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# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty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, 129 ms]
(4) AND
    (5) IRSwT
        (6) IntTRSCompressionProof [EQUIVALENT, 0 ms]
        (7) IRSwT
        (8) IntTRSUnneededArgumentFilterProof [EQUIVALENT, 0 ms]
        (9) IRSwT
        (10) TempFilterProof [SOUND, 10 ms]
        (11) IntTRS
        (12) PolynomialOrderProcessor [EQUIVALENT, 0 ms]
        (13) YES
    (14) IRSwT
        (15) IntTRSCompressionProof [EQUIVALENT, 0 ms]
        (16) IRSwT
        (17) IntTRSUnneededArgumentFilterProof [EQUIVALENT, 0 ms]
        (18) IRSwT
        (19) FilterProof [EQUIVALENT, 0 ms]
        (20) IntTRS
        (21) IntTRSPeriodicNontermProof [COMPLETE, 6 ms]
        (22) NO


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(0)
Obligation:
Rules:
f1_0_main_Load(arg1, arg2) -> f53_0_loop_EQ(arg1P, arg2P) :|: arg2 = arg1P && -1 <= arg2 - 1 && 0 <= arg1 - 1
f53_0_loop_EQ(x, x1) -> f53_0_loop_EQ(x2, x3) :|: x = x2 && 4 <= x - 1
f53_0_loop_EQ(x4, x5) -> f53_0_loop_EQ(x6, x7) :|: x4 - 1 = x6 && x4 <= 4 && 0 <= x4 - 1
__init(x8, x9) -> f1_0_main_Load(x10, x11) :|: 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) -> f53_0_loop_EQ(arg1P, arg2P) :|: arg2 = arg1P && -1 <= arg2 - 1 && 0 <= arg1 - 1
f53_0_loop_EQ(x, x1) -> f53_0_loop_EQ(x2, x3) :|: x = x2 && 4 <= x - 1
f53_0_loop_EQ(x4, x5) -> f53_0_loop_EQ(x6, x7) :|: x4 - 1 = x6 && x4 <= 4 && 0 <= x4 - 1
__init(x8, x9) -> f1_0_main_Load(x10, x11) :|: 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) -> f53_0_loop_EQ(arg1P, arg2P) :|: arg2 = arg1P && -1 <= arg2 - 1 && 0 <= arg1 - 1
(2) f53_0_loop_EQ(x, x1) -> f53_0_loop_EQ(x2, x3) :|: x = x2 && 4 <= x - 1
(3) f53_0_loop_EQ(x4, x5) -> f53_0_loop_EQ(x6, x7) :|: x4 - 1 = x6 && x4 <= 4 && 0 <= x4 - 1
(4) __init(x8, x9) -> f1_0_main_Load(x10, x11) :|: 0 <= 0

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

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

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(5)
Obligation:

Termination digraph:
Nodes:
(1) f53_0_loop_EQ(x4, x5) -> f53_0_loop_EQ(x6, x7) :|: x4 - 1 = x6 && x4 <= 4 && 0 <= x4 - 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:
f53_0_loop_EQ(x4:0, x5:0) -> f53_0_loop_EQ(x4:0 - 1, x7:0) :|: x4:0 > 0 && x4:0 < 5

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

   f53_0_loop_EQ(x1, x2) -> f53_0_loop_EQ(x1)

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(9)
Obligation:
Rules:
f53_0_loop_EQ(x4:0) -> f53_0_loop_EQ(x4:0 - 1) :|: x4:0 > 0 && x4:0 < 5

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(10) TempFilterProof (SOUND)
Used the following sort dictionary for filtering: 
f53_0_loop_EQ(INTEGER)
Replaced non-predefined constructor symbols by 0.
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(11)
Obligation:
Rules:
f53_0_loop_EQ(x4:0) -> f53_0_loop_EQ(c) :|: c = x4:0 - 1 && (x4:0 > 0 && x4:0 < 5)

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(12) PolynomialOrderProcessor (EQUIVALENT)
Found the following polynomial interpretation:
[f53_0_loop_EQ(x)] = x

The following rules are decreasing:
f53_0_loop_EQ(x4:0) -> f53_0_loop_EQ(c) :|: c = x4:0 - 1 && (x4:0 > 0 && x4:0 < 5)
The following rules are bounded:
f53_0_loop_EQ(x4:0) -> f53_0_loop_EQ(c) :|: c = x4:0 - 1 && (x4:0 > 0 && x4:0 < 5)

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(13)
YES

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(14)
Obligation:

Termination digraph:
Nodes:
(1) f53_0_loop_EQ(x, x1) -> f53_0_loop_EQ(x2, x3) :|: x = x2 && 4 <= x - 1

Arcs:
(1) -> (1)

This digraph is fully evaluated!

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(15) IntTRSCompressionProof (EQUIVALENT)
Compressed rules.
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(16)
Obligation:
Rules:
f53_0_loop_EQ(x2:0, x1:0) -> f53_0_loop_EQ(x2:0, x3:0) :|: x2:0 > 4

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

   f53_0_loop_EQ(x1, x2) -> f53_0_loop_EQ(x1)

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(18)
Obligation:
Rules:
f53_0_loop_EQ(x2:0) -> f53_0_loop_EQ(x2:0) :|: x2:0 > 4

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(19) FilterProof (EQUIVALENT)
Used the following sort dictionary for filtering: 
f53_0_loop_EQ(INTEGER)
Replaced non-predefined constructor symbols by 0.
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(20)
Obligation:
Rules:
f53_0_loop_EQ(x2:0) -> f53_0_loop_EQ(x2:0) :|: x2:0 > 4

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(21) IntTRSPeriodicNontermProof (COMPLETE)
Normalized system to the following form:
f(pc, x2:0) -> f(1, x2:0) :|: pc = 1 && x2:0 > 4
Witness term starting non-terminating reduction: f(1, 5)
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(22)
NO