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# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty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) IRSwT
(5) IntTRSCompressionProof [EQUIVALENT, 0 ms]
(6) IRSwT
(7) IntTRSUnneededArgumentFilterProof [EQUIVALENT, 0 ms]
(8) IRSwT
(9) TempFilterProof [SOUND, 41 ms]
(10) IntTRS
(11) PolynomialOrderProcessor [EQUIVALENT, 6 ms]
(12) YES


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(0)
Obligation:
Rules:
f1_0_main_Load(arg1, arg2) -> f31_0_fib_NE(arg1P, arg2P) :|: arg2 = arg1P && -1 <= arg2 - 1 && 0 <= arg1 - 1
f31_0_fib_NE(x, x1) -> f31_0_fib_NE(x2, x3) :|: x - 1 = x2 && x - 1 <= x - 1 && 1 <= x - 1
f31_0_fib_NE(x4, x5) -> f142_0_fib_InvokeMethod(x6, x7) :|: 0 = x7 && 2 = x6 && 2 = x4
f31_0_fib_NE(x8, x9) -> f142_0_fib_InvokeMethod(x10, x11) :|: x8 - 2 = x11 && x8 = x10 && x8 - 1 <= x8 - 1 && 1 <= x8 - 1
f81_0_fib_Return(x12, x13) -> f142_0_fib_InvokeMethod(x14, x15) :|: x12 - 2 = x15 && x12 = x14 && 0 <= x12 - 1
f142_0_fib_InvokeMethod(x16, x17) -> f31_0_fib_NE(x18, x19) :|: x17 = x18 && x17 <= x16 - 1 && 1 <= x16 - 1
__init(x20, x21) -> f1_0_main_Load(x22, x23) :|: 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) -> f31_0_fib_NE(arg1P, arg2P) :|: arg2 = arg1P && -1 <= arg2 - 1 && 0 <= arg1 - 1
f31_0_fib_NE(x, x1) -> f31_0_fib_NE(x2, x3) :|: x - 1 = x2 && x - 1 <= x - 1 && 1 <= x - 1
f31_0_fib_NE(x4, x5) -> f142_0_fib_InvokeMethod(x6, x7) :|: 0 = x7 && 2 = x6 && 2 = x4
f31_0_fib_NE(x8, x9) -> f142_0_fib_InvokeMethod(x10, x11) :|: x8 - 2 = x11 && x8 = x10 && x8 - 1 <= x8 - 1 && 1 <= x8 - 1
f81_0_fib_Return(x12, x13) -> f142_0_fib_InvokeMethod(x14, x15) :|: x12 - 2 = x15 && x12 = x14 && 0 <= x12 - 1
f142_0_fib_InvokeMethod(x16, x17) -> f31_0_fib_NE(x18, x19) :|: x17 = x18 && x17 <= x16 - 1 && 1 <= x16 - 1
__init(x20, x21) -> f1_0_main_Load(x22, x23) :|: 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) -> f31_0_fib_NE(arg1P, arg2P) :|: arg2 = arg1P && -1 <= arg2 - 1 && 0 <= arg1 - 1
(2) f31_0_fib_NE(x, x1) -> f31_0_fib_NE(x2, x3) :|: x - 1 = x2 && x - 1 <= x - 1 && 1 <= x - 1
(3) f31_0_fib_NE(x4, x5) -> f142_0_fib_InvokeMethod(x6, x7) :|: 0 = x7 && 2 = x6 && 2 = x4
(4) f31_0_fib_NE(x8, x9) -> f142_0_fib_InvokeMethod(x10, x11) :|: x8 - 2 = x11 && x8 = x10 && x8 - 1 <= x8 - 1 && 1 <= x8 - 1
(5) f81_0_fib_Return(x12, x13) -> f142_0_fib_InvokeMethod(x14, x15) :|: x12 - 2 = x15 && x12 = x14 && 0 <= x12 - 1
(6) f142_0_fib_InvokeMethod(x16, x17) -> f31_0_fib_NE(x18, x19) :|: x17 = x18 && x17 <= x16 - 1 && 1 <= x16 - 1
(7) __init(x20, x21) -> f1_0_main_Load(x22, x23) :|: 0 <= 0

