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# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty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, 279 ms]
(4) AND
    (5) IRSwT
        (6) IntTRSCompressionProof [EQUIVALENT, 0 ms]
        (7) IRSwT
        (8) IntTRSUnneededArgumentFilterProof [EQUIVALENT, 0 ms]
        (9) IRSwT
        (10) TempFilterProof [SOUND, 21 ms]
        (11) IntTRS
        (12) RankingReductionPairProof [EQUIVALENT, 0 ms]
        (13) YES
    (14) IRSwT
        (15) IntTRSCompressionProof [EQUIVALENT, 0 ms]
        (16) IRSwT
        (17) TempFilterProof [SOUND, 42 ms]
        (18) IntTRS
        (19) PolynomialOrderProcessor [EQUIVALENT, 0 ms]
        (20) IntTRS
        (21) RankingReductionPairProof [EQUIVALENT, 0 ms]
        (22) YES


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(0)
Obligation:
Rules:
f169_0_createList_Return(arg1, arg2, arg3) -> f236_0_main_InvokeMethod(arg1P, arg2P, arg3P) :|: -1 <= arg2P - 1 && 0 <= arg1P - 1 && -1 <= arg2 - 1 && 0 <= arg1 - 1 && arg2P <= arg2 && arg1P - 1 <= arg2 && arg1P <= arg1
f1_0_main_Load(x, x1, x2) -> f236_0_main_InvokeMethod(x3, x4, x5) :|: -1 <= x4 - 1 && 0 <= x3 - 1 && 0 <= x - 1 && x3 <= x
f1_0_main_Load(x6, x7, x8) -> f208_0_createList_LE(x9, x10, x11) :|: 0 <= x6 - 1 && -1 <= x9 - 1 && -1 <= x7 - 1
f208_0_createList_LE(x12, x14, x15) -> f208_0_createList_LE(x16, x17, x18) :|: x12 - 1 = x16 && 0 <= x12 - 1
f236_0_main_InvokeMethod(x19, x20, x21) -> f358_0_duplicate_NULL(x22, x23, x24) :|: x22 <= x20 && 0 <= x25 - 1 && x24 <= x20 && 0 <= x19 - 1 && -1 <= x20 - 1 && -1 <= x22 - 1 && -1 <= x24 - 1 && 1 = x23
f358_0_duplicate_NULL(x26, x27, x28) -> f358_0_duplicate_NULL(x29, x30, x31) :|: 1 = x30 && 0 = x27 && -1 <= x31 - 1 && -1 <= x29 - 1 && 0 <= x28 - 1 && 0 <= x26 - 1 && x31 + 1 <= x28 && x31 + 1 <= x26 && x29 + 1 <= x28 && x29 + 1 <= x26
f358_0_duplicate_NULL(x32, x33, x34) -> f358_0_duplicate_NULL(x35, x36, x37) :|: 0 = x36 && 1 = x33 && 0 <= x37 - 1 && 0 <= x35 - 1 && 0 <= x34 - 1 && 0 <= x32 - 1 && x37 <= x34 && x37 <= x32 && x35 <= x34 && x35 <= x32
__init(x38, x39, x40) -> f1_0_main_Load(x41, x42, x43) :|: 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 %).
----------------------------------------

