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# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty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, 191 ms]
(4) AND
    (5) IRSwT
        (6) IntTRSCompressionProof [EQUIVALENT, 11 ms]
        (7) IRSwT
        (8) IntTRSUnneededArgumentFilterProof [EQUIVALENT, 0 ms]
        (9) IRSwT
        (10) TempFilterProof [SOUND, 42 ms]
        (11) IntTRS
        (12) RankingReductionPairProof [EQUIVALENT, 0 ms]
        (13) YES
    (14) IRSwT
        (15) IntTRSCompressionProof [EQUIVALENT, 41 ms]
        (16) IRSwT
        (17) FilterProof [EQUIVALENT, 0 ms]
        (18) IntTRS
        (19) IntTRSCompressionProof [EQUIVALENT, 0 ms]
        (20) IntTRS
        (21) PolynomialOrderProcessor [EQUIVALENT, 6 ms]
        (22) YES


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(0)
Obligation:
Rules:
f1_0_main_Load(arg1, arg2, arg3) -> f172_0_appendNewList_LE(arg1P, arg2P, arg3P) :|: 0 = arg2P && 0 <= arg1 - 1 && -1 <= arg1P - 1 && 0 <= arg2 - 1
f172_0_appendNewList_LE(x, x1, x2) -> f172_0_appendNewList_LE(x3, x4, x5) :|: x - 1 = x3 && 0 <= x1 - 1 && 1 <= x - 1 && x1 <= x4 - 1
f172_0_appendNewList_LE(x6, x7, x8) -> f172_0_appendNewList_LE(x9, x10, x11) :|: 1 = x10 && x6 - 1 = x9 && 1 <= x6 - 1
f172_0_appendNewList_LE(x12, x13, x14) -> f282_0_copy_NULL(x15, x16, x17) :|: 0 = x17 && x12 <= 1 && 3 <= x16 - 1 && 1 <= x15 - 1
f282_0_copy_NULL(x18, x19, x20) -> f282_0_copy_NULL(x21, x22, x23) :|: x23 + 2 <= x19 && x20 + 2 <= x18 && -1 <= x22 - 1 && 1 <= x21 - 1 && 0 <= x19 - 1 && 1 <= x18 - 1 && x22 + 1 <= x19 && x21 - 1 <= x19
__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) -> f172_0_appendNewList_LE(arg1P, arg2P, arg3P) :|: 0 = arg2P && 0 <= arg1 - 1 && -1 <= arg1P - 1 && 0 <= arg2 - 1
f172_0_appendNewList_LE(x, x1, x2) -> f172_0_appendNewList_LE(x3, x4, x5) :|: x - 1 = x3 && 0 <= x1 - 1 && 1 <= x - 1 && x1 <= x4 - 1
f172_0_appendNewList_LE(x6, x7, x8) -> f172_0_appendNewList_LE(x9, x10, x11) :|: 1 = x10 && x6 - 1 = x9 && 1 <= x6 - 1
f172_0_appendNewList_LE(x12, x13, x14) -> f282_0_copy_NULL(x15, x16, x17) :|: 0 = x17 && x12 <= 1 && 3 <= x16 - 1 && 1 <= x15 - 1
f282_0_copy_NULL(x18, x19, x20) -> f282_0_copy_NULL(x21, x22, x23) :|: x23 + 2 <= x19 && x20 + 2 <= x18 && -1 <= x22 - 1 && 1 <= x21 - 1 && 0 <= x19 - 1 && 1 <= x18 - 1 && x22 + 1 <= x19 && x21 - 1 <= x19
__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) -> f172_0_appendNewList_LE(arg1P, arg2P, arg3P) :|: 0 = arg2P && 0 <= arg1 - 1 && -1 <= arg1P - 1 && 0 <= arg2 - 1
(2) f172_0_appendNewList_LE(x, x1, x2) -> f172_0_appendNewList_LE(x3, x4, x5) :|: x - 1 = x3 && 0 <= x1 - 1 && 1 <= x - 1 && x1 <= x4 - 1
(3) f172_0_appendNewList_LE(x6, x7, x8) -> f172_0_appendNewList_LE(x9, x10, x11) :|: 1 = x10 && x6 - 1 = x9 && 1 <= x6 - 1
(4) f172_0_appendNewList_LE(x12, x13, x14) -> f282_0_copy_NULL(x15, x16, x17) :|: 0 = x17 && x12 <= 1 && 3 <= x16 - 1 && 1 <= x15 - 1
(5) f282_0_copy_NULL(x18, x19, x20) -> f282_0_copy_NULL(x21, x22, x23) :|: x23 + 2 <= x19 && x20 + 2 <= x18 && -1 <= x22 - 1 && 1 <= x21 - 1 && 0 <= x19 - 1 && 1 <= x18 - 1 && x22 + 1 <= x19 && x21 - 1 <= x19
(6) __init(x24, x25, x26) -> f1_0_main_Load(x27, x28, x29) :|: 0 <= 0

