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# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty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, 208 ms]
(4) IRSwT
(5) IntTRSCompressionProof [EQUIVALENT, 35 ms]
(6) IRSwT
(7) IntTRSUnneededArgumentFilterProof [EQUIVALENT, 0 ms]
(8) IRSwT
(9) TempFilterProof [SOUND, 48 ms]
(10) IntTRS
(11) PolynomialOrderProcessor [EQUIVALENT, 0 ms]
(12) IntTRS
(13) PolynomialOrderProcessor [EQUIVALENT, 0 ms]
(14) YES


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(0)
Obligation:
Rules:
f1_0_main_Load(arg1, arg2, arg3) -> f135_0_f_LE(arg1P, arg2P, arg3P) :|: arg2 = arg2P && arg2 = arg1P && -1 <= arg2 - 1 && 0 <= arg1 - 1
f135_0_f_LE(x, x1, x2) -> f182_0_f_LE(x3, x4, x5) :|: x1 = x5 && 2 = x4 && x = x3 && 0 <= x - 1
f182_0_f_LE(x6, x7, x8) -> f135_0_f_LE(x9, x10, x11) :|: x8 - 1 = x10 && x6 - x7 = x9 && 0 <= x6 - 1 && 0 <= x7 - 1
f182_0_f_LE(x12, x13, x14) -> f182_0_f_LE(x15, x16, x17) :|: x14 - 1 = x17 && x13 - 1 = x16 && x12 = x15 && 0 <= x12 - 1 && 0 <= x13 - 1
f182_0_f_LE(x18, x19, x20) -> f182_0_f_LE(x21, x22, x23) :|: x19 - 1 = x22 && x18 = x21 && 0 <= x18 - 1 && 0 <= x19 - 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) -> f135_0_f_LE(arg1P, arg2P, arg3P) :|: arg2 = arg2P && arg2 = arg1P && -1 <= arg2 - 1 && 0 <= arg1 - 1
f135_0_f_LE(x, x1, x2) -> f182_0_f_LE(x3, x4, x5) :|: x1 = x5 && 2 = x4 && x = x3 && 0 <= x - 1
f182_0_f_LE(x6, x7, x8) -> f135_0_f_LE(x9, x10, x11) :|: x8 - 1 = x10 && x6 - x7 = x9 && 0 <= x6 - 1 && 0 <= x7 - 1
f182_0_f_LE(x12, x13, x14) -> f182_0_f_LE(x15, x16, x17) :|: x14 - 1 = x17 && x13 - 1 = x16 && x12 = x15 && 0 <= x12 - 1 && 0 <= x13 - 1
f182_0_f_LE(x18, x19, x20) -> f182_0_f_LE(x21, x22, x23) :|: x19 - 1 = x22 && x18 = x21 && 0 <= x18 - 1 && 0 <= x19 - 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) -> f135_0_f_LE(arg1P, arg2P, arg3P) :|: arg2 = arg2P && arg2 = arg1P && -1 <= arg2 - 1 && 0 <= arg1 - 1
(2) f135_0_f_LE(x, x1, x2) -> f182_0_f_LE(x3, x4, x5) :|: x1 = x5 && 2 = x4 && x = x3 && 0 <= x - 1
(3) f182_0_f_LE(x6, x7, x8) -> f135_0_f_LE(x9, x10, x11) :|: x8 - 1 = x10 && x6 - x7 = x9 && 0 <= x6 - 1 && 0 <= x7 - 1
(4) f182_0_f_LE(x12, x13, x14) -> f182_0_f_LE(x15, x16, x17) :|: x14 - 1 = x17 && x13 - 1 = x16 && x12 = x15 && 0 <= x12 - 1 && 0 <= x13 - 1
(5) f182_0_f_LE(x18, x19, x20) -> f182_0_f_LE(x21, x22, x23) :|: x19 - 1 = x22 && x18 = x21 && 0 <= x18 - 1 && 0 <= x19 - 1
(6) __init(x24, x25, x26) -> f1_0_main_Load(x27, x28, x29) :|: 0 <= 0

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

This digraph is fully evaluated!
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(4)
Obligation:

Termination digraph:
Nodes:
(1) f135_0_f_LE(x, x1, x2) -> f182_0_f_LE(x3, x4, x5) :|: x1 = x5 && 2 = x4 && x = x3 && 0 <= x - 1
(2) f182_0_f_LE(x6, x7, x8) -> f135_0_f_LE(x9, x10, x11) :|: x8 - 1 = x10 && x6 - x7 = x9 && 0 <= x6 - 1 && 0 <= x7 - 1
(3) f182_0_f_LE(x12, x13, x14) -> f182_0_f_LE(x15, x16, x17) :|: x14 - 1 = x17 && x13 - 1 = x16 && x12 = x15 && 0 <= x12 - 1 && 0 <= x13 - 1
(4) f182_0_f_LE(x18, x19, x20) -> f182_0_f_LE(x21, x22, x23) :|: x19 - 1 = x22 && x18 = x21 && 0 <= x18 - 1 && 0 <= x19 - 1

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

This digraph is fully evaluated!

