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# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty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, 75 ms]
(4) IRSwT
(5) IntTRSCompressionProof [EQUIVALENT, 0 ms]
(6) IRSwT
(7) FilterProof [EQUIVALENT, 0 ms]
(8) IntTRS
(9) IntTRSCompressionProof [EQUIVALENT, 0 ms]
(10) IntTRS
(11) PolynomialOrderProcessor [EQUIVALENT, 0 ms]
(12) YES


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(0)
Obligation:
Rules:
f1_0_main_New(arg1) -> f176_0_iter_NULL(arg1P) :|: 3 <= arg1P - 1
f176_0_iter_NULL(x) -> f176_0_iter_NULL(x1) :|: -1 <= x1 - 1 && 0 <= x - 1 && x1 + 1 <= x
__init(x2) -> f1_0_main_New(x3) :|: 0 <= 0
Start term: __init(arg1)

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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_New(arg1) -> f176_0_iter_NULL(arg1P) :|: 3 <= arg1P - 1
f176_0_iter_NULL(x) -> f176_0_iter_NULL(x1) :|: -1 <= x1 - 1 && 0 <= x - 1 && x1 + 1 <= x
__init(x2) -> f1_0_main_New(x3) :|: 0 <= 0
Start term: __init(arg1)

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(3) IRSwTTerminationDigraphProof (EQUIVALENT)
Constructed termination digraph!
Nodes:
(1) f1_0_main_New(arg1) -> f176_0_iter_NULL(arg1P) :|: 3 <= arg1P - 1
(2) f176_0_iter_NULL(x) -> f176_0_iter_NULL(x1) :|: -1 <= x1 - 1 && 0 <= x - 1 && x1 + 1 <= x
(3) __init(x2) -> f1_0_main_New(x3) :|: 0 <= 0

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

This digraph is fully evaluated!
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(4)
Obligation:

Termination digraph:
Nodes:
(1) f176_0_iter_NULL(x) -> f176_0_iter_NULL(x1) :|: -1 <= x1 - 1 && 0 <= x - 1 && x1 + 1 <= x

Arcs:
(1) -> (1)

This digraph is fully evaluated!

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(5) IntTRSCompressionProof (EQUIVALENT)
Compressed rules.
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(6)
Obligation:
Rules:
f176_0_iter_NULL(x:0) -> f176_0_iter_NULL(x1:0) :|: x1:0 > -1 && x:0 > 0 && x:0 >= x1:0 + 1

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(7) FilterProof (EQUIVALENT)
Used the following sort dictionary for filtering: 
f176_0_iter_NULL(INTEGER)
Replaced non-predefined constructor symbols by 0.
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(8)
Obligation:
Rules:
f176_0_iter_NULL(x:0) -> f176_0_iter_NULL(x1:0) :|: x1:0 > -1 && x:0 > 0 && x:0 >= x1:0 + 1

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(9) IntTRSCompressionProof (EQUIVALENT)
Compressed rules.
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(10)
Obligation:
Rules:
f176_0_iter_NULL(x:0:0) -> f176_0_iter_NULL(x1:0:0) :|: x1:0:0 > -1 && x:0:0 > 0 && x:0:0 >= x1:0:0 + 1

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(11) PolynomialOrderProcessor (EQUIVALENT)
Found the following polynomial interpretation:
[f176_0_iter_NULL(x)] = x

The following rules are decreasing:
f176_0_iter_NULL(x:0:0) -> f176_0_iter_NULL(x1:0:0) :|: x1:0:0 > -1 && x:0:0 > 0 && x:0:0 >= x1:0:0 + 1
The following rules are bounded:
f176_0_iter_NULL(x:0:0) -> f176_0_iter_NULL(x1:0:0) :|: x1:0:0 > -1 && x:0:0 > 0 && x:0:0 >= x1:0:0 + 1

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(12)
YES