Arcs:
(1) -> (2), (3), (4)
(2) -> (2), (3), (4)
(3) -> (6)
(4) -> (6)
(5) -> (6)
(6) -> (2), (3), (4)
(7) -> (1)

This digraph is fully evaluated!
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(4)
Obligation:

Termination digraph:
Nodes:
(1) f31_0_fib_NE(x, x1) -> f31_0_fib_NE(x2, x3) :|: x - 1 = x2 && x - 1 <= x - 1 && 1 <= x - 1
(2) f142_0_fib_InvokeMethod(x16, x17) -> f31_0_fib_NE(x18, x19) :|: x17 = x18 && x17 <= x16 - 1 && 1 <= x16 - 1
(3) f31_0_fib_NE(x8, x9) -> f142_0_fib_InvokeMethod(x10, x11) :|: x8 - 2 = x11 && x8 = x10 && x8 - 1 <= x8 - 1 && 1 <= x8 - 1
(4) f31_0_fib_NE(x4, x5) -> f142_0_fib_InvokeMethod(x6, x7) :|: 0 = x7 && 2 = x6 && 2 = x4

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

This digraph is fully evaluated!

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(5) IntTRSCompressionProof (EQUIVALENT)
Compressed rules.
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(6)
Obligation:
Rules:
f31_0_fib_NE(x:0, x1:0) -> f31_0_fib_NE(x:0 - 1, x3:0) :|: x:0 > 1
f31_0_fib_NE(x10:0, x9:0) -> f31_0_fib_NE(x10:0 - 2, x19:0) :|: x10:0 - 2 <= x10:0 - 1 && x10:0 > 1
f31_0_fib_NE(x, x1) -> f31_0_fib_NE(0, x2) :|: TRUE && x = 2

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

   f31_0_fib_NE(x1, x2) -> f31_0_fib_NE(x1)

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(8)
Obligation:
Rules:
f31_0_fib_NE(x:0) -> f31_0_fib_NE(x:0 - 1) :|: x:0 > 1
f31_0_fib_NE(x10:0) -> f31_0_fib_NE(x10:0 - 2) :|: x10:0 - 2 <= x10:0 - 1 && x10:0 > 1
f31_0_fib_NE(x) -> f31_0_fib_NE(0) :|: TRUE && x = 2

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(9) TempFilterProof (SOUND)
Used the following sort dictionary for filtering: 
f31_0_fib_NE(VARIABLE)
Replaced non-predefined constructor symbols by 0.
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(10)
Obligation:
Rules:
f31_0_fib_NE(x:0) -> f31_0_fib_NE(c) :|: c = x:0 - 1 && x:0 > 1
f31_0_fib_NE(x10:0) -> f31_0_fib_NE(c1) :|: c1 = x10:0 - 2 && (x10:0 - 2 <= x10:0 - 1 && x10:0 > 1)
f31_0_fib_NE(c2) -> f31_0_fib_NE(c3) :|: c3 = 0 && c2 = 2 && (TRUE && x = 2)

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(11) PolynomialOrderProcessor (EQUIVALENT)
Found the following polynomial interpretation:
[f31_0_fib_NE(x)] = x

The following rules are decreasing:
f31_0_fib_NE(x:0) -> f31_0_fib_NE(c) :|: c = x:0 - 1 && x:0 > 1
f31_0_fib_NE(x10:0) -> f31_0_fib_NE(c1) :|: c1 = x10:0 - 2 && (x10:0 - 2 <= x10:0 - 1 && x10:0 > 1)
f31_0_fib_NE(c2) -> f31_0_fib_NE(c3) :|: c3 = 0 && c2 = 2 && (TRUE && x = 2)
The following rules are bounded:
f31_0_fib_NE(x:0) -> f31_0_fib_NE(c) :|: c = x:0 - 1 && x:0 > 1
f31_0_fib_NE(x10:0) -> f31_0_fib_NE(c1) :|: c1 = x10:0 - 2 && (x10:0 - 2 <= x10:0 - 1 && x10:0 > 1)
f31_0_fib_NE(c2) -> f31_0_fib_NE(c3) :|: c3 = 0 && c2 = 2 && (TRUE && x = 2)

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(12)
YES