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# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty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, 144 ms]
(4) AND
    (5) IRSwT
        (6) IntTRSCompressionProof [EQUIVALENT, 6 ms]
        (7) IRSwT
        (8) TempFilterProof [SOUND, 36 ms]
        (9) IntTRS
        (10) RankingReductionPairProof [EQUIVALENT, 0 ms]
        (11) YES
    (12) IRSwT
        (13) IntTRSCompressionProof [EQUIVALENT, 30 ms]
        (14) IRSwT
        (15) IntTRSUnneededArgumentFilterProof [EQUIVALENT, 0 ms]
        (16) IRSwT
        (17) FilterProof [EQUIVALENT, 0 ms]
        (18) IntTRS
        (19) IntTRSCompressionProof [EQUIVALENT, 0 ms]
        (20) IntTRS
        (21) RankingReductionPairProof [EQUIVALENT, 5 ms]
        (22) YES


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(0)
Obligation:
Rules:
f1_0_main_Load(arg1, arg2) -> f152_0_createList_LE(arg1P, arg2P) :|: 0 <= arg2 - 1 && -1 <= x3 - 1 && arg1P + 1 <= arg1 && 0 <= arg1 - 1 && -1 <= arg1P - 1 && x3 - 1 = arg2P
f152_0_createList_LE(x, x1) -> f196_0_reverse_NULL(x2, x4) :|: -1 <= x2 - 1 && -1 <= x - 1 && x1 <= 0 && x2 <= x
f152_0_createList_LE(x5, x6) -> f152_0_createList_LE(x7, x8) :|: x6 - 1 = x8 && 0 <= x7 - 1 && -1 <= x5 - 1 && 0 <= x6 - 1 && x7 - 2 <= x5
f196_0_reverse_NULL(x9, x10) -> f196_0_reverse_NULL(x11, x12) :|: -1 <= x11 - 1 && 0 <= x9 - 1 && x11 + 1 <= x9
__init(x13, x14) -> f1_0_main_Load(x15, x16) :|: 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:
f1_0_main_Load(arg1, arg2) -> f152_0_createList_LE(arg1P, arg2P) :|: 0 <= arg2 - 1 && -1 <= x3 - 1 && arg1P + 1 <= arg1 && 0 <= arg1 - 1 && -1 <= arg1P - 1 && x3 - 1 = arg2P
f152_0_createList_LE(x, x1) -> f196_0_reverse_NULL(x2, x4) :|: -1 <= x2 - 1 && -1 <= x - 1 && x1 <= 0 && x2 <= x
f152_0_createList_LE(x5, x6) -> f152_0_createList_LE(x7, x8) :|: x6 - 1 = x8 && 0 <= x7 - 1 && -1 <= x5 - 1 && 0 <= x6 - 1 && x7 - 2 <= x5
f196_0_reverse_NULL(x9, x10) -> f196_0_reverse_NULL(x11, x12) :|: -1 <= x11 - 1 && 0 <= x9 - 1 && x11 + 1 <= x9
__init(x13, x14) -> f1_0_main_Load(x15, x16) :|: 0 <= 0
Start term: __init(arg1, arg2)

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(3) IRSwTTerminationDigraphProof (EQUIVALENT)
Constructed termination digraph!
Nodes:
(1) f1_0_main_Load(arg1, arg2) -> f152_0_createList_LE(arg1P, arg2P) :|: 0 <= arg2 - 1 && -1 <= x3 - 1 && arg1P + 1 <= arg1 && 0 <= arg1 - 1 && -1 <= arg1P - 1 && x3 - 1 = arg2P
(2) f152_0_createList_LE(x, x1) -> f196_0_reverse_NULL(x2, x4) :|: -1 <= x2 - 1 && -1 <= x - 1 && x1 <= 0 && x2 <= x
(3) f152_0_createList_LE(x5, x6) -> f152_0_createList_LE(x7, x8) :|: x6 - 1 = x8 && 0 <= x7 - 1 && -1 <= x5 - 1 && 0 <= x6 - 1 && x7 - 2 <= x5
(4) f196_0_reverse_NULL(x9, x10) -> f196_0_reverse_NULL(x11, x12) :|: -1 <= x11 - 1 && 0 <= x9 - 1 && x11 + 1 <= x9
(5) __init(x13, x14) -> f1_0_main_Load(x15, x16) :|: 0 <= 0

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

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

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(5)
Obligation:

Termination digraph:
Nodes:
(1) f152_0_createList_LE(x5, x6) -> f152_0_createList_LE(x7, x8) :|: x6 - 1 = x8 && 0 <= x7 - 1 && -1 <= x5 - 1 && 0 <= x6 - 1 && x7 - 2 <= x5

Arcs:
(1) -> (1)

This digraph is fully evaluated!

