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# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty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, 206 ms]
(4) IRSwT
(5) IntTRSCompressionProof [EQUIVALENT, 0 ms]
(6) IRSwT
(7) IntTRSUnneededArgumentFilterProof [EQUIVALENT, 0 ms]
(8) IRSwT
(9) TempFilterProof [SOUND, 75 ms]
(10) IntTRS
(11) PolynomialOrderProcessor [EQUIVALENT, 0 ms]
(12) IntTRS
(13) PolynomialOrderProcessor [EQUIVALENT, 0 ms]
(14) IntTRS
(15) PolynomialOrderProcessor [EQUIVALENT, 0 ms]
(16) YES


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(0)
Obligation:
Rules:
f1_0_main_Load(arg1, arg2) -> f115_0_main_EQ(arg1P, arg2P) :|: 0 <= arg1 - 1 && -1 <= arg2 - 1 && 0 <= arg1P - 1
f115_0_main_EQ(x, x1) -> f115_0_main_EQ'(x2, x3) :|: x - 2 * x4 = 1 && 0 <= x - 1 && x = x2
f115_0_main_EQ'(x5, x7) -> f115_0_main_EQ(x9, x12) :|: 0 <= x5 - 1 && x5 - 2 * x13 = 1 && x5 - 2 * x13 <= 1 && 0 <= x5 - 2 * x13 && x5 - 1 = x9
f115_0_main_EQ(x15, x16) -> f115_0_main_EQ'(x17, x18) :|: x15 - 2 * x19 = 0 && x20 <= x15 - 1 && 0 <= x15 - 1 && x15 = x17
f115_0_main_EQ'(x21, x22) -> f115_0_main_EQ(x23, x24) :|: x21 - 2 * x25 = 0 && 0 <= x21 - 1 && x23 <= x21 - 1 && 0 <= x21 - 2 * x25 && x21 - 2 * x25 <= 1 && x21 - 2 * x23 <= 1 && 0 <= x21 - 2 * x23
__init(x26, x27) -> f1_0_main_Load(x28, x29) :|: 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) -> f115_0_main_EQ(arg1P, arg2P) :|: 0 <= arg1 - 1 && -1 <= arg2 - 1 && 0 <= arg1P - 1
f115_0_main_EQ(x, x1) -> f115_0_main_EQ'(x2, x3) :|: x - 2 * x4 = 1 && 0 <= x - 1 && x = x2
f115_0_main_EQ'(x5, x7) -> f115_0_main_EQ(x9, x12) :|: 0 <= x5 - 1 && x5 - 2 * x13 = 1 && x5 - 2 * x13 <= 1 && 0 <= x5 - 2 * x13 && x5 - 1 = x9
f115_0_main_EQ(x15, x16) -> f115_0_main_EQ'(x17, x18) :|: x15 - 2 * x19 = 0 && x20 <= x15 - 1 && 0 <= x15 - 1 && x15 = x17
f115_0_main_EQ'(x21, x22) -> f115_0_main_EQ(x23, x24) :|: x21 - 2 * x25 = 0 && 0 <= x21 - 1 && x23 <= x21 - 1 && 0 <= x21 - 2 * x25 && x21 - 2 * x25 <= 1 && x21 - 2 * x23 <= 1 && 0 <= x21 - 2 * x23
__init(x26, x27) -> f1_0_main_Load(x28, x29) :|: 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) -> f115_0_main_EQ(arg1P, arg2P) :|: 0 <= arg1 - 1 && -1 <= arg2 - 1 && 0 <= arg1P - 1
(2) f115_0_main_EQ(x, x1) -> f115_0_main_EQ'(x2, x3) :|: x - 2 * x4 = 1 && 0 <= x - 1 && x = x2
(3) f115_0_main_EQ'(x5, x7) -> f115_0_main_EQ(x9, x12) :|: 0 <= x5 - 1 && x5 - 2 * x13 = 1 && x5 - 2 * x13 <= 1 && 0 <= x5 - 2 * x13 && x5 - 1 = x9
(4) f115_0_main_EQ(x15, x16) -> f115_0_main_EQ'(x17, x18) :|: x15 - 2 * x19 = 0 && x20 <= x15 - 1 && 0 <= x15 - 1 && x15 = x17
(5) f115_0_main_EQ'(x21, x22) -> f115_0_main_EQ(x23, x24) :|: x21 - 2 * x25 = 0 && 0 <= x21 - 1 && x23 <= x21 - 1 && 0 <= x21 - 2 * x25 && x21 - 2 * x25 <= 1 && x21 - 2 * x23 <= 1 && 0 <= x21 - 2 * x23
(6) __init(x26, x27) -> f1_0_main_Load(x28, x29) :|: 0 <= 0

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

This digraph is fully evaluated!
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(4)
Obligation:

