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# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty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, 271 ms]
(4) AND
    (5) IRSwT
        (6) IntTRSCompressionProof [EQUIVALENT, 0 ms]
        (7) IRSwT
        (8) TempFilterProof [SOUND, 26 ms]
        (9) IntTRS
        (10) PolynomialOrderProcessor [EQUIVALENT, 10 ms]
        (11) YES
    (12) IRSwT
        (13) IntTRSCompressionProof [EQUIVALENT, 0 ms]
        (14) IRSwT
        (15) TempFilterProof [SOUND, 77 ms]
        (16) IntTRS
        (17) PolynomialOrderProcessor [EQUIVALENT, 0 ms]
        (18) IntTRS
        (19) PolynomialOrderProcessor [EQUIVALENT, 3 ms]
        (20) YES


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(0)
Obligation:
Rules:
f1_0_main_Load(arg1, arg2, arg3) -> f306_0_main_LE(arg1P, arg2P, arg3P) :|: arg1P + arg2P = arg3P && 0 <= arg1 - 1 && -1 <= arg1P - 1 && -1 <= arg2 - 1 && -1 <= arg2P - 1
f306_0_main_LE(x, x1, x2) -> f306_0_main_LE(x3, x4, x5) :|: x - 1 + x1 = x5 && x1 = x4 && x - 1 = x3 && x1 <= x - 1 && 0 <= x2 - 1
f306_0_main_LE(x6, x7, x8) -> f306_0_main_LE(x9, x10, x11) :|: x6 + x7 - 1 = x11 && x7 - 1 = x10 && x6 = x9 && x6 <= x7 - 1 && 0 <= x8 - 1
f306_0_main_LE(x12, x13, x14) -> f306_0_main_LE(x15, x16, x17) :|: x12 - 1 + x12 = x17 && x12 = x16 && x12 - 1 = x15 && x12 = x13 && 0 <= x14 - 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) -> f306_0_main_LE(arg1P, arg2P, arg3P) :|: arg1P + arg2P = arg3P && 0 <= arg1 - 1 && -1 <= arg1P - 1 && -1 <= arg2 - 1 && -1 <= arg2P - 1
f306_0_main_LE(x, x1, x2) -> f306_0_main_LE(x3, x4, x5) :|: x - 1 + x1 = x5 && x1 = x4 && x - 1 = x3 && x1 <= x - 1 && 0 <= x2 - 1
f306_0_main_LE(x6, x7, x8) -> f306_0_main_LE(x9, x10, x11) :|: x6 + x7 - 1 = x11 && x7 - 1 = x10 && x6 = x9 && x6 <= x7 - 1 && 0 <= x8 - 1
f306_0_main_LE(x12, x13, x14) -> f306_0_main_LE(x15, x16, x17) :|: x12 - 1 + x12 = x17 && x12 = x16 && x12 - 1 = x15 && x12 = x13 && 0 <= x14 - 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) -> f306_0_main_LE(arg1P, arg2P, arg3P) :|: arg1P + arg2P = arg3P && 0 <= arg1 - 1 && -1 <= arg1P - 1 && -1 <= arg2 - 1 && -1 <= arg2P - 1
(2) f306_0_main_LE(x, x1, x2) -> f306_0_main_LE(x3, x4, x5) :|: x - 1 + x1 = x5 && x1 = x4 && x - 1 = x3 && x1 <= x - 1 && 0 <= x2 - 1
(3) f306_0_main_LE(x6, x7, x8) -> f306_0_main_LE(x9, x10, x11) :|: x6 + x7 - 1 = x11 && x7 - 1 = x10 && x6 = x9 && x6 <= x7 - 1 && 0 <= x8 - 1
(4) f306_0_main_LE(x12, x13, x14) -> f306_0_main_LE(x15, x16, x17) :|: x12 - 1 + x12 = x17 && x12 = x16 && x12 - 1 = x15 && x12 = x13 && 0 <= x14 - 1
(5) __init(x18, x19, x20) -> f1_0_main_Load(x21, x22, x23) :|: 0 <= 0

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

This digraph is fully evaluated!
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(4)
Complex Obligation (AND)

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(5)
Obligation:

Termination digraph:
Nodes:
(1) f306_0_main_LE(x, x1, x2) -> f306_0_main_LE(x3, x4, x5) :|: x - 1 + x1 = x5 && x1 = x4 && x - 1 = x3 && x1 <= x - 1 && 0 <= x2 - 1

Arcs:
(1) -> (1)

This digraph is fully evaluated!

