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# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty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, 308 ms]
(4) AND
    (5) IRSwT
        (6) IntTRSCompressionProof [EQUIVALENT, 13 ms]
        (7) IRSwT
        (8) IntTRSUnneededArgumentFilterProof [EQUIVALENT, 0 ms]
        (9) IRSwT
        (10) TempFilterProof [SOUND, 35 ms]
        (11) IntTRS
        (12) RankingReductionPairProof [EQUIVALENT, 0 ms]
        (13) YES
    (14) IRSwT
        (15) IntTRSCompressionProof [EQUIVALENT, 7 ms]
        (16) IRSwT
        (17) TempFilterProof [SOUND, 34 ms]
        (18) IntTRS
        (19) PolynomialOrderProcessor [EQUIVALENT, 10 ms]
        (20) IntTRS
        (21) RankingReductionPairProof [EQUIVALENT, 0 ms]
        (22) YES


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(0)
Obligation:
Rules:
f169_0_createList_Return(arg1, arg2, arg3) -> f229_0_random_ArrayAccess(arg1P, arg2P, arg3P) :|: -1 <= arg1P - 1 && -1 <= arg2 - 1 && 0 <= arg1 - 1 && arg1P <= arg2
f1_0_main_Load(x, x1, x2) -> f229_0_random_ArrayAccess(x3, x4, x5) :|: x1 = x4 && -1 <= x3 - 1 && 0 <= x - 1
f1_0_main_Load(x6, x7, x8) -> f194_0_createList_LE(x9, x10, x11) :|: 0 <= x6 - 1 && -1 <= x9 - 1 && -1 <= x7 - 1
f194_0_createList_LE(x12, x13, x14) -> f194_0_createList_LE(x15, x16, x17) :|: x12 - 1 = x15 && 0 <= x12 - 1
f229_0_random_ArrayAccess(x18, x19, x20) -> f298_0_appE_NONNULL(x21, x22, x23) :|: -1 <= x23 - 1 && 0 <= x21 - 1 && 0 <= x18 - 1 && x23 + 1 <= x18 && x21 <= x18 && 1 <= x19 - 1 && -1 <= x22 - 1
f298_0_appE_NONNULL(x24, x25, x26) -> f298_0_appE_NONNULL(x27, x28, x29) :|: x25 = x28 && -1 <= x29 - 1 && 0 <= x27 - 1 && 0 <= x26 - 1 && 2 <= x24 - 1 && x29 + 1 <= x26 && x29 + 3 <= x24 && x27 <= x26 && x27 + 2 <= x24
f298_0_appE_NONNULL(x30, x31, x32) -> f298_0_appE_NONNULL(x33, x34, x35) :|: x31 - 1 = x34 && -1 <= x35 - 1 && 1 <= x33 - 1 && -1 <= x32 - 1 && 1 <= x30 - 1 && x35 <= x32 && x35 + 2 <= x30 && x33 - 2 <= x32 && 0 <= x31 - 1 && x33 <= x30
__init(x36, x37, x38) -> f1_0_main_Load(x39, x40, x41) :|: 0 <= 0
Start term: __init(arg1, arg2, arg3)

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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:
f169_0_createList_Return(arg1, arg2, arg3) -> f229_0_random_ArrayAccess(arg1P, arg2P, arg3P) :|: -1 <= arg1P - 1 && -1 <= arg2 - 1 && 0 <= arg1 - 1 && arg1P <= arg2
f1_0_main_Load(x, x1, x2) -> f229_0_random_ArrayAccess(x3, x4, x5) :|: x1 = x4 && -1 <= x3 - 1 && 0 <= x - 1
f1_0_main_Load(x6, x7, x8) -> f194_0_createList_LE(x9, x10, x11) :|: 0 <= x6 - 1 && -1 <= x9 - 1 && -1 <= x7 - 1
f194_0_createList_LE(x12, x13, x14) -> f194_0_createList_LE(x15, x16, x17) :|: x12 - 1 = x15 && 0 <= x12 - 1
f229_0_random_ArrayAccess(x18, x19, x20) -> f298_0_appE_NONNULL(x21, x22, x23) :|: -1 <= x23 - 1 && 0 <= x21 - 1 && 0 <= x18 - 1 && x23 + 1 <= x18 && x21 <= x18 && 1 <= x19 - 1 && -1 <= x22 - 1
f298_0_appE_NONNULL(x24, x25, x26) -> f298_0_appE_NONNULL(x27, x28, x29) :|: x25 = x28 && -1 <= x29 - 1 && 0 <= x27 - 1 && 0 <= x26 - 1 && 2 <= x24 - 1 && x29 + 1 <= x26 && x29 + 3 <= x24 && x27 <= x26 && x27 + 2 <= x24
f298_0_appE_NONNULL(x30, x31, x32) -> f298_0_appE_NONNULL(x33, x34, x35) :|: x31 - 1 = x34 && -1 <= x35 - 1 && 1 <= x33 - 1 && -1 <= x32 - 1 && 1 <= x30 - 1 && x35 <= x32 && x35 + 2 <= x30 && x33 - 2 <= x32 && 0 <= x31 - 1 && x33 <= x30
__init(x36, x37, x38) -> f1_0_main_Load(x39, x40, x41) :|: 0 <= 0
Start term: __init(arg1, arg2, arg3)

