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# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty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, 86 ms]
(4) IRSwT
(5) IntTRSCompressionProof [EQUIVALENT, 36 ms]
(6) IRSwT
(7) IRSwTChainingProof [EQUIVALENT, 0 ms]
(8) IRSwT
(9) IRSwTTerminationDigraphProof [EQUIVALENT, 0 ms]
(10) IRSwT
(11) IntTRSCompressionProof [EQUIVALENT, 1 ms]
(12) IRSwT
(13) TempFilterProof [SOUND, 48 ms]
(14) IntTRS
(15) PolynomialOrderProcessor [EQUIVALENT, 0 ms]
(16) YES


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(0)
Obligation:
Rules:
f1_0_main_Load(arg1, arg2) -> f210_0_main_LE(arg1P, arg2P) :|: -1 <= x3 - 1 && 1 <= arg2 - 1 && -1 <= x2 - 1 && 0 <= arg1 - 1 && 0 - x2 = arg1P && 0 - x3 = arg2P
f210_0_main_LE(x, x1) -> f210_0_main_LE(x4, x5) :|: x1 + x = x5 && x + 1 = x4 && x1 <= x - 1
__init(x6, x7) -> f1_0_main_Load(x8, x9) :|: 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) -> f210_0_main_LE(arg1P, arg2P) :|: -1 <= x3 - 1 && 1 <= arg2 - 1 && -1 <= x2 - 1 && 0 <= arg1 - 1 && 0 - x2 = arg1P && 0 - x3 = arg2P
f210_0_main_LE(x, x1) -> f210_0_main_LE(x4, x5) :|: x1 + x = x5 && x + 1 = x4 && x1 <= x - 1
__init(x6, x7) -> f1_0_main_Load(x8, x9) :|: 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) -> f210_0_main_LE(arg1P, arg2P) :|: -1 <= x3 - 1 && 1 <= arg2 - 1 && -1 <= x2 - 1 && 0 <= arg1 - 1 && 0 - x2 = arg1P && 0 - x3 = arg2P
(2) f210_0_main_LE(x, x1) -> f210_0_main_LE(x4, x5) :|: x1 + x = x5 && x + 1 = x4 && x1 <= x - 1
(3) __init(x6, x7) -> f1_0_main_Load(x8, x9) :|: 0 <= 0

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

This digraph is fully evaluated!
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(4)
Obligation:

Termination digraph:
Nodes:
(1) f210_0_main_LE(x, x1) -> f210_0_main_LE(x4, x5) :|: x1 + x = x5 && x + 1 = x4 && x1 <= x - 1

Arcs:
(1) -> (1)

This digraph is fully evaluated!

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(5) IntTRSCompressionProof (EQUIVALENT)
Compressed rules.
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(6)
Obligation:
Rules:
f210_0_main_LE(x:0, x1:0) -> f210_0_main_LE(x:0 + 1, x1:0 + x:0) :|: x:0 - 1 >= x1:0

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(7) IRSwTChainingProof (EQUIVALENT)
Chaining!
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(8)
Obligation:
Rules:
f210_0_main_LE(x, x1) -> f210_0_main_LE(x + 2, x1 + 2 * x + 1) :|: TRUE && x + -1 * x1 >= 1 && -1 * x1 >= 0

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(9) IRSwTTerminationDigraphProof (EQUIVALENT)
Constructed termination digraph!
Nodes:
(1) f210_0_main_LE(x, x1) -> f210_0_main_LE(x + 2, x1 + 2 * x + 1) :|: TRUE && x + -1 * x1 >= 1 && -1 * x1 >= 0

Arcs:
(1) -> (1)

This digraph is fully evaluated!
----------------------------------------

(10)
Obligation:

Termination digraph:
Nodes:
(1) f210_0_main_LE(x, x1) -> f210_0_main_LE(x + 2, x1 + 2 * x + 1) :|: TRUE && x + -1 * x1 >= 1 && -1 * x1 >= 0

Arcs:
(1) -> (1)

This digraph is fully evaluated!

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

(12)
Obligation:
Rules:
f210_0_main_LE(x:0, x1:0) -> f210_0_main_LE(x:0 + 2, x1:0 + 2 * x:0 + 1) :|: 0 <= -1 * x1:0 && x:0 + -1 * x1:0 >= 1

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(13) TempFilterProof (SOUND)
Used the following sort dictionary for filtering: 
f210_0_main_LE(INTEGER, INTEGER)
Replaced non-predefined constructor symbols by 0.
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(14)
Obligation:
Rules:
f210_0_main_LE(x:0, x1:0) -> f210_0_main_LE(c, c1) :|: c1 = x1:0 + 2 * x:0 + 1 && c = x:0 + 2 && (0 <= -1 * x1:0 && x:0 + -1 * x1:0 >= 1)

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(15) PolynomialOrderProcessor (EQUIVALENT)
Found the following polynomial interpretation:
[f210_0_main_LE(x, x1)] = 1 - 2*x + x^2 - 2*x1

The following rules are decreasing:
f210_0_main_LE(x:0, x1:0) -> f210_0_main_LE(c, c1) :|: c1 = x1:0 + 2 * x:0 + 1 && c = x:0 + 2 && (0 <= -1 * x1:0 && x:0 + -1 * x1:0 >= 1)
The following rules are bounded:
f210_0_main_LE(x:0, x1:0) -> f210_0_main_LE(c, c1) :|: c1 = x1:0 + 2 * x:0 + 1 && c = x:0 + 2 && (0 <= -1 * x1:0 && x:0 + -1 * x1:0 >= 1)

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(16)
YES