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(6) IntTRSCompressionProof (EQUIVALENT)
Compressed rules.
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(7)
Obligation:
Rules:
f306_0_main_LE(x:0, x1:0, x2:0) -> f306_0_main_LE(x:0 - 1, x1:0, x:0 - 1 + x1:0) :|: x2:0 > 0 && x:0 - 1 >= x1:0

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(8) TempFilterProof (SOUND)
Used the following sort dictionary for filtering: 
f306_0_main_LE(INTEGER, INTEGER, INTEGER)
Replaced non-predefined constructor symbols by 0.
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(9)
Obligation:
Rules:
f306_0_main_LE(x:0, x1:0, x2:0) -> f306_0_main_LE(c, x1:0, c1) :|: c1 = x:0 - 1 + x1:0 && c = x:0 - 1 && (x2:0 > 0 && x:0 - 1 >= x1:0)

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(10) PolynomialOrderProcessor (EQUIVALENT)
Found the following polynomial interpretation:
[f306_0_main_LE(x, x1, x2)] = x - x1

The following rules are decreasing:
f306_0_main_LE(x:0, x1:0, x2:0) -> f306_0_main_LE(c, x1:0, c1) :|: c1 = x:0 - 1 + x1:0 && c = x:0 - 1 && (x2:0 > 0 && x:0 - 1 >= x1:0)
The following rules are bounded:
f306_0_main_LE(x:0, x1:0, x2:0) -> f306_0_main_LE(c, x1:0, c1) :|: c1 = x:0 - 1 + x1:0 && c = x:0 - 1 && (x2:0 > 0 && x:0 - 1 >= x1:0)

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(11)
YES

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(12)
Obligation:

Termination digraph:
Nodes:
(1) f306_0_main_LE(x12, x13, x14) -> f306_0_main_LE(x15, x16, x17) :|: x12 - 1 + x12 = x17 && x12 = x16 && x12 - 1 = x15 && x12 = x13 && 0 <= x14 - 1
(2) f306_0_main_LE(x6, x7, x8) -> f306_0_main_LE(x9, x10, x11) :|: x6 + x7 - 1 = x11 && x7 - 1 = x10 && x6 = x9 && x6 <= x7 - 1 && 0 <= x8 - 1

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

This digraph is fully evaluated!

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(13) IntTRSCompressionProof (EQUIVALENT)
Compressed rules.
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(14)
Obligation:
Rules:
f306_0_main_LE(x12:0, x12:0, x14:0) -> f306_0_main_LE(x12:0 - 1, x12:0, x12:0 - 1 + x12:0) :|: x14:0 > 0
f306_0_main_LE(x6:0, x7:0, x8:0) -> f306_0_main_LE(x6:0, x7:0 - 1, x6:0 + x7:0 - 1) :|: x8:0 > 0 && x7:0 - 1 >= x6:0

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(15) TempFilterProof (SOUND)
Used the following sort dictionary for filtering: 
f306_0_main_LE(VARIABLE, VARIABLE, INTEGER)
Replaced non-predefined constructor symbols by 0.
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(16)
Obligation:
Rules:
f306_0_main_LE(x12:0, x12:0, x14:0) -> f306_0_main_LE(c, x12:0, c1) :|: c1 = x12:0 - 1 + x12:0 && c = x12:0 - 1 && x14:0 > 0
f306_0_main_LE(x6:0, x7:0, x8:0) -> f306_0_main_LE(x6:0, c2, c3) :|: c3 = x6:0 + x7:0 - 1 && c2 = x7:0 - 1 && (x8:0 > 0 && x7:0 - 1 >= x6:0)

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(17) PolynomialOrderProcessor (EQUIVALENT)
Found the following polynomial interpretation:
[f306_0_main_LE(x, x1, x2)] = -x + x^2 - x1 + x1^2 + x2

The following rules are decreasing:
f306_0_main_LE(x6:0, x7:0, x8:0) -> f306_0_main_LE(x6:0, c2, c3) :|: c3 = x6:0 + x7:0 - 1 && c2 = x7:0 - 1 && (x8:0 > 0 && x7:0 - 1 >= x6:0)
The following rules are bounded:
f306_0_main_LE(x6:0, x7:0, x8:0) -> f306_0_main_LE(x6:0, c2, c3) :|: c3 = x6:0 + x7:0 - 1 && c2 = x7:0 - 1 && (x8:0 > 0 && x7:0 - 1 >= x6:0)

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(18)
Obligation:
Rules:
f306_0_main_LE(x12:0, x12:0, x14:0) -> f306_0_main_LE(c, x12:0, c1) :|: c1 = x12:0 - 1 + x12:0 && c = x12:0 - 1 && x14:0 > 0

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(19) PolynomialOrderProcessor (EQUIVALENT)
Found the following polynomial interpretation:
[f306_0_main_LE(x, x1, x2)] = x - x1

The following rules are decreasing:
f306_0_main_LE(x12:0, x12:0, x14:0) -> f306_0_main_LE(c, x12:0, c1) :|: c1 = x12:0 - 1 + x12:0 && c = x12:0 - 1 && x14:0 > 0
The following rules are bounded:
f306_0_main_LE(x12:0, x12:0, x14:0) -> f306_0_main_LE(c, x12:0, c1) :|: c1 = x12:0 - 1 + x12:0 && c = x12:0 - 1 && x14:0 > 0

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(20)
YES