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(3) IRSwTTerminationDigraphProof (EQUIVALENT)
Constructed termination digraph!
Nodes:
(1) f169_0_createList_Return(arg1, arg2, arg3) -> f229_0_random_ArrayAccess(arg1P, arg2P, arg3P) :|: -1 <= arg1P - 1 && -1 <= arg2 - 1 && 0 <= arg1 - 1 && arg1P <= arg2
(2) f1_0_main_Load(x, x1, x2) -> f229_0_random_ArrayAccess(x3, x4, x5) :|: x1 = x4 && -1 <= x3 - 1 && 0 <= x - 1
(3) f1_0_main_Load(x6, x7, x8) -> f194_0_createList_LE(x9, x10, x11) :|: 0 <= x6 - 1 && -1 <= x9 - 1 && -1 <= x7 - 1
(4) f194_0_createList_LE(x12, x13, x14) -> f194_0_createList_LE(x15, x16, x17) :|: x12 - 1 = x15 && 0 <= x12 - 1
(5) f229_0_random_ArrayAccess(x18, x19, x20) -> f298_0_appE_NONNULL(x21, x22, x23) :|: -1 <= x23 - 1 && 0 <= x21 - 1 && 0 <= x18 - 1 && x23 + 1 <= x18 && x21 <= x18 && 1 <= x19 - 1 && -1 <= x22 - 1
(6) f298_0_appE_NONNULL(x24, x25, x26) -> f298_0_appE_NONNULL(x27, x28, x29) :|: x25 = x28 && -1 <= x29 - 1 && 0 <= x27 - 1 && 0 <= x26 - 1 && 2 <= x24 - 1 && x29 + 1 <= x26 && x29 + 3 <= x24 && x27 <= x26 && x27 + 2 <= x24
(7) f298_0_appE_NONNULL(x30, x31, x32) -> f298_0_appE_NONNULL(x33, x34, x35) :|: x31 - 1 = x34 && -1 <= x35 - 1 && 1 <= x33 - 1 && -1 <= x32 - 1 && 1 <= x30 - 1 && x35 <= x32 && x35 + 2 <= x30 && x33 - 2 <= x32 && 0 <= x31 - 1 && x33 <= x30
(8) __init(x36, x37, x38) -> f1_0_main_Load(x39, x40, x41) :|: 0 <= 0

Arcs:
(1) -> (5)
(2) -> (5)
(3) -> (4)
(4) -> (4)
(5) -> (6), (7)
(6) -> (6), (7)
(7) -> (6), (7)
(8) -> (2), (3)

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

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(5)
Obligation:

Termination digraph:
Nodes:
(1) f194_0_createList_LE(x12, x13, x14) -> f194_0_createList_LE(x15, x16, x17) :|: x12 - 1 = x15 && 0 <= x12 - 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:
f194_0_createList_LE(x12:0, x13:0, x14:0) -> f194_0_createList_LE(x12:0 - 1, x16:0, x17:0) :|: x12:0 > 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:

   f194_0_createList_LE(x1, x2, x3) -> f194_0_createList_LE(x1)

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(9)
Obligation:
Rules:
f194_0_createList_LE(x12:0) -> f194_0_createList_LE(x12:0 - 1) :|: x12:0 > 0

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(10) TempFilterProof (SOUND)
Used the following sort dictionary for filtering: 
f194_0_createList_LE(INTEGER)
Replaced non-predefined constructor symbols by 0.
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(11)
Obligation:
Rules:
f194_0_createList_LE(x12:0) -> f194_0_createList_LE(c) :|: c = x12:0 - 1 && x12:0 > 0

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(12) RankingReductionPairProof (EQUIVALENT)
Interpretation:
[ f194_0_createList_LE ] = f194_0_createList_LE_1

The following rules are decreasing:
f194_0_createList_LE(x12:0) -> f194_0_createList_LE(c) :|: c = x12:0 - 1 && x12:0 > 0

The following rules are bounded:
f194_0_createList_LE(x12:0) -> f194_0_createList_LE(c) :|: c = x12:0 - 1 && x12:0 > 0


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(13)
YES

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(14)
Obligation:

Termination digraph:
Nodes:
(1) f298_0_appE_NONNULL(x24, x25, x26) -> f298_0_appE_NONNULL(x27, x28, x29) :|: x25 = x28 && -1 <= x29 - 1 && 0 <= x27 - 1 && 0 <= x26 - 1 && 2 <= x24 - 1 && x29 + 1 <= x26 && x29 + 3 <= x24 && x27 <= x26 && x27 + 2 <= x24
(2) f298_0_appE_NONNULL(x30, x31, x32) -> f298_0_appE_NONNULL(x33, x34, x35) :|: x31 - 1 = x34 && -1 <= x35 - 1 && 1 <= x33 - 1 && -1 <= x32 - 1 && 1 <= x30 - 1 && x35 <= x32 && x35 + 2 <= x30 && x33 - 2 <= x32 && 0 <= x31 - 1 && x33 <= x30

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

This digraph is fully evaluated!