Termination digraph:
Nodes:
(1) f115_0_main_EQ(x, x1) -> f115_0_main_EQ'(x2, x3) :|: x - 2 * x4 = 1 && 0 <= x - 1 && x = x2
(2) f115_0_main_EQ'(x21, x22) -> f115_0_main_EQ(x23, x24) :|: x21 - 2 * x25 = 0 && 0 <= x21 - 1 && x23 <= x21 - 1 && 0 <= x21 - 2 * x25 && x21 - 2 * x25 <= 1 && x21 - 2 * x23 <= 1 && 0 <= x21 - 2 * x23
(3) f115_0_main_EQ(x15, x16) -> f115_0_main_EQ'(x17, x18) :|: x15 - 2 * x19 = 0 && x20 <= x15 - 1 && 0 <= x15 - 1 && x15 = x17
(4) f115_0_main_EQ'(x5, x7) -> f115_0_main_EQ(x9, x12) :|: 0 <= x5 - 1 && x5 - 2 * x13 = 1 && x5 - 2 * x13 <= 1 && 0 <= x5 - 2 * x13 && x5 - 1 = x9

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

This digraph is fully evaluated!

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(5) IntTRSCompressionProof (EQUIVALENT)
Compressed rules.
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(6)
Obligation:
Rules:
f115_0_main_EQ'(x5:0, x7:0) -> f115_0_main_EQ(x5:0 - 1, x12:0) :|: x5:0 - 2 * x13:0 <= 1 && x5:0 - 2 * x13:0 >= 0 && x5:0 - 2 * x13:0 = 1 && x5:0 > 0
f115_0_main_EQ(x2:0, x1:0) -> f115_0_main_EQ'(x2:0, x3:0) :|: x2:0 - 2 * x4:0 = 1 && x2:0 > 0
f115_0_main_EQ'(x21:0, x22:0) -> f115_0_main_EQ(x23:0, x24:0) :|: x21:0 - 2 * x23:0 <= 1 && x21:0 - 2 * x23:0 >= 0 && x21:0 - 2 * x25:0 <= 1 && x21:0 - 2 * x25:0 >= 0 && x23:0 <= x21:0 - 1 && x21:0 > 0 && x21:0 - 2 * x25:0 = 0
f115_0_main_EQ(x15:0, x16:0) -> f115_0_main_EQ'(x15:0, x18:0) :|: x15:0 - 2 * x19:0 = 0 && x20:0 <= x15:0 - 1 && x15:0 > 0

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

   f115_0_main_EQ'(x1, x2) -> f115_0_main_EQ'(x1)
   f115_0_main_EQ(x1, x2) -> f115_0_main_EQ(x1)

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(8)
Obligation:
Rules:
f115_0_main_EQ'(x5:0) -> f115_0_main_EQ(x5:0 - 1) :|: x5:0 - 2 * x13:0 <= 1 && x5:0 - 2 * x13:0 >= 0 && x5:0 - 2 * x13:0 = 1 && x5:0 > 0
f115_0_main_EQ(x2:0) -> f115_0_main_EQ'(x2:0) :|: x2:0 - 2 * x4:0 = 1 && x2:0 > 0
f115_0_main_EQ'(x21:0) -> f115_0_main_EQ(x23:0) :|: x21:0 - 2 * x23:0 <= 1 && x21:0 - 2 * x23:0 >= 0 && x21:0 - 2 * x25:0 <= 1 && x21:0 - 2 * x25:0 >= 0 && x23:0 <= x21:0 - 1 && x21:0 > 0 && x21:0 - 2 * x25:0 = 0
f115_0_main_EQ(x15:0) -> f115_0_main_EQ'(x15:0) :|: x15:0 - 2 * x19:0 = 0 && x20:0 <= x15:0 - 1 && x15:0 > 0

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(9) TempFilterProof (SOUND)
Used the following sort dictionary for filtering: 
f115_0_main_EQ'(INTEGER)
f115_0_main_EQ(INTEGER)
Replaced non-predefined constructor symbols by 0.
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(10)
Obligation:
Rules:
f115_0_main_EQ'(x5:0) -> f115_0_main_EQ(c) :|: c = x5:0 - 1 && (x5:0 - 2 * x13:0 <= 1 && x5:0 - 2 * x13:0 >= 0 && x5:0 - 2 * x13:0 = 1 && x5:0 > 0)
f115_0_main_EQ(x2:0) -> f115_0_main_EQ'(x2:0) :|: x2:0 - 2 * x4:0 = 1 && x2:0 > 0
f115_0_main_EQ'(x21:0) -> f115_0_main_EQ(x23:0) :|: x21:0 - 2 * x23:0 <= 1 && x21:0 - 2 * x23:0 >= 0 && x21:0 - 2 * x25:0 <= 1 && x21:0 - 2 * x25:0 >= 0 && x23:0 <= x21:0 - 1 && x21:0 > 0 && x21:0 - 2 * x25:0 = 0
f115_0_main_EQ(x15:0) -> f115_0_main_EQ'(x15:0) :|: x15:0 - 2 * x19:0 = 0 && x20:0 <= x15:0 - 1 && x15:0 > 0

