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# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty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, 15 ms]
(6) IRSwT
(7) TempFilterProof [SOUND, 19 ms]
(8) IntTRS
(9) PolynomialOrderProcessor [EQUIVALENT, 0 ms]
(10) YES


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(0)
Obligation:
Rules:
f1_0_main_New(arg1, arg2, arg3) -> f169_0_exampleMethods_LE(arg1P, arg2P, arg3P) :|: 10 = arg3P && 0 = arg2P && 13 <= arg1P - 1
f169_0_exampleMethods_LE(x, x1, x2) -> f169_0_exampleMethods_LE(x3, x4, x5) :|: x2 - 1 = x5 && x1 + 10 = x4 && x2 + 4 <= x && 2 <= x3 - 1 && 2 <= x - 1 && x3 <= x && 0 <= x2 - 1 && -1 <= x1 - 1
__init(x6, x7, x8) -> f1_0_main_New(x9, x10, x11) :|: 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_New(arg1, arg2, arg3) -> f169_0_exampleMethods_LE(arg1P, arg2P, arg3P) :|: 10 = arg3P && 0 = arg2P && 13 <= arg1P - 1
f169_0_exampleMethods_LE(x, x1, x2) -> f169_0_exampleMethods_LE(x3, x4, x5) :|: x2 - 1 = x5 && x1 + 10 = x4 && x2 + 4 <= x && 2 <= x3 - 1 && 2 <= x - 1 && x3 <= x && 0 <= x2 - 1 && -1 <= x1 - 1
__init(x6, x7, x8) -> f1_0_main_New(x9, x10, x11) :|: 0 <= 0
Start term: __init(arg1, arg2, arg3)

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(3) IRSwTTerminationDigraphProof (EQUIVALENT)
Constructed termination digraph!
Nodes:
(1) f1_0_main_New(arg1, arg2, arg3) -> f169_0_exampleMethods_LE(arg1P, arg2P, arg3P) :|: 10 = arg3P && 0 = arg2P && 13 <= arg1P - 1
(2) f169_0_exampleMethods_LE(x, x1, x2) -> f169_0_exampleMethods_LE(x3, x4, x5) :|: x2 - 1 = x5 && x1 + 10 = x4 && x2 + 4 <= x && 2 <= x3 - 1 && 2 <= x - 1 && x3 <= x && 0 <= x2 - 1 && -1 <= x1 - 1
(3) __init(x6, x7, x8) -> f1_0_main_New(x9, x10, x11) :|: 0 <= 0

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

This digraph is fully evaluated!
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(4)
Obligation:

Termination digraph:
Nodes:
(1) f169_0_exampleMethods_LE(x, x1, x2) -> f169_0_exampleMethods_LE(x3, x4, x5) :|: x2 - 1 = x5 && x1 + 10 = x4 && x2 + 4 <= x && 2 <= x3 - 1 && 2 <= x - 1 && x3 <= x && 0 <= x2 - 1 && -1 <= x1 - 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:
f169_0_exampleMethods_LE(x:0, x1:0, x2:0) -> f169_0_exampleMethods_LE(x3:0, x1:0 + 10, x2:0 - 1) :|: x2:0 > 0 && x1:0 > -1 && x:0 >= x3:0 && x:0 > 2 && x:0 >= x2:0 + 4 && x3:0 > 2

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(7) TempFilterProof (SOUND)
Used the following sort dictionary for filtering: 
f169_0_exampleMethods_LE(INTEGER, INTEGER, INTEGER)
Replaced non-predefined constructor symbols by 0.
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(8)
Obligation:
Rules:
f169_0_exampleMethods_LE(x:0, x1:0, x2:0) -> f169_0_exampleMethods_LE(x3:0, c, c1) :|: c1 = x2:0 - 1 && c = x1:0 + 10 && (x2:0 > 0 && x1:0 > -1 && x:0 >= x3:0 && x:0 > 2 && x:0 >= x2:0 + 4 && x3:0 > 2)

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(9) PolynomialOrderProcessor (EQUIVALENT)
Found the following polynomial interpretation:
[f169_0_exampleMethods_LE(x, x1, x2)] = x2

The following rules are decreasing:
f169_0_exampleMethods_LE(x:0, x1:0, x2:0) -> f169_0_exampleMethods_LE(x3:0, c, c1) :|: c1 = x2:0 - 1 && c = x1:0 + 10 && (x2:0 > 0 && x1:0 > -1 && x:0 >= x3:0 && x:0 > 2 && x:0 >= x2:0 + 4 && x3:0 > 2)
The following rules are bounded:
f169_0_exampleMethods_LE(x:0, x1:0, x2:0) -> f169_0_exampleMethods_LE(x3:0, c, c1) :|: c1 = x2:0 - 1 && c = x1:0 + 10 && (x2:0 > 0 && x1:0 > -1 && x:0 >= x3:0 && x:0 > 2 && x:0 >= x2:0 + 4 && x3:0 > 2)

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(10)
YES