Arcs:
(1) -> (3), (4)
(2) -> (2), (3), (4)
(3) -> (2), (3), (4)
(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) f172_0_appendNewList_LE(x6, x7, x8) -> f172_0_appendNewList_LE(x9, x10, x11) :|: 1 = x10 && x6 - 1 = x9 && 1 <= x6 - 1
(2) f172_0_appendNewList_LE(x, x1, x2) -> f172_0_appendNewList_LE(x3, x4, x5) :|: x - 1 = x3 && 0 <= x1 - 1 && 1 <= x - 1 && x1 <= x4 - 1

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

This digraph is fully evaluated!

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(6) IntTRSCompressionProof (EQUIVALENT)
Compressed rules.
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(7)
Obligation:
Rules:
f172_0_appendNewList_LE(x6:0, x7:0, x8:0) -> f172_0_appendNewList_LE(x6:0 - 1, 1, x11:0) :|: x6:0 > 1
f172_0_appendNewList_LE(x:0, x1:0, x2:0) -> f172_0_appendNewList_LE(x:0 - 1, x4:0, x5:0) :|: x:0 > 1 && x1:0 > 0 && x4:0 - 1 >= x1:0

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

   f172_0_appendNewList_LE(x1, x2, x3) -> f172_0_appendNewList_LE(x1, x2)

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(9)
Obligation:
Rules:
f172_0_appendNewList_LE(x6:0, x7:0) -> f172_0_appendNewList_LE(x6:0 - 1, 1) :|: x6:0 > 1
f172_0_appendNewList_LE(x:0, x1:0) -> f172_0_appendNewList_LE(x:0 - 1, x4:0) :|: x:0 > 1 && x1:0 > 0 && x4:0 - 1 >= x1:0

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(10) TempFilterProof (SOUND)
Used the following sort dictionary for filtering: 
f172_0_appendNewList_LE(INTEGER, VARIABLE)
Replaced non-predefined constructor symbols by 0.
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(11)
Obligation:
Rules:
f172_0_appendNewList_LE(x6:0, x7:0) -> f172_0_appendNewList_LE(c, c1) :|: c1 = 1 && c = x6:0 - 1 && x6:0 > 1
f172_0_appendNewList_LE(x:0, x1:0) -> f172_0_appendNewList_LE(c2, x4:0) :|: c2 = x:0 - 1 && (x:0 > 1 && x1:0 > 0 && x4:0 - 1 >= x1:0)

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(12) RankingReductionPairProof (EQUIVALENT)
Interpretation:
[ f172_0_appendNewList_LE ] = f172_0_appendNewList_LE_1

The following rules are decreasing:
f172_0_appendNewList_LE(x6:0, x7:0) -> f172_0_appendNewList_LE(c, c1) :|: c1 = 1 && c = x6:0 - 1 && x6:0 > 1
f172_0_appendNewList_LE(x:0, x1:0) -> f172_0_appendNewList_LE(c2, x4:0) :|: c2 = x:0 - 1 && (x:0 > 1 && x1:0 > 0 && x4:0 - 1 >= x1:0)

