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# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty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, 149 ms]
(4) AND
    (5) IRSwT
        (6) IntTRSCompressionProof [EQUIVALENT, 0 ms]
        (7) IRSwT
        (8) TempFilterProof [SOUND, 95 ms]
        (9) IntTRS
        (10) PolynomialOrderProcessor [EQUIVALENT, 12 ms]
        (11) YES
    (12) IRSwT
        (13) IntTRSCompressionProof [EQUIVALENT, 0 ms]
        (14) IRSwT
        (15) IntTRSUnneededArgumentFilterProof [EQUIVALENT, 2 ms]
        (16) IRSwT
        (17) FilterProof [EQUIVALENT, 0 ms]
        (18) IntTRS
        (19) IntTRSCompressionProof [EQUIVALENT, 0 ms]
        (20) IntTRS
        (21) RankingReductionPairProof [EQUIVALENT, 0 ms]
        (22) YES


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(0)
Obligation:
Rules:
f1_0_main_New(arg1, arg2, arg3) -> f160_0_main_LE(arg1P, arg2P, arg3P) :|: 28 = arg3P && 27 = arg2P && 0 <= arg1P - 1
f160_0_main_LE(x, x1, x2) -> f160_0_main_LE(x3, x4, x5) :|: x1 = x5 && x1 - 1 = x4 && 3 <= x3 - 1 && 0 <= x - 1 && 0 <= x2 - 1 && x3 - 3 <= x
f160_0_main_LE(x6, x7, x8) -> f167_0_length_InvokeMethod(x9, x10, x11) :|: -1 <= x9 - 1 && 0 <= x6 - 1 && x8 <= 0 && x9 + 1 <= x6
f167_0_length_InvokeMethod(x12, x13, x14) -> f167_0_length_InvokeMethod(x15, x16, x17) :|: -1 <= x15 - 1 && 0 <= x12 - 1 && x15 + 1 <= x12
__init(x18, x19, x20) -> f1_0_main_New(x21, x22, x23) :|: 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_New(arg1, arg2, arg3) -> f160_0_main_LE(arg1P, arg2P, arg3P) :|: 28 = arg3P && 27 = arg2P && 0 <= arg1P - 1
f160_0_main_LE(x, x1, x2) -> f160_0_main_LE(x3, x4, x5) :|: x1 = x5 && x1 - 1 = x4 && 3 <= x3 - 1 && 0 <= x - 1 && 0 <= x2 - 1 && x3 - 3 <= x
f160_0_main_LE(x6, x7, x8) -> f167_0_length_InvokeMethod(x9, x10, x11) :|: -1 <= x9 - 1 && 0 <= x6 - 1 && x8 <= 0 && x9 + 1 <= x6
f167_0_length_InvokeMethod(x12, x13, x14) -> f167_0_length_InvokeMethod(x15, x16, x17) :|: -1 <= x15 - 1 && 0 <= x12 - 1 && x15 + 1 <= x12
__init(x18, x19, x20) -> f1_0_main_New(x21, x22, x23) :|: 0 <= 0
Start term: __init(arg1, arg2, arg3)

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(3) IRSwTTerminationDigraphProof (EQUIVALENT)
Constructed termination digraph!
Nodes:
(1) f1_0_main_New(arg1, arg2, arg3) -> f160_0_main_LE(arg1P, arg2P, arg3P) :|: 28 = arg3P && 27 = arg2P && 0 <= arg1P - 1
(2) f160_0_main_LE(x, x1, x2) -> f160_0_main_LE(x3, x4, x5) :|: x1 = x5 && x1 - 1 = x4 && 3 <= x3 - 1 && 0 <= x - 1 && 0 <= x2 - 1 && x3 - 3 <= x
(3) f160_0_main_LE(x6, x7, x8) -> f167_0_length_InvokeMethod(x9, x10, x11) :|: -1 <= x9 - 1 && 0 <= x6 - 1 && x8 <= 0 && x9 + 1 <= x6
(4) f167_0_length_InvokeMethod(x12, x13, x14) -> f167_0_length_InvokeMethod(x15, x16, x17) :|: -1 <= x15 - 1 && 0 <= x12 - 1 && x15 + 1 <= x12
(5) __init(x18, x19, x20) -> f1_0_main_New(x21, x22, x23) :|: 0 <= 0

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

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

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(5)
Obligation:

Termination digraph:
Nodes:
(1) f160_0_main_LE(x, x1, x2) -> f160_0_main_LE(x3, x4, x5) :|: x1 = x5 && x1 - 1 = x4 && 3 <= x3 - 1 && 0 <= x - 1 && 0 <= x2 - 1 && x3 - 3 <= x

Arcs:
(1) -> (1)

This digraph is fully evaluated!