(2)
Obligation:
Rules:
f169_0_createList_Return(arg1, arg2, arg3) -> f236_0_main_InvokeMethod(arg1P, arg2P, arg3P) :|: -1 <= arg2P - 1 && 0 <= arg1P - 1 && -1 <= arg2 - 1 && 0 <= arg1 - 1 && arg2P <= arg2 && arg1P - 1 <= arg2 && arg1P <= arg1
f1_0_main_Load(x, x1, x2) -> f236_0_main_InvokeMethod(x3, x4, x5) :|: -1 <= x4 - 1 && 0 <= x3 - 1 && 0 <= x - 1 && x3 <= x
f1_0_main_Load(x6, x7, x8) -> f208_0_createList_LE(x9, x10, x11) :|: 0 <= x6 - 1 && -1 <= x9 - 1 && -1 <= x7 - 1
f208_0_createList_LE(x12, x14, x15) -> f208_0_createList_LE(x16, x17, x18) :|: x12 - 1 = x16 && 0 <= x12 - 1
f236_0_main_InvokeMethod(x19, x20, x21) -> f358_0_duplicate_NULL(x22, x23, x24) :|: x22 <= x20 && 0 <= x25 - 1 && x24 <= x20 && 0 <= x19 - 1 && -1 <= x20 - 1 && -1 <= x22 - 1 && -1 <= x24 - 1 && 1 = x23
f358_0_duplicate_NULL(x26, x27, x28) -> f358_0_duplicate_NULL(x29, x30, x31) :|: 1 = x30 && 0 = x27 && -1 <= x31 - 1 && -1 <= x29 - 1 && 0 <= x28 - 1 && 0 <= x26 - 1 && x31 + 1 <= x28 && x31 + 1 <= x26 && x29 + 1 <= x28 && x29 + 1 <= x26
f358_0_duplicate_NULL(x32, x33, x34) -> f358_0_duplicate_NULL(x35, x36, x37) :|: 0 = x36 && 1 = x33 && 0 <= x37 - 1 && 0 <= x35 - 1 && 0 <= x34 - 1 && 0 <= x32 - 1 && x37 <= x34 && x37 <= x32 && x35 <= x34 && x35 <= x32
__init(x38, x39, x40) -> f1_0_main_Load(x41, x42, x43) :|: 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) -> f236_0_main_InvokeMethod(arg1P, arg2P, arg3P) :|: -1 <= arg2P - 1 && 0 <= arg1P - 1 && -1 <= arg2 - 1 && 0 <= arg1 - 1 && arg2P <= arg2 && arg1P - 1 <= arg2 && arg1P <= arg1
(2) f1_0_main_Load(x, x1, x2) -> f236_0_main_InvokeMethod(x3, x4, x5) :|: -1 <= x4 - 1 && 0 <= x3 - 1 && 0 <= x - 1 && x3 <= x
(3) f1_0_main_Load(x6, x7, x8) -> f208_0_createList_LE(x9, x10, x11) :|: 0 <= x6 - 1 && -1 <= x9 - 1 && -1 <= x7 - 1
(4) f208_0_createList_LE(x12, x14, x15) -> f208_0_createList_LE(x16, x17, x18) :|: x12 - 1 = x16 && 0 <= x12 - 1
(5) f236_0_main_InvokeMethod(x19, x20, x21) -> f358_0_duplicate_NULL(x22, x23, x24) :|: x22 <= x20 && 0 <= x25 - 1 && x24 <= x20 && 0 <= x19 - 1 && -1 <= x20 - 1 && -1 <= x22 - 1 && -1 <= x24 - 1 && 1 = x23
(6) f358_0_duplicate_NULL(x26, x27, x28) -> f358_0_duplicate_NULL(x29, x30, x31) :|: 1 = x30 && 0 = x27 && -1 <= x31 - 1 && -1 <= x29 - 1 && 0 <= x28 - 1 && 0 <= x26 - 1 && x31 + 1 <= x28 && x31 + 1 <= x26 && x29 + 1 <= x28 && x29 + 1 <= x26
(7) f358_0_duplicate_NULL(x32, x33, x34) -> f358_0_duplicate_NULL(x35, x36, x37) :|: 0 = x36 && 1 = x33 && 0 <= x37 - 1 && 0 <= x35 - 1 && 0 <= x34 - 1 && 0 <= x32 - 1 && x37 <= x34 && x37 <= x32 && x35 <= x34 && x35 <= x32
(8) __init(x38, x39, x40) -> f1_0_main_Load(x41, x42, x43) :|: 0 <= 0

Arcs:
(1) -> (5)
(2) -> (5)
(3) -> (4)
(4) -> (4)
(5) -> (7)
(6) -> (7)
(7) -> (6)
(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) f208_0_createList_LE(x12, x14, x15) -> f208_0_createList_LE(x16, x17, x18) :|: x12 - 1 = x16 && 0 <= x12 - 1

Arcs:
(1) -> (1)

This digraph is fully evaluated!

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

(7)
Obligation:
Rules:
f208_0_createList_LE(x12:0, x14:0, x15:0) -> f208_0_createList_LE(x12:0 - 1, x17:0, x18:0) :|: 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:

   f208_0_createList_LE(x1, x2, x3) -> f208_0_createList_LE(x1)

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(9)
Obligation:
Rules:
f208_0_createList_LE(x12:0) -> f208_0_createList_LE(x12:0 - 1) :|: x12:0 > 0

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(10) TempFilterProof (SOUND)
Used the following sort dictionary for filtering: 
f208_0_createList_LE(INTEGER)
Replaced non-predefined constructor symbols by 0.
----------------------------------------

(11)
Obligation:
Rules:
f208_0_createList_LE(x12:0) -> f208_0_createList_LE(c) :|: c = x12:0 - 1 && x12:0 > 0

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(12) RankingReductionPairProof (EQUIVALENT)
Interpretation:
[ f208_0_createList_LE ] = f208_0_createList_LE_1

The following rules are decreasing:
f208_0_createList_LE(x12:0) -> f208_0_createList_LE(c) :|: c = x12:0 - 1 && x12:0 > 0

The following rules are bounded:
f208_0_createList_LE(x12:0) -> f208_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) f358_0_duplicate_NULL(x32, x33, x34) -> f358_0_duplicate_NULL(x35, x36, x37) :|: 0 = x36 && 1 = x33 && 0 <= x37 - 1 && 0 <= x35 - 1 && 0 <= x34 - 1 && 0 <= x32 - 1 && x37 <= x34 && x37 <= x32 && x35 <= x34 && x35 <= x32
(2) f358_0_duplicate_NULL(x26, x27, x28) -> f358_0_duplicate_NULL(x29, x30, x31) :|: 1 = x30 && 0 = x27 && -1 <= x31 - 1 && -1 <= x29 - 1 && 0 <= x28 - 1 && 0 <= x26 - 1 && x31 + 1 <= x28 && x31 + 1 <= x26 && x29 + 1 <= x28 && x29 + 1 <= x26

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

This digraph is fully evaluated!