The following rules are bounded:
f172_0_appendNewList_LE(x6:0, x7:0) -> f172_0_appendNewList_LE(c, c1) :|: c1 = 1 && c = x6:0 - 1 && x6:0 > 1
f172_0_appendNewList_LE(x:0, x1:0) -> f172_0_appendNewList_LE(c2, x4:0) :|: c2 = x:0 - 1 && (x:0 > 1 && x1:0 > 0 && x4:0 - 1 >= x1:0)


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(13)
YES

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(14)
Obligation:

Termination digraph:
Nodes:
(1) f282_0_copy_NULL(x18, x19, x20) -> f282_0_copy_NULL(x21, x22, x23) :|: x23 + 2 <= x19 && x20 + 2 <= x18 && -1 <= x22 - 1 && 1 <= x21 - 1 && 0 <= x19 - 1 && 1 <= x18 - 1 && x22 + 1 <= x19 && x21 - 1 <= x19

Arcs:
(1) -> (1)

This digraph is fully evaluated!

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

(16)
Obligation:
Rules:
f282_0_copy_NULL(x18:0, x19:0, x20:0) -> f282_0_copy_NULL(x21:0, x22:0, x23:0) :|: x22:0 + 1 <= x19:0 && x21:0 - 1 <= x19:0 && x18:0 > 1 && x19:0 > 0 && x21:0 > 1 && x22:0 > -1 && x20:0 + 2 <= x18:0 && x23:0 + 2 <= x19:0

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(17) FilterProof (EQUIVALENT)
Used the following sort dictionary for filtering: 
f282_0_copy_NULL(INTEGER, INTEGER, INTEGER)
Replaced non-predefined constructor symbols by 0.
----------------------------------------

(18)
Obligation:
Rules:
f282_0_copy_NULL(x18:0, x19:0, x20:0) -> f282_0_copy_NULL(x21:0, x22:0, x23:0) :|: x22:0 + 1 <= x19:0 && x21:0 - 1 <= x19:0 && x18:0 > 1 && x19:0 > 0 && x21:0 > 1 && x22:0 > -1 && x20:0 + 2 <= x18:0 && x23:0 + 2 <= x19:0

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(19) IntTRSCompressionProof (EQUIVALENT)
Compressed rules.
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(20)
Obligation:
Rules:
f282_0_copy_NULL(x18:0:0, x19:0:0, x20:0:0) -> f282_0_copy_NULL(x21:0:0, x22:0:0, x23:0:0) :|: x20:0:0 + 2 <= x18:0:0 && x23:0:0 + 2 <= x19:0:0 && x22:0:0 > -1 && x21:0:0 > 1 && x19:0:0 > 0 && x18:0:0 > 1 && x21:0:0 - 1 <= x19:0:0 && x22:0:0 + 1 <= x19:0:0

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(21) PolynomialOrderProcessor (EQUIVALENT)
Found the following polynomial interpretation:
[f282_0_copy_NULL(x, x1, x2)] = x1

The following rules are decreasing:
f282_0_copy_NULL(x18:0:0, x19:0:0, x20:0:0) -> f282_0_copy_NULL(x21:0:0, x22:0:0, x23:0:0) :|: x20:0:0 + 2 <= x18:0:0 && x23:0:0 + 2 <= x19:0:0 && x22:0:0 > -1 && x21:0:0 > 1 && x19:0:0 > 0 && x18:0:0 > 1 && x21:0:0 - 1 <= x19:0:0 && x22:0:0 + 1 <= x19:0:0
The following rules are bounded:
f282_0_copy_NULL(x18:0:0, x19:0:0, x20:0:0) -> f282_0_copy_NULL(x21:0:0, x22:0:0, x23:0:0) :|: x20:0:0 + 2 <= x18:0:0 && x23:0:0 + 2 <= x19:0:0 && x22:0:0 > -1 && x21:0:0 > 1 && x19:0:0 > 0 && x18:0:0 > 1 && x21:0:0 - 1 <= x19:0:0 && x22:0:0 + 1 <= x19:0:0

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(22)
YES