NO
proof of prog.inttrs
# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty


Termination of the given IRSwT could be disproven:

(0) IRSwT
(1) IRSFormatTransformerProof [EQUIVALENT, 0 ms]
(2) IRSwT
(3) IRSwTTerminationDigraphProof [EQUIVALENT, 196 ms]
(4) AND
    (5) IRSwT
        (6) IntTRSCompressionProof [EQUIVALENT, 0 ms]
        (7) IRSwT
        (8) TempFilterProof [SOUND, 101 ms]
        (9) IntTRS
        (10) PolynomialOrderProcessor [EQUIVALENT, 12 ms]
        (11) YES
    (12) IRSwT
        (13) IntTRSCompressionProof [EQUIVALENT, 0 ms]
        (14) IRSwT
        (15) IntTRSUnneededArgumentFilterProof [EQUIVALENT, 0 ms]
        (16) IRSwT
        (17) FilterProof [EQUIVALENT, 0 ms]
        (18) IntTRS
        (19) IntTRSPeriodicNontermProof [COMPLETE, 5 ms]
        (20) NO


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(0)
Obligation:
Rules:
f1_0_main_Load(arg1, arg2, arg3) -> f181_0_main_LE(arg1P, arg2P, arg3P) :|: arg2 = arg3P && arg2 - 1 = arg1P && 1 <= arg2P - 1 && 0 <= arg1 - 1 && -1 <= arg2 - 1 && arg2P - 1 <= arg1
f181_0_main_LE(x, x1, x2) -> f181_0_main_LE(x3, x4, x5) :|: x = x5 && x - 1 = x3 && 2 <= x4 - 1 && 0 <= x1 - 1 && 0 <= x2 - 1 && x4 - 2 <= x1
f181_0_main_LE(x6, x7, x8) -> f168_0_visit_NULL(x9, x10, x11) :|: 0 <= x9 - 1 && x8 <= 0 && 0 <= x7 - 1
f181_0_main_LE(x12, x13, x14) -> f168_0_visit_NULL(x15, x16, x17) :|: 2 <= x15 - 1 && 1 <= x13 - 1 && x14 <= 0 && x15 - 1 <= x13
f168_0_visit_NULL(x18, x19, x20) -> f168_0_visit_NULL(x21, x22, x23) :|: -1 <= x21 - 1 && 0 <= x18 - 1
__init(x24, x25, x26) -> f1_0_main_Load(x27, x28, x29) :|: 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:
f1_0_main_Load(arg1, arg2, arg3) -> f181_0_main_LE(arg1P, arg2P, arg3P) :|: arg2 = arg3P && arg2 - 1 = arg1P && 1 <= arg2P - 1 && 0 <= arg1 - 1 && -1 <= arg2 - 1 && arg2P - 1 <= arg1
f181_0_main_LE(x, x1, x2) -> f181_0_main_LE(x3, x4, x5) :|: x = x5 && x - 1 = x3 && 2 <= x4 - 1 && 0 <= x1 - 1 && 0 <= x2 - 1 && x4 - 2 <= x1
f181_0_main_LE(x6, x7, x8) -> f168_0_visit_NULL(x9, x10, x11) :|: 0 <= x9 - 1 && x8 <= 0 && 0 <= x7 - 1
f181_0_main_LE(x12, x13, x14) -> f168_0_visit_NULL(x15, x16, x17) :|: 2 <= x15 - 1 && 1 <= x13 - 1 && x14 <= 0 && x15 - 1 <= x13
f168_0_visit_NULL(x18, x19, x20) -> f168_0_visit_NULL(x21, x22, x23) :|: -1 <= x21 - 1 && 0 <= x18 - 1
__init(x24, x25, x26) -> f1_0_main_Load(x27, x28, x29) :|: 0 <= 0
Start term: __init(arg1, arg2, arg3)