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(11) PolynomialOrderProcessor (EQUIVALENT)
Found the following polynomial interpretation:
[f115_0_main_EQ'(x)] = -1 + x
[f115_0_main_EQ(x1)] = -1 + x1

The following rules are decreasing:
f115_0_main_EQ'(x5:0) -> f115_0_main_EQ(c) :|: c = x5:0 - 1 && (x5:0 - 2 * x13:0 <= 1 && x5:0 - 2 * x13:0 >= 0 && x5:0 - 2 * x13:0 = 1 && x5:0 > 0)
The following rules are bounded:
f115_0_main_EQ'(x5:0) -> f115_0_main_EQ(c) :|: c = x5:0 - 1 && (x5:0 - 2 * x13:0 <= 1 && x5:0 - 2 * x13:0 >= 0 && x5:0 - 2 * x13:0 = 1 && x5:0 > 0)
f115_0_main_EQ(x2:0) -> f115_0_main_EQ'(x2:0) :|: x2:0 - 2 * x4:0 = 1 && x2:0 > 0
f115_0_main_EQ(x15:0) -> f115_0_main_EQ'(x15:0) :|: x15:0 - 2 * x19:0 = 0 && x20:0 <= x15:0 - 1 && x15:0 > 0

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(12)
Obligation:
Rules:
f115_0_main_EQ(x2:0) -> f115_0_main_EQ'(x2:0) :|: x2:0 - 2 * x4:0 = 1 && x2:0 > 0
f115_0_main_EQ'(x21:0) -> f115_0_main_EQ(x23:0) :|: x21:0 - 2 * x23:0 <= 1 && x21:0 - 2 * x23:0 >= 0 && x21:0 - 2 * x25:0 <= 1 && x21:0 - 2 * x25:0 >= 0 && x23:0 <= x21:0 - 1 && x21:0 > 0 && x21:0 - 2 * x25:0 = 0
f115_0_main_EQ(x15:0) -> f115_0_main_EQ'(x15:0) :|: x15:0 - 2 * x19:0 = 0 && x20:0 <= x15:0 - 1 && x15:0 > 0

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(13) PolynomialOrderProcessor (EQUIVALENT)
Found the following polynomial interpretation:
[f115_0_main_EQ(x)] = -1 + 2*x
[f115_0_main_EQ'(x1)] = -1 + x1

The following rules are decreasing:
f115_0_main_EQ(x2:0) -> f115_0_main_EQ'(x2:0) :|: x2:0 - 2 * x4:0 = 1 && x2:0 > 0
f115_0_main_EQ(x15:0) -> f115_0_main_EQ'(x15:0) :|: x15:0 - 2 * x19:0 = 0 && x20:0 <= x15:0 - 1 && x15:0 > 0
The following rules are bounded:
f115_0_main_EQ(x2:0) -> f115_0_main_EQ'(x2:0) :|: x2:0 - 2 * x4:0 = 1 && x2:0 > 0
f115_0_main_EQ(x15:0) -> f115_0_main_EQ'(x15:0) :|: x15:0 - 2 * x19:0 = 0 && x20:0 <= x15:0 - 1 && x15:0 > 0

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(14)
Obligation:
Rules:
f115_0_main_EQ'(x21:0) -> f115_0_main_EQ(x23:0) :|: x21:0 - 2 * x23:0 <= 1 && x21:0 - 2 * x23:0 >= 0 && x21:0 - 2 * x25:0 <= 1 && x21:0 - 2 * x25:0 >= 0 && x23:0 <= x21:0 - 1 && x21:0 > 0 && x21:0 - 2 * x25:0 = 0

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(15) PolynomialOrderProcessor (EQUIVALENT)
Found the following polynomial interpretation:
[f115_0_main_EQ'(x)] = 0
[f115_0_main_EQ(x1)] = -1

The following rules are decreasing:
f115_0_main_EQ'(x21:0) -> f115_0_main_EQ(x23:0) :|: x21:0 - 2 * x23:0 <= 1 && x21:0 - 2 * x23:0 >= 0 && x21:0 - 2 * x25:0 <= 1 && x21:0 - 2 * x25:0 >= 0 && x23:0 <= x21:0 - 1 && x21:0 > 0 && x21:0 - 2 * x25:0 = 0
The following rules are bounded:
f115_0_main_EQ'(x21:0) -> f115_0_main_EQ(x23:0) :|: x21:0 - 2 * x23:0 <= 1 && x21:0 - 2 * x23:0 >= 0 && x21:0 - 2 * x25:0 <= 1 && x21:0 - 2 * x25:0 >= 0 && x23:0 <= x21:0 - 1 && x21:0 > 0 && x21:0 - 2 * x25:0 = 0

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(16)
YES