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# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty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, 146 ms]
(4) IRSwT
(5) IntTRSCompressionProof [EQUIVALENT, 34 ms]
(6) IRSwT
(7) TempFilterProof [SOUND, 51 ms]
(8) IntTRS
(9) PolynomialOrderProcessor [EQUIVALENT, 0 ms]
(10) IntTRS
(11) RankingReductionPairProof [EQUIVALENT, 0 ms]
(12) YES


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(0)
Obligation:
Rules:
f1_0_main_Load(arg1, arg2, arg3) -> f213_0_main_LE(arg1P, arg2P, arg3P) :|: 0 <= arg1 - 1 && -1 <= arg1P - 1 && -1 <= arg3P - 1 && -1 <= arg2 - 1 && -1 <= arg2P - 1
f213_0_main_LE(x, x1, x2) -> f227_0_main_LE(x3, x4, x5) :|: x2 = x5 && x = x4 && x1 = x3 && x2 <= x1 - 1
f227_0_main_LE(x6, x7, x8) -> f213_0_main_LE(x9, x10, x11) :|: x8 = x11 && x6 - 1 = x10 && x7 = x9 && x7 <= x8
f227_0_main_LE(x12, x13, x14) -> f227_0_main_LE(x15, x16, x17) :|: x14 = x17 && x13 - 1 = x16 && x12 = x15 && x14 <= x13 - 1
__init(x18, x19, x20) -> f1_0_main_Load(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_Load(arg1, arg2, arg3) -> f213_0_main_LE(arg1P, arg2P, arg3P) :|: 0 <= arg1 - 1 && -1 <= arg1P - 1 && -1 <= arg3P - 1 && -1 <= arg2 - 1 && -1 <= arg2P - 1
f213_0_main_LE(x, x1, x2) -> f227_0_main_LE(x3, x4, x5) :|: x2 = x5 && x = x4 && x1 = x3 && x2 <= x1 - 1
f227_0_main_LE(x6, x7, x8) -> f213_0_main_LE(x9, x10, x11) :|: x8 = x11 && x6 - 1 = x10 && x7 = x9 && x7 <= x8
f227_0_main_LE(x12, x13, x14) -> f227_0_main_LE(x15, x16, x17) :|: x14 = x17 && x13 - 1 = x16 && x12 = x15 && x14 <= x13 - 1
__init(x18, x19, x20) -> f1_0_main_Load(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_Load(arg1, arg2, arg3) -> f213_0_main_LE(arg1P, arg2P, arg3P) :|: 0 <= arg1 - 1 && -1 <= arg1P - 1 && -1 <= arg3P - 1 && -1 <= arg2 - 1 && -1 <= arg2P - 1
(2) f213_0_main_LE(x, x1, x2) -> f227_0_main_LE(x3, x4, x5) :|: x2 = x5 && x = x4 && x1 = x3 && x2 <= x1 - 1
(3) f227_0_main_LE(x6, x7, x8) -> f213_0_main_LE(x9, x10, x11) :|: x8 = x11 && x6 - 1 = x10 && x7 = x9 && x7 <= x8
(4) f227_0_main_LE(x12, x13, x14) -> f227_0_main_LE(x15, x16, x17) :|: x14 = x17 && x13 - 1 = x16 && x12 = x15 && x14 <= x13 - 1
(5) __init(x18, x19, x20) -> f1_0_main_Load(x21, x22, x23) :|: 0 <= 0

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

This digraph is fully evaluated!
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(4)
Obligation:

Termination digraph:
Nodes:
(1) f213_0_main_LE(x, x1, x2) -> f227_0_main_LE(x3, x4, x5) :|: x2 = x5 && x = x4 && x1 = x3 && x2 <= x1 - 1
(2) f227_0_main_LE(x6, x7, x8) -> f213_0_main_LE(x9, x10, x11) :|: x8 = x11 && x6 - 1 = x10 && x7 = x9 && x7 <= x8
(3) f227_0_main_LE(x12, x13, x14) -> f227_0_main_LE(x15, x16, x17) :|: x14 = x17 && x13 - 1 = x16 && x12 = x15 && x14 <= x13 - 1

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

This digraph is fully evaluated!

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(5) IntTRSCompressionProof (EQUIVALENT)
Compressed rules.
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(6)
Obligation:
Rules:
f227_0_main_LE(x6:0, x4:0, x11:0) -> f227_0_main_LE(x6:0 - 1, x4:0, x11:0) :|: x4:0 <= x11:0 && x6:0 - 2 >= x11:0
f227_0_main_LE(x12:0, x13:0, x14:0) -> f227_0_main_LE(x12:0, x13:0 - 1, x14:0) :|: x14:0 <= x13:0 - 1

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(7) TempFilterProof (SOUND)
Used the following sort dictionary for filtering: 
f227_0_main_LE(VARIABLE, INTEGER, INTEGER)
Replaced non-predefined constructor symbols by 0.
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(8)
Obligation:
Rules:
f227_0_main_LE(x6:0, x4:0, x11:0) -> f227_0_main_LE(c, x4:0, x11:0) :|: c = x6:0 - 1 && (x4:0 <= x11:0 && x6:0 - 2 >= x11:0)
f227_0_main_LE(x12:0, x13:0, x14:0) -> f227_0_main_LE(x12:0, c1, x14:0) :|: c1 = x13:0 - 1 && x14:0 <= x13:0 - 1

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(9) PolynomialOrderProcessor (EQUIVALENT)
Found the following polynomial interpretation:
[f227_0_main_LE(x, x1, x2)] = -1 + x1 - x2

The following rules are decreasing:
f227_0_main_LE(x12:0, x13:0, x14:0) -> f227_0_main_LE(x12:0, c1, x14:0) :|: c1 = x13:0 - 1 && x14:0 <= x13:0 - 1
The following rules are bounded:
f227_0_main_LE(x12:0, x13:0, x14:0) -> f227_0_main_LE(x12:0, c1, x14:0) :|: c1 = x13:0 - 1 && x14:0 <= x13:0 - 1

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(10)
Obligation:
Rules:
f227_0_main_LE(x6:0, x4:0, x11:0) -> f227_0_main_LE(c, x4:0, x11:0) :|: c = x6:0 - 1 && (x4:0 <= x11:0 && x6:0 - 2 >= x11:0)

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(11) RankingReductionPairProof (EQUIVALENT)
Interpretation:
[ f227_0_main_LE ] = -1*f227_0_main_LE_3 + f227_0_main_LE_1

The following rules are decreasing:
f227_0_main_LE(x6:0, x4:0, x11:0) -> f227_0_main_LE(c, x4:0, x11:0) :|: c = x6:0 - 1 && (x4:0 <= x11:0 && x6:0 - 2 >= x11:0)

The following rules are bounded:
f227_0_main_LE(x6:0, x4:0, x11:0) -> f227_0_main_LE(c, x4:0, x11:0) :|: c = x6:0 - 1 && (x4:0 <= x11:0 && x6:0 - 2 >= x11:0)


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(12)
YES