT)
Constructed termination digraph!
Nodes:
(1) f161_0_createList_Return(arg1, arg2) -> f227_0_isCyclic_NONNULL(arg1P, arg2P) :|: -1 <= arg1P - 1 && -1 <= arg1 - 1 && arg1P <= arg1
(2) f1_0_main_Load(x, x1) -> f227_0_isCyclic_NONNULL(x2, x3) :|: -1 <= x2 - 1 && 0 <= x - 1
(3) f227_0_isCyclic_NONNULL(x4, x5) -> f331_0_isCyclic_NULL(x6, x7) :|: -1 <= x7 - 1 && 0 <= x6 - 1 && 0 <= x4 - 1 && x7 + 1 <= x4 && x6 <= x4
(4) f331_0_isCyclic_NULL(x8, x9) -> f331_0_isCyclic_NULL(x10, x11) :|: -1 <= x11 - 1 && -1 <= x10 - 1 && 2 <= x9 - 1 && 0 <= x8 - 1 && x11 + 3 <= x9 && x10 + 1 <= x8
(5) f1_0_main_Load(x12, x13) -> f201_0_createList_LE(x14, x15) :|: 0 <= x12 - 1 && -1 <= x14 - 1 && -1 <= x13 - 1
(6) f201_0_createList_LE(x16, x17) -> f201_0_createList_LE(x18, x19) :|: x16 - 1 = x18 && 0 <= x16 - 1
(7) __init(x20, x21) -> f1_0_main_Load(x22, x23) :|: 0 <= 0

Arcs:
(1) -> (3)
(2) -> (3)
(3) -> (4)
(4) -> (4)
(5) -> (6)
(6) -> (6)
(7) -> (2), (5)

This digraph is fully evaluated!
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(4)
Complex Obligation (AND)

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(5)
Obligation:

Termination digraph:
Nodes:
(1) f201_0_createList_LE(x16, x17) -> f201_0_createList_LE(x18, x19) :|: x16 - 1 = x18 && 0 <= x16 - 1

Arcs:
(1) -> (1)

This digraph is fully evaluated!

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(6) IntTRSCompressionProof (EQUIVALENT)
Compressed rules.
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(7)
Obligation:
Rules:
f201_0_createList_LE(x16:0, x17:0) -> f201_0_createList_LE(x16:0 - 1, x19:0) :|: x16:0 > 0

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

   f201_0_createList_LE(x1, x2) -> f201_0_createList_LE(x1)

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(9)
Obligation:
Rules:
f201_0_createList_LE(x16:0) -> f201_0_createList_LE(x16:0 - 1) :|: x16:0 > 0

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(10) TempFilterProof (SOUND)
Used the following sort dictionary for filtering: 
f201_0_createList_LE(INTEGER)
Replaced non-predefined constructor symbols by 0.
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(11)
Obligation:
Rules:
f201_0_createList_LE(x16:0) -> f201_0_createList_LE(c) :|: c = x16:0 - 1 && x16:0 > 0

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(12) PolynomialOrderProcessor (EQUIVALENT)
Found the following polynomial interpretation:
[f201_0_createList_LE(x)] = x

The following rules are decreasing:
f201_0_createList_LE(x16:0) -> f201_0_createList_LE(c) :|: c = x16:0 - 1 && x16:0 > 0
The following rules are bounded:
f201_0_createList_LE(x16:0) -> f201_0_createList_LE(c) :|: c = x16:0 - 1 && x16:0 > 0

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(13)
YES

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(14)
Obligation:

Termination digraph:
Nodes:
(1) f331_0_isCyclic_NULL(x8, x9) -> f331_0_isCyclic_NULL(x10, x11) :|: -1 <= x11 - 1 && -1 <= x10 - 1 && 2 <= x9 - 1 && 0 <= x8 - 1 && x11 + 3 <= x9 && x10 + 1 <= x8

Arcs:
(1) -> (1)

This digraph is fully evaluated!

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

(16)
Obligation:
Rules:
f331_0_isCyclic_NULL(x8:0, x9:0) -> f331_0_isCyclic_NULL(x10:0, x11:0) :|: x9:0 >= x11:0 + 3 && x8:0 >= x10:0 + 1 && x8:0 > 0 && x9:0 > 2 && x10:0 > -1 && x11:0 > -1

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(17) FilterProof (EQUIVALENT)
Used the following sort dictionary for filtering: 
f331_0_isCyclic_NULL(INTEGER, INTEGER)
Replaced non-predefined constructor symbols by 0.
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(18)
Obligation:
Rules:
f331_0_isCyclic_NULL(x8:0, x9:0) -> f331_0_isCyclic_NULL(x10:0, x11:0) :|: x9:0 >= x11:0 + 3 && x8:0 >= x10:0 + 1 && x8:0 > 0 && x9:0 > 2 && x10:0 > -1 && x11:0 > -1

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

(20)
Obligation:
Rules:
f331_0_isCyclic_NULL(x8:0:0, x9:0:0) -> f331_0_isCyclic_NULL(x10:0:0, x11:0:0) :|: x10:0:0 > -1 && x11:0:0 > -1 && x9:0:0 > 2 && x8:0:0 > 0 && x8:0:0 >= x10:0:0 + 1 && x9:0:0 >= x11:0:0 + 3

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(21) PolynomialOrderProcessor (EQUIVALENT)
Found the following polynomial interpretation:
[f331_0_isCyclic_NULL(x, x1)] = x

The following rules are decreasing:
f331_0_isCyclic_NULL(x8:0:0, x9:0:0) -> f331_0_isCyclic_NULL(x10:0:0, x11:0:0) :|: x10:0:0 > -1 && x11:0:0 > -1 && x9:0:0 > 2 && x8:0:0 > 0 && x8:0:0 >= x10:0:0 + 1 && x9:0:0 >= x11:0:0 + 3
The following rules are bounded:
f331_0_isCyclic_NULL(x8:0:0, x9:0:0) -> f331_0_isCyclic_NULL(x10:0:0, x11:0:0) :|: x10:0:0 > -1 && x11:0:0 > -1 && x9:0:0 > 2 && x8:0:0 > 0 && x8:0:0 >= x10:0:0 + 1 && x9:0:0 >= x11:0:0 + 3

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(22)
YES