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(6) IntTRSCompressionProof (EQUIVALENT)
Compressed rules.
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(7)
Obligation:
Rules:
f160_0_main_LE(x:0, x1:0, x2:0) -> f160_0_main_LE(x3:0, x1:0 - 1, x1:0) :|: x2:0 > 0 && x:0 >= x3:0 - 3 && x3:0 > 3 && x:0 > 0

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(8) TempFilterProof (SOUND)
Used the following sort dictionary for filtering: 
f160_0_main_LE(INTEGER, VARIABLE, VARIABLE)
Replaced non-predefined constructor symbols by 0.
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(9)
Obligation:
Rules:
f160_0_main_LE(x:0, x1:0, x2:0) -> f160_0_main_LE(x3:0, c, x1:0) :|: c = x1:0 - 1 && (x2:0 > 0 && x:0 >= x3:0 - 3 && x3:0 > 3 && x:0 > 0)

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(10) PolynomialOrderProcessor (EQUIVALENT)
Found the following polynomial interpretation:
[f160_0_main_LE(x, x1, x2)] = x1^2 + 2*x2

The following rules are decreasing:
f160_0_main_LE(x:0, x1:0, x2:0) -> f160_0_main_LE(x3:0, c, x1:0) :|: c = x1:0 - 1 && (x2:0 > 0 && x:0 >= x3:0 - 3 && x3:0 > 3 && x:0 > 0)
The following rules are bounded:
f160_0_main_LE(x:0, x1:0, x2:0) -> f160_0_main_LE(x3:0, c, x1:0) :|: c = x1:0 - 1 && (x2:0 > 0 && x:0 >= x3:0 - 3 && x3:0 > 3 && x:0 > 0)

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(11)
YES

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(12)
Obligation:

Termination digraph:
Nodes:
(1) f167_0_length_InvokeMethod(x12, x13, x14) -> f167_0_length_InvokeMethod(x15, x16, x17) :|: -1 <= x15 - 1 && 0 <= x12 - 1 && x15 + 1 <= x12

Arcs:
(1) -> (1)

This digraph is fully evaluated!

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(13) IntTRSCompressionProof (EQUIVALENT)
Compressed rules.
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(14)
Obligation:
Rules:
f167_0_length_InvokeMethod(x12:0, x13:0, x14:0) -> f167_0_length_InvokeMethod(x15:0, x16:0, x17:0) :|: x15:0 > -1 && x12:0 > 0 && x15:0 + 1 <= x12:0

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

   f167_0_length_InvokeMethod(x1, x2, x3) -> f167_0_length_InvokeMethod(x1)

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(16)
Obligation:
Rules:
f167_0_length_InvokeMethod(x12:0) -> f167_0_length_InvokeMethod(x15:0) :|: x15:0 > -1 && x12:0 > 0 && x15:0 + 1 <= x12:0

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(17) FilterProof (EQUIVALENT)
Used the following sort dictionary for filtering: 
f167_0_length_InvokeMethod(INTEGER)
Replaced non-predefined constructor symbols by 0.
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(18)
Obligation:
Rules:
f167_0_length_InvokeMethod(x12:0) -> f167_0_length_InvokeMethod(x15:0) :|: x15:0 > -1 && x12:0 > 0 && x15:0 + 1 <= x12:0

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(19) IntTRSCompressionProof (EQUIVALENT)
Compressed rules.
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(20)
Obligation:
Rules:
f167_0_length_InvokeMethod(x12:0:0) -> f167_0_length_InvokeMethod(x15:0:0) :|: x15:0:0 > -1 && x12:0:0 > 0 && x15:0:0 + 1 <= x12:0:0

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(21) RankingReductionPairProof (EQUIVALENT)
Interpretation:
[ f167_0_length_InvokeMethod ] = f167_0_length_InvokeMethod_1

The following rules are decreasing:
f167_0_length_InvokeMethod(x12:0:0) -> f167_0_length_InvokeMethod(x15:0:0) :|: x15:0:0 > -1 && x12:0:0 > 0 && x15:0:0 + 1 <= x12:0:0

The following rules are bounded:
f167_0_length_InvokeMethod(x12:0:0) -> f167_0_length_InvokeMethod(x15:0:0) :|: x15:0:0 > -1 && x12:0:0 > 0 && x15:0:0 + 1 <= x12:0:0


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(22)
YES