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# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty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, 149 ms]
(4) IRSwT
(5) IntTRSCompressionProof [EQUIVALENT, 40 ms]
(6) IRSwT
(7) TempFilterProof [SOUND, 46 ms]
(8) IntTRS
(9) RankingReductionPairProof [EQUIVALENT, 0 ms]
(10) YES


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(0)
Obligation:
Rules:
f1_0_main_Load(arg1, arg2) -> f168_0_main_EQ(arg1P, arg2P) :|: 0 <= arg1 - 1 && 0 <= arg2P - 1 && 0 <= arg1P - 1 && -1 <= arg2 - 1
f168_0_main_EQ(x, x1) -> f168_0_main_EQ(x2, x3) :|: x1 - x = x3 && x = x2 && x <= x1 - 1 && 0 <= x - 1 && 0 <= x1 - 1
f168_0_main_EQ(x4, x5) -> f168_0_main_EQ(x6, x7) :|: x5 = x7 && x4 - x5 = x6 && x5 <= x4 - 1 && 0 <= x5 - 1 && 0 <= x4 - 1
__init(x8, x9) -> f1_0_main_Load(x10, x11) :|: 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) -> f168_0_main_EQ(arg1P, arg2P) :|: 0 <= arg1 - 1 && 0 <= arg2P - 1 && 0 <= arg1P - 1 && -1 <= arg2 - 1
f168_0_main_EQ(x, x1) -> f168_0_main_EQ(x2, x3) :|: x1 - x = x3 && x = x2 && x <= x1 - 1 && 0 <= x - 1 && 0 <= x1 - 1
f168_0_main_EQ(x4, x5) -> f168_0_main_EQ(x6, x7) :|: x5 = x7 && x4 - x5 = x6 && x5 <= x4 - 1 && 0 <= x5 - 1 && 0 <= x4 - 1
__init(x8, x9) -> f1_0_main_Load(x10, x11) :|: 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) -> f168_0_main_EQ(arg1P, arg2P) :|: 0 <= arg1 - 1 && 0 <= arg2P - 1 && 0 <= arg1P - 1 && -1 <= arg2 - 1
(2) f168_0_main_EQ(x, x1) -> f168_0_main_EQ(x2, x3) :|: x1 - x = x3 && x = x2 && x <= x1 - 1 && 0 <= x - 1 && 0 <= x1 - 1
(3) f168_0_main_EQ(x4, x5) -> f168_0_main_EQ(x6, x7) :|: x5 = x7 && x4 - x5 = x6 && x5 <= x4 - 1 && 0 <= x5 - 1 && 0 <= x4 - 1
(4) __init(x8, x9) -> f1_0_main_Load(x10, x11) :|: 0 <= 0

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

This digraph is fully evaluated!
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(4)
Obligation:

Termination digraph:
Nodes:
(1) f168_0_main_EQ(x, x1) -> f168_0_main_EQ(x2, x3) :|: x1 - x = x3 && x = x2 && x <= x1 - 1 && 0 <= x - 1 && 0 <= x1 - 1
(2) f168_0_main_EQ(x4, x5) -> f168_0_main_EQ(x6, x7) :|: x5 = x7 && x4 - x5 = x6 && x5 <= x4 - 1 && 0 <= x5 - 1 && 0 <= x4 - 1

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

This digraph is fully evaluated!

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(5) IntTRSCompressionProof (EQUIVALENT)
Compressed rules.
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(6)
Obligation:
Rules:
f168_0_main_EQ(x2:0, x1:0) -> f168_0_main_EQ(x2:0, x1:0 - x2:0) :|: x2:0 > 0 && x2:0 <= x1:0 - 1 && x1:0 > 0
f168_0_main_EQ(x4:0, x5:0) -> f168_0_main_EQ(x4:0 - x5:0, x5:0) :|: x5:0 > 0 && x5:0 <= x4:0 - 1 && x4:0 > 0

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(7) TempFilterProof (SOUND)
Used the following sort dictionary for filtering: 
f168_0_main_EQ(INTEGER, INTEGER)
Replaced non-predefined constructor symbols by 0.
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(8)
Obligation:
Rules:
f168_0_main_EQ(x2:0, x1:0) -> f168_0_main_EQ(x2:0, c) :|: c = x1:0 - x2:0 && (x2:0 > 0 && x2:0 <= x1:0 - 1 && x1:0 > 0)
f168_0_main_EQ(x4:0, x5:0) -> f168_0_main_EQ(c1, x5:0) :|: c1 = x4:0 - x5:0 && (x5:0 > 0 && x5:0 <= x4:0 - 1 && x4:0 > 0)

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(9) RankingReductionPairProof (EQUIVALENT)
Interpretation:
[ f168_0_main_EQ ] = f168_0_main_EQ_1 + f168_0_main_EQ_2

The following rules are decreasing:
f168_0_main_EQ(x2:0, x1:0) -> f168_0_main_EQ(x2:0, c) :|: c = x1:0 - x2:0 && (x2:0 > 0 && x2:0 <= x1:0 - 1 && x1:0 > 0)
f168_0_main_EQ(x4:0, x5:0) -> f168_0_main_EQ(c1, x5:0) :|: c1 = x4:0 - x5:0 && (x5:0 > 0 && x5:0 <= x4:0 - 1 && x4:0 > 0)

The following rules are bounded:
f168_0_main_EQ(x2:0, x1:0) -> f168_0_main_EQ(x2:0, c) :|: c = x1:0 - x2:0 && (x2:0 > 0 && x2:0 <= x1:0 - 1 && x1:0 > 0)
f168_0_main_EQ(x4:0, x5:0) -> f168_0_main_EQ(c1, x5:0) :|: c1 = x4:0 - x5:0 && (x5:0 > 0 && x5:0 <= x4:0 - 1 && x4:0 > 0)


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(10)
YES