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(3) IRSwTTerminationDigraphProof (EQUIVALENT)
Constructed termination digraph!
Nodes:
(1) f1_0_main_Load(arg1, arg2, arg3) -> f181_0_main_LE(arg1P, arg2P, arg3P) :|: arg2 = arg3P && arg2 - 1 = arg1P && 1 <= arg2P - 1 && 0 <= arg1 - 1 && -1 <= arg2 - 1 && arg2P - 1 <= arg1
(2) f181_0_main_LE(x, x1, x2) -> f181_0_main_LE(x3, x4, x5) :|: x = x5 && x - 1 = x3 && 2 <= x4 - 1 && 0 <= x1 - 1 && 0 <= x2 - 1 && x4 - 2 <= x1
(3) f181_0_main_LE(x6, x7, x8) -> f168_0_visit_NULL(x9, x10, x11) :|: 0 <= x9 - 1 && x8 <= 0 && 0 <= x7 - 1
(4) f181_0_main_LE(x12, x13, x14) -> f168_0_visit_NULL(x15, x16, x17) :|: 2 <= x15 - 1 && 1 <= x13 - 1 && x14 <= 0 && x15 - 1 <= x13
(5) f168_0_visit_NULL(x18, x19, x20) -> f168_0_visit_NULL(x21, x22, x23) :|: -1 <= x21 - 1 && 0 <= x18 - 1
(6) __init(x24, x25, x26) -> f1_0_main_Load(x27, x28, x29) :|: 0 <= 0

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

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

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(5)
Obligation:

Termination digraph:
Nodes:
(1) f181_0_main_LE(x, x1, x2) -> f181_0_main_LE(x3, x4, x5) :|: x = x5 && x - 1 = x3 && 2 <= x4 - 1 && 0 <= x1 - 1 && 0 <= x2 - 1 && x4 - 2 <= x1

Arcs:
(1) -> (1)

This digraph is fully evaluated!

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(6) IntTRSCompressionProof (EQUIVALENT)
Compressed rules.
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(7)
Obligation:
Rules:
f181_0_main_LE(x5:0, x1:0, x2:0) -> f181_0_main_LE(x5:0 - 1, x4:0, x5:0) :|: x2:0 > 0 && x4:0 - 2 <= x1:0 && x4:0 > 2 && x1:0 > 0

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(8) TempFilterProof (SOUND)
Used the following sort dictionary for filtering: 
f181_0_main_LE(VARIABLE, INTEGER, VARIABLE)
Replaced non-predefined constructor symbols by 0.
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(9)
Obligation:
Rules:
f181_0_main_LE(x5:0, x1:0, x2:0) -> f181_0_main_LE(c, x4:0, x5:0) :|: c = x5:0 - 1 && (x2:0 > 0 && x4:0 - 2 <= x1:0 && x4:0 > 2 && x1:0 > 0)

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(10) PolynomialOrderProcessor (EQUIVALENT)
Found the following polynomial interpretation:
[f181_0_main_LE(x, x1, x2)] = x^2 + 2*x2

The following rules are decreasing:
f181_0_main_LE(x5:0, x1:0, x2:0) -> f181_0_main_LE(c, x4:0, x5:0) :|: c = x5:0 - 1 && (x2:0 > 0 && x4:0 - 2 <= x1:0 && x4:0 > 2 && x1:0 > 0)
The following rules are bounded:
f181_0_main_LE(x5:0, x1:0, x2:0) -> f181_0_main_LE(c, x4:0, x5:0) :|: c = x5:0 - 1 && (x2:0 > 0 && x4:0 - 2 <= x1:0 && x4:0 > 2 && x1:0 > 0)

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(11)
YES

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(12)
Obligation:

Termination digraph:
Nodes:
(1) f168_0_visit_NULL(x18, x19, x20) -> f168_0_visit_NULL(x21, x22, x23) :|: -1 <= x21 - 1 && 0 <= x18 - 1

Arcs:
(1) -> (1)

This digraph is fully evaluated!

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(13) IntTRSCompressionProof (EQUIVALENT)
Compressed rules.
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(14)
Obligation:
Rules:
f168_0_visit_NULL(x18:0, x19:0, x20:0) -> f168_0_visit_NULL(x21:0, x22:0, x23:0) :|: x21:0 > -1 && x18:0 > 0

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

   f168_0_visit_NULL(x1, x2, x3) -> f168_0_visit_NULL(x1)

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(16)
Obligation:
Rules:
f168_0_visit_NULL(x18:0) -> f168_0_visit_NULL(x21:0) :|: x21:0 > -1 && x18:0 > 0

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(17) FilterProof (EQUIVALENT)
Used the following sort dictionary for filtering: 
f168_0_visit_NULL(INTEGER)
Replaced non-predefined constructor symbols by 0.
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(18)
Obligation:
Rules:
f168_0_visit_NULL(x18:0) -> f168_0_visit_NULL(x21:0) :|: x21:0 > -1 && x18:0 > 0

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(19) IntTRSPeriodicNontermProof (COMPLETE)
Normalized system to the following form:
f(pc, x18:0) -> f(1, x21:0) :|: pc = 1 && (x21:0 > -1 && x18:0 > 0)
Witness term starting non-terminating reduction: f(1, 15)
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(20)
NO