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(6) IntTRSCompressionProof (EQUIVALENT)
Compressed rules.
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(7)
Obligation:
Rules:
f152_0_createList_LE(x5:0, x6:0) -> f152_0_createList_LE(x7:0, x6:0 - 1) :|: x6:0 > 0 && x7:0 - 2 <= x5:0 && x7:0 > 0 && x5:0 > -1

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(8) TempFilterProof (SOUND)
Used the following sort dictionary for filtering: 
f152_0_createList_LE(INTEGER, INTEGER)
Replaced non-predefined constructor symbols by 0.
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(9)
Obligation:
Rules:
f152_0_createList_LE(x5:0, x6:0) -> f152_0_createList_LE(x7:0, c) :|: c = x6:0 - 1 && (x6:0 > 0 && x7:0 - 2 <= x5:0 && x7:0 > 0 && x5:0 > -1)

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(10) RankingReductionPairProof (EQUIVALENT)
Interpretation:
[ f152_0_createList_LE ] = f152_0_createList_LE_2

The following rules are decreasing:
f152_0_createList_LE(x5:0, x6:0) -> f152_0_createList_LE(x7:0, c) :|: c = x6:0 - 1 && (x6:0 > 0 && x7:0 - 2 <= x5:0 && x7:0 > 0 && x5:0 > -1)

The following rules are bounded:
f152_0_createList_LE(x5:0, x6:0) -> f152_0_createList_LE(x7:0, c) :|: c = x6:0 - 1 && (x6:0 > 0 && x7:0 - 2 <= x5:0 && x7:0 > 0 && x5:0 > -1)


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(11)
YES

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(12)
Obligation:

Termination digraph:
Nodes:
(1) f196_0_reverse_NULL(x9, x10) -> f196_0_reverse_NULL(x11, x12) :|: -1 <= x11 - 1 && 0 <= x9 - 1 && x11 + 1 <= x9

Arcs:
(1) -> (1)

This digraph is fully evaluated!

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(13) IntTRSCompressionProof (EQUIVALENT)
Compressed rules.
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(14)
Obligation:
Rules:
f196_0_reverse_NULL(x9:0, x10:0) -> f196_0_reverse_NULL(x11:0, x12:0) :|: x11:0 > -1 && x9:0 > 0 && x9:0 >= x11:0 + 1

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

   f196_0_reverse_NULL(x1, x2) -> f196_0_reverse_NULL(x1)

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(16)
Obligation:
Rules:
f196_0_reverse_NULL(x9:0) -> f196_0_reverse_NULL(x11:0) :|: x11:0 > -1 && x9:0 > 0 && x9:0 >= x11:0 + 1

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(17) FilterProof (EQUIVALENT)
Used the following sort dictionary for filtering: 
f196_0_reverse_NULL(INTEGER)
Replaced non-predefined constructor symbols by 0.
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(18)
Obligation:
Rules:
f196_0_reverse_NULL(x9:0) -> f196_0_reverse_NULL(x11:0) :|: x11:0 > -1 && x9:0 > 0 && x9:0 >= x11:0 + 1

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(19) IntTRSCompressionProof (EQUIVALENT)
Compressed rules.
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(20)
Obligation:
Rules:
f196_0_reverse_NULL(x9:0:0) -> f196_0_reverse_NULL(x11:0:0) :|: x11:0:0 > -1 && x9:0:0 > 0 && x9:0:0 >= x11:0:0 + 1

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(21) RankingReductionPairProof (EQUIVALENT)
Interpretation:
[ f196_0_reverse_NULL ] = f196_0_reverse_NULL_1

The following rules are decreasing:
f196_0_reverse_NULL(x9:0:0) -> f196_0_reverse_NULL(x11:0:0) :|: x11:0:0 > -1 && x9:0:0 > 0 && x9:0:0 >= x11:0:0 + 1

The following rules are bounded:
f196_0_reverse_NULL(x9:0:0) -> f196_0_reverse_NULL(x11:0:0) :|: x11:0:0 > -1 && x9:0:0 > 0 && x9:0:0 >= x11:0:0 + 1


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(22)
YES