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

(16)
Obligation:
Rules:
f358_0_duplicate_NULL(x26:0, cons_0, x28:0) -> f358_0_duplicate_NULL(x29:0, 1, x31:0) :|: x29:0 + 1 <= x28:0 && x29:0 + 1 <= x26:0 && x31:0 + 1 <= x26:0 && x31:0 + 1 <= x28:0 && x26:0 > 0 && x28:0 > 0 && x31:0 > -1 && x29:0 > -1 && cons_0 = 0
f358_0_duplicate_NULL(x32:0, cons_1, x34:0) -> f358_0_duplicate_NULL(x35:0, 0, x37:0) :|: x35:0 <= x34:0 && x35:0 <= x32:0 && x37:0 <= x32:0 && x37:0 <= x34:0 && x32:0 > 0 && x34:0 > 0 && x37:0 > 0 && x35:0 > 0 && cons_1 = 1

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(17) TempFilterProof (SOUND)
Used the following sort dictionary for filtering: 
f358_0_duplicate_NULL(INTEGER, VARIABLE, INTEGER)
Replaced non-predefined constructor symbols by 0.
----------------------------------------

(18)
Obligation:
Rules:
f358_0_duplicate_NULL(x26:0, c, x28:0) -> f358_0_duplicate_NULL(x29:0, c1, x31:0) :|: c1 = 1 && c = 0 && (x29:0 + 1 <= x28:0 && x29:0 + 1 <= x26:0 && x31:0 + 1 <= x26:0 && x31:0 + 1 <= x28:0 && x26:0 > 0 && x28:0 > 0 && x31:0 > -1 && x29:0 > -1 && cons_0 = 0)
f358_0_duplicate_NULL(x32:0, c2, x34:0) -> f358_0_duplicate_NULL(x35:0, c3, x37:0) :|: c3 = 0 && c2 = 1 && (x35:0 <= x34:0 && x35:0 <= x32:0 && x37:0 <= x32:0 && x37:0 <= x34:0 && x32:0 > 0 && x34:0 > 0 && x37:0 > 0 && x35:0 > 0 && cons_1 = 1)

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(19) PolynomialOrderProcessor (EQUIVALENT)
Found the following polynomial interpretation:
[f358_0_duplicate_NULL(x, x1, x2)] = -2 + x1 + x2

The following rules are decreasing:
f358_0_duplicate_NULL(x32:0, c2, x34:0) -> f358_0_duplicate_NULL(x35:0, c3, x37:0) :|: c3 = 0 && c2 = 1 && (x35:0 <= x34:0 && x35:0 <= x32:0 && x37:0 <= x32:0 && x37:0 <= x34:0 && x32:0 > 0 && x34:0 > 0 && x37:0 > 0 && x35:0 > 0 && cons_1 = 1)
The following rules are bounded:
f358_0_duplicate_NULL(x32:0, c2, x34:0) -> f358_0_duplicate_NULL(x35:0, c3, x37:0) :|: c3 = 0 && c2 = 1 && (x35:0 <= x34:0 && x35:0 <= x32:0 && x37:0 <= x32:0 && x37:0 <= x34:0 && x32:0 > 0 && x34:0 > 0 && x37:0 > 0 && x35:0 > 0 && cons_1 = 1)

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(20)
Obligation:
Rules:
f358_0_duplicate_NULL(x26:0, c, x28:0) -> f358_0_duplicate_NULL(x29:0, c1, x31:0) :|: c1 = 1 && c = 0 && (x29:0 + 1 <= x28:0 && x29:0 + 1 <= x26:0 && x31:0 + 1 <= x26:0 && x31:0 + 1 <= x28:0 && x26:0 > 0 && x28:0 > 0 && x31:0 > -1 && x29:0 > -1 && cons_0 = 0)

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(21) RankingReductionPairProof (EQUIVALENT)
Interpretation:
[ f358_0_duplicate_NULL ] = f358_0_duplicate_NULL_3

The following rules are decreasing:
f358_0_duplicate_NULL(x26:0, c, x28:0) -> f358_0_duplicate_NULL(x29:0, c1, x31:0) :|: c1 = 1 && c = 0 && (x29:0 + 1 <= x28:0 && x29:0 + 1 <= x26:0 && x31:0 + 1 <= x26:0 && x31:0 + 1 <= x28:0 && x26:0 > 0 && x28:0 > 0 && x31:0 > -1 && x29:0 > -1 && cons_0 = 0)

The following rules are bounded:
f358_0_duplicate_NULL(x26:0, c, x28:0) -> f358_0_duplicate_NULL(x29:0, c1, x31:0) :|: c1 = 1 && c = 0 && (x29:0 + 1 <= x28:0 && x29:0 + 1 <= x26:0 && x31:0 + 1 <= x26:0 && x31:0 + 1 <= x28:0 && x26:0 > 0 && x28:0 > 0 && x31:0 > -1 && x29:0 > -1 && cons_0 = 0)


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(22)
YES