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

(16)
Obligation:
Rules:
f298_0_appE_NONNULL(x24:0, x25:0, x26:0) -> f298_0_appE_NONNULL(x27:0, x25:0, x29:0) :|: x27:0 <= x26:0 && x27:0 + 2 <= x24:0 && x29:0 + 3 <= x24:0 && x29:0 + 1 <= x26:0 && x24:0 > 2 && x26:0 > 0 && x29:0 > -1 && x27:0 > 0
f298_0_appE_NONNULL(x30:0, x31:0, x32:0) -> f298_0_appE_NONNULL(x33:0, x31:0 - 1, x35:0) :|: x31:0 > 0 && x33:0 <= x30:0 && x33:0 - 2 <= x32:0 && x35:0 + 2 <= x30:0 && x35:0 <= x32:0 && x30:0 > 1 && x32:0 > -1 && x35:0 > -1 && x33:0 > 1

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(17) TempFilterProof (SOUND)
Used the following sort dictionary for filtering: 
f298_0_appE_NONNULL(INTEGER, VARIABLE, INTEGER)
Replaced non-predefined constructor symbols by 0.
----------------------------------------

(18)
Obligation:
Rules:
f298_0_appE_NONNULL(x24:0, x25:0, x26:0) -> f298_0_appE_NONNULL(x27:0, x25:0, x29:0) :|: x27:0 <= x26:0 && x27:0 + 2 <= x24:0 && x29:0 + 3 <= x24:0 && x29:0 + 1 <= x26:0 && x24:0 > 2 && x26:0 > 0 && x29:0 > -1 && x27:0 > 0
f298_0_appE_NONNULL(x30:0, x31:0, x32:0) -> f298_0_appE_NONNULL(x33:0, c, x35:0) :|: c = x31:0 - 1 && (x31:0 > 0 && x33:0 <= x30:0 && x33:0 - 2 <= x32:0 && x35:0 + 2 <= x30:0 && x35:0 <= x32:0 && x30:0 > 1 && x32:0 > -1 && x35:0 > -1 && x33:0 > 1)

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(19) PolynomialOrderProcessor (EQUIVALENT)
Found the following polynomial interpretation:
[f298_0_appE_NONNULL(x, x1, x2)] = -1 + x2

The following rules are decreasing:
f298_0_appE_NONNULL(x24:0, x25:0, x26:0) -> f298_0_appE_NONNULL(x27:0, x25:0, x29:0) :|: x27:0 <= x26:0 && x27:0 + 2 <= x24:0 && x29:0 + 3 <= x24:0 && x29:0 + 1 <= x26:0 && x24:0 > 2 && x26:0 > 0 && x29:0 > -1 && x27:0 > 0
The following rules are bounded:
f298_0_appE_NONNULL(x24:0, x25:0, x26:0) -> f298_0_appE_NONNULL(x27:0, x25:0, x29:0) :|: x27:0 <= x26:0 && x27:0 + 2 <= x24:0 && x29:0 + 3 <= x24:0 && x29:0 + 1 <= x26:0 && x24:0 > 2 && x26:0 > 0 && x29:0 > -1 && x27:0 > 0

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

(20)
Obligation:
Rules:
f298_0_appE_NONNULL(x30:0, x31:0, x32:0) -> f298_0_appE_NONNULL(x33:0, c, x35:0) :|: c = x31:0 - 1 && (x31:0 > 0 && x33:0 <= x30:0 && x33:0 - 2 <= x32:0 && x35:0 + 2 <= x30:0 && x35:0 <= x32:0 && x30:0 > 1 && x32:0 > -1 && x35:0 > -1 && x33:0 > 1)

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(21) RankingReductionPairProof (EQUIVALENT)
Interpretation:
[ f298_0_appE_NONNULL ] = f298_0_appE_NONNULL_2

The following rules are decreasing:
f298_0_appE_NONNULL(x30:0, x31:0, x32:0) -> f298_0_appE_NONNULL(x33:0, c, x35:0) :|: c = x31:0 - 1 && (x31:0 > 0 && x33:0 <= x30:0 && x33:0 - 2 <= x32:0 && x35:0 + 2 <= x30:0 && x35:0 <= x32:0 && x30:0 > 1 && x32:0 > -1 && x35:0 > -1 && x33:0 > 1)

The following rules are bounded:
f298_0_appE_NONNULL(x30:0, x31:0, x32:0) -> f298_0_appE_NONNULL(x33:0, c, x35:0) :|: c = x31:0 - 1 && (x31:0 > 0 && x33:0 <= x30:0 && x33:0 - 2 <= x32:0 && x35:0 + 2 <= x30:0 && x35:0 <= x32:0 && x30:0 > 1 && x32:0 > -1 && x35:0 > -1 && x33:0 > 1)


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(22)
YES