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# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty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, 199 ms]
(4) AND
    (5) IRSwT
        (6) IntTRSCompressionProof [EQUIVALENT, 3 ms]
        (7) IRSwT
        (8) TempFilterProof [SOUND, 4 ms]
        (9) IntTRS
        (10) PolynomialOrderProcessor [EQUIVALENT, 0 ms]
        (11) YES
    (12) IRSwT
        (13) IntTRSCompressionProof [EQUIVALENT, 23 ms]
        (14) IRSwT
        (15) TempFilterProof [SOUND, 12 ms]
        (16) IntTRS
        (17) RankingReductionPairProof [EQUIVALENT, 0 ms]
        (18) YES
    (19) IRSwT
        (20) IntTRSCompressionProof [EQUIVALENT, 5 ms]
        (21) IRSwT
        (22) IntTRSUnneededArgumentFilterProof [EQUIVALENT, 0 ms]
        (23) IRSwT
        (24) TempFilterProof [SOUND, 16 ms]
        (25) IntTRS
        (26) RankingReductionPairProof [EQUIVALENT, 0 ms]
        (27) YES


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(0)
Obligation:
Rules:
f297_0_createIntList_Return(arg1, arg2) -> f508_0_random_ArrayAccess(arg1P, arg2P) :|: -1 <= arg1P - 1 && -1 <= arg1 - 1 && arg1P <= arg1
f1_0_main_Load(x, x1) -> f508_0_random_ArrayAccess(x2, x3) :|: -1 <= x2 - 1 && 0 <= x - 1
f508_0_random_ArrayAccess(x4, x5) -> f698_0_nth_LE(x6, x8) :|: -1 <= x8 - 1 && 0 <= x9 - 1 && x6 <= x4 && 0 <= x4 - 1 && 0 <= x6 - 1
f698_0_nth_LE(x10, x11) -> f746_0_main_LE(x12, x13) :|: x12 + 2 <= x10 && x11 <= 1 && 0 <= x10 - 1
f698_0_nth_LE(x14, x15) -> f698_0_nth_LE(x16, x17) :|: x15 - 1 = x17 && -1 <= x16 - 1 && 0 <= x14 - 1 && 1 <= x15 - 1 && x16 + 1 <= x14
f746_0_main_LE(x18, x19) -> f746_0_main_LE(x20, x21) :|: x18 - 1 = x20 && 0 <= x18 - 1
f1_0_main_Load(x22, x23) -> f658_0_createIntList_LE(x24, x25) :|: 1 = x25 && 0 <= x22 - 1 && -1 <= x24 - 1 && -1 <= x23 - 1
f658_0_createIntList_LE(x26, x27) -> f658_0_createIntList_LE(x28, x29) :|: x27 + 1 = x29 && x26 - 1 = x28 && 0 <= x27 - 1 && 0 <= x26 - 1
__init(x30, x31) -> f1_0_main_Load(x32, x33) :|: 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:
f297_0_createIntList_Return(arg1, arg2) -> f508_0_random_ArrayAccess(arg1P, arg2P) :|: -1 <= arg1P - 1 && -1 <= arg1 - 1 && arg1P <= arg1
f1_0_main_Load(x, x1) -> f508_0_random_ArrayAccess(x2, x3) :|: -1 <= x2 - 1 && 0 <= x - 1
f508_0_random_ArrayAccess(x4, x5) -> f698_0_nth_LE(x6, x8) :|: -1 <= x8 - 1 && 0 <= x9 - 1 && x6 <= x4 && 0 <= x4 - 1 && 0 <= x6 - 1
f698_0_nth_LE(x10, x11) -> f746_0_main_LE(x12, x13) :|: x12 + 2 <= x10 && x11 <= 1 && 0 <= x10 - 1
f698_0_nth_LE(x14, x15) -> f698_0_nth_LE(x16, x17) :|: x15 - 1 = x17 && -1 <= x16 - 1 && 0 <= x14 - 1 && 1 <= x15 - 1 && x16 + 1 <= x14
f746_0_main_LE(x18, x19) -> f746_0_main_LE(x20, x21) :|: x18 - 1 = x20 && 0 <= x18 - 1
f1_0_main_Load(x22, x23) -> f658_0_createIntList_LE(x24, x25) :|: 1 = x25 && 0 <= x22 - 1 && -1 <= x24 - 1 && -1 <= x23 - 1
f658_0_createIntList_LE(x26, x27) -> f658_0_createIntList_LE(x28, x29) :|: x27 + 1 = x29 && x26 - 1 = x28 && 0 <= x27 - 1 && 0 <= x26 - 1
__init(x30, x31) -> f1_0_main_Load(x32, x33) :|: 0 <= 0
Start term: __init(arg1, arg2)

