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# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty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, 153 ms]
(4) IRSwT
(5) IntTRSCompressionProof [EQUIVALENT, 0 ms]
(6) IRSwT
(7) TempFilterProof [SOUND, 40 ms]
(8) IntTRS
(9) PolynomialOrderProcessor [EQUIVALENT, 16 ms]
(10) YES


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(0)
Obligation:
Rules:
f1_0_main_Load(arg1, arg2) -> f91_0_divBy_LE(arg1P, arg2P) :|: 0 = arg1P && 0 <= arg1 - 1 && 0 <= arg2 - 1 && -1 <= arg2P - 1
f91_0_divBy_LE(x, x1) -> f91_0_divBy_LE'(x2, x3) :|: 0 <= x1 - 1 && x4 <= x1 - 1 && -1 <= x4 - 1 && -1 <= x - 1 && x = x2 && x1 = x3
f91_0_divBy_LE'(x5, x6) -> f91_0_divBy_LE(x8, x9) :|: x5 + x9 = x8 && 0 <= x6 - 2 * x9 && x6 - 2 * x9 <= 1 && -1 <= x9 - 1 && -1 <= x5 - 1 && x9 <= x6 - 1 && 0 <= x6 - 1
__init(x10, x11) -> f1_0_main_Load(x12, x13) :|: 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) -> f91_0_divBy_LE(arg1P, arg2P) :|: 0 = arg1P && 0 <= arg1 - 1 && 0 <= arg2 - 1 && -1 <= arg2P - 1
f91_0_divBy_LE(x, x1) -> f91_0_divBy_LE'(x2, x3) :|: 0 <= x1 - 1 && x4 <= x1 - 1 && -1 <= x4 - 1 && -1 <= x - 1 && x = x2 && x1 = x3
f91_0_divBy_LE'(x5, x6) -> f91_0_divBy_LE(x8, x9) :|: x5 + x9 = x8 && 0 <= x6 - 2 * x9 && x6 - 2 * x9 <= 1 && -1 <= x9 - 1 && -1 <= x5 - 1 && x9 <= x6 - 1 && 0 <= x6 - 1
__init(x10, x11) -> f1_0_main_Load(x12, x13) :|: 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) -> f91_0_divBy_LE(arg1P, arg2P) :|: 0 = arg1P && 0 <= arg1 - 1 && 0 <= arg2 - 1 && -1 <= arg2P - 1
(2) f91_0_divBy_LE(x, x1) -> f91_0_divBy_LE'(x2, x3) :|: 0 <= x1 - 1 && x4 <= x1 - 1 && -1 <= x4 - 1 && -1 <= x - 1 && x = x2 && x1 = x3
(3) f91_0_divBy_LE'(x5, x6) -> f91_0_divBy_LE(x8, x9) :|: x5 + x9 = x8 && 0 <= x6 - 2 * x9 && x6 - 2 * x9 <= 1 && -1 <= x9 - 1 && -1 <= x5 - 1 && x9 <= x6 - 1 && 0 <= x6 - 1
(4) __init(x10, x11) -> f1_0_main_Load(x12, x13) :|: 0 <= 0

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

This digraph is fully evaluated!
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(4)
Obligation:

Termination digraph:
Nodes:
(1) f91_0_divBy_LE(x, x1) -> f91_0_divBy_LE'(x2, x3) :|: 0 <= x1 - 1 && x4 <= x1 - 1 && -1 <= x4 - 1 && -1 <= x - 1 && x = x2 && x1 = x3
(2) f91_0_divBy_LE'(x5, x6) -> f91_0_divBy_LE(x8, x9) :|: x5 + x9 = x8 && 0 <= x6 - 2 * x9 && x6 - 2 * x9 <= 1 && -1 <= x9 - 1 && -1 <= x5 - 1 && x9 <= x6 - 1 && 0 <= x6 - 1

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

This digraph is fully evaluated!

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(5) IntTRSCompressionProof (EQUIVALENT)
Compressed rules.
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(6)
Obligation:
Rules:
f91_0_divBy_LE(x2:0, x1:0) -> f91_0_divBy_LE(x2:0 + x9:0, x9:0) :|: x9:0 <= x1:0 - 1 && x2:0 > -1 && x4:0 > -1 && x9:0 > -1 && x1:0 > 0 && x4:0 <= x1:0 - 1 && x1:0 - 2 * x9:0 >= 0 && x1:0 - 2 * x9:0 <= 1

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(7) TempFilterProof (SOUND)
Used the following sort dictionary for filtering: 
f91_0_divBy_LE(INTEGER, INTEGER)
Replaced non-predefined constructor symbols by 0.
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(8)
Obligation:
Rules:
f91_0_divBy_LE(x2:0, x1:0) -> f91_0_divBy_LE(c, x9:0) :|: c = x2:0 + x9:0 && (x9:0 <= x1:0 - 1 && x2:0 > -1 && x4:0 > -1 && x9:0 > -1 && x1:0 > 0 && x4:0 <= x1:0 - 1 && x1:0 - 2 * x9:0 >= 0 && x1:0 - 2 * x9:0 <= 1)

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(9) PolynomialOrderProcessor (EQUIVALENT)
Found the following polynomial interpretation:
[f91_0_divBy_LE(x, x1)] = x1

The following rules are decreasing:
f91_0_divBy_LE(x2:0, x1:0) -> f91_0_divBy_LE(c, x9:0) :|: c = x2:0 + x9:0 && (x9:0 <= x1:0 - 1 && x2:0 > -1 && x4:0 > -1 && x9:0 > -1 && x1:0 > 0 && x4:0 <= x1:0 - 1 && x1:0 - 2 * x9:0 >= 0 && x1:0 - 2 * x9:0 <= 1)
The following rules are bounded:
f91_0_divBy_LE(x2:0, x1:0) -> f91_0_divBy_LE(c, x9:0) :|: c = x2:0 + x9:0 && (x9:0 <= x1:0 - 1 && x2:0 > -1 && x4:0 > -1 && x9:0 > -1 && x1:0 > 0 && x4:0 <= x1:0 - 1 && x1:0 - 2 * x9:0 >= 0 && x1:0 - 2 * x9:0 <= 1)

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(10)
YES