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(5) IntTRSCompressionProof (EQUIVALENT)
Compressed rules.
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(6)
Obligation:
Rules:
f182_0_f_LE(x18:0, x19:0, x20:0) -> f182_0_f_LE(x18:0, x19:0 - 1, x23:0) :|: x19:0 > 0 && x18:0 > 0
f182_0_f_LE(x12:0, x13:0, x14:0) -> f182_0_f_LE(x12:0, x13:0 - 1, x14:0 - 1) :|: x13:0 > 0 && x12:0 > 0
f182_0_f_LE(x6:0, x7:0, x8:0) -> f182_0_f_LE(x6:0 - x7:0, 2, x8:0 - 1) :|: x6:0 - x7:0 > 0 && x6:0 > 0 && x7:0 > 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:

   f182_0_f_LE(x1, x2, x3) -> f182_0_f_LE(x1, x2)

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(8)
Obligation:
Rules:
f182_0_f_LE(x18:0, x19:0) -> f182_0_f_LE(x18:0, x19:0 - 1) :|: x19:0 > 0 && x18:0 > 0
f182_0_f_LE(x6:0, x7:0) -> f182_0_f_LE(x6:0 - x7:0, 2) :|: x6:0 - x7:0 > 0 && x6:0 > 0 && x7:0 > 0

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(9) TempFilterProof (SOUND)
Used the following sort dictionary for filtering: 
f182_0_f_LE(INTEGER, VARIABLE)
Replaced non-predefined constructor symbols by 0.
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(10)
Obligation:
Rules:
f182_0_f_LE(x18:0, x19:0) -> f182_0_f_LE(x18:0, c) :|: c = x19:0 - 1 && (x19:0 > 0 && x18:0 > 0)
f182_0_f_LE(x6:0, x7:0) -> f182_0_f_LE(c1, c2) :|: c2 = 2 && c1 = x6:0 - x7:0 && (x6:0 - x7:0 > 0 && x6:0 > 0 && x7:0 > 0)

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(11) PolynomialOrderProcessor (EQUIVALENT)
Found the following polynomial interpretation:
[f182_0_f_LE(x, x1)] = x + x1

The following rules are decreasing:
f182_0_f_LE(x18:0, x19:0) -> f182_0_f_LE(x18:0, c) :|: c = x19:0 - 1 && (x19:0 > 0 && x18:0 > 0)
The following rules are bounded:
f182_0_f_LE(x18:0, x19:0) -> f182_0_f_LE(x18:0, c) :|: c = x19:0 - 1 && (x19:0 > 0 && x18:0 > 0)
f182_0_f_LE(x6:0, x7:0) -> f182_0_f_LE(c1, c2) :|: c2 = 2 && c1 = x6:0 - x7:0 && (x6:0 - x7:0 > 0 && x6:0 > 0 && x7:0 > 0)

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(12)
Obligation:
Rules:
f182_0_f_LE(x6:0, x7:0) -> f182_0_f_LE(c1, c2) :|: c2 = 2 && c1 = x6:0 - x7:0 && (x6:0 - x7:0 > 0 && x6:0 > 0 && x7:0 > 0)

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(13) PolynomialOrderProcessor (EQUIVALENT)
Found the following polynomial interpretation:
[f182_0_f_LE(x, x1)] = x

The following rules are decreasing:
f182_0_f_LE(x6:0, x7:0) -> f182_0_f_LE(c1, c2) :|: c2 = 2 && c1 = x6:0 - x7:0 && (x6:0 - x7:0 > 0 && x6:0 > 0 && x7:0 > 0)
The following rules are bounded:
f182_0_f_LE(x6:0, x7:0) -> f182_0_f_LE(c1, c2) :|: c2 = 2 && c1 = x6:0 - x7:0 && (x6:0 - x7:0 > 0 && x6:0 > 0 && x7:0 > 0)

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(14)
YES