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(3) IRSwTTerminationDigraphProof (EQUIVALENT)
Constructed termination digraph!
Nodes:
(1) f297_0_createIntList_Return(arg1, arg2) -> f508_0_random_ArrayAccess(arg1P, arg2P) :|: -1 <= arg1P - 1 && -1 <= arg1 - 1 && arg1P <= arg1
(2) f1_0_main_Load(x, x1) -> f508_0_random_ArrayAccess(x2, x3) :|: -1 <= x2 - 1 && 0 <= x - 1
(3) f508_0_random_ArrayAccess(x4, x5) -> f698_0_nth_LE(x6, x8) :|: -1 <= x8 - 1 && 0 <= x9 - 1 && x6 <= x4 && 0 <= x4 - 1 && 0 <= x6 - 1
(4) f698_0_nth_LE(x10, x11) -> f746_0_main_LE(x12, x13) :|: x12 + 2 <= x10 && x11 <= 1 && 0 <= x10 - 1
(5) f698_0_nth_LE(x14, x15) -> f698_0_nth_LE(x16, x17) :|: x15 - 1 = x17 && -1 <= x16 - 1 && 0 <= x14 - 1 && 1 <= x15 - 1 && x16 + 1 <= x14
(6) f746_0_main_LE(x18, x19) -> f746_0_main_LE(x20, x21) :|: x18 - 1 = x20 && 0 <= x18 - 1
(7) f1_0_main_Load(x22, x23) -> f658_0_createIntList_LE(x24, x25) :|: 1 = x25 && 0 <= x22 - 1 && -1 <= x24 - 1 && -1 <= x23 - 1
(8) f658_0_createIntList_LE(x26, x27) -> f658_0_createIntList_LE(x28, x29) :|: x27 + 1 = x29 && x26 - 1 = x28 && 0 <= x27 - 1 && 0 <= x26 - 1
(9) __init(x30, x31) -> f1_0_main_Load(x32, x33) :|: 0 <= 0

Arcs:
(1) -> (3)
(2) -> (3)
(3) -> (4), (5)
(4) -> (6)
(5) -> (4), (5)
(6) -> (6)
(7) -> (8)
(8) -> (8)
(9) -> (2), (7)

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

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(5)
Obligation:

Termination digraph:
Nodes:
(1) f658_0_createIntList_LE(x26, x27) -> f658_0_createIntList_LE(x28, x29) :|: x27 + 1 = x29 && x26 - 1 = x28 && 0 <= x27 - 1 && 0 <= x26 - 1

Arcs:
(1) -> (1)

This digraph is fully evaluated!

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

(7)
Obligation:
Rules:
f658_0_createIntList_LE(x26:0, x27:0) -> f658_0_createIntList_LE(x26:0 - 1, x27:0 + 1) :|: x26:0 > 0 && x27:0 > 0

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(8) TempFilterProof (SOUND)
Used the following sort dictionary for filtering: 
f658_0_createIntList_LE(INTEGER, INTEGER)
Replaced non-predefined constructor symbols by 0.
----------------------------------------

(9)
Obligation:
Rules:
f658_0_createIntList_LE(x26:0, x27:0) -> f658_0_createIntList_LE(c, c1) :|: c1 = x27:0 + 1 && c = x26:0 - 1 && (x26:0 > 0 && x27:0 > 0)

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(10) PolynomialOrderProcessor (EQUIVALENT)
Found the following polynomial interpretation:
[f658_0_createIntList_LE(x, x1)] = x

The following rules are decreasing:
f658_0_createIntList_LE(x26:0, x27:0) -> f658_0_createIntList_LE(c, c1) :|: c1 = x27:0 + 1 && c = x26:0 - 1 && (x26:0 > 0 && x27:0 > 0)
The following rules are bounded:
f658_0_createIntList_LE(x26:0, x27:0) -> f658_0_createIntList_LE(c, c1) :|: c1 = x27:0 + 1 && c = x26:0 - 1 && (x26:0 > 0 && x27:0 > 0)

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(11)
YES

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

(12)
Obligation:

Termination digraph:
Nodes:
(1) f698_0_nth_LE(x14, x15) -> f698_0_nth_LE(x16, x17) :|: x15 - 1 = x17 && -1 <= x16 - 1 && 0 <= x14 - 1 && 1 <= x15 - 1 && x16 + 1 <= x14

Arcs:
(1) -> (1)

This digraph is fully evaluated!

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

(14)
Obligation:
Rules:
f698_0_nth_LE(x14:0, x15:0) -> f698_0_nth_LE(x16:0, x15:0 - 1) :|: x15:0 > 1 && x16:0 + 1 <= x14:0 && x16:0 > -1 && x14:0 > 0

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(15) TempFilterProof (SOUND)
Used the following sort dictionary for filtering: 
f698_0_nth_LE(INTEGER, INTEGER)
Replaced non-predefined constructor symbols by 0.
----------------------------------------

(16)
Obligation:
Rules:
f698_0_nth_LE(x14:0, x15:0) -> f698_0_nth_LE(x16:0, c) :|: c = x15:0 - 1 && (x15:0 > 1 && x16:0 + 1 <= x14:0 && x16:0 > -1 && x14:0 > 0)

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(17) RankingReductionPairProof (EQUIVALENT)
Interpretation:
[ f698_0_nth_LE ] = f698_0_nth_LE_1

The following rules are decreasing:
f698_0_nth_LE(x14:0, x15:0) -> f698_0_nth_LE(x16:0, c) :|: c = x15:0 - 1 && (x15:0 > 1 && x16:0 + 1 <= x14:0 && x16:0 > -1 && x14:0 > 0)

The following rules are bounded:
f698_0_nth_LE(x14:0, x15:0) -> f698_0_nth_LE(x16:0, c) :|: c = x15:0 - 1 && (x15:0 > 1 && x16:0 + 1 <= x14:0 && x16:0 > -1 && x14:0 > 0)


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(18)
YES

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

(19)
Obligation:

Termination digraph:
Nodes:
(1) f746_0_main_LE(x18, x19) -> f746_0_main_LE(x20, x21) :|: x18 - 1 = x20 && 0 <= x18 - 1

Arcs:
(1) -> (1)

This digraph is fully evaluated!

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

(21)
Obligation:
Rules:
f746_0_main_LE(x18:0, x19:0) -> f746_0_main_LE(x18:0 - 1, x21:0) :|: x18:0 > 0

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

   f746_0_main_LE(x1, x2) -> f746_0_main_LE(x1)

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(23)
Obligation:
Rules:
f746_0_main_LE(x18:0) -> f746_0_main_LE(x18:0 - 1) :|: x18:0 > 0

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(24) TempFilterProof (SOUND)
Used the following sort dictionary for filtering: 
f746_0_main_LE(INTEGER)
Replaced non-predefined constructor symbols by 0.
----------------------------------------

(25)
Obligation:
Rules:
f746_0_main_LE(x18:0) -> f746_0_main_LE(c) :|: c = x18:0 - 1 && x18:0 > 0

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(26) RankingReductionPairProof (EQUIVALENT)
Interpretation:
[ f746_0_main_LE ] = f746_0_main_LE_1

The following rules are decreasing:
f746_0_main_LE(x18:0) -> f746_0_main_LE(c) :|: c = x18:0 - 1 && x18:0 > 0

The following rules are bounded:
f746_0_main_LE(x18:0) -> f746_0_main_LE(c) :|: c = x18:0 - 1 && x18:0 